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Thesis on

COMPARATIVE STUDY OF FORECASTING MODELS

BASED ON WEATHER PARAMETERS

Submitted for the award of

DOCTOR OF PHILOSOPHY

Degree in

STATISTICS

SUBMITTED BY

Mohita Anand Sharma

UNDER THE SUPERVISION OF

Dr. J.B. Singh

Senior Professor Statistics

SHOBHIT INSTITUTE OF ENGINEERING & TECHNOLOGY

A DEEMED-TO-BE UNIVERSITY

MODIPURAM, MEERUT– 250110 (INDIA)

2012

ShobhitUniversity

University Campus :NH-58, Modipuram,

Meerut 250110, INDIA

T. : + 91-121-2575091/92F. : + 91-121-2575724

E. : [email protected]. : www.shobhituniversity.ac.in

Certificate

This is to certify that the thesis, entitled “Comparative Study Of Forecasting

Models Based On Weather Parameters” which is being submitted by Ms. Mohita

Anand Sharma for the award of Degree of Doctor of Philosophy in Statistics to the

Faculty of Humanities, Physical and Mathematical Sciences of Shobhit University,

Meerut, a deemed-to-be University, established by GOI u/s 3 of UGC Act 1956, is a record

of bonafide investigations and extensions of the problems carried out by her under my

supervision and guidance.

To the best of my knowledge, the matter embodied in this thesis is the original work

of the candidate herself and has not been submitted for the award of any other degree or

diploma of any University or Institution.

It is further certified that she has worked with me for the required period in the

Faculty of Humanities, Physical and Mathematical Sciences, Shobhit University, Meerut,

(U.P.), India.

Prof. J.B. Singh

(Supervisor)

Senior Professor Statistics

DECLARATION

I, hereby, declare that the work presented in this thesis, entitled “Comparative

Study Of Forecasting Models Based On Weather Parameters” in fulfillment of the

requirements for the award of Degree of Doctor of Philosophy, submitted in the Faculty

of Humanities, Physical and Mathematical Sciences at Shobhit University, Meerut, a

deemed-to-be University, established by GOI u/s 3 of UGC Act 1956, is an authentic

record of my own research work carried out under the supervision of Prof. J.B. Singh.

I also declare that the work embodied in the present thesis

(i) Is my original work and has not been copied from any Journal/Thesis/Book, and

(ii) Has not been submitted by me for any other Degree or Diploma of any

University/ Institution.

[Mohita Anand Sharma]

ACKNOWLEDGEMENT

Research is well versed with booms and hiccups. But despite these, one

relishes at the faint end where one comes out of this entrenched mundane. During

this mercurial period one gets along with innumerable individuals to whom you owe

something or more. This is my endeavor here to figure at least few people who lent

their support for smooth accomplishment of my doctoral work.

Foremost, I would like to express my heartiest gratitude to Prof. J. B. Singh, my

guide, for providing his openhanded driving force behind my research activities. It is

the great opportunity to complete my doctoral program under his scholarly and

innovative guidance. I owe him for his efficient supervision, constant inspiration,

encouragement and stimulating discussions throughout the research work.

I am thankful to the Chancellor Dr. Shobhit Kumar, Pro-vice chancellor Kuwar

Shekhar Vijendra, Vice-Chancellor Prof. R. P. Agarwal and Dean Prof. S. C. Agarwal for

providing amiable environment for concluding research in the University. I would

like to acknowledge the significant contribution of Prof. Irene Sarkar and Prof. Sanjay

Sharma for their valuable guidance, encouragement and providing necessary support

in the development of models in this research work.

Sincere gratified to my parents Late Mohan Swaroop Anand and Late Nisha

Anand, who instilled inspiration in my life, who brought me up, nurtured and

imparted me the real virtues of humanity, empathy and kindness. They are my real

motivators who sacrificed in order to bring me to the present position and blessed

me with their grace and affection. I wish they were alive to see me achieve this goal

but I am sure they must be blessing me from heaven.

Especial express thanks to my loving and caring younger brother Surya Anand,

for his gigantic over all support from day one. I am profoundly thankful for the love

and extreme affection of my elder sister Sanchita, collogue Anshul Ujaliyan and best

friends Mahima & Nida to encourage me constantly. Special thanks to Ms. Rajni

Nayyar and her family to devote the imperative time to facilitate me direct/indirect.

I am immensely thankful to Indian Methodology Department, Dehradun for

providing the valuable data for this research work. This murky world is a difficult

place to walk without blessings and teachings of some people. I am grateful to my

teachers who bestowed upon me the real lessons of life. I would like to thank all my

family and friends who have directly or indirectly contributed in my research

endeavor.

Honest recognition to my In-laws Shree Madan Pal Sharma and Smt. Brijbala

Sharma along with complete family to encourage me and taking care of my

children.

With an endless word of thanks to my backbone - my lovable husband Mr.

Prashant Kumar Sharma, as he toughs each and every aspect of my life. The credible

love of my kids had boosted me with their charming smiles and activities throughout

the day since birth.

Finally, but most importantly, I pay my reverence to the GOD, preserver and

protector with whose grace I stand tall at present. He showed me the right path in all

ups and downs and moments of despair throughout the tenure of this work. I bow my

head in complete submission before Him.

CONTENTS

Proem (i)

List of Tables (ii)-(iii)

List of Figures (iv)-(vii)

Chapter 1: Introduction 1-6

1.1 Scope 1

1.2 Motivation 1

1.3 Overview 2

1.4 Contribution 3

1.5 Objectives 4

1.6 Study area 5

Chapter 2: Review of literature 7-25

2.1 Probability Distribution 8

2.2 Multiple Regression (MR) 13

2.3 Autoregressive Integrated Moving Average (ARIMA) 16

2.4 Artificial Neural Network (ANN) 17

2.5 Comparison among MR, ARIMA and ANN 20

Chapter 3: Fitting of Probability Distribution 26-85

3.1 Introduction 26

3.2 Descriptive Statistics 26

3.3 Methodology 48

3.3.1 Fitting the probability distribution 48

3.3.2 Testing the goodness of fit 48

3.3.3 Identification of best fit probability distribution 52

3.4 Probability Distribution Pattern 54

3.4.1 Introduction 54

3.4.2 Rainfall 54

3.4.3 Maximum temperature 58

3.4.4 Minimum temperature 62

3.4.5 Relative humidity at 7 AM 66

3.4.6 Relative humidity at 2 PM 70

3.4.7 Pan evaporation 74

3.4.8 Bright sunshine 78

3.5 Conclusion 82

Chapter 4: Weather forecasting models 86-94

4.1 Introduction 86

4.2 Correlation Analysis 86

4.3 Methodology for forecasting models 87

4.3.1 Multiple Linear Regressions 87

4.3.2 Autoregressive Integrated Moving Average 88

4.3.3 Artificial Neural Network 90

4.3.4 Hybrid Approach 92

4.3.5 Performance Evaluation Criteria 93

4.4 Development of forecasting model for weather parameters 95


4.4.2 Rainfall 95



4.4.5. Relative humidity at 7 A.M. 108

4.4.6 Relative humidity at 2 P.M. 111



4.5 Comparison of prediction ability of forecasting models 124


4.5.2 Rainfall 124







4.6 Conclusion 148

Chapter 5: Identification of precise weather forecasting model 149-180

5.1 Introduction 149

5.2 Validation of weather forecasting model 149

5.2.1 Rainfall 149







5.3 Conclusion 180

Chapter 6: Summary and future scope 181-183

6.1 Summary 181

6.2 Future scope 183

Bibliography 184-200

APPENDICES

(A) Procedure followed for Stepwise Regression Analysis 201-224

(B) Programs for developing ANN models 225-238

(C) Programs for developing Hybrid MLR_ANN models 239-246

List of Reprints (Attached) of Publications

PROEM

Copious business and economic time series are non-stationary and contains

trend and seasonal discrepancy. So an accurate forecasting of such time series will

always be an important chore for effective decisions in marketing, production,

weather forecasting and many other sectors. Since weather forecasting is the most

crucial and challenging operational errands accepted worldwide. There are many

methodologies that decompose a time series linear and non-linear forms which will

always require forecasting.

We have indicated the open problems and scope of the research work in the

first chapter, which delineate enthuse that drive us to identify the anticipated

methods. Further, summing a concise contribution of the toil, intimating the objective

implicated for the study area. Literature assess is briefly illustrated with recent

progress in prediction of weather parameters in the second chapter.

Third chapter presents the descriptive statistics of the weather data seasonally

and weekly considered for the monsoon months for the study, moreover presenting

the methodology for fitting the probability distribution for each weather parameter

using goodness of fit tests.

The fourth chapter dwells with the methodology of traditional and proposed

hybrid forecasting models developed for each weather parameter, ensuring their

predictive ability graphically.

Chapter fifth corroborates the precise weather forecasting model for each

parameter by comparing the models through performance evaluation criteria. Finally,

the last chapter windups along with the future span of work.

LIST OF TABLES

Table

No.

Title Page

No.

3.1 Summary of statistics for Rainfall. 27

3.2 Summary of statistics for Maximum Temperature. 31

3.3 Summary of statistics for Minimum Temperature. 34

3.4 Summary of statistics for Relative Humidity at 7 AM. 37

3.5 Summary of statistics for Relative Humidity at 2 PM. 40

3.6 Summary of statistics for Pan Evaporation. 43

3.7 Summary of statistics for Bright Sunshine. 46

3.8 Description of various probability distribution functions. 49

3.9(a) Distributions fitted for Rainfall data sets. 56

3.9(b) Distributions with highest score for Rainfall data sets. 56

3.9(c) Parameters of the distributions fitted for Rainfall data sets. 57

3.9(d) Best fit probability distribution for Rainfall. 58

3.10(a) Distributions fitted for Maximum Temperature data sets. 59

3.10(b) Distributions with highest score for Maximum Temperature data

sets.

60

3.10(c) Parameters of the distributions fitted for Maximum Temperature

data sets.

60

3.10(d) Best fit probability distribution for Maximum Temperature data

sets.

62

3.11(a) Distributions fitted by the tests for Minimum Temperature data

sets.

63

3.11(b) Distributions with highest score for Minimum Temperature data

sets.

63

3.11(c) Parameters of the distributions fitted for Minimum Temperature

data sets.

64

3.11(d) Best fit probability distribution for Minimum Temperature. 66

3.12(a) Distributions fitted for Relative Humidity at 7 AM data sets. 67

3.12(b) Distributions with highest score for Relative Humidity at 7 AM

data sets.

67

3.12(c) Parameters of the distributions fitted for Relative Humidity at

7 AM data sets.

68

3.12(d) Best fit probability distribution for Relative Humidity at 7 AM. 69

3.13(a) Distributions fitted for Relative Humidity at 2 PM data sets. 71

3.13(b) Distributions with highest score for Relative Humidity at 2 PM

data sets.

72

LIST OF TABLES (CONTINUED)

Table

No.

Title Page

No.

3.13(c) Parameters of the distributions fitted for Relative Humidity at 2 PM

data sets.

72

3.13(d) Best fit probability distribution for Relative Humidity at 2 PM. 74

3.14(a) Distributions fitted for Pan Evaporation data sets. 75

3.14(b) Distributions with highest score for Pan Evaporation data sets. 75

3.14(c) Parameters of the distributions fitted for Pan Evaporation data sets. 76

3.14(d) Best fit probability distribution for Pan Evaporation. 77

3.15(a) Distributions fitted for Bright Sunshine data sets. 79

3.15(b) Distributions with highest score for Bright Sunshine data sets. 80

3.15(c) Parameters of the distributions fitted for Bright Sunshine data sets. 80

3.15(d) Best fit probability distribution for Bright Sunshine. 81

4.1 Inter correlation coefficient between weather parameters for total data

set.

87

5.1 Comparison of the performance of forecasting models for Rainfall. 153

5.2 Comparison of the performance of forecasting models for Maximum

Temperature.

158

5.3 Comparison of the performance of forecasting models for Minimum

Temperature.

162

5.4 Comparison of the performance of forecasting models for Relative

Humidity at 7 AM.

163

5.5 Comparison of the performance of forecasting models for Relative

Humidity at 2 PM.

167

5.6 Comparison of the performance of forecasting models for Pan

Evaporation.

171

5.7 Comparison of the performance of forecasting models for Bright

Sunshine.

176

LIST OF FIGURES

Figure

No.

Title Page

No.

3.1 Mean, standard deviation and range of weekly Rainfall. 28

3.2 50 years weekly Rainfall for monsoon period. 29

3.3 Mean, standard deviation and range of weekly Maximum

Temperature.

31

3.4 50 years weekly Maximum Temperature for monsoon period 32

3.5 Mean, standard deviation and range of weekly Minimum

Temperature.

34

3.6 50 years weekly Minimum Temperature for monsoon period 35

3.7 Mean, standard deviation and range of weekly Relative Humidity

at 7 AM.

37

3.8 50 years weekly Relative Humidity at 7 AM for monsoon period 38

3.9 Mean, standard deviation and range of weekly Relative Humidity

at 2 PM.

40

3.10 50 years weekly Relative Humidity at 2 PM for monsoon period 41

3.11 Mean, standard deviation and range of weekly Pan Evaporation. 43

3.12 50 years weekly Pan Evaporation for monsoon period 44

3.13 Mean, standard deviation and range of weekly Bright Sunshine. 46

3.14 50 years weekly Bright Sunshine for monsoon period 47

4.1 An (m x n x o) artificial neural network structure, showing a

multilayer perceptron.

91

4.2 Plots of autocorrelation and partial autocorrelation coefficients and

time lags of weakly average Rainfall parameter.

96

4.3 Artificial neural network structure for weekly average Rainfall

prediction parameter

97

4.4 Mapping of the number of epochs obtained for desired goal for

ANN model for Rainfall parameter

97

4.5 Hybrid MLR_ANN structure for weekly average Rainfall

prediction parameter.

99


hybrid MLR_ANN model for Rainfall parameter.

99


time lags of weakly average Maximum Temperature parameter.

101

LIST OF FIGURES (CONTINUED)

Figure

No.

Title Page

No.

4.8 Artificial neural network structure for weekly average Maximum

Temperature prediction parameter

102


ANN model for Maximum Temperature

102


time lags of weakly average Minimum Temperature parameter.

104

4.11 Artificial neural network structure for weekly average Minimum

Temperature prediction parameter.

105


ANN model for Minimum Temperature parameter.

105

4.13 Hybrid MLR_ANN structure for weekly average Minimum

Temperature prediction parameter

106


Hybrid MLR_ANN model for Minimum Temperature parameter.

107


time lags of weakly average Relative Humidity at 7 AM

parameter.

109

4.16 Artificial neural network structure for weekly average Relative

Humidity at 7 AM prediction parameter.

110


ANN model for Relative Humidity at 7 AM parameter.

110


time lags of weakly average Relative Humidity at 2 PM parameter.

112

4.19 Artificial neural network structure for weekly average Relative

Humidity at 2 PM prediction parameter.

113


ANN model for Relative Humidity at 2 PM parameter.

113


time lags of weakly average Pan Evaporation parameter.

115

4.22 Artificial neural network structure for weekly average Pan

Evaporation prediction parameter.

116


ANN model for Pan Evaporation parameter.

117

4.24 Hybrid MLR_ANN structure for weekly average Pan Evaporation


118


hybrid MLR_ANN model for Pan Evaporation parameter.

118


Figure

No.

Title Page

No.


time lags of weakly average Bright Sunshine parameter.

120

4.27 Artificial neural network structure for weekly average Bright

Sunshine prediction parameter.

121


ANN model for Bright Sunshine parameter.

121

4.29 Hybrid MLR_ANN structure for weekly average Bright Sunshine


122


hybrid MLR_ANN Bright Sunshine parameter.

123

4.31 Plots of the Actual and Predicted weekly average Rainfall for

training data set using Multiple Linear Regression, ARIMA,

Artificial Neural Network, Hybrid MLR_ARIMA and Hybrid

MLR_ANN models.

125

4.32 Plots of the Actual and Predicted weekly average Maximum

Temperature for training data set using Multiple Linear Regression,

ARIMA and Artificial Neural Network models.

129

4.33 Plots of the Actual and Predicted weekly average Minimum

Temperature for training data set using Multiple Regression,

ARIMA, Artificial Neural Network, Hybrid MLR_ARIMA and

Hybrid MLR_ANN models.

132

4.34 Plots of the Actual and Predicted weekly average Relative Humidity

7AM for training data set using Multiple Linear Regression,


135

4.35 Plots of the Actual and Predicted weekly average Relative Humidity

2PM for training data set using Multiple Linear Regression,


139

4.36 Plots of the Actual and Predicted weekly average Pan Evaporation

for training data set using Multiple Linear Regression, ARIMA,


MLR_ANN models.

142

4.37 Plots of the Actual and Predicted weekly average Bright Sunshine

for training data set using Multiple Linear Regression, ARIMA,


MLR_ANN models.

145


Figure

No.

Title Page

No.

5.1 Plots of the Actual and Predicted weekly average Rainfall for

testing data set using Multiple Linear Regression, ARIMA,


MLR_ANN models.

150

5.2 Plots of the Actual and Predicted weekly average Maximum

Temperature (OC) for testing data set using Multiple Linear

Regression, ARIMA and Artificial Neural Network models.

155

5.3 Plots of the Actual and Predicted weekly average Minimum

Temperature (OC) for testing data set using Multiple Linear

Regression, ARIMA, and Artificial Neural Network, Hybrid

MLR_ARIMA and Hybrid MLR_ANN models.

159

5.4 Plots of the Actual and Predicted weekly average Relative

Humidity 7AM for testing data set using Multiple Linear

Regression, ARIMA and Artificial Neural Network models.

164

5.5 Plots of the Actual and Predicted weekly average Relative

Humidity 2PM for testing data set using Multiple Linear

Regression, ARIMA, Artificial Neural Network models.

168

5.6 Plots of the Actual and Predicted weekly average Pan Evaporation

for testing data set using Multiple Linear Regression, ARIMA,


MLR_ANN models.

172

5.7 Plots of the Actual and Predicted weekly average Bright Sunshine

for testing data set using Multiple Linear Regression, ARIMA,


MLR_ANN models.

177

INTRODUCTION

1.1 Scope

1.2 Motivation

1.3 Overview

1.4 Contribution

1.5 Objectives

1.6 Study area

CHAPTER 1

INTRODUCTION

1.1 Scope

Weather forecasting is an important issue in the field of meteorology all over the world.

Several factors contribute significantly to increase the forecasting accuracy; one among them is

the development of statistical methods for enhancing the scope and accuracy of model predictions.

Numerous efforts have been devoted to develop and improve the existing time series weather

forecasting models by using different techniques. The role of statistical methodology for

predicting the weather parameters is considered to be most important for their precise estimates.

Although, high-speed computers, meteorological satellites, and weather radars are tools that had

played major roles in improving weather forecasts. But the improvement in initial conditions is the

result of an increased number of observations and better use of the observations in computational

techniques. Since, many efforts have been made by researchers to identify the best precise weather

forecasting models. The combinations of linear and non-linear models are one of the most popular

and widely used hybrid models for improving the forecasting accuracy. The present study is

planned to investigate the potential for using the existing Multiple Linear Regression,

Autoregressive Integrated Moving Average and Artificial Neural Network models to forecast

weather parameters. A comparative study of the existing and proposed weather forecasting models

is performed to identify the precise and reliable weather forecasting models.

1.2 Motivation

The prediction of weather conditions can have significant impacts on various sectors of

society in different parts of the country. Forecasts are used by government and industry to protect

life and property and to improve the efficiency of operations, and by individuals to plan a wide

range of daily activities. The notable improvement in forecast accuracy has been achieved since

the 1950s, that is, a direct outgrowth of technological developments, basic and applied research,

and the application of new knowledge and methods by weather forecasters. The advance

knowledge of weather parameters in a particular region is advantageous in effective planning.

Several studies on forecasting weather variables based on time series data in reference to a

particular region have been carried out at national and international level both in the farm and non-

farm sectors. It was observed that the combination of two or more computational models/ hybrid

models decomposes a time series into linear and non-linear form and prove to be better approach

in comparison to single models for the reason that hybrid model produces small forecasting error

in terms of accuracy.

In contrary some of the studies also mentioned that hybrid approaches are not always

better. Such uncertainty in weather forecasting models open up new opportunities for the selection

of precise forecasting model. These aspects motivate this thesis, to explore the existing

opportunities to identify the precise weather forecasting model. Predictions of weather parameters

provide by such identify models based on time series data will be of particular interest to the

weather forecasters.

1.3 Overview

The prime contribution of this thesis is to compare the existing weather forecasting model

and to select the precise model based on their predictive ability. The methodology consists of four

stages for each study period data of weather parameters which are

(i) Computation of descriptive statistics.

(ii) Statistical analysis to identify the best fit probability distribution.

(iii) Development of weather forecasting models and comparison of their predictive

ability.

(iv) Identification of precise and reliable weather forecasting model.

These four components correspond to reduce forecasting errors by relaxing certain

assumptions of traditional forecasting techniques. These components are interlinked to each other.

The first component of the methodology explains the details of different measures of

general statistics of time series data to explore the real situation of the different weather

parameters. The objective of this phase is to understand the distribution pattern of weather data.

The second component of the methodology is concerned with the fitting of the suitable probability

distribution to each weather parameter independently by using different goodness of fit test. This

methodology further establishes the analytical devised and testing procedure for future application.

Weather forecasting models were developed using time series weather data and their predictive

ability was compared using graphical and numerical performance evaluation criteria in the third

stage of the methodology. Finally hybrid models were developed and appropriate forecasting

model was identifying for future application of researchers in the related field.

1.4 Contribution

This thesis is divided into six chapters starting with acknowledgement, table of contents

and appendix in the end. Chapter first consists of a brief introduction which outlines the scope,

motivation, describes overview of the proposed methodology with objectives and study area of

the thesis as well as summarizing the major contributions in brief. Chapter second describes the

recent advances in predicting weather parameters which provide the brief review of literature in

the related field. Chapter third explain the descriptive statistics of seasonal and weekly weather

data and presents the methodology for fitting the probability distribution to weather parameters

using testing of goodness of fit. The procedure of identifying the best fit probability distribution

was explained for each weather parameter. Chapter fourth describes the methodology of Multiple

Linear Regression (MLR), Autoregressive Integrated Moving Average (ARIMA), Artificial

Neural Network (ANN) and hybrid forecasting models in brief. Forecasting models developed for

each weather variables are presented and hybrid model of Multiple Linear Regression with

ARIMA and ANN is proposed. Finally a comparison of prediction ability of forecasting model is

presented graphically for each weather parameters.

Chapter fifth describes a comparison of models which are designed to identify the precise

weather forecasting model. Finally the precise weather forecasting model is identified based on

minimum value of mean error, mean absolute error, root mean square error and prediction error

and maximum value of correlation coefficients. The last chapter presents the main summary of

the thesis and discusses direction for future work. Bibliography is presented just after the last

chapter. Graphs for the comparison of the actual and predicted weekly average weather

parameters for training and testing data set using all the statistical techniques are inclusive in the

respective chapters. The details of the computer programme used through standard software’s

SAS and Matlab in the thesis are presented in the form of appendix at the end along with the

research papers already published in related to our research work on weather forecasting models.

1.5 Objective

The advance knowledge of weather parameters in a particular region is very helpful in

sound planning. A reliable prediction of Indian monsoon in a region on seasonal and inter-

seasonal time series is not only scientifically challenging but also important for the future

planning. The role of statistical techniques for predicting the weather parameters at a particular

place and time depends on an understanding of the past time series data. The transient behavior of

weather parameters over a particular period of time makes difficult to predict correctly and

consistently. Indian economy in general and especially in the field of Agriculture and Industry

depends upon weather conditions. The frequent fluctuation in weather parameters in different part

of India is being faced by the government/ non-government planning agencies.

In recent time, the concept of combined models/ hybrid weather forecasting model is

introduced to increase the accuracy in prediction. The problem to identify the precise weather

forecasting model seems to be interesting. Thus providing reliable prediction and forecasting of

weather parameters in the Himalaya in particular and of India in general is an important challenge

for planners and scientists. Keeping in view a comparative study of weather forecasting models

and to propose hybrid model for seasonal and inter-seasonal time series data is planned with the

following objectives:

(i) To study the distribution pattern of weather parameters.

(ii) Development of weather forecasting models.

(iii) To compare the predictive ability of the developed model.

(iv) To identify the precise and reliable weather forecasting model.

1.6 Study Area

The present study is based on a time series weather data of 50 years observed at the

Pantnagar station and collected from the IMD approved meteorology observatory Dehradun, India.

India is situated in the east direction of earth and lies between latitude 220°00’N and longitude

770°00’W. Pantnagar station is located at 29°N latitude and 79°3’ E longitudes approximately

243.89 meters above mean sea level, in the Tarai region of Uttarakhand and lies within the

Shivalik Ranges of the Himalayan foothills. On an average the region has a humid subtropical

climate having hot summers (40-42OC) and cold winters (2-4OC) with monsoon rains occurring

from June to September. July is the rainiest month followed by August. In September, due to the

depression from the Bay of Bengal, the local weather is affected, causing heavy rains. With the

withdrawal of monsoon in September, the intensity of rainfall rapidly decreases till in November it

becomes practically rainless. Rain Gauge Station suggest that the annual average rainfall in and

around Pantnagar is of the order of 1400 mm. More than 80% of the rain is received from south-

west monsoon during these four month period from June to September, and the rainfall of rainy

season is significantly different from that of dry season. Winter precipitation in the region,

associated with the passage of Western disturbances, is in the form of snowfall in the Higher

Central Himalaya.

The average monsoon season in and around Pantnagar region, ranges between 15 to 20

weeks. 17 weeks data set from 4th June to 30th September of each year is considered as inter

season monsoon periods for our study. The data comprises of seven parameters viz. rainfall,

maximum and minimum temperature, relative humidity at 7:00 AM and 2:00 PM, pan evaporation

and bright sunshine hours, collected during the monsoon months on a time series weather data of

850 weeks from 1961 to 2010.

REVIEW OF LITERATURE

Probability Distribution

Multiple Regression

Autoregressive Integrated Moving Average

Artificial Neural Network

Comparison among MR, ARIMA and ANN

CHAPTER 2

REVIEW OF LITERATURE

Weather is a continuous, data-intensive, multi-dimensional, dynamic and chaotic process,

and these properties make weather forecasting a formidable challenge. It is one of the most

imperative and demanding operational responsibilities carried out by meteorological services all

over the world. At present, the assessment of the nature and causes of seasonal climate variability

is still conception. Since, it is a complicated procedure that includes numerous specialized fields of

know-how (Guhathakurata, 2006); therefore, in the field of meteorology all decisions are to be

taken in the visage of uncertainty associated with local of and global climatic variables. Several

authors have discussed the vagueness associated with the weather systems. Chaotic features

associated with the atmospheric phenomena also have attracted the attention of the modern

scientists (Sivakumar 2001; Sivakumar et al. 1999; Men et al. 2004). Different scientists over the

globe have developed stochastic weather models. It is often used to predict and warn about natural

disasters that are caused by abrupt change in climate conditions. The variables defining weather

conditions vary continuously with time, forming time series of each parameter and can be used to

develop a forecasting model either statistically or using some other means that uses this time series

data (Chatfield 1994; Montgomery and Lynwood 1996).

Weather prediction modeling involves a combination of computer models, observation and

knowledge of trends and patterns. Generally, two methods are used to forecast weather: (a) the

empirical approach and (b) the dynamical approach (Lorenz, 1969). The first approach is based

upon the occurrence of analogues and is often referred to by meteorologists as analogue

forecasting. This approach is useful for predicting local-scale weather if recorded cases are

plentiful. The second approach is based upon equations and forward simulations of the

atmosphere, and is often referred to as computer modeling. Because of the grid coarseness, the

dynamical approach is only useful for modeling large-scale weather phenomena and may not

predict short-term weather efficiently. Many weather prediction systems use a combination of

empirical and dynamical techniques. At macro level, weather forecasting is usually done using the

data gathered by remote sensing satellites. Weather parameters like maximum temperature,

minimum temperature, extent of rainfall, cloud conditions, wind streams and their directions, are

projected using images taken by these meteorological satellites to asses future trends. The

satellites-based systems are inherently costlier and require complete support system. Moreover,

such systems are capable of providing only such information, which is usually generalized over a

larger geographical area.

The successful weather predictions are performed since early 1920’s. The practical use of

numerical weather prediction starts in the middle of nineteenth century. A number of forecast

models, both global and regional are being used to create forecasts. This chapter is intended to

provide the brief review of literature in the field of weather forecasting models in general and

specially for comparative study of weather forecasts models. Thus, the chapter is divided into five

parts and the research work done in each field is being reviewed in different sections.

2.1 Probability distribution

Analysis of weather data strongly depends on its probability distribution pattern.

Establishing a probability distribution that provides a good fit to the weather parameter has long

been a topic of interest in the fields of hydrology, meteorology and other fields. Several studies

have been conducted in India and abroad on weather analysis and best fit probability distribution

function such as normal, log-normal, gumbel, weibull and Pearson type distribution were

identified.

Fisher (1924) studied the influence of rainfall on the yield of wheat in Rothamasted. He

showed that it is the distribution of rainfall during a season rather than its total amount which

influence the crop yield. Tippet (1929) subsequently applied the technique on sunshine

distribution and found that sunshine has beneficial effect throughout the year on wheat crop.

Another useful line of work relating to the study of rainfall distribution was introduced by

Manning (1956). He transformed the skew frequency distribution of rainfall to approximate

closely to the theoretical normal distribution showing that fifteen observations were enough to get

a reasonable good estimate of the distribution and confidence limit.

Further, Rao et al. (1963) have used the extreme value distribution (EVD) on rainfall

(Chow, 1964) to predict the flood and drought situations in the parts of India. Abraham (1965) has

applied the fisher’s method to see the joint relationship of crop yield and weather variable (rainfall

and temperature). Rai and Jay (1966) studied humidity and upper winds temperature over Madras

in relation to precipitation occurrence and found the vertical distribution of temperature and

humidity associated with dry or wet days over the same area.

Benson (1968) adopted a large scale planning for improved flood plain management and

expending water resources development and he suggested adopting a procedure where records are

available for all government agencies. Along with Pearson type I, Gumble’s and log normal

distribution, the log Pearson type III distribution has been selected as the based method with

provision for departure from the base method were justified continuing study leading towards

improvements or revision of method is recommended. Kulkarni and Pant (1969) studied the

cumulative frequency distribution of rainfall of different intensities during south-west monsoon

for 20 stations in India. The distribution was found to be exponential and curves were fitted to

observed date by the method of least square.

Mooley et al. (1970) have studied statistical distribution of rainfall during south-west and

north-east monsoon season at representative stations in India. Gamma distribution has been fitted

to rainfall data and has been tested by Chi-square test. Bhargava et al. (1974) showed that for a

number of crops the distribution of rainfall over the season has a great influence on the yield.

Krishnan and Kushwaha (1972) studied the mathematical distribution of accumulated rainfall for

2 pentads, 4 pentads,…., 20 pentads commencing from the onset of monsoon in respect of a

typical arid zone stations. In case of Jaipur the distribution beyond a month is normal while for

Jodhpur, distribution is not normal at all. Raman Rao et al. (1975) analyzed the daily rainfall data

collected at Bijapur for the year from 1921 to 1970.

Parthsarthy and Dhar (1976) have studied the trends and periodicities in the annual

rainfall of the metrological subdivisions of Madhya Pradesh for the 60 years period. It was seen

that the frequency distribution of annual rainfall of east and west M.P. is normal. Significant

increase of 15% of the mean annual rainfall per 30 years was observed in west M.P. Cunnane et

al. (1978) and Gringorten (1963) have used plotting rule for extreme probability paper to study

the extreme value analysis. Mukherjee et al. (1979) made the studies to improve the weather

bulletin and a beginning in this direction is made with a detailed study of rainfall even within same

district. They observed that there is a wide variation in the intensity and distribution of rainfall.

Mukherjee et al. (1980) have conducted the study of monthly seasonal and annual rainfall

distribution for 16 stations in Pune and 11 stations in Ahmed Nagar for 50 years period. Combined

study of two district shows that the rainfall distribution in western part of Pune is same as western

part of Ahmed Nagar while rainfall in eastern part of Pune is same as that in eastern part of

Ahmed Nagar. Kulandaivelu (1984) analyzed the daily precipitation data of Coimbatore for a

period of 70 years for weekly totals by fitting incomplete Gamma distribution model. The data

indicate the likely commencement of rains, period of drought length of growing season and end of

growing season. Based on the assured rainfall at (50%) probability level, suitable cropping system

was suggested for Coimbatore.

Phien and Ajirajah (1984) showed that for the annual flood, annual maximum rainfall,

annual stream flow and annual rainfall, the log-Pearson type III distribution was highly suitable

after evaluating by Chi-square and Kolmogorov- Smirnov tests. Biswas and Khambete (1989)

computed the lowest amount of rainfall at different probability level by fitting gamma distribution

probability model to week by week total rainfall of 82 stations in dry farming tract of Maharashtra.

Rao and Singh (1990) studied the distributions of weather variables and developed methodology

for forecasting extreme values of weather variables at Pantnagar. They observed that the square

root model ( )y a b x cx is approximate to predict wheat yield based on metrological

observations.

Gumble distribution was applied by Mukherjee et al. (1991) to estimate return period of

reoccurrence of highest one day rainfall. Lin et al. (1993) stated that in accordance with the

probability distribution all stations in same area can be classified in different clusters and special

characteristic among a clusters can have spatial relationship to a certain extent in that cluster.

Chapman (1994) evaluated five daily rainfall generating models with several methods and

analyzed that Srikanthan-McMahon model performed well when calibrated with long rainfall

records. Nese and Jon (1994) estimated the potential effect of the biases on the mean and standard

deviation of a temperature distribution; biasing simulations were performed on various normal

distributions. In addition, it is shown that these biases can affect other relevant climatic statistics.

Duan et al. (1995) suggested that for modeling daily rainfall amounts, the Weibull and to a lesser

extent the exponential distribution is suitable.

Extreme value analysis was done for seven stations of Krishna Godavari agro-climatic

zone in Andhra Pradesh of India, Kulshrestha et al. (1995). Similar analysis was made for 14

stations of Gujarat state of India and ascertained the most suitable type of distribution,

Kulshrestha et al. (1999). Statistical distribution has been used to define extremes with given

return periods (Aggarwal et al. 1988, Bhatt et al. 1996). Upadhaya and Singh (1998) stated that it

is possible to predict rainfall fairly accurate using various probability distributions for certain

returns periods although the rainfall varies with space, time and have erratic nature. Sen and

Eljadid (1999) reported that for monthly rainfall in arid regions, gamma probability distribution is

best fit hence which enables one to construct regional maps for the area of two gamma parameters,

shape and scale.

Rai and Chandrahas (1996) studied the effect of intensity and distribution pattern of

weather parameters at different stages of crop growth and rice yield. They found that temperature

and sunshine hours are effective at the growing phase, whereas, sunshine hours were found

ineffective during early growth phase. Ogunlela (2001) studied the stochastic analysis of rainfall

event in Illorin using probability distribution functions. He concluded that the log-Peasson type III

distribution best described the peak daily rainfall data for Ilorin. Kar (2002) has predicted the

extreme rainfall for mid central table zone of Orissa using Extreme value Type-I distribution and

was concluded that extreme value type-I distribution was a good fit for predicting the one day

maximum rainfall. Tao et al. (2002) recommended generalized extreme value model as the most

suitable distribution after a systematic assessment procedure for representing extreme-value

process and its relatively simple parameter estimation.

Topaloglu (2002) reported that gumbel probability model estimated by the method of

moments and evaluated by chi-square tests was found to be the best model in the Seyhan river

basin. Fowler et al. (2003) have used two methods to assess rainfall extremes and their

probabilities. One of the method comprised percentile approach (Kar, 2002) and the other used the

statistical distributions of rainfall (Hennersy et al., 1997). Salami (2004) studied the

meteorological data for Texas and found that Gumbel distribution fits adequately for both

evaporation and temperature data, while for precipitation data log-Pearson type III distribution

conforms more accurate. Lee (2005) indicated that log-Pearson type III distribution fits for 50% of

total station number for the rainfall distribution characteristics of Chia-Nan plain area. Bhakar et

al. (2006) observed the frequency analysis of consecutive days peaked rainfall at Banswara,

Rajasthan, India, and found gamma distribution as the best fit as compared by other distribution

and tested by Chi-square value. Deidda and Puliga (2006) found for left-censored records of

Sardinia, that some weak are evident for the generalized Pareto distribution.

Kwaku et al. (2007) revealed that the log-normal distribution was the best fit probability

distribution for one to five consecutive days’ maximum rainfall for Accra, Ghana. Hanson et al.,

(2008) analysis indicated that Pearson type III distribution fits the full record of daily precipitation

data and Kappa distribution best describes the observed distribution of wet-day daily rainfall.

Olofintoye et al. (2009) examined that 50% of the total station number in Nigeria follows log-

Pearson type III distribution for peak daily rainfall, while 40% and 10% of the total station follows

Pearson type III and log-Gumbel distribution respectively. Sharma and Singh (2010) studied the

distribution pattern for extreme value rainfall for Pantnagar data. Generalized extreme value

distribution was observed in most of the weekly period as best fit probability distribution.

There are a wide variety of previous studies which have explored the probability

distribution of daily rainfall for the purpose of rainfall frequency analysis. However, we are

unaware of any studies that have used recent developments. This research seeks to reexamine the

question of which continuous distribution best fits the weekly average monsoon weather variable.

Our primary objective is to determine a suitable distribution of monsoon season for each weekly

average weather variables using different probability distributions.

2.2 Multiple Regression (MR)

Forecasting models based on time series data are being developed for prediction of the

different variables. Regression is a statistical empirical technique and is widely used in business,

the social and behavioral sciences, the biological sciences, climate prediction, and many other

areas. Linear and non-linear multiple regression models of different orders are also being used for

predicting purpose based on the time series data. These models can consider more than one

predictor for rainfall prediction. There are some limitations of multiple regression approach such

as multiple collinearly, inter relation, extreme observation and non-linear relationship between

dependent and independent variables.

Goulden (1962) found the relationship between monthly average of weather parameters

and crop yield using multiple regression technique. Ramchandran (1967) made an analysis of the

normal rainfall of 167 observatory station distributed over India and the neighborhood country,

using regression equation representing monthly and annually rainfall as a linear function of

latitude, longitude and elevation above sea level. Bali (1970) found more precise results with the

help of regression method for calculating average yields and explained the inadequacy of currently

employed methods for forecasting crop yield in India. Huda et al. (1975) reported that a second

degree multiple regression can be employed for studying the relationship between rice yield and

weather variables.

Huda et al. (1976) applied second degree multiple regression equation to quantify the

relationship between maize yield and meteorological data and it was found that maize yield was

affected differently by different weather variables during different stages of growth. Singh et al.

(1979) gave hints for forecasting the yield rate by traditional and objective methods, on the basis

of biometrical character as well as weather parameters. Regression studies on the relationship of

crop yield with weather factor have been made. Agrawal et al. (1980) studied regression models

for forecasting the yield of rice in Raipur district on weekly data using weather variables. Khatri et

al. (1983) used regression analysis for crop estimation surveys on historic and rainfall data for

developing the forecasting model with the help of stepwise regression analysis. Hastenrath (1988)

developed statistical model using regression method to predict Indian summer monsoon rainfall

anomaly.

Singh (1988) developed a suitable pre-harvesting forecasting model with the help of

multiple regression techniques for sugarcane yield. Singh and Bapat (1988) developed a pre-

harvest forecast model using stepwise regression for selection of yield attribute to entering finally

in forecast model. Pal (1995) studied the relationship between weather parameters and yields

using linear multiple regression and second degree multiple regression equation based on time

series weather data. Sparks (1997) developed a multiple regression model for time series data to

predict a production in arid climate at high evaluations. Shashi Kumar et al. (1998) showed that

the principal components regression gives better precision for the estimates than ordinary least

square regression analysis.

Vaccari et al. (1999) modeled plant motion time-series and nutrient recovery data for

advanced life support using multi variable polynomial regression. Hassani et al. (2003) proposed

human height prediction model based on multiple polynomial regression that was used

successfully to forecast the growth potentials of height with precision and was helpful in children

growth study. Sen (2003) has presented long-range summer monsoon rainfall forecast model

based on power regression technique with the use of Ei Nino, Eurasian snow cover, North West

Europe temperature, Europe pressure gradient, 50 hpa Wind pattern, Arabian sea SST, east Asia

pressure and south Indian ocean temperature in previous year. The experimental results showed

that the model error was 4%. Nkrintra et al. (2005) described the development of a statistical

forecasting method for SMR over Thailand using multiple linear regression and local polynomial

based non-parametric approaches. SST, sea level pressure (SLP), wind speed, EiNino Southern

Oscillation (ENSO), IOD was chosen as predictors. The experiments indicated that the correlation

between observed and forecast rainfall was 0.6.

Sohn et al. (2005) has developed a prediction model for the occurrence of heavy rain in

South Korea using multiple linear regression, decision tree and artificial neural network. They

used 45 synoptic factors generated by the numerical model as potential predictors. Anderson et al.

(2006) examines the possibility of forecasting traffic volumes by using a multiple linear regression

model to perform what is termed direct demand forecasting and obtained consistent results from

the traditional four-step methodology. Zaw and Naing (2008) performed the modeling of monthly

rainfall prediction over Myanmar by applying the polynomial regression equation and compared

with multiple linear regression model. Experiments indicated that the prediction model based on

MPR has higher accuracy than MLR.

Radhika and Shashi (2009) used time series data of daily maximum temperature and

found non-linear regression method suitable to train support vector machines (SVMs) for weather

prediction. Kannan et al. (2010) computed values for rainfall fall in the ground level using five

years input data by Karl Pearson correlation coefficient and predicted for future years rainfall fall

in ground level by multiple linear regression. Ghani and Ahmad (2010) applied six types of linear

regression including stepwise multiple regression to select the suitable controlled variables in

forecast fish landing.

2.3 Autoregressive Integrated Moving Average (ARIMA)

Two popular models for seasonal time series are multiplicative seasonal ARIMA

(autoregressive-integrated-moving average) models (Box and Jenkins 1976) and ARIMA

component (structural) models. Despite the rising popularity of ARIMA component models in the

time series literature of recent years, empirical studies comparing these models with seasonal

ARIMA models have been relatively rare.

Cottrell et al. (1995) proposed a systematic methodology to determine which weights are

nonsignificant and to eliminate them to simplify the architecture. They tried to combine the

statistical techniques of linear and nonlinear time series with the connectionist approach. Zhang

and Qi (2003) investigated as how to effectively model time series with both seasonal and trend

patterns. They found that combined detrending and deseasonalization is the most effective data

preprocessing approach. Campbell and Diebold (2005) used simple time-series approach to

modeling and forecasting daily average temperature in U.S. cities and found it useful for the

vantage point of participants in the weather derivatives market.

Iqbal et al. (2005) made the study on ARIMA to forecast the area and production of wheat

in Pakistan. Further, suggesting that the scope of higher area and production lies in adequate

availability of inputs, educating and training the farming community, soil conservation and

reclamation, and especially the supportive government policies regarding wheat cultivation in the

country. Zhou and Hu (2008) proposed a hybrid modeling and forecasting approach based on the

grey and the Box–Jenkins autoregressive moving average (ARMA) models to forecast the

gyrodrift concluding that the hybrid method has a higher forecasting precision to the complex

problems than the single method. Kal et al. (2010) developed a framework to determine the

optimal inventory policy under the environment that the leadtime demand is generated by the

ARIMA process.

Alnaa and Ahiakpor (2011) considered ARIMA model to predict inflation in Ghana.

Inflation was predicted highest for the months of March, April and May. Further, suggesting that

inflation has a long memory and that once the inflation spiral is set in motion, it will take at least

12 periods (months) to bring it to a stable state. Badmus and Ariyo (2011) utilized ARIMA for

forecasting the cultivated area and production of maize in Nigeria. They concluded that the total

cropped area can be increased in future, if land reclamation and conservation measures are

adopted.

Saima et al. (2011) explained a hybrid fuzzy time series model is proposed that will

develop an Interval type 2 fuzzy model based on ARIMA. IT2-FLS is utilized here for handling

the uncertainty in the time series data to obtain accurate forecasting result. Ghosh et al. (2012)

developed a model based on ARIMA to depict the future prospects of coal based thermal power

sector of India. The evidence showed that India needs to identify alternative sources of power

generation to grow without damaging world and maintaining sustainability.

2.4 Artificial Neural Network (ANN)

An Artificial Neural Network is a powerful data modeling tool that provides a

methodology for solving many types of non-linear problems that are difficult to solve by

traditional techniques. Neural Network makes very few assumptions as opposed to normality

assumptions commonly found in statistical methods. From a statistician’s point of view neural

networks are analogous to nonparametric, nonlinear and regression models. The ANN approach

has several advantages over conventional phenomenological or semi-empirical models, since they

require known input data set without any assumptions (Gardner and Dorling, 1998; Nagendra

and khare, 2006). It exhibits rapid information processing and is able to develop a mapping of the

input and output variables. Such a mapping can subsequently be used to predict desired outputs as

a function of suitable inputs (Nagendra and Khare, 2006).

The ANNs use many simplifications over actual biological neurons that help us to use the

computational principles employed in the massively parallel machine (Haykin 1999). The neural

networks adaptively change their synaptic weights through the process of learning. Feed Forward

Neural Networks with Back-propagation (BKP) of error have been used in past for modeling and

forecasting various parameters of interest using time series data, Cottrell el al. (1995) used the

time series modeling to provide a method for weight elimination in ANNs. Since the last few

decades, ANN a voluminous development in the application field of ANN has opened up new

avenues to the forecasting task involving atmosphere related phenomena (Gardner and Dorling,

1998; Hsieh and Tang, 1998). The prediction in an artificial neural network method (ANN)

always takes place according to any data situation (without limitation) based on initial training as

indicated by Adielsson (2005).

Thus, for forecasting, certain statistical techniques can be combined with the connectionist

approach of ANN exploiting the information contained in linear or nonlinear time series. The

knowledge that an ANN gains about a problem domain is encoded in the weights assigned to the

connections of the ANN. The ANN can then be thought as a black box, taking in and giving out

information (Roadknight et al. 1997). ANN non-linear models have been widely used for

resolving forecast problem as identified by Hill et al. (1996), Faraway and Chatfield (1998),

Kaashoek and Van Dijk (2001), Tseng et al. (2002), Altun et al. (2007), Fallah-Ghalhary

(2009), Wu et al. (2010) and El-Shafie et al.(2011).

Hu (1964) initiated the implementation of ANN, an important Soft Computing

methodology in weather forecasting. Forecasting the behavior of complex system has been a broad

application domain for neural networks. In particular such as electric load forecasting (Park et al.

1991), economic forecasting (Refenes et al. 1994), forecasting natural physical phenomena

(Weigend et al., 1994), river flow forecasting (Atiya et al. 1996) and forecasting student

admission in colleges (Puri et al. 2007) have been widely studied.

A successful application of ANN to rainfall forecasting has been done by French et al.

(1992) who applied a neural network to forecast one-hour-ahead, two-dimensional rainfall fields

on a regular grid. Moro et al. (1994) applied a neural network approach for weather forecasting for

local data. Kalogirou et al. (1997) implemented ANN to reconstruct the rainfall time series over

Cyprus. Kuligowski and Barros (1998) analyzed a precipitation forecasts model using neural

network approach. Lee et al. (1998) applied Artificial Neural Network in rainfall prediction by

splitting the available data into homogeneous subpopulations. Wong et al. (1999) constructed

fuzzy rule bases with the aid of SOM and back propagation neural networks and then with the help

of the rule base developed predictive model for rainfall over Switzerland using spatial

interpolation.

Atiya et al. (1997) studied the generalization performance in large network, which means

producing appropriate outputs for those input samples not encountered during training process of

the network is best described by training data size and number of synaptic weights. Share prices

are also taken as time series to forecast stock prices of the future (Mathur et al. 1998). Maqsood

et al. (2002a, 2002b) used neurocomputing based weather monitoring and analysis models.

Anmala et al. (2000) reported that recurrent networks may perform better than standard feed

forward networks in predicting monthly runoff. Sahai et al. (2000) applied the ANN technique to

five time series of June, July, August, September monthly and seasonal rainfall. The previous five

years values from all the five time series were used to train the ANN to predict for the next year.

They found good performance in predicting rainfall.

Toth et al. (2000) investigated the capability of ANN in short-term rainfall forecasting

using historical rainfall data as the only input information. Kishtawal et al. (2003) assessed the

feasibility of a nonlinear technique based on genetic algorithm, an Artificial Intelligence technique

for the prediction of summer rainfall over India. Guhathakurta (2006) was the first ever to

implement ANN technique to predict summer monsoon rainfall over a state of India. Miao et al.

(2006) developed almost seven different methods for ANN and each one can be used in a different

analysis rather than classical statistical methods in identifying factors influencing corn yield and

grain quality variability. Chattopadhyay (2007) analyzed that neural network with three nodes in

the hidden layer is found to be the best predictive model for possibility of predicting average

summer-monsoon rainfall over India. Paras el al. (2007) concluded that neural networks are

capable of modeling a weather forecast system. Statistical indicators chosen are capable of

extracting the trends, which can be considered as features for developing the models.

Hayati et al. (2007) showed that multi-layer perceptron (MLP) network has minimum

forecasting error for each season and can be considered as a good method for temperature

forecasting. Kumar et al. (2007) presented reasonably good Artificial Intelligence approaches for

regional rainfall forecasting for Orissa state, India on monthly and seasonal time series scale. The

study emphasizes the value of using large scale climate teleconnections for regional rainfall

forecasting and the significance of Artificial intelligence approaches in predicting the uncertain

rainfall. Hung et al. (2009) developed the ANN model and applied for real time rainfall

forecasting and flood management is Bangkok, Thailand, Resulting that ANN forecasts have

superiorly over the ones obtained by the persistent model. Rainfall forecast for Bangkok from 1 to

3 h ahead were found highly satisfactory. Sharma et al. (2011) proposed hybrid MR with ANN

for the Himalayan monsoon data, suggesting that hybrid techniques can be used as a reliable

rainfall forecasting tool in the Himalaya.

2.5 Comparison among of MR, ARIMA and ANN

The comparative evaluation of the performance of all the three models has been conducted

using forecasting weather variable and many other time series data. Many studies have been

conducted by Lek (1996), Starett (1998), Manel (1999), Salt (1999), Ozesmi (1999), Gail (2005),

Diane (2007) with their co-authors and Pastor (2005) to compare two methods to show that in

predicting the dependent variable, the ANN method results are more accurate than Multiple Linear

Regression. The performance of ANN and traditional statistical methods were also compared and

discussed by Kumar (2005), Pao (2006), Wang and Elhag (2007), Zhang (2001) and Wang et al.

(2010).

Dutta and Shekhar (1988) compared neural networks to a multiple regression model in an

application to predict bond ratings based on ten financial variables. The neural network model

consistently outperformed the regression model yielding a success rate of 88.3 percent versus 64.7

percent for regression. In addition, the neural network was never off by more than one rating while

the regression model was often off by several ratings.

Marquez et al. (1991) compared the performance of neural network models to various

regression models. They tested the data with the correct regression model, two other regression

models that were one-step in either direction away from the correct model on a ladder of

expression, and two neural network models. The neural network was generally within two percent

of the mean absolute percentage error of the current model for the linear and inverse cases. The

neural network performance on the log data was poorer and in general the neural networks did not

perform quite as well as the other regression models but the authors concluded that neural

networks have considerable potential as an alternative to regression.

Specht and Donald (1991) examined the use of neural network to perform the function of

multiple linear regressions. They compared the use of neural networks with standard linear

regression in four cases: the regression model was correctly specified with all assumptions valid;

the regression model was correctly specified, but the data contained an outlier; the regression

variables exhibited multicollinearity; and the regression model was incorrectly specified by

omitting an interaction. The authors concluded that neural networks are robust and relatively

insensitive to problems with bad data, model assumptions, and faulty model construction.

Chang et al. (1991) used neural networks to forecast rainfall based on time series data. The

data showed both seasonal and cyclical components that were incorporated into the input data set.

The forecasts were for one month into its future. Based on the mean square error, neural network

outperformed the unnamed statistical approach. Dulibar (1991) performed a study to predict the

performance of carriers in a particular segment of the transportation sector. In the study, she

compared a neural network model to several regression models. The measure of performance used

was percent capacity utilization. The neural network performed better than some of the regression

models and not as well as others. According to the author, a possible reason for the poorer

performance of the neural network model is that he firms involved in the study were explicitly

included in the regression models but not in the Neural Network model.

Raghupathi et al. (1991) found that a neural network provided 86 % correct classifications

and would therefore likely provide a good model for the bankruptcy prediction process.

Salchenberger et al. (1992) used neural networks to predict thrift failures. They compared a

neural network to the more traditional logit model. They used five financial ratios in attempting to

classify an institution as one that would fail or not fail. For each data set created, the neural

network performed at least as well as the logit model. Also, as the classification cutoff was

lowered, the neural network committed less type I errors than the logit model.

Tam and Kiang (1992), neural networks were compared to several popular discriminant

analysis methods in a bank failure classification application. The sample consisted of 59 matched

pairs of Texas banks. The neural network model showed better predictive accuracy on the test set

than the other methods. Wu and Yen (1992) proposed a neural network structure and the

associated design-oriented procedure for neural network development for regression applications.

The proposed methodology is illustrated by two practical applications. One is a linear regression

case concerning the relation between marginal cost and cumulative production; the other is a

nonlinear regression case concerning the yield of wheat corresponding to the application rate of

fertilizer. They compared the results of the regression techniques with those of neural networks.

Fletcher and Goss (1993) used financial ratios to compare neural networks to a logit

model. The output of the models represented the probability that a particular firm would fail. The

neural network was 82.4 % accurate with a 0.5 cutoff versus 77 % for the logit model. Similar

results were found at other cutoff values. Also, the neural network had less error variance and

lower prediction risk than the logit model.

Yi and Prybutok (1996) compared neural networks to multiple regression and ARIMA

models in an application to predict the maximum ozone concentration in a large metropolitan area.

The independent variables consisted of nine meteorological and auto emission measures. The

neural network model was statistically superior to both the regression and ARIMA models.

Empirical results have shown that Neural Networks outperform linear regression as data quality

varies was described by Bansal et al. (1993) and Marquez et al. (1991). Michaelides et al. (1995)

compared the performance of ANN with multiple linear regression in estimating missing rainfall

data over Cyprus.

Goh (1996, 1998) studied the comparison between ANN and multiple regressions in

construction, management, and engineering. Comrie (1997) studied multiple regression models

and neural networks are examined for a range of cities under different climate and ozone regimes,

enabling a comparative study of the two approaches, resulting neural network techniques are better

than regression models for daily ozone prediction. Ostensibly, Neural Network is simply an

extension of regression modeling which can be referred to a flexible non-linear regression models,

Menamin and Stuart (1997).

Venkatesan et al. (1997) have used neural network technique to predict monsoon rainfall

of India using few predictions and compared the results with linear regression techniques, showing

that the model based on neural network technique performed better. Baker (1998) compared linear

regression and neural network methods for forecasting educational spending and found Neural

Network provide comparable prediction accuracy. Man-Chung et al. (1998) proposed conjugate

gradient with multiple linear regression (MLR) weight initialization requires a lower computation

cost and learns better than steepest decent with random initialization for financial time series data

collected from Shanghai Stock Exchange.

Ranasinghe et al. (1999) compared ANN and multiple regression analysis in estimating

willingness to pay for urban water supply found that forecasting error of the best ANN model was

about half of the best multiple regression model. Hippert et al. (2000) proposed a hybrid

forecasting system that combines linear models and multilayer neural networks to forecast hourly

temperatures based on the past observed temperatures and the maximum and minimum forecast

temperatures supplied by the weather service.

Mathur et al. (2001) made a comparative study of neural network and regression models

for predicting stock prices and results verified the suitability and superiority of neural network

model over regression model. Victor (2001) described a neural network forecasting model as an

alternative to regression model for messy data problems and limitations in variable structure

specification. Zhang (2003) applied a hybrid methodology that combines both ARIMA and ANN

models to take advantage of the unique strength of ARIMA and ANN models in linear and

nonlinear modeling.

Maqsood et al. (2004) developed neural network based ensemble models and applied for

hourly weather forecasting of solution Saskatchewan. The experimental results show that the

ensemble network can be trained effectively without excessively compromising the performance.

Further, compared to the regression models, the ensemble networks forecast the weather parameter

with higher accuracy. Suhartono et al. (2005) made a comparative study of forecasting models for

trend and seasonal time series, concluding that the more complex model does not always yield

better forecast than the simpler one, especially on the testing samples. Also showing FFNN model

always yields better forecast in training data and it indicates an over fitting problem.

Taskaya-Temizel et al. (2005) suggested that the use of a nonlinear component may

degenerate the performance of hybrid methods and that a simpler hybrid comprising linear AR

model with a TDNN outperforms the more complex hybrid in tests on benchmark economic and

financial time series. Somvanshi et al. (2006) examined two fundamental different approaches

ARIMA and ANN for designing a model and predict the behavioral pattern in rainfall phenomena

based on past behavior. The study revealed that ANN model can be used as an approximate

forecasting tool to predict the rainfall, that out performs the ARIMA model.

Pandey et al. (2008) made a comparative study of neural network & fuzzy time series

forecasting techniques for crop yield and observed that neural network produces more accurate

results in comparison of fuzzy time series methods. Pao (2008) studied multiple linear regressions

and neural networks models with seven explanatory variables of corporation’s feature and three

external macro-economic control variables to analyze the important determinants of capital

structures of high-tech and traditional industries in Taiwan, respectively. The ANN models

achieve a better fit and forecast than the regression models for debt ratio. Aladag et al. (2009)

studied a new hybrid approach combining ERNN and ARIMA modes to time series data resulting

with best forecasting accuracy. Kulshrestha et al. (2009) examined that ANN gives more accurate

results to predict the probability of extreme rainfall than the probability by Fisher-Tippet Type II

distribution.

Khashei and Bijari (2011) proposed hybrid methodologies c

ombining linear models such as ARIMA and nonlinear models such as ANNs together and

found them more effective than traditional hybrid methodologies. Sharma and Singh (2011)

studied the forecasting models to make comparison among the model to identify the appropriate

model for the prediction of rainfall, concluding that ANN approach is better than other models.

Zaefizadeh et al. (2011) showed that in the ANN technique the mean deviation index of

estimation significantly was one-third of its rate in the MLR, because there was a significant

interaction between genotype and environment and its impact on estimation by MLR method.

Therefore, they recommended ANN approach as better predictor of yield in Barley than in

multiple linear regression. Recently, Ghodsi and Zakerinia (2012) used ARIMA, ANN and fuzzy

regression to analyse price forecasting. Fuzzy regression was found to be best method in

forecasting. Teri and Onal (2012) conducted MLR and ANN to forecast monthly river flow in

Turkey. The performance of the models suggested that the flow could be forecasting easily from

available flow data using ANN.

After a thorough review about the forecasting model, it becomes necessary to compare the

efficiency of different weather forecasting model and to identify the more precise weather

forecasting model. In the present work MLR, ARIMA and ANN models were used and hybrid

model of MLR with ARIMA and ANN, and a comparative analysis was done using different

analytical methods for weekly average time series data so as to identify the appropriate model.

FITTING OF PROBABILITY DISTRIBUTION

Introduction

Descriptive Statistics

Methodology

Probability Distribution Pattern

Conclusion

CHAPTER 3 FITTING OF PROBABILITY DISTRIBUTION

3.1 Introduction

Establishing a best fit probability distribution for different parameter has long been a topic

of interest in the field of meterology. The investigation of weather parameter distribution strongly

depends upon their distribution pattern. The present study is planned to identify the best fit

probability distribution based on distribution pattern for different data set. The 16 probability

distribution are identified out of large number of commonly used probability distribution for such

type of study. The descriptive statistics are computed first for each weather parameters for

different study periods. The parameters are discussed and explained through tables and graphs.

The test statistics D, 2A and 2 are computed for all 16 probability distribution. The best fit

probability distribution is identified based on highest ranks computed through all the three tests

independently. The best fit probability distribution so obtained is presented with their test statistic

value in each study period. It was further weighted using highest scores of selected probability

distribution for each study period. The combination of total test score of all the three test statistics

was computed for all the 16 probability distributions. The distribution having the maximum score

obtained from 18 set of data is identified. The parameters of the best fit probability distribution for

different data set of each weather parameters are presented. Fitted distribution was used to

generate random number for each data set. Finally, the best fit probability distribution for each

weather parameters was identified using the least square method.

3.2 Descriptive Statistics

The weekly data of seven parameters viz. rainfall, maximum and maximum temperature,

relative humidity at 7.00 am and 2.00 pm. bright sunshine hours and pan evaporation for the four

monsoon months were recorded. The monsoon season in this region lies between 15 to 20 weeks.

Keeping this point in view 17 weeks weather data from 4th June to 30th September of each year

was considered for the present study.

The descriptive statistics of the seasonal and weekly weather data set was computed

resulting the mean, standard deviation, skewness coefficient and coefficient of variation for all the

seven parameters. Minimum and maximum weekly value is also presented for each weather

parameter. The standard deviation indicate about the fluctuation of the parameter. The coefficient

of skewness are computed for all parameter which explain about the shape of the curve. The

coefficient of variation was computed for each parameters which explain the variability in the

data. The details of each parameter was also presented in the form of graphs. The period-wise

details of each weather parameters is presented and discussed in the subsequent sections.

Rainfall (mm)

The study period-wise summary of rainfall is presented in table 3.1 along with mean,

standard deviation, skewness, coefficient of variation, maximum and minimum values.

Table 3.1. Summary of statistics for Rainfall.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation

Skewness Coefficient

of

variation

Maximum

(weekly

total)

Minimum

(weekly

total)

Seasonal 4June-30Sep 73.20 80.40 0.4402 0.4022 443.20 0.00

1 week 4 june-10 june 29.07 53.92 3.1482 1.8546 263.60 0.00

2 week 11 june-17 june 41.21 57.11 2.3822 1.3859 291.80 0.00

3 week 18 june-24 june 46.32 51.61 1.941 1.1142 217.40 0.00

4 week 25 june-1 july 57.17 52.47 1.4613 0.9177 245.20 0.00

5 week 2 july-8 july 75.44 86.77 1.8573 1.1501 361.60 0.00

6 week 9 july-15 july 102.50 88.37 1.1507 0.8621 355.10 0.00

7 week 16 july-22 july 96.77 78.75 1.6998 0.8139 396.00 1.80

8 week 23 july-29 july 109.31 98.87 1.2496 0.9045 434.20 0.00

9 week 30 july-5 aug 74.83 59.20 0.8854 0.7911 223.30 0.90

10 week 6 aug-12 aug 93.99 87.17 1.8554 0.9275 443.20 0.00

11 week 13 aug-19 aug 104.07 94.04 1.4451 0.9036 422.80 3.10

12 week 20 aug-26 aug 99.55 87.83 1.4541 0.8823 395.20 0.00

13 week 27 aug-2 sep 75.55 82.09 1.8776 1.0865 413.20 0.00

14 week 3 sep-9 sep 79.26 81.90 1.5354 1.0333 353.20 0.00

15 week 10 sep-16 sep 60.43 68.96 1.5986 1.1411 296.20 0.00

16 week 17sep-23 sep 62.25 92.92 1.7897 1.4926 347.60 0.00

17 week 24 sep-30 sep 36.59 62.98 1.9873 1.7211 230.40 0.00

Where, the mean of seasonal rainfall of 50 years was 73.20 mm and mean of weekly

rainfall was varying from 29.07 mm in first week of June to 109.31 mm in last week of July. The

maximum value of weekly rainfall lies between 217.4 mm in third week of June in year 1975 to

443.20 mm in fourth week of August in year 2000.

The weekly minimum value of the rainfall in most of the weeks in 50 years was zero

except for the year 2002 which was the highest in third week of August. The standard deviation

for seasonal rainfall for 50 years was 80.40 mm while the weekly variation of standard deviation

ranging from 51.61 mm in third week of June to 98.87 mm in fourth week of July. The graphical

representation of the weekly rainfall is shown in figure 3.1 and weekly rainfall statistics for

seasonal 850 weeks of total 50 years is also presented in figure 3.2.

Figure 3.1 Mean, standard deviation and range of weekly Rainfall.

The maximum value of coefficient of variation for weekly data was observed 1.8546 in the

first week of June which indicates maximum fluctuation in the rainfall data set, that is, a large

variation in the occurrence of rainfall during the 50 years was observed. The measure of skewness

in seasonal data was 0.4402 and ranging from 0.8854 in first week of August to 3.1482 in first

week of June, which further shows a large degree of asymmetry of a distribution around its mean.

Maximum Temperature (OC)

The summary of statistics for maximum temperature were presented in table 3.2 along with

mean, standard deviation, skewness, coefficient of variation, maximum and minimum values,

where, the mean of weekly maximum temperature of 50 years seasonally was 33.08 OC and mean

of weekly maximum temperature was varying from 31.78 OC in second week of September to

37.75 OC in first week of June. The maximum value of seasonal maximum temperature was 43.20

OC in two years 1966 and 1967 and that of weekly maximum temperature was lying between

34.00 OC in third week of September in year 2007 to 43.20 OC in first week of June in two years

1966 and 1967.

It was moreover observed that the minimum value of the seasonal maximum temperature

was 23.60 OC in the year 2005. The weekly minimum value of the maximum temperature was

between 23.60 OC in fourth week of June in year 2005 to 30.20 OC in first week of June in year

2002. The standard deviation for seasonal maximum temperature for 50 years was 2.67 OC while

the weekly variation of standard deviation ranging from 1.19 OC in fourth week of September to

3.33 OC in first week of June.

The maximum value of coefficient of variation for weekly data was observed as 0.1018 in

the second week of June which indicates a large fluctuation in the maximum temperature data set.

The measure of skewness in seasonal data was -0.2994 and ranging from -1.3196 in first week of

September to 0.9789 in second week of July indicating the degree of asymmetry of a distribution

around its mean. The graphical representation of the weekly maximum temperature is shown in

figure 3.3 and weekly maximum temperature statistics for seasonal 850 weeks of total 50 years is

also presented in figure 3.4.

Table 3.2. Summary of statistics for Maximum Temperature.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum

Seasonal 4June-30Sep 33.08 2.67 -0.2994 0.0239 43.20 23.60

1 week 4 june-10 June 37.75 3.33 -0.3093 0.0881 43.20 30.20

2 week 11 june-17 june 36.44 3.71 -0.5374 0.1018 42.70 24.40

3 week 18 june-24 june 35.53 3.01 0.4169 0.0847 42.90 29.30

4 week 25 june-1 july 33.83 2.55 -0.7018 0.0754 41.90 23.60

5 week 2 july-8 july 33.80 1.89 0.5194 0.0558 39.30 29.80

6 week 9 july-15 july 32.84 1.95 0.9789 0.0591 39.70 29.60

7 week 16 july-22 july 32.25 1.48 -0.2906 0.0459 34.90 28.60

8 week 23 july-29 july 32.10 1.43 -0.1761 0.0444 34.90 28.20

9 week 30 july-5 aug 32.23 1.39 0.1455 0.0433 36.00 29.20

10 week 6 aug-12 aug 32.35 1.26 0.5677 0.0390 36.00 29.90

11 week 13 aug-19 aug 31.88 1.35 -0.3130 0.0425 34.10 28.50

12 week 20 aug-26 aug 31.89 1.19 0.0737 0.0373 35.40 29.10

13 week 27 aug-2 sep 32.11 1.32 -0.5801 0.0411 35.10 28.60

14 week 3 sep-9 sep 31.79 1.79 -1.3196 0.0565 35.30 24.80

15 week 10 sep-16 sep 31.78 1.39 -0.3879 0.0436 34.80 28.20

16 week 17sep-23 sep 31.85 1.26 -0.9759 0.0397 34.00 28.20

17 week 24 sep-30 sep 31.91 1.46 -0.7634 0.0458 34.20 28.10

Figure 3.3 Mean, standard deviation and range of weekly Maximum Temperature.

Minimum Temperature (OC)

The summary of statistics for minimum temperature in different period is presented in

table 3.3 along with mean, standard deviation, skewness, coefficient of variation, maximum and

minimum values, where, the mean of minimum temperature of 50 years seasonally was 24.43 OC

and mean of weekly minimum temperature was varying from 21.57 OC in last week of September

to 25.28 OC in fourth week of June. The maximum value of seasonal minimum temperature was

29.20 OC in year 1995 and for weekly minimum temperature lies between 24.50 OC in third week

of September in year 1998 to 29.20 OC in second week of June in year 1995.

It was further observed that the minimum value of the seasonal minimum temperature was

17.20 OC in the year 1984. The weekly minimum value of the minimum temperature was between

17.20OC in last week of September in year 1984 to 23.20 OC in second, third and fourth week of

July in years 1976, 1979 and 1981, respectively and also in first week of August in year 1975.

The standard deviation for seasonal minimum temperature for 50 years was 1.51OC while

the weekly variation of standard deviation ranging from 0.68 OC in third week of July to 1.81 OC

in third week of June. The maximum value of coefficient of variation for weekly data was

observed to be 0.0714 in the last week of September which indicates fluctuation in the minimum

temperature data set.

The measure of skewness in seasonal data was -0.6409 and ranging from -2.3776 in third

week of August to 0.0132 in third week of June indicating the degree of asymmetry of a

distribution around its mean. The graphical representation of the weekly minimum temperature is

shown in figure 3.5 and weekly minimum temperature statistics for seasonal 850 weeks of total 50

years is also presented in figure 3.6.

Table 3.3. Summary of statistics for Minimum Temperature.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum


1 week 4 june-10 June 24.07 1.75 -0.2130 0.0727 28.30 18.80

2 week 11 june-17 june 24.70 1.54 0.0132 0.0625 29.20 21.50

3 week 18 june-24 june 25.01 1.81 2.0372 0.0723 33.50 21.50

4 week 25 june-1 july 25.28 1.70 2.1990 0.0673 34.00 20.00

5 week 2 july-8 july 25.22 1.17 -1.4444 0.0463 27.90 20.10

6 week 9 july-15 july 25.25 0.89 -0.0492 0.0350 27.30 23.20

7 week 16 july-22 july 25.04 0.68 -0.4531 0.0272 26.20 23.20

8 week 23 july-29 july 25.17 0.93 0.2800 0.0367 27.20 23.20

9 week 30 july-5 aug 25.11 0.80 0.1502 0.0318 27.10 23.20

10 week 6 aug-12 aug 25.01 0.77 -0.3468 0.0308 26.60 22.70

11 week 13 aug-19 aug 24.80 0.90 -2.3776 0.0361 26.40 20.30

12 week 20 aug-26 aug 24.65 0.70 -0.5365 0.0284 26.50 22.60

13 week 27 aug-2 sep 24.45 0.81 -2.1114 0.0331 25.80 20.70

14 week 3 sep-9 sep 24.01 0.70 -0.4238 0.0292 25.30 22.20

15 week 10 sep-16 sep 23.44 0.80 -1.4321 0.0341 24.80 20.10

16 week 17sep-23 sep 22.54 1.34 -1.4456 0.0596 24.50 17.60

17 week 24 sep-30 sep 21.57 1.54 -0.7552 0.0714 24.90 17.20

Figure 3.5 Mean, standard deviation and range of weekly Minimum Temperature.

Relative Humidity at 7AM (%)

The study period wise summary of Relative Humidity at 7 AM is presented in table 3.4

along with mean, standard deviation, skewness, coefficient of variation, maximum and minimum

values, where, the mean of Relative Humidity at 7 AM of 50 years seasonally was 86.71 % and

mean of weekly average Relative Humidity at 7 AM was varying from 66.80 % in first week of

June to 91.65 % in fourth week of August.

The maximum value of seasonal average Relative Humidity at 7 AM was 98 % in four

years 1988, 1995, 1996 and 1998 and for weekly average Relative Humidity at 7 AM lies between

87 % in first week of June in years 1971 and 1984 to 98 % in second and third week of August in

year 1988 and 1995 respectively and also in first week of September in the two years 1996 and

1998.

It was moreover pragmatic that the least value of the seasonal average Relative Humidity

at 7 AM was 38 % in the year 1966. The weekly minimum value of the Relative Humidity at 7

AM was between 38 % in first week of June in year 1966 to 85 % in second week of September in

two years 1981 and 2008. The standard deviation for seasonal average Relative Humidity at 7 AM

for 50 years was 9.35 % while the weekly variation of standard deviation ranging from 2.97 % in

second week of September to 12.16 % in first week of June.

The maximum value of coefficient of variation for weekly data was observed as 0.1820 in

the first week of June which indicates fluctuation in the Relative Humidity at 7 AM data set. The

measure of skewness in seasonal data was -0.6237 and ranging from -1.7081 in fourth week of

July to 0.0113 in first week of June indicating the degree of asymmetry of a distribution around its

mean. The graphical representation of the weekly Relative Humidity at 7 AM is shown in figure

3.7 and weekly Relative Humidity at 7 AM statistics for seasonal 850 weeks of total 50 years is

also presented in figure 3.8.

Table 3.4. Summary of statistics for Relative Humidity at 7 AM.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum

Seasonal 4June-30Sep 86.71 9.35 -0.6237 0.0286 98 38

1 week 4 june-10 June 66.80 12.16 0.0113 0.1820 87 38

2 week 11 june-17 june 72.75 11.87 -0.1560 0.1632 94 46

3 week 18 june-24 june 78.93 10.79 -0.9558 0.1367 95 50

4 week 25 june-1 july 84.49 6.26 -1.0339 0.0740 95 63

5 week 2 july-8 july 86.64 4.85 -0.4663 0.0560 96 75

6 week 9 july-15 july 87.71 6.23 -1.3950 0.0710 96 63

7 week 16 july-22 july 89.65 4.00 -0.4663 0.0446 95 80

8 week 23 july-29 july 90.20 5.05 -1.7081 0.0560 97 69

9 week 30 july-5 aug 90.67 3.20 -0.1685 0.0353 96 84

10 week 6 aug-12 aug 90.64 3.59 -0.48305 0.0396 98 81

11 week 13 aug-19 aug 90.96 3.39 -0.2252 0.0373 98 83

12 week 20 aug-26 aug 91.65 3.10 -0.6682 0.0338 97 83

13 week 27 aug-2 sep 90.89 3.65 -0.6405 0.0402 97 79

14 week 3 sep-9 sep 91.60 3.29 -0.7568 0.0359 98 80

15 week 10 sep-16 sep 91.61 2.97 -0.2493 0.0325 96 85

16 week 17sep-23 sep 90.44 3.25 -0.6645 0.0359 96 80

17 week 24 sep-30 sep 88.43 5.23 -1.4181 0.0591 95 69

Figure 3.7 Mean, standard deviation and range of weekly average Relative Humidity at 7 AM.

Relative Humidity at 2 PM (%)

The summary of statistics for Relative Humidity at 2 PM for different study periods is

presented in table 3.5 along with mean, standard deviation, skewness, coefficient of variation,

maximum and minimum values, where, the mean of Relative Humidity at 2 PM of 50 years

seasonally was 65.70 % and mean of weekly average Relative Humidity at 2 PM was varying

from 39.98 % in first week of June to 74.45 % in eleventh week, that is, third week of August.

The maximum value of coefficient of variation for seasonal average Relative Humidity at 2

PM was 0.0537. The maximum value of coefficient of variation for weekly data was observed to

be 0.3903 in the first week of June which indicates a relatively high fluctuation in the Relative

Humidity at 2 PM data set.

The measure of skewness in seasonal data was -0.4778 and ranging from -1.0936 in fourth

week of June to 0.3368 in fourth week of August indicating the degree of asymmetry of a

distribution around its mean. The maximum value of seasonal average Relative Humidity at 2 PM

was 92 % in year 1988 and for weekly average Relative Humidity at 2 PM lies between 72 % in

first week of June in year 1962 to 92 % in second week of August in year 1988.

It was further observed that the minimum value of the seasonal average Relative Humidity

at 2 PM was 16 % in the two years 1965 and 2005. The weekly minimum value of the Relative

Humidity at 2 PM was between 16 % in second week of June in two years 1965 and 2005 to 62 %

in fourth week of August in year 1973. The standard deviation for seasonal average Relative

Humidity at 2 PM for 50 years was 13.60 % whereas the weekly variation of standard deviation

ranging from 5.01 % in twelfth week, that is, fourth week of August to 16.47 % in second week of

June.

The graphical representation of the weekly average Relative Humidity at 2 PM is shown in

figure 3.9 and weekly Relative Humidity at 2 PM statistics for seasonal 850 weeks of total 50

years is also presented in figure 3.10.

Table 3.5. Summary of statistics for Relative Humidity at 2 PM.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum

Seasonal 4June-30Sep 65.70 13.60 -0.4778 0.0537 92 16

1 week 4 june-10 June 39.98 15.60 0.3116 0.3903 72 17

2 week 11 june-17 june 47.15 16.47 0.1156 0.3494 80 16

3 week 18 june-24 june 55.70 13.92 -0.7909 0.2499 79 20

4 week 25 june-1 july 63.29 10.33 -1.0936 0.1632 82 24

5 week 2 july-8 july 65.18 10.44 -0.3999 0.1602 88 35

6 week 9 july-15 july 70.70 8.91 -0.4001 0.1260 85 50

7 week 16 july-22 july 72.09 6.77 0.2866 0.0938 85 61

8 week 23 july-29 july 73.34 7.36 -0.1840 0.1004 88 56.6

9 week 30 july-5 aug 73.13 6.43 -1.0567 0.0879 85 50

10 week 6 aug-12 aug 72.63 7.10 0.1527 0.0978 92 59

11 week 13 aug-19 aug 74.45 6.21 0.1023 0.0834 89 61

12 week 20 aug-26 aug 72.97 5.01 0.3368 0.0686 87 62

13 week 27 aug-2 sep 70.54 7.22 -0.3006 0.1024 83 53

14 week 3 sep-9 sep 70.26 7.86 -0.2773 0.1118 85.6 50

15 week 10 sep-16 sep 69.85 8.03 0.1521 0.1150 89 54

16 week 17sep-23 sep 65.63 8.02 -0.5267 0.1221 81 42

17 week 24 sep-30 sep 60.01 9.66 -0.4532 0.1610 78 36

Figure 3.9 Mean, standard deviation and range of weekly average Relative Humidity at 2 PM.

Pan Evaporation (mm)

The summary of statistics for Pan Evaporation is presented in table 3.6 along with mean,

standard deviation, skewness, coefficient of variation, maximum and minimum values, where, the

mean of Pan Evaporation of 50 years seasonally was 5.31 mm and mean of weekly average Pan

Evaporation was varying from 3.83 mm in third week of September to 10.59 mm in first week of

June.

The maximum value of seasonal average Pan Evaporation was 18.50 mm in year 1967 and

for weekly average Pan Evaporation lies between 5.20 mm in third week of September in year

1996 to 18.50 mm in first week of June in year 1967. It was also observed that the minimum value

of the seasonal average Pan Evaporation was zero. The weekly minimum value of the Pan

Evaporation was between zero mm in second and fourth week of July in the same year 1976 and

second week of August in year 1972 to 4.20 mm in first week of June in year 1971.

The standard deviation for seasonal average Pan Evaporation for 50 years was 2.67mm

with the weekly variation of standard deviation ranging from 0.76 mm in last week of September

to 2.94 mm in second week of June. The maximum value of coefficient of variation for weekly

data was observed as 0.4402 in the fourth week of July indicating fluctuation in the Pan

Evaporation data set.

The measure of skewness in seasonal data was -0.4380 and weekly average is ranging from

-0.0483 in last week of September to 0.9927 in third week of August indicating the degree of

asymmetry of a distribution around its mean. The graphical representation of the weekly average

Pan Evaporation is shown in figure 3.11 and weekly Pan Evaporation statistics for seasonal 850

weeks of total 50 years is also presented in figure 3.12.

Table 3.6. Summary of statistics for Pan Evaporation.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum


1 week 4 june-10 June 10.59 2.78 0.1739 0.2623 18.50 4.20

2 week 11 june-17 june 9.00 2.94 0.5974 0.3264 17.20 2.90

3 week 18 june-24 june 8.19 2.67 0.7330 0.3259 15.90 3.60

4 week 25 june-1 july 6.26 1.83 0.7244 0.2921 12.10 2.10

5 week 2 july-8 july 5.89 2.17 0.4327 0.3676 11.40 1.40

6 week 9 july-15 july 5.29 1.92 -0.2450 0.3633 9.50 0.00

7 week 16 july-22 july 4.88 1.75 0.7634 0.3585 10.20 1.50

8 week 23 july-29 july 4.12 1.82 0.3147 0.4402 9.80 0.00

9 week 30 july-5 aug 4.15 1.23 -0.2453 0.2957 6.50 1.60

10 week 6 aug-12 aug 4.04 1.45 -0.6027 0.3573 6.70 0.00

11 week 13 aug-19 aug 4.14 1.49 0.9927 0.3608 9.50 1.60

12 week 20 aug-26 aug 4.04 1.20 -0.0942 0.2958 6.40 1.50

13 week 27 aug-2 sep 4.07 1.26 0.2181 0.3087 7.60 1.80

14 week 3 sep-9 sep 3.97 1.04 -0.2653 0.2608 6.30 1.60

15 week 10 sep-16 sep 3.88 0.91 -0.9353 0.2332 5.50 1.20

16 week 17sep-23 sep 3.83 0.83 -0.3608 0.2163 5.20 1.90

17 week 24 sep-30 sep 3.85 0.76 -0.0483 0.1978 5.60 2.30

Figure 3.11 Mean, standard deviation and range of weekly average Pan Evaporation.

Bright Sunshine (hours)

The summary of Bright Sunshine for different study period is presented in table 3.6 along

with mean, standard deviation, skewness, coefficient of variation, maximum and minimum values,

where, the mean of Bright Sunshine of 50 years seasonally was 6.38 hours and mean of weekly

average Bright Sunshine was varying from 5.30 hours in fourth week of July to 8.60 hours in first

week of June.

The maximum value of seasonal average Bright Sunshine was 11.60 hours in the two years

1986 and 2009 and for weekly average Bright Sunshine was between 8.80 hours in third week of

August in year 1965 to 11.60 hours in first and third week of June in years 1986 and 2009

respectively. It was also observed that the minimum among the seasonal average Bright Sunshine

was 0.70 hours in the year 2009. The weekly minimum value of the Bright Sunshine was between

0.70 hours in third week of August in the year 2009 to 4.20 hours in first week of June in year

2000.

The standard deviation for seasonal average Bright Sunshine for 50 years was 2.16 hours

with the weekly variation of standard deviation ranging from 1.72 hours in fourth week of July to

2.21 hours in third week of June. The maximum value of coefficient of variation for weekly data

was observed in the second week of August which indicates a reasonable fluctuation in the Bright

Sunshine data set.

The measure of skewness in seasonal data was 0.1935 and weekly average is ranging from

-1.2914 in second week of June to 0.4982 in second week of august indicating the measure of

asymmetry of a distribution around its mean. The graphical representation of the weekly average

Bright Sunshine is shown in figure 3.13 and weekly Bright Sunshine statistics for seasonal 850

weeks of total 50 years is also presented in figure 3.14.

Table 3.7. Summary of statistics for Bright Sunshine.

Study

Period

Data

(From – To)

Parameters

Mean Standard

Deviation


of

variation

Maximum Minimum

Seasonal 4June-30Sep 6.38 2.16 0.1935 0.1060 11.60 0.70

1 week 4 june-10 June 8.60 1.80 -0.5602 0.2092 11.60 4.20

2 week 11 june-17 june 7.79 1.97 -1.2914 0.2528 11.30 1.30

3 week 18 june-24 june 7.12 2.21 -0.4005 0.3107 11.60 1.70

4 week 25 june-1 july 6.19 1.74 0.0237 0.2812 10.00 2.40

5 week 2 july-8 july 6.15 1.92 -0.4944 0.3126 9.30 1.70

6 week 9 july-15 july 5.66 2.10 -0.2721 0.3705 9.70 0.90

7 week 16 july-22 july 5.54 1.77 -0.3552 0.3197 8.90 1.40

8 week 23 july-29 july 5.30 1.72 0.2553 0.3255 9.10 1.20

9 week 30 july-5 aug 5.66 1.80 -0.1008 0.3186 9.30 2.00

10 week 6 aug-12 aug 5.56 2.07 0.4982 0.3711 11.30 0.90

11 week 13 aug-19 aug 5.41 1.73 -0.6706 0.3192 8.80 0.70

12 week 20 aug-26 aug 5.52 1.89 0.1980 0.3429 9.30 1.60

13 week 27 aug-2 sep 6.02 2.02 0.0019 0.3360 9.90 1.60

14 week 3 sep-9 sep 6.03 2.20 -0.1090 0.3644 9.50 2.40

15 week 10 sep-16 sep 6.49 2.06 -0.1149 0.3166 10.70 2.60

16 week 17sep-23 sep 7.31 1.90 -0.4497 0.2601 10.60 2.30

17 week 24 sep-30 sep 8.02 1.87 -0.9273 0.2328 10.30 3.20

Figure 3.13 Mean, standard deviation and range of weekly average Bright Sunshine.

3.3 Methodology

Weather parameter data was analyzed to identify the best fit probability distribution for

each period of study. Three statistical goodness of fit test were carried out in order to select the

best fit probability distribution on the basis of highest rank with minimum value of test statistic.

The appropriate probability distributions are identified for the different dataset using maximum

overall score based on sum of individual point score obtained from three selected goodness of fit

test. Random numbers were generated for actual and estimated weekly weather parameters for

each period of study using the parameters of selected distributions.

3.3.1 Fitting the probability distribution

The probability distributions viz. normal, lognormal, gamma, weibull, pearson, generalized

extreme value were fitted to the data for evaluating the best fit probability distribution for weather

parameters. In addition, the different forms of these distributions were also tried and thus total 16

probability distributions viz. normal, lognormal (2P, 3P), gamma (2P, 3P), generalized gamma

(3P, 4P), log-gamma, weibull (2P, 3P), pearson 5 (2P, 3P), pearson 6 (3P, 4P), log-pearson 3,

generalized extreme value were applied to find out the best fit probability distribution The

description of various probability distribution functions viz. density function, range and the

parameter involved are presented in table 3.8.

3.3.2 Testing the goodness of fit

The goodness of fit test measures the compatibility of random sample with the theoretical

probability distribution. The goodness of fit tests is applied for testing the following null

hypothesis:

HO: the weather parameter data follow the specified distribution

HA: the weather parameter data does not follow the specified distribution.

Table 3.8. Description of various probability distribution functions.

Distribution Probability density function Range Parameters

Gamma

(3P) 1

( ) exp( )

x xf x

x

( 0)

( 0)

( 0

)

shape parameter

scale parameter

location parameter

yields thetwo parameter

gamma distribution

Gamma function

Gamma

(2P) 1

( ) exp( )

x xf x

Generalized

Extreme

Value

11 1 1exp 1 1 0

( )1

exp exp 0

kkkz kz k

f x

z z k

1 0 0

0

xk for k

x for k

( 0)scale parameter

k shape parameter

location parameter

xwhere z

Generalized

Gamma

(4P)

1

( ) exp( )

k kk x x

f xk

x

( 0)

( 0)

( 0)

( 0

)

k shape parameter k

shape parameter

scale parameter

location parameter

yields the three parameter

Generalized gamma distribution

Generalized

Gamma

(3P)

1

( ) exp( )

k kkx xf xk

Log-

Gamma 1

( ) exp( )

In x In xf x

x

0 x

( 0)

( 0)

shape parameter

scale parameter

Lognormal

(3P)

21 ( )

exp2

( )2

In x

f xx

x

( 0)

( 0)

( 0

)

scale parameter

shapeparameter

location parameter


lognormaldistribution

Lognormal

(2P)

21 ( )

exp2

( )2

In x

f xx

Log-

Pearson 3 exp

11 ( ) ( )

( )( )

In x In xf x

x

0 0

0

x e

e x

( 0)

( 0)

shape parameter

scale parameter

location parameter

Table 3.8. Continued

The following goodness-of-fit tests viz. Kolmogorov-Smirnov test and Anderson-Darling

test were used along with the chi-square test at (0.01) level of significance for the selection of

the best fit probability distribution. The distribution function of these tests is explained in brief in

the next section.

Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov statistic (D) is defined as the largest vertical difference between

the theoretical and the empirical cumulative distribution function (ECDF):

i 1 iD max (x ) , (x )

i i1 i n n nF F

(1)

Where,

Xi = random sample, i =1, 2,….., n.

Distribution Probability density function Range Parameters

Normal 21 1

( ) exp22

xf x

x ( 0)

mean

standard Deviation

Pearson 5

(3P)

exp( )

1( ) ( )

xf x

x

x

( 0)

( 0)

( 0

5 )

shape parameter

scale parameter

location parameter


pearson distribution

Pearson 5

(2P)

exp( )

1( )

xf x

x

Pearson 6

(4P)

11

( )1 2( , ) 1 ( )

1 2

xf x

B x

x

1 1

2 2

( 0)

( 0)

( 0)

( 0

6 )

shape parameter

shape parameter

scale parameter

location parameter

yields thethree parameter

pearson distribution

Pearson 6

(3P)

11

( )1 2( , ) 1

1 2

xf x

B x

Weibull (3P) 1

( ) expx x

P x

x

( 0)

( 0)

( 0

)

shape parameter

scale parameter

location parameter

yieldsthetwo parameter

weibull distribution

Weibull (2P) 1

( ) expx x

P x

CDF= 1

F (x)= . Number of observations xnn

(2)

This test is used to decide if a sample comes from a hypothesized continuous distribution.

Anderson-Darling Test

The Anderson-Darling statistic (A2) is defined as

i=1

n

+n-i 1

12A = - n - (2i -1).[In F(X ) + In(1- F(X ))]

in (3)

It is a test to compare the fit of an observed cumulative distribution function to an expected

cumulative distribution function. This test gives more weight to the tails then the Kolmogorov-

Smirnov test.

Chi-Squared Test

The Chi-Squared statistic is defined as

i=1

k

2O - E2 i i

χ =E

i

(4)

Where,

Oi = observed frequency,

Ei = expected frequency,

‘i’= number of observations (1, 2, …….k)

Ei is calculated by the following computation

i 2 1E =F(x ) - F(x ) (5)

F is the CDF of the probability distribution being tested.

The observed number of observation (k) in interval ‘i’ is computed from equation given below

k 1 log n2

(6)

Where, n is the sample size.

This test is for continuous sample data only and is used to determine if a sample comes from a

population with a specific distribution.

3.3.3 Identification of best fit probability distribution

The three goodness of fit test mentioned above were fitted to the weather parameters data

treating different data set. The test statistic of each test were computed and tested at ( =0.01)

level of significance. Accordingly the ranking of different probability distributions were marked

from 1 to 16 based on minimum test statistic value. The distribution holding the first rank was

selected for all the three tests independently. The assessments of all the probability distribution

were made on the bases of total test score obtained by combining the entire three tests. Maximum

score 16 was awarded to rank first probability distribution based on the test statistic and further

less score were awarded to the distribution having rank more than 1, that is 2 to 16. Thus the total

score of the entire three tests were summarized to identify the fourth distribution on the bases of

highest score obtained.

The probability distribution having the maximum score was included as a fourth

probability distribution in addition to three probability distributions which were previously

identified. Thus on the bases of the four identified probability distribution the procedure for

obtaining the best fitted probability distribution is explained below:

Generating random numbers

The four probability distributions identified for each data set were used to select the best

probability distribution. The parameters of these four probability distributions were used to

generate the random numbers.

Least square method

The least square method was used to identify the best fit probability. The random numbers

were generated for the distributions and residuals (R) were computed for each observation of the

data set.

n

i ii=1

Y YR

(7)

Where, iY = the actual observation

iY

= the estimated observation ( i = 1, 2,….., n )

The distribution having minimum sum of residuals was considered to be the best fit

probability distribution for that particular data set. Finally the best fit probability distributions for

weather parameters on different sets of data were obtained and the best fit distribution for each set

of data was identified. The convergence and performance of the best fit probability distribution

had been evaluated on the basis of the Easy fit 5.5 version of software.

3.4 Probability Distribution

3.4.1 Introduction

The methodology presented above is applied to the 50 years weather data as classified into 18

data sets. These 18 data sets are classified as 1 seasonal and 17 weekly to study the distribution pattern

at different levels. The test statistic D, 2A and 2 for each data set are computed and the

combination of total test score are obtained for each data set for all probability distributions.

The distribution is identified using maximum overall score based on sum of individual point

score obtained from three selected goodness of fit test. The distributions identified which are having

highest score and the best fit are listed in counter form where the parameter of these identified

distribution for each data set of weather parameters are also mentioned.

These values of the parameter are used to generate random numbers for each data set and

the least square method is used for the weather parameters analysis. The residuals are computed

for each data set of weather parameters. Sum of the deviation are obtained for all identified

distribution. The probability distribution having minimum deviation is treated as the best selected

probability distribution for the individual data set. The discussions on the results for the individual

parameters are mentioned below.

3.4.2 Rainfall

The test statistic D, 2A and 2 for each data set, of rainfall is computed for 16 probability

distribution. The probability distribution having the first rank along with their test statistic is

presented in table 3.9(a). It is observed that for seasonal rainfall, Pearson 5 (3P) distribution is

fitted using Kolmogorov Smirnov and Anderson Darling tests based on first rank. Similarly,

Normal distribution is fitted using Chi-square test for seasonal rainfall. Thus these probability

distributions are identified as the best fit based on these three tests independently.

The fourth probability distribution identified, which is having highest score, is presented in

table 3.9(b) with their scores. Those distributions which are having same highest score are also

included in the selected probability distribution. The probability distributions fitted for the fourth

week data set, that is, last week of June are Generalized Extreme Value distribution and

Lognormal (3P) distribution based on highest score. While for ninth week data set, that is, in first

week of August, Weibull (2P, 3P) distributions are fitted having 36 as highest score and for the

thirteenth week, that is, in last week of August, Gamma (2P) distribution and Generalized Extreme

Value distribution, having 38 as the highest score are selected.

It is observed that Generalized Extreme Value probability distribution is fitted in more than

50% weeks. The distributions identified are thus listed in table 3.9(c) where the parameter of these

identified distribution for each data set is mentioned.


the least square method is used for selecting the best fit probability distribution. The probability

distribution having minimum deviation is treated as the best selected probability distribution for

the individual data set for the rainfall as presented in \table 3.9(d).

Normal distribution represents the best fitted distribution for seasonal rainfall and is also

observed in the sixth week data set, that is, second week of July. Generalized Extreme Value is

observed six times in the weekly data sets, means, first, second, tenth, fifteenth, twelfth and thirteenth

weeks, that is, first week of June, second week of June, August and September, and last two weeks of

august, respectively, indicating the highest contribution of the distribution.

Further, we observe that Gamma (3P) distribution, Log-Pearson 3 distribution, Pearson 6

(3P) distribution and Lognormal (3P) distribution are found as the best fitted probability

distributions for the weekly rainfall data sets.

Table 3.9(a). Distributions fitted for Rainfall data sets.

Table 3.9(b). Distributions with highest score for Rainfall data sets.

Study Period Distributions with highest Score

Distribution Score

Seasonal Pearson 5 (3P) 42

1 Week Gen. Extreme 42



4 Week Gen. Extreme and Lognormal (3P) 39

5 Week Gamma (2P) 34



8 Week Lognormal (3P) 42

9 Week Weibull (2P) and Weibull (3P) 36




13 Week Gamma (2P) and Gen. Extreme 38


15 Week Pearson 6 (3P) 38



Study

period

Test ranking first position

Kolmogorov Smirnov Anderson Darling Chi-square

Distribution Statistic Distribution Statistic Distribution Statistic

Seasonal Pearson 5 (3P) 0.0681 Pearson 5 (3P) 0.2976 Normal 1.3956

1 Week Gen. Extreme 0.1762 Gen. Extreme 2.1837 Gen. Extreme 8.3819


3 Week Gen. Extreme 0.0717 Gen. Extreme 0.3888 Pearson 6 (3P) 2.5659

4 Week Gen. Extreme 0.0735 Gen. Extreme 0.3163 Lognormal (3P) 0.8179

5 Week Pearson 6 (3P) 0.0912 Pearson 6 (3P) 0.5049 Gen. Gamma (4P) 0.4317

6 Week Gen. Gamma (4P) 0.0777 Gen. Extreme 0.4326 Normal 2.5651

7 Week Gen. Extreme 0.0931 Gen. Extreme 0.4959 Gen. Gamma (3P) 0.7131

8 Week Lognormal (3P) 0.0702 Lognormal (3P) 0.3222 Lognormal (3P) 1.0616

9 Week Weibull (2P) 0.0719 Log-Pearson 3 0.2836 Lognormal (3P) 0.9351

10 Week Lognormal (3P) 0.0584 Gen. Extreme 0.1731 Gen. Extreme 0.9746

11 Week Gamma (2P) 0.0689 Log- Pearson 3 0.2110 Log-Pearson 3 0.0377


13 Week Gamma (2P) 0.1048 Gen. Extreme 0.8361 Gamma (2P) 2.1996


15 Week Weibull (3P) 0.0852 Pearson 6 (3P) 0.5338 Lognormal (3P) 1.0211

16 Week Gamma (3P) 0.1337 Pearson 6 (3P) 1.1343 Pearson 6 (3P) 2.4038

17 Week Gamma (3P) 0.1596 Pearson 6 (3P) 2.4205 Gen. Extreme 3.9689

Table 3.9(c). Parameters of the distributions fitted for Rainfall data sets.

Study Period Distributions Parameters

Seasonal Normal =29.44 =73.195

Pearson 5(3P) =112.06 =33906.0 =--232.11

1 week Gen. Extreme Value k=0.57245 =12.389 =5.8622



Pearson 6 (3P) 1=0.57131 2=2.8403E+8 =2.2429E+10


Lognormal (3P) =0.80884 =3.9156 =-10.454

5 week Gamma (2P) =0.75596 =99.791

Gen. Gamma (4P) k=1.5013 =0.31028 =197.88 =0.01

Pearson 6 (3P) 1=0.63702 2=4.6454E+7 =5.5026E+9


Gen. Gamma (4P) k=3.1445 =0.19166 =270.22 =0.01

Normal =88.365 =102.5


Gen. Gamma (3P) k=0.98854 =1.4834 =64.094

8 week Lognormal (3P) =0.79731 =4.5807 =-21.17

9 week Log-Pearson 3 =2.7764 =-0.64072 =5.6879

Lognormal (3P) =0.73597 =4.2048 =-11.153

Weibull (2P) =1.0891 =79.975

Weibull (3P) =1.2235 =79.444 =0.33401


Lognormal (3P) =0.73608 =4.4708 =-19.338

11 week Gamma (2P) =1.2248 =84.969

Log-Pearson 3 =12.356 =-0.30187 =7.9246


13 week Gamma (2P) =0.84718 =89.184

Gen. Extreme Value k=0.24491 =45.313 =35.09


15 week Lognormal (3P) =1.4308 =3.3643 =-1.496

Pearson 6 (3P) 1=0.53587 2=2.4525E+5 =2.7661E+7

Weibull (3P) =0.64552 =61.049 =0.01

16 week Gamma (3P) =0.23816 =209.68 =0.01

Pearson 6 (3P) 1=0.27517 2=7.8369E+7 =1.7728E+10

17 week Gamma (3P) =0.18131 =139.08 =0.01

Gen. Extreme Value k=0.56455 =16.239 =6.8411

Pearson 6 (3P) 1=0.21422 2=3.4782E+7 =5.9395E+9

Table 3.9(d). Best fit probability distribution for Rainfall.

STUDY PERIOD BEST-FIT

Seasonal Normal

1 Week Gen. Extreme Value


3 Week Pearson 6 (3P)

4 Week Lognormal (3P)


6 Week Normal

7 Week Gen. Gamma (3P)

8 Week Lognormal (3P)

9 Week Log-Pearson 3








17 Week Gamma (3P)

3.4.3 Maximum Temperature

The test statistic D, 2A and 2 for each data set, of maximum temperature is computed for

16 probability distribution. The probability distribution having the first rank along with their test

statistic is presented in table 3.10(a). It is observed that for seasonal maximum temperature,

Weibull (2P) distribution is fitted using Kolmogorov Smirnov and Chi-square tests based on first

rank. Similarly, Log Pearson 3 is fitted using Anderson Darling test for seasonal maximum

temperature. Thus these probability distributions are identified as the best fit based on these three

tests independently.

The fourth probability distribution identified, which is having highest score, is presented in

table 3.10(b) with their scores. The Log-Pearson 3 distribution is fitted having the highest score as

43 for the seasonal maximum temperature data set and also for third and seventh week with the

score of 44 and 47 respectively. Pearson 5 (3P) is observed consecutively in the fifth and sixth

week with the score 40 and 44, respectively, and is also in the eleventh week with a score of 37.

Similarly, Weibull (3P) is observed successively in the last two weeks and also in the fourteenth

week, that is, first week of September. The distributions identified are thus listed in table 3.10(c),

where, the parameter of these identified distribution for each data set are mentioned.

To generate random numbers for each data set these values of the parameter are used and

the least square method is used for selecting the best fit probability distribution. The probability


the individual data set for the maximum temperature as presented in table 3.10(d).

Table 3.10(a). Distributions fitted for Maximum Temperature data sets.

Weibull (2P) distribution represents the best fitted probability distribution for seasonal

maximum temperature and is also observed in the ninth and twelfth week data set, that is, first and

fourth week of August, respectively. Further, Log-Pearson 3 is observed consecutively in the sixth

and seventh week data set, that is, second and third week of July, respectively.

Study

period




Seasonal Weibull (2P) 0.0746 Log Pearson 3 0.4322 Weibull (2P) 2.2681


2 Week Pearson 6 (4P) 0.0647 Pearson 6 (4P) 0.3325 Pearson 5 (2P) 0.2498

3 Week Pearson 5 (3P) 0.0785 Log Pearson 3 0.2532 Log Pearson 3 1.4033

4 Week Gen. Extreme 0.1139 Log Normal (3P) 1.1606 Log Normal (3P) 4.2667

5 Week Pearson 5 ( 3P) 0.0710 Pearson 5 (3P) 0.1673 Weibull (3P) 0.3935

6 Week Gen. Extreme 0.0500 Pearson 5 (3P) 0.1330 Log Pearson 3 0.7313

7 Week Weibull (3P) 0.0611 Log Pearson 3 0.1767 Log Pearson 3 1.0778

8 Week Pearson 5 (3P) 0.0667 Gen. Extreme 0.2656 Gen. Extreme 0.3337

9 Week Pearson 6 (3P) 0.0871 Gen. Gamma (3P) 0.2481 Weibull (2P) 1.7466

10 Week Weibull (3P) 0.0775 Gen. Extreme 0.3098 Gen. Extreme 1.6984

11 Week Pearson 5 (3P) 0.0754 Log Normal (3P) 0.4681 Log Pearson 3 0.3302

12 Week Weibull (2P) 0.0631 Pearson 6 (4P) 0.3450 Pearson 5 (3P) 2.3160

13 Week Gen. Extreme 0.0736 Weibull (2P) 0.2783 Weibull (2P) 2.7992


15 Week Weibull (3P) 0.0557 Log Pearson 3 0.1649 Pearson 5 (2P) 1.3690

16 Week Weibull (3P) 0.0691 Weibull (3P) 0.2918 Weibull (3P) 1.9102


Similarly, Weibull (3P) is observed successively in the sixteenth and seventeenth week,

that is, in last two weeks of September and also in the fifth and fourteenth week, that is, in first

week of July and September, respectively. Further, we observe that Generalized Gamma (3P)

distribution, Generalized Extreme Value, Pearson 5 (2P, 3P) distribution and Lognormal (3P)

distribution are found as the best fitted probability distributions for the weekly maximum

temperature data sets.

Table 3.10(b). Distributions with highest score for Maximum Temperature data sets.


Distribution Score

Seasonal Log Pearson 3 43

1 Week Gen. Extreme Value 44


3 Week Log Pearson 3 44

4 Week Log Normal (3P) 44



7 Week Log Pearson 3 47

8 Week Log Gamma 45





13 Week Weibull (2P) 44


15 Week Gen. Gamma (3P) 36



Table 3.10(c). Parameters of the distributions fitted for Maximum Temperature data sets.


Seasonal Log-Pearson 3 =25.842 =-0.00473 =3.6207

Weibull (2P) =50.1 =33.395

1 week Gen. Extreme Value k=-0.41111 =3.5854 =36.758

Pearson 6 (4P) 1=1530.1 2=2181.0 =142.91 =-62.564

2 week Pearson 5 (2P) =87.078 =3137.7

Pearson 6 (4P) 1=1.5633E+6 2=2.6044E+5 =289.5 =-

1701.3

3 week Log-Pearson 3 =100.04 =0.00839 =2.7275

Pearson 5 (3P) =68.215 =1634.5 =11.209


Lognormal (3P) =0.02174 =4.7508 =-81.87

Table 3.10(c). Continued


5 week Pearson 5 (3P) =69.116 =1040.8 =18.517

Weibull (3P) =2.597 =5.1432 =29.223


Log-Pearson 3 =7.0789 =0.02172 =3.3361

Pearson 5 (3P) =21.315 =171.78 =24.383

7 week Log-Pearson 3 =23.506 =-0.00954 =3.6966

Weibull (3P) =5.0304 =6.9459 =25.872


Log-Gamma =6023.6 =5.7572E-4

Pearson 5 (3P) =331.76 =8638.6 =5.9613

9 week Gen. Gamma (3P) k=1.0026 =543.3 =0.0603

Pearson 6 (3P) 1=2984.7 2=668.66 =7.2093

Pearson 6 (4P) 1=3429.0 2=950.5 =10.414 =-5.3796

Weibull (2P) =28.524 =32.749


Weibull (3P) =2.4245 =3.2317 =29.477

11 week Log-Pearson 3 =22.25 =-0.00908 =3.6631

Lognormal (3P) =0.03054 =3.792 =-12.46

Pearson 5 (3P) =319.73 =7827.0 =7.3207

12 week Pearson 5 (3P) =308.23 =6381.1 =11.114

Pearson 6 (4P) 1=4004.5 2=10022.0 =157.47 =-31.041

Weibull (2P) =33.748 =32.317


Weibull (2P) =29.152 =32.632


Weibull (3P) =20.417 =29.794 =2.7487

15 week Gen. Gamma (3P) k=1.0009 =528.38 =0.0605

Log-Pearson 3 =14.722 =-0.01149 =3.6271

Pearson 5 (2P) =521.2 =16533.0

Weibull (3P) =5.682 =7.2696 =25.058

16 week Weibull (3P) =1.5522E+7 =1.5144E+7 =-1.5144E+7

17 week Weibull (3P) =41.383 =48.366 =-15.802

Table 3.10(d). Best fit probability distribution for Maximum Temperature data sets.


Seasonal Weibull (2P)




4 Week Log Normal (3P)

5 Week Weibull (3P)

6 Week Log Pearson 3

7 Week Log Pearson 3


9 Week Weibull (2P)

10 Week Gen Extreme value


12 Week Weibull (2P)

13 Week Gen. Extreme





3.4.4 Minimum Temperature

The test statistic D, 2A and 2 for each data set, of minimum temperature is computed for


statistic is presented in table 3.11(a). It is observed that for seasonal minimum temperature,

Lognormal (3P) distribution is fitted using Kolmogorov Smirnov, Weibull (3P) distribution is

fitted using Anderson Darling tests and Gamma (3P) distribution is fitted using Chi-square test

based on first rank. Thus these probability distributions are identified as the best fit based on these

three tests independently.

The fourth probability distribution identified which is having highest score, is presented in


included in the selected probability distribution.

Table 3.11(a). Distributions fitted by the tests for Minimum Temperature data sets.

Table 3.11(b). Distributions with highest score for Minimum Temperature data sets.


Distribution Score

Seasonal Lognormal (3P) 36

1 Week Lognormal (3P) and Pearson 6 (4P) 42

2 Week Log-Pearson 3 42

3 Week Gen. Extreme and Gen. Gamma (4P) 39



6 Week Log-Gamma 39

7 Week Normal 38


9 Week Normal 39









Study

period




Seasonal Lognormal (3P) 0.0606 Weibull (3P) 0.1507 Gamma (3P) 0.2870

1 Week Lognormal (3P) 0.0667 Lognormal (3P) 0.2740 Weibull (3P) 3.4178

2 Week Weibull (2P) 0.0883 Log- Pearson 3 0.6013 Weibull (2P) 2.0818

3 Week Gamma (3P) 0.0785 Gen. Extreme 0.3980 Weibull (2P) 1.6087

4 Week Gen. Extreme 0.1264 Gen. Extreme 1.7918 Weibull (2P) 2.5028

5 Week Gen. Extreme 0.1562 Weibull (3P) 1.4011 Gamma (2P) 4.8657

6 Week Gamma (3P) 0.0886 Gen. Extreme 0.3089 Gen. Extreme 3.6625

7 Week Gen. Gamma (4P) 0.0712 Gen. Extreme 0.4068 Pearson 6 (4P) 1.5668


9 Week Normal 0.0710 Lognormal (2P) 0.3093 Normal 0.7627

10 Week Gen. Extreme 0.0751 Normal 0.2347 Pearson 5 (3P) 0.4314



13 Week Weibull (2P) 0.1203 Weibull (3P) 0.3749 Pearson 5 (3P) 3.3573

14 Week Weibull (2P) 0.0887 Log-Pearson 3 0.3438 Gen. Extreme 3.2287

15 Week Weibull (3P) 0.0896 Weibull (3P) 0.6710 Gamma (2P) 1.2229

16 Week Gen. Extreme 0.0723 Weibull (3P) 0.3678 Pearson 5 (2P) 1.5721


The probability distributions fitted for the first week data set, that is, first week of June are

Lognormal (3P) and Pearson 6 (4P) distributions based on the highest score as 42. While for the

third week of June, Generalized Extreme value and Generalized Gamma (4P) distribution having 39

as the highest score are selected. The distributions identified are thus listed in table 3.11(c) where

the parameter of these identified distribution for each data set are mentioned.

These values of the parameter are further utilized to generate random numbers for each

data set and the least square method is worn for selecting the best fit probability distribution. The

probability distribution having minimum deviation is treated as the best selected probability

distribution for the individual data set for the minimum temperature as presented in table 3.11(d).

Weibull (3P) distribution represents the best fitted distribution for seasonal minimum temperature

and is also observed in the fifteenth week data set, that is, second week of September.

Further, we observe that Generalized Extreme Value is observed repetitively in sixth,

seventh and eighth week, that is, in second, third and fourth week of July, respectively, also in the

fourth week, that is, last week of June. Further, it was observed, Weibull (2P) appeared four times

among the 17 weeks, that is, in second, twelfth, fourteenth and seventeenth week. Moreover,

Gamma (2P, 3P) distributions, Normal distribution, Pearson 5 (2P, 3P) distributions and Pearson 6

(4P) distribution are obtained as the best fitted probability distributions for the weekly minimum


Table 3.11(c). Parameters of the distributions fitted for Minimum Temperature data sets.


Seasonal Gamma (3P) =417.41 =0.02319 =14.749

Lognormal (3P) =0.02324 =2.9883 =4.5826

Weibull (3P) =9.1205 =3.7283 =20.897

1 week Lognormal (3P) =0.02953 =4.0735 =-34.706

Pearson 6 (4P) 1=2.1298E+6 2=4.1770E+5 =201.05 =-1001.1

Weibull (3P) =4.9884 =8.4329 =16.313

2 week Log-Pearson 3 =91.984 =-0.00655 =3.8076

Weibull (2P) =19.808 =25.259

3 week Gamma (3P) =8.8541 =0.56396 =20.017




Gen. Gamma (4P) k=0.66312 =23.481 =0.04531 =19.635

Weibull (2P) =21.248 =25.449


Weibull (2P) =22.761 =25.681

5 week Gamma (2P) =466.36 =0.05407

Gen. Extreme Value k=-0.43734 =1.1081 =24.927

Weibull (3P) =10.717 =11.146 =14.543

6 week Gamma (3P) =189.35 =0.06414 =13.095


Log-Gamma =8468.0 =3.8121E-4


Gen. Gamma (4P) k=1.2543 =139.59 =0.19991 =14.797

Normal =0.68218 =25.044

Pearson 6 (4P) 1=7.5120E+5 2=1.0096E+6 =596.77 =-418.98


9 week Lognormal (2P) =0.03148 =3.2226

Normal =0.79908 =25.106


Normal =0.77006 =25.008

Pearson 5 (3P) =402.16 =6301.9 =9.2897


Pearson 6 (4P) 1=6.8551E+7 2=1.2744E+7 =541.25 =-2886.7

Weibull (3P) =29.428 =20.996 =4.1713

12 week Weibull (2P) =42.758 =24.921

13 week Pearson 5 (3P) =1117.8 =31616.0 =-3.9018

Weibull (2P) =32.235 =24.83

Weibull (3P) =8.6285E+7 =5.0049E+7 =-5.0049E+7


Log-Pearson 3 =15.75 =-0.00742 =3.295

Weibull (2P) =40.329 =24.304

15 week Gamma (2P) =859.04 =0.02729

Weibull (3P) =24.716 =16.039 =7.7422


Pearson 5 (2P) =251.28 =5641.6

Pearson 6 (4P) 1=5.4524E+6 2=2.5203E+6 =806.8 =-1722.9

Weibull (3P) =2.2691E+8 =2.1828E+8 =-2.1828E+8

17 week Weibull (2P) =16.24 =22.186

Weibull (3P) =10.426 =13.7 =8.5124

Table 3.11(d). Best fit probability distribution for Minimum Temperature.




2 Week Weibull (2P)

3 Week Gamma (3P)


5 Week Gamma (2P)




9 Week Normal









3.4.5 Relative Humidity at 7AM

The test statistic D, 2A and 2 for each data set, of average relative humidity at 7 AM is

computed for 16 probability distribution. The probability distribution having the first rank along

with their test statistic is presented in table 3.12(a). It is observed that for seasonal average relative

humidity at 7 AM, Weibull (3P) distribution is fitted using Kolmogorov Smirnov and Anderson

Darling tests and Log-Pearson 3 distribution is fitted using Chi-square test based on first rank.

Thus these probability distributions are identified as the best fit based on these three tests

independently.

The fourth probability distribution identified which is having the highest score, is presented

in table 3.12(b) with their scores. Those distributions which are having same highest score are also

included in the selected probability distribution. The probability distributions fitted for the second

week of June are Generalized Extreme value and Generalized Gamma distribution based on

highest score as 46. While for fifteenth week data set, that is, second week of September, Log-

Pearson 3 and Normal distributions are fitted having 37 as the highest score are selected.

Table 3.12(a). Distributions fitted for Relative Humidity at 7 AM data sets.

Table 3.12(b). Distributions with highest score for Relative Humidity at 7 AM data sets.


Distribution Score

Seasonal Weibull (3P) 42

1 Week Log-Gamma 42

2 Week Gen. Extreme and Gen. Gamma (4P) 46









11 Week Normal 41




15 Week Log-Pearson 3 and Normal 37

16 Week Normal 38


Study

period




Seasonal Weibull (3P) 0.0644 Weibull (3P) 0.2282 Log-Pearson 3 2.4455


2 Week Gen. Gamma (4P) 0.0817 Gen. Gamma (4P) 0.3753 Gen. Extreme 2.5458




6 Week Weibull (3P) 0.1003 Gen. Extreme 0.5463 Log-gamma 0.9289



9 Week Weibull (2P) 0.0953 Gen. Extreme 0.5176 Weibull (2P) 1.2428

10 Week Gen. Extreme 0.0917 Log-Pearson 3 0.3199 Gen. Extreme 0.7057






16 Week Gen. Extreme 0.1110 Gen. Extreme 0.5244 Normal 1.6683

17 Week Log-Pearson 3 0.1056 Log-Pearson 3 0.5137 Log-Pearson 3 2.3387

Table 3.12(c). Parameters of the distributions fitted for Relative Humidity at 7 AM data sets.


Seasonal Log-Pearson 3 =8.5299 =-0.00988 =4.5465

Weibull (3P) =16.171 =33.249 =54.544

1 week Gen. Extreme k=-0.23796 =12.069 =62.183

Gen. Gamma (3P) k=0.99828 =29.997 =2.2137

Log-Gamma =496.56 =0.00843

Pearson 5 (2P) =28.444 =1835.6


Gen. Gamma (4P) k=6.8227 =0.27514 =46.462 =42.88


Weibull (3P) =6.1010E+7 =4.8501E+8 =-4.8501E+8

4 week Weibull (3P) =25.424 =128.95 =-41.738


Log-Pearson 3 =11.507 =-0.01676 =4.653

Pearson5 (3P) =300.04 =25916.0 =-0.02967

Weibull (3P) =7.4322 =31.766 =56.865


Log-gamma =3503.9 =0.00128

Weibull (3P) =4.1714E+7 =1.9459E+8 =-1.9459E+8

7 week Gamma (3P) =209.79 =0.27939 =31.041


Pearson 5 (3P) =376.91 =29802.0 =10.246

8 week Weibull (2P) =18.978 =92.549

8 week Weibull (3P) =6.1271E+7 =2.2208E+8 =-2.2208E+8


Weibull (2P) =32.887 =92.033


Log-Pearson 3 =10.99 =-0.01207 =4.6388


Normal =3.3925 =90.956

Pearson 6 (4P) 1=5.3567E+5 2=58139.0 =83.891 =-682.0

12 week Weibull (2P) =34.646 =92.952

Weibull (3P) =17.665 =45.223 =47.786


Pearson 5 (3P) =498.21 =41655.0 =7.1435

Weibull (3P) =8.8403 =28.468 =63.942

14 week Pearson 5 (3P) =618.57 =51280.0 =8.5325

Weibull (3P) =9.0316 =26.325 =66.64


Log-Pearson 3 =41.964 =-0.00503 =4.7282

Normal =2.9724 =91.606

Weibull (2P) =35.621 =92.888


Normal =3.2459 =90.442

17 week Log-Pearson 3 =1.4453 =-0.05156 =4.5549

Table 3.12(d). Best fit probability distribution for Relative Humidity at 7 AM.


Seasonal Log-Pearson 3



3 Week Weibull (3P)

4 Week Weibull (3P)


6 Week Weibull (3P)

7 Week Gamma (3P)

8 Week Weibull (2P)








16 Week Normal


The distributions identified are thus listed in table 3.12(c), where the parameter of these

identified distribution for each data set are mentioned. Random numbers are generated using the

parametric values for each data set and the least square method is worn for selecting the best fit

probability distribution. The probability distribution having minimum deviation is treated as the

best selected probability distribution for the individual data set for the average relative humidity at

7 AM as presented in table 3.12(d). Log-Pearson 3 distribution represents the best fitted

distribution for seasonal average relative humidity at 7 AM and is also observed in the fifth,

fifteenth and seventeenth week data set, that is, first week of July, second and last week of

September, respectively.

Further, we observe that Generalized Extreme Value is obtained in recurrence form in the

ninth, tenth and eleventh week, that is, in first three weeks of August and also in the thirteenth week,

that is, in last week of August. Moreover, Gamma (3P) distribution, Generalized Gamma (3P, 4P)

distributions, Normal distribution, Pearson 5 (3P) distribution, Weibull (2P, 3P) distributions are

found as the best fitted probability distributions for the weekly average relative humidity at 7 AM

data sets.

3.4.6 Relative Humidity at 2 PM

The test statistic D, 2A and 2 for each data set, of average relative humidity at 2 PM is

computed for 16 probability distribution. The probability distribution having the first rank along

with their test statistic is presented in table 3.13(a). It is observed that for seasonal average relative

humidity at 2 PM, Generalized Extreme Value is fitted using Kolmogorov Smirnov test, Weibull

(2P) distribution is fitted using Anderson Darling test and Weibull (3P) distribution is fitted using

Chi-square test based on first rank. Thus these probability distributions are identified as the best fit

based on these three tests independently.

The fourth probability distribution identified which is having highest score is presented in


included in the selected probability distribution. The Probability distributions with the highest

score as 45 for seasonal average relative humidity at 2 PM are three, that is, Generalized Extreme

Value and Weibull (2P, 3P) distributions. Moreover, the probability distributions fitted, based on

the highest score as 41 for the third week data set, that is, third week of June are Pearson 6 (4P)

and Weibull (3P) distributions.

While for fourth week data set, that is, fourth week of June, Generalized Gamma (4P) and

Lognormal (3P) distributions are fitted having 37 as the highest score. Similarly, for the sixth

week, that is, second week of July, Generalized Extreme Value and Log-Pearson 3 distributions

are having 38 as the highest score are selected. Further, for ninth week data set, that is, first week

of August, Normal and Pearson 6 (4P) distributions are selected having 40 as the highest score fit.

Also, with the highest score of 37 in the twelfth week, that is, fourth week of August, Lognormal

(2P) and Pearson 5 (2P) distributions are selected. The distributions identified are thus listed in

table 3.13(c) where the parameter of these identified distribution for each data set are mentioned.


the least square method is considered for selecting the best fit probability distribution. The

probability distribution having minimum deviation is treated as the best selected probability

distribution for the individual data set for the average relative humidity at 2 PM as presented in

table 3.13(d). Weibull (2P) distribution represents the best fitted distribution for seasonal average

relative humidity at 2 PM and is also observed in the tenth and fifteenth week data set, that is,

second week of August and September, respectively.

Further, we observe Weibull (3P) distribution as recurrence in the sixteenth and seventeenth

week, which are the last two weeks of September, and is also observed in the fifth and ninth week,

that is, first week of July and August, respectively. Besides, Generalized Extreme Value,

Generalized Gamma (4P) distribution, Log-Pearson 3 distribution, Normal distribution, Pearson 5

(2P) distribution, Pearson 6 (4P) distributions are found as the best fitted probability distributions for

the weekly average relative humidity at 2 PM data sets.

Table 3.13(a). Distributions fitted for Relative Humidity at 2 PM data sets.

Study

period




Seasonal Gen. Extreme 0.0712 Weibull (2P) 0.3734 Weibull (3P) 2.0229


2 Week Log-Pearson 3 0.0758 Log-Pearson 3 0.3080 Lognormal (2P) 1.2683


4 Week Gen. Extreme 0.0995 Weibull (3P) 0.3831 Gamma (2P) 1.2667

5 Week Log-Pearson 3 0.0779 Weibull (3P) 0.2791 Log-Pearson 3 2.4067



8 Week Normal 0.0749 Gen. Extreme 0.2524 Weibull (2P) 1.0323


10 Week Weibull (2P) 0.1026 Normal 0.4765 Weibull (2P) 2.2518

11 Week Log-Pearson 3 0.0672 Weibull (3P) 0.2489 Gen. Extreme 2.5681

12 Week Pearson 5 (2P) 0.0826 Lognormal (2P) 0.3529 Normal 1.5444

13 Week Gamma (3P) 0.0915 Gen. Extreme 0.3627 Gen. Extreme 4.0812

14 Week Gen. Gamma (4P) 0.0692 Normal 0.2625 Weibull (3P) 0.6171

15 Week Weibull (2P) 0.0842 Gen. Extreme 0.2904 Gen. Gamma (4P) 0.7798



Table 3.13(b). Distributions with highest score for Relative Humidity at 2 PM data sets.


Distribution Score

Seasonal Gen. Extreme and Weibull (2P, 3P) 45



3 Week Pearson 6 (4P) and Weibull (3P) 41

4 Week Gen. Gamma (4P) and Lognormal (3P) 37


6 Week Gen. Extreme Value and Log-Pearson 3 38


8 Week Normal 42

9 Week Normal and Pearson 6 (4P) 40

10 Week Normal 44


12 Week Lognormal (2P) and Pearson 5 (2P) 37






Table 3.13(c). Parameters of the distributions fitted for Relative Humidity at 2 PM data sets.


Seasonal Gen. Extreme Value k=-0.52018 =3.8843 =64.855

Weibull (2P) =21.659 =67.161

Weibull (3P) =8.9724 =27.338 =39.844


Weibull (2P) =2.8071 =44.234


Log-Pearson 3 =9.6018 =-0.12479 =4.984

Lognormal (2P) =0.3828 =3.7858

3 week Gamma (3P) =240.94 =0.92958 =-168.36

Pearson 6 (4P) 1=10915.0 2=5529.0 =426.83 =-786.39

Weibull (3P) =26.877 =302.82 =-240.97

4 week Gamma (2P) =37.54 =1.6858


Gen. Gamma (4P) k=3.107 =191.32 =80.612 =-373.76

Lognormal (3P) =0.02348 =6.0912 =-378.52

Weibull (3P) =24.603 =206.72 =-138.95

5 week Log-Pearson 3 =3.9062 =-0.08697 =4.5031

Weibull (3P) =6.1205 =58.567 =10.777




Log-Pearson 3 =8.1847 =-0.04591 =4.626

Pearson 6 (4P) 1=22.745 2=4.5724E+7 =9.2934E+7 =24.56



Normal =7.3628 =73.338

Weibull (2P) =11.591 =76.209

9 week Normal =6.4266 =73.128

Pearson 6 (4P) 1=2.4128E+6 2=3.6815E+5 =550.43 =-3534.3

Weibull (3P) =22.354 =116.68 =-40.776

10 week Normal =7.1024 =72.632

Weibull (2P) =12.408 =75.186


Log-Pearson 3 =443.55 =-0.00397 =6.069

Weibull (3P) =3.0131 =18.909 =57.572

12 week Lognormal (2P) =0.06761 =4.2878

Normal =5.0058 =72.972

Pearson 5 (2P) =219.35 =15934.0

13 week Gamma (3P) =154.71 =0.58977 =-20.676

Gen. Extreme k=-0.36324 =7.5515 =68.249

14 week Gen. Gamma (4P) k=18.772 =8.3733 =368.55 =-341.2

Normal =7.855 =70.258

Weibull (3P) =4.9722 =36.763 =36.513


Gen. Gamma (4P) k=2.7474 =0.73193 =22.975 =52.467

Weibull (2P) =10.378 =72.787

16 week Gamma (3P) =173.41 =0.61761 =-41.471

Weibull (3P) =7.3775 =52.812 =16.088

17 week Weibull (3P) =6.7096 =58.176 =5.7427

Table 3.13(d). Best fit probability distribution for Relative Humidity at 2 PM.







5 Week Weibull (3P)



8 Week Normal

9 Week Weibull (3P)





14 Week Normal




3.4.7 Pan Evaporation

The test statistic D, 2A and 2 for each data set, of average Pan Evaporation is computed

for 16 probability distribution. The probability distribution having the first rank along with their

test statistic is presented in table 3.14(a). It is observed that for seasonal average pan evaporation,

Log-Pearson 3 is fitted using Kolmogorov Smirnov test, Generalized Extreme Value is fitted using

Anderson Darling test and Gamma (3P) distribution is fitted using Chi-square test based on first

rank. Thus these probability distributions are identified as the best fit based on these three tests

independently.

The fourth probability distribution identified, which is having the highest score, is presented

in table 3.14(b) with their scores. Those distributions which are having same highest score are also

included in the selected probability distribution. The probability distributions selected, based on

the highest fit score as 35 for the sixth week data set, that is, second week of July are Pearson 6

(3P) and Generalized Gamma (3P) distributions.

Table 3.14(a). Distributions fitted for Pan Evaporation data sets.

Table 3.14(b). Distributions with highest score for Pan Evaporation data sets.


Distribution Score

Seasonal Weibull (3P) 45

1 Week Normal 43




5 Week Lognormal (3P) 34

6 Week Pearson 6 (3P) and Gen. Gamma (3P) 35


8 Week Normal 42





13 Week Normal 48




17 Week Gen. Extreme and Weibull (2P) 43

Study

period




Seasonal Log-Pearson 3 0.0537 Gen. Extreme 0.1404 Gamma (3P) 1.0902

1 Week Log-Pearson 3 0.0596 Normal 0.1440 Log-Pearson 3 0.6899

2 Week Pearson 6 (3P) 0.0626 Pearson 5 (3P) 0.1640 Lognormal (2P) 1.0880

3 Week Weibull (3P) 0.1092 Gamma (3P) 0.3565 Gen. Extreme 0.6749

4 Week Pearson 5 (3P) 0.0744 Gen. Extreme 0.3528 Log-Gamma 1.2723

5 Week Lognormal (3P) 0.0982 Pearson 5 (3P) 0.3740 Gen. Gamma (3P) 0.5740

6 Week Pearson 6 (3P) 0.0845 Normal 0.4287 Pearson 6 (3P) 1.3534

7 Week Weibull (2P) 0.0985 Pearson 5 (3P) 0.5113 Gamma (2P) 2.3513

8 Week Normal 0.0659 Normal 0.2283 Normal 1.4441


10 Week Gen. Gamma (4P) 0.0842 Gen. Extreme 0.2938 Weibull (3P) 0.7581

11 Week Normal 0.0984 Gen. Extreme 0.5842 Normal 0.3715

12 Week Gen. Extreme 0.1113 Normal 0.7725 Gen. Gamma (4P) 2.7838

13 Week Normal 0.0955 Normal 0.4676 Normal 1.2692


15 Week Weibull (3P) 0.0870 Weibull (3P) 0.3080 Gamma(2P) 1.7154

16 Week Gen. Gamma (4P) 0.0708 Gen. Gamma (4P) 0.1977 Lognormal (2P) 0.6335

17 Week Weibull (2P) 0.0911 Gen. Extreme 0.3561 Log Gamma 0.4024

Table 3.14(c). Parameters of the distributions fitted for Pan Evaporation data sets.


Seasonal Gamma (3P) =176.56 =0.06712 =-6.5449


Log-Pearson 3 =5.3329 =-0.07625 =2.0609

Weibull (3P) =6.9709 =5.419 =0.24249

1 week Log-Pearson 3 =7.448 =-0.10341 =3.0937

Normal =2.7786 =10.592

2 week Lognormal (2P) =0.33983 =2.1431

Pearson 5 (3P) =53.101 =1082.3 =-11.77

Pearson 6 (3P) 1=9.4905 2=3.9194E+5 =3.7171E+5

3 week Gamma (3P) =4.3022 =1.2891 =2.6401


Weibull (3P) =1.9242 =5.4847 =3.3242


Log-Gamma =34.648 =0.0517

Pearson 5 (3P) =66.621 =946.37 =-8.1624

5 week Gen. Gamma (3P) k=0.9814 =7.1148 =0.79576

Lognormal (3P) =0.13189 =2.7742 =-10.276

Pearson 5 (3P) =100.88 =2128.3 =-15.42

6 week Gen. Gamma (3P) k=1.0123 =7.9218 =0.69854

Normal =1.9227 =5.292

Pearson 6 (3P) 1=8.5077 2=1.7018E+8 =1.0754E+8

7 week Gamma (2P) =7.7831 =0.62674


Pearson 5 (3P) =49.793 =579.97 =-7.0088

Weibull (2P) =3.1558 =5.3419

8 week Normal =1.8146 =4.122

9 week Gen. Extreme value k=-0.41167 =1.3065 =3.7864

10 week Gen. Gamma (4P) k=7.3788 =0.35509 =5.7372


Weibull (3P) =3.5914 =4.5836


Normal =1.493 =4.138


Gen. Gamma (4P) k=4.1478 =0.48649 =4.1257 =1.1742

Normal =1.1954 =4.042

Weibull (3P) =3.5458 =4.1785 =0.27855

13 week Normal =1.2559 =4.068


Pearson 6 (4P) 1=4.3526E+5 2=2.8077E+5 =273.81 =-420.49

Weibull (3P) =4.6889 =4.5552 =-0.1909

15 week Gamma (2P) =18.396 =0.21092

Weibull (3P) =20.584 =15.172 =-10.901

16 week Gen. Gamma (4P) k=16.334 =0.11141 =3.3346 =1.7597

Lognormal (2P) =0.23367 =1.3172


Log-Gamma =41.233 =0.03222

Weibull (2P) =5.7794 =4.117

While for the seventeenth week, that is, last week of September, Generalized Extreme

Value and Weibull (2P) distributions are having 43 as the highest fit score are selected. The

distributions identified are thus listed in table 3.14(c) where the parameter of these identified

distribution for each data set are mentioned. The least square method is utilized for selecting the

best fit probability distribution after generating random number for each data set with the help of

the parametric values obtained. The probability distribution having minimum deviation is treated

as the best selected probability distribution for the individual data set for the average pan

evaporation as presented in table 3.14(d).

Table 3.14(d). Best fit probability distribution for Pan Evaporation.


Seasonal Gamma (3P)



3 Week Weibull (3P)



6 Week Normal


8 Week Normal

9 Week Gen. Extreme


11 Week Normal


13 Week Normal


15 Week Gamma (2P)



Gamma (3P) distribution represents the best fitted distribution for seasonal average pan

evaporation. Further, we observe Normal distribution plays a vital role by appearing four times as

the best fit in the weekly data set in sixth, eighth, eleventh and thirteenth week, that is, in second and

last week of July and also in third and last week of August, respectively. Further, Generalized

Gamma (4P) distribution is observed thrice in the weekly data set, means, in the tenth, twelfth and

sixteenth week, that is, second and fourth week of August and third week of September,

respectively.

Besides, Gamma (2P), Generalized Extreme Value, Generalized Gamma (3P) distribution,

Log-Pearson 3 distribution, Normal distribution, Pearson 5 (3P) distribution, Pearson 6 (3P)

distribution and Weibull (2P, 3P) are obtained as the best fitted probability distributions for the

weekly average pan evaporation data sets.

3.4.8 Bright Sunshine

The test statistic D, 2A and 2 for each data set, of average bright sunshine is computed for


statistic is presented in table 3.15(a). It is observed that for seasonal average Bright Sunshine,

Generalized Extreme Value is fitted using Kolmogorov Smirnov and Anderson Darling tests and

Log-Gamma distribution is fitted using Chi-square test based on first rank. Thus these probability

distributions are identified as the best fit based on these three tests independently.

The fourth probability distribution identified which is having highest score is presented in


included in the selected probability distribution. The Probability distributions with the highest

score as 45 for seasonal average bright sunshine are Generalized Extreme Value and Log-Gamma

distributions. Moreover, the probability distributions selected, based on the highest fit score as 36

for the eleventh week data set, that is, third week of August are Weibull (3P) and Generalized

Gamma (4P) distributions. While for last week of August, Generalized Extreme Value and

Pearson 6 (4P) distributions having 36 as the highest score are selected. The distributions

identified are thus listed in table 3.15(c) where the parameter of these identified distribution for

each data set are mentioned.

Random numbers are generated using the parameter values for each data set and the least

square method is inculcated for selecting the best fit probability distribution. The probability


the individual data set for the average bright sunshine and is presented in table 3.15(d).

Log-Gamma distribution represents the best fitted distribution for seasonal average bright

sunshine. Further, we observe Generalized Extreme Value plays an essential role by appearing six

times in the weekly data set, means, first, fifth, fourteenth, eleventh, eighth and seventeenth week,

that is, in the first week of June, July and September, third week of August, last week of July and

September, respectively.

Further, Log-Pearson 3 distribution is observed thrice in the weekly data set, in the sixth,

thirteenth and sixteenth week, that is, second week of July, last week of August and third week of

September, respectively. In addition, Gamma (2P, 3P) distributions, Generalized Gamma (3P)

distribution, Pearson 5 (2P) distribution, Pearson 6 (4P) distribution and Weibull (2P, 3P)

distributions are found as the best fitted probability distributions for the weekly average bright

sunshine data sets.

Table 3.15(a). Distributions fitted for Bright Sunshine data sets.

Study

period




Seasonal Gen. Extreme 0.0709 Gen. Extreme 0.1333 Log Gamma 0.0521


2 Week Weibull (3P) 0.0852 Weibull (3P) 0.4103 Normal 3.8528


4 Week Gamma (3P) 0.0536 Gen. Extreme 0.1784 Pearson 5 (3P) 0.9833


6 Week Log-Pearson 3 0.0546 Log-Pearson 3 0.1712 Log-Pearson 3 0.4699


8 Week Gen. Gamma (3P) 0.1039 Gen. Extreme 0.7188 Weibull (2P) 2.6207


10 Week Pearson 6 (4P) 0.0674 Gen. Extreme 0.3030 Pearson 5 (2P) 0.7440

11 Week Gen. Extreme 0.0752 Gen. Gamma (4P) 0.4851 Pearson 6 (3P) 0.8082

12 Week Lognormal (2P) 0.0719 Log-Pearson 3 0.3558 Gamma (2P) 0.9435

13 Week Gen. Gamma (4P) 0.0725 Log-Pearson 3 0.2354 Pearson 6 (4P) 2.1774



16 Week Log-Pearson 3 0.0647 Log-Pearson 3 0.1547 Gen. Extreme 0.2675

17 Week Gen. Extreme 0.0704 Gen. Extreme 0.2137 Log-Pearson 3 0.1466

Table 3.15(b). Distributions with highest score for Bright Sunshine data sets.


Distribution Score

Seasonal Gen. Extreme and Log-Gamma 45











11 Week Weibull (3P) and Gen. Gamma (4P) 36


13 Week Gen. Extreme and Pearson 6 (4P) 36





Table 3.15(c). Parameters of the distributions fitted for Bright Sunshine data sets.

Table 3.15(c). Parameters of the distributions fitted for Bright Sunshine data sets.


Seasonal Gen. Extreme Value k=-0.19099 =0.65064 =6.104

Log-Gamma =302.87 =0.0061


2 week Gen. Gamma (4P) k=4.5392E+7 =3.6791 =1.4983E+8 =-1.4983E+8

Normal =1.9683 =7.786

Weibull (3P) =42.341 =65.371 =-56.739

3 week Gamma (2P) =10.361 =0.6874

Weibull (3P) =5.5538 =11.374 =-3.388

4 week Gamma (3P) =99.275 =0.17517 =-11.197


Pearson 5 (3P) =186.61 =4420.6 =-17.66


6 week Log-Pearson 3 =1.9754 =-0.33814 =2.3106

7 week Pearson 6 (4P) 1=34811.0 2=84746.0 =671.58 =-270.33

Weibull (3P) =5.5963 =9.1116 =-2.8707


Gen. Gamma (3P) k=0.98179 =9.0462 =0.56163

Weibull (2P) =3.1144 =5.8867


Gen. Gamma (4P) k=7.6955 =0.18288 =6.6278 =1.865

Pearson 5 (2P) =7.2398 =35.996


Table 3.15(d). Best fit probability distribution for Bright Sunshine.


Seasonal Log-Gamma


2 Week Weibull (3P)

3 Week Gamma (2P)

4 Week Gamma (3P)



7 Week Weibull (3P)

8 Week Gen. Extreme











10 week Pearson 5 (2P) =4.645 =21.397

Pearson 6 (4P) 1=71.073 2=88.787 =15.705 =-7.1531


Gen. Gamma (4P) k=9.3109 =0.41883 =9.1555 =-1.8893

Pearson 6 (3P) 1=7.1213 2=1.1589E+8 =8.7428E+7

Weibull (3P) =12.653 =18.439 =-12.289

12 week Gamma (2P) =8.5054 =0.64853

Gen. Gamma (3P) k=0.98578 =8.2386 =0.64853

Lognormal (2P) =0.37365 =1.6434

Log-Pearson 3 =8.0178 =-0.1333 =2.7121


Gen. Gamma (4P) k=6.1523 =0.28286 =7.5382 =1.3255

Log-Pearson 3 =4.8308 =-0.17463 =2.5739

Pearson 6 (4P) 1=9730.1 2=13924.0 =216.92 =-145.57


Weibull (2P) =2.7832 =6.7243


Weibull (2P) =3.3259 =7.1501


Log-Pearson 3 =2.1593 =-0.20805 =2.3975


Log-Pearson 3 =1.7031 =-0.21526 =2.4145

3.5 Conclusion

In this chapter before identifying the best fit probability distribution, the descriptive

statistics are computed for each weather parameters for different study period. The result of

weather parameters analysis for identifying the best fit probability distribution revealed that the

distribution pattern for different data set can be identified out of a large number of commonly used

probability distributions by using different goodness of fit tests. After observing the weekly

weather parameters independently we can conclude them as follows:

Rainfall

The data represented that the maximum value of weekly rainfall is 443.20 mm in fourth

week of August in year 2000. Normal distribution represents the best fitted probability distribution

for seasonal rainfall and is also observed in the second week of July. Moreover, Generalized

Extreme Value is observed six times in the weekly data, in the first week of June, second week of

June, August and September, and also, in the last two weeks of August, indicating the highest

contribution of the distribution. In addition, Gamma (3P) distribution, Log-Pearson 3 distribution,

Pearson 6 (3P) distribution and Lognormal (3P) distribution are pragmatic as the best fitted

probability distributions for the weekly rainfall data sets.

Maximum Temperature

The data offered that the seasonal maximum temperature ranged between 23.6 OC in the

year 2005 to 43.2 OC in the years 1966 and 1967. Weibull (2P) distribution represents the best

fitted distribution for seasonal maximum temperature and is also observed in the ninth and twelfth

week, that is, first and fourth week of August. Further, Log-Pearson 3 is observed consecutively in

the second and third week of July. Similarly, Weibull (3P) is observed successively in the last two

weeks of September and also in the first week of July and September. Moreover, we observe that

Generalized Gamma (3P) distribution, Generalized Extreme Value, Pearson 5 (2P, 3P) distributions

and Lognormal (3P) distribution as the best fitted probability distributions for the weekly maximum


Minimum Temperature

The seasonal minimum temperature ranged between 17.2 OC in the year 1984 to 29.2 OC in

the year 1995. Weibull (3P) distribution represents the best fitted distribution for seasonal

minimum temperature and is also observed in the second week of September. Further, we observe

that Generalized Extreme Value is obtained repetitively in three weeks, that is, second, third and

fourth week of July, also in the last week of June. Also, Weibull (2P) appeared four times among

the 17 weeks, that is, second week of June, first week of September and fourth week of August

and September. Besides, Gamma (2P, 3P) distributions, Normal distribution, Pearson 5 (2P, 3P)

distributions and Pearson 6 (4P) distribution are obtained as the best fitted probability distributions

for the weekly minimum temperature data sets.

Relative Humidity at 7AM

The data explains the seasonal average relative humidity at 7 AM ranged between 38%

(minimum) to 98% (maximum). Log-Pearson 3 distribution represents the best fitted distribution

for seasonal average relative humidity at 7 AM and is also observed in the first week of July,

second and last week of September. Likewise, we observe that Generalized Extreme Value is

obtained in recurrence form in the first three weeks of August and also in the last week of August.

Moreover, Gamma (3P) distribution, Generalized Gamma (3P, 4P) distributions, Normal

distribution, Pearson 5 (3P) distribution, Weibull (2P, 3P) distributions are found as the best fitted

probability distributions for the weekly average relative humidity at 7 AM data sets.

Relative Humidity at 2 PM

The seasonal average relative humidity at 2 PM ranged between 16 % in the years 1965 and

2005 to 92 % in the year 1988. Weibull (2P) distribution represents the best fitted distribution for

seasonal average relative humidity at 2 PM and is also observed in the tenth and fifteenth week data

set, that is, second week of August and September respectively. As well as, we observe Weibull

(3P) distribution as recurrence in last two weeks of September, and are also observed in first week

of July and August. Besides, Generalized Extreme Value, Generalized Gamma (4P) distribution,

Log-Pearson 3 distribution, Normal distribution, Pearson 5 (2P) distribution, Pearson 6 (4P)

distributions are obtained as best fitted probability distributions for the weekly average relative

humidity at 2 PM data sets.

Pan Evaporation

The data shows the seasonal average Pan Evaporation ranged between zero mm to 18.5

mm in the year 1967. Gamma (3P) distribution represents the best fitted distribution for seasonal

average pan evaporation. Additionally, we observe Normal distribution plays a vital role by

appearing four times as best fit in the weekly data set, that is, in second and last week of July and

also in third and last week of August. Moreover, Generalized Gamma (4P) distribution is observed

thrice in the weekly data set, that is, second and fourth week of August and third week of

September respectively. Besides, Gamma (2P), Generalized Extreme Value, Generalized Gamma

(3P) distribution, Log-Pearson 3 distribution, Normal distribution, Pearson 5 (3P) distribution,

Pearson 6 (3P) distribution and Weibull (2P, 3P) are obtained as the best fitted probability

distributions for the weekly average pan evaporation data sets.

Bright Sunshine

The seasonal average bright sunshine ranged between 0.70 hours in the year 2009 to 11.6

hours in the years 1986 and 2009. Log-Gamma distribution represents the best fitted distribution

for seasonal average bright sunshine. Further, we observe Generalized Extreme Value plays an

essential role by appearing six times in the weekly data set, in the first week of June, July and

September and also in third week of August as well as the last week of July and September.

Moreover, Log-Pearson 3 distribution is observed thrice in the weekly data set, that is, second

week of July, last week of August and third week of September, respectively. In addition, Gamma

(2P, 3P) distributions, Generalized Gamma (3P) distribution, Pearson 5 (2P) distribution, Pearson

6 (4P) distribution and Weibull (2P, 3P) distributions are found as the best fitted probability

distributions for the weekly average bright sunshine data sets.

The best fit probability distributions for seasonal and weekly weather parameters for

different study period are different. In general, Generalized Extreme Value, Weibull (2P, 3P)

distributions are most commonly the best fit probability distribution for most of the weeks among

the different weather parameters.

WEATHER FORECASTING MODELS

Introduction

Correlation Analysis

Methodology for forecasting models

Development of forecasting model for weather

parameters

Comparison of prediction ability of forecasting models

Conclusion

CHAPTER 4

WEATHER FORECASTING MODELS

4.1 Introduction

The early civilizations used reoccurring astronomical and meteorological events to monitor

seasonal changes in the weather. In arid and non-arid regions, the most dominant meteorological

parameter is rainfall which reflects wet and dry period characteristics and is measured at point

locations but assumed to represent the surrounding areas. The occurrence of rainfall depends on

several other weather parameters.

This chapter describes all the seven weather parameters under study for which the weekly

data of seven parameter viz. rainfall, maximum and maximum temperature, relative humidity at

7.00 am and 2.00 pm, bright sunshine hours and pan evaporation for the four monsoon months

were recorded. The monsoon season in this region ranges between 15 to 20 weeks, thus 17 weeks

weather data from 4th June to 30th September of 50 years (1961-2010) was considered for the

present study.

4.2 Correlation Analysis

The Inter correlation coefficient between different parameters based on 50 years data set is

computed and presented in table 4.1. From table 4.1 it was observed that maximum temperature is

positively correlated with minimum temperature but highly positively correlated with pan

evaporation and bright sunshine. Relative humidity at 7 am is also highly positively correlated

with relative humidity at 2 pm.

There is negative correlation between rainfall and maximum temperature, minimum

temperature, pan evaporation and bright sunshine hours while rainfall is positively correlated with

relative humidity at 7 am and 2 pm. It can be observed that there is highest correlation between

relative humidity at 7 am and relative humidity at 2 pm and lowest correlation among rainfall

and minimum temperature.

Table 4.1. Inter correlation coefficient between weather parameters for total data set.

Maximum Minimum Relative Relative Pan Bright Rainfall

Parameters Temperature Temperature humidity humidity Evaporation Sunshine

(7 am) (2 pm)

Maximum temperature 1.00000 0.19218 -0.82554 -0.83234 0.77801 0.57776 -0.47340

Minimum Temperature 0.19218 1.00000 -0.11481 0.07373 0.14586 -0.14112 -0.00666

Relative Humidity (7 am) -0.82554 -0.11481 1.00000 0.83736 -0.77588 -0.48374 0.36459

Relative Humidity (2 pm) -0.83234 0.07373 0.83736 1.00000 -0.73340 -0.66752 0.52490

Pan Evaporation 0.77801 0.14586 -0.77588 -0.73340 1.00000 0.43054 -0.28520

Bright Sunshine 0.57776 -0.14112 -0.48374 -0.66752 0.43054 1.00000 -0.53330

Rainfall -0.47340 -0.00666 0.36459 0.52490 -0.28520 -0.53330 1.00000

4.3 Methodology for forecasting Models The methodology for forecasting models viz. Multiple Linear Regression (MLR),

Autoregressive Integrated Moving Average (ARIMA) and Artificial Neural Network (ANN)

models are given in brief in the next subsections. Hybrid approach for developing weather

forecasting model and their performance evaluation criteria is also discussed in this section.

4.3.1 Multiple Linear regression model

Multiple regression analysis is to include a number of independent parameters at the same

time for predicting the significance of a dependent parameter, (Snedecor and Cochran, 1967). In

the study, the multiple linear regression equation fitted to the weekly weather parameters treating

one as independent parameter and six other as independent parameters are given below in

generalized form.

0 1 1 2 2 6 6Y = β +β X +β X + ............... + β X + ε (4.1)

Where:

0 = Intercept,

i = regression coefficient of ith independent parameters, ( i = 1,2,…, 6),

= error term,

Xi = ith weather parameter.

To identify the significant parameters for predicting the dependent parameter based on the

six independent parameters, stepwise regression analysis was used. Stepwise process starts with a

simple regression model in which most extremely correlated one independent parameter was only

incorporated at first in the company of a dependent parameter. Correlation coefficient is further

examined in the practice to find an additional independent parameter that explains the major

portion of the error remaining from the initial regression model. Until the model includes all the

significant contributing parameters, the procedure keeps on repeating. The possible bias in the

stepwise regression procedure fallout from the consideration of only one parameter at a time.

4.3.2 Autoregressive Integrated Moving Average Model

The equally spaced univariate time series data, transfer function data, and intervention data

are analyzed and forecast using the Autoregressive Integrated Moving-Average (ARIMA) or

autoregressive moving-average (ARMA) model. An ARIMA model predicts a value in a response

time series as a linear combination of its own past values, past errors (also called shocks or

innovations), and current and past values of other time series.

The ARIMA approach was first popularized by Box and Jenkins (1976), and ARIMA

models are often referred to as Box-Jenkins models. ARIMA (p, d, q) models are the extension of

AR model that use three components for modeling the serial correlation in the time series data.

The first component is the autoregressive (AR) term, where there is a memory of past events and it

uses the ‘p’ lags of the time series. The second component is the integration (I) term which

accounts for stabilizing or making the data stationary, making it easier to forecast. Each

integration order corresponds to differencing the time series. I (d) means differencing the data‘d’

times. The third component is the moving average (MA) term of the forecast errors, such that the

longer the historical data, the more accurate the forecasts will be, as it learns overtime. The MA

(q) model uses the ‘q’ lags of the forecast errors to improve the forecast.

A dependent weekly parameter time series data, tY :1 t n , mathematically the pure

ARIMA model is written as:

t t

BW

Ba

(4.2)

where,

t = indexes time of weekly parameter.

tW = the response series tY = d

t1 B Y .

= the mean time of weekly parameter.

B = the backshift operator, that is, t t 1BX X

1 p

1 pB 1 B ............ B = the autoregressive operator.

1 q

1 qB 1 B ............ B = the moving average operator.

ta = the independent disturbance (random error).

d = the degree of differencing.

In this ARIMA (p, d, q) modeling, the foremost step is to decide whether the time series is

stationary or non-stationary. If it is non-stationary, it is transformed into a stationary time series by

applying appropriate degree of differencing by selecting suitable value of ‘d’. The appropriate

values of p and q are chosen by examining the autocorrelation function (ACF) and partial

autocorrelation function (PACF) of the time series data set.

4.3.3 Artificial Neural Network Model

Artificial Neural Networks are massively parallel adaptive networks of simple non-linear

computing elements called neurons which are intended to abstract and model some of the

functionality of the human nervous system in an attempt to partially capture some of its

computational strengths. Artificial Neural Network (ANN) is loosely based on biological neural

systems, in that; they are made up of an interconnected system of neurons. Also, a neural network

can identify patterns adaptively between input and output data set in a somewhat analogous

fashion to the learning process. Neural networks are highly robust with respect to underlying data

distributions and no assumptions are made about relationships between parameters.

Artificial Neural Networks (ANNs) provide a methodology for solving many types of non-

linear problems that are difficult to solve by traditional techniques. In Artificial Neural Network

Software all inputs and outputs are normalized between 0 and 1. Appropriate process of

normalization and denormalization of data is needed before and after the program execution. The

best and the simplest way is to divide it by the maximum for normalization and after the program

execution the result is to be multiplied by the same amount.

There are many neural network models, but the basic structure involves a system of

layered, interconnected nodes and neurons are presented in figure 4.1. The nodes are arranged to

form an input layer, with neurons in each hidden layer connected to all neurons in neighboring

layers. The input layer supplies data to the hidden layer and does not contain activation or transfer

functions. A typical feed-forward network might use a dot-product activation function that, for

each neuron Bj ( j = 1, 2, …..,n) in the hidden layer, is computed as:

m

j ij i oj o

i 1

B w A w A

(4.3)

with input nodes Ai ( i = 1, 2,…., m) and weights Wij between nodes Ai and neurons Bj. The bias

node (Ao) typically has a constant input of 1, with a matching weight Woj. A similar

calculation is made for each neuron Ck ( k = 1, 2, …., o) in the output layer (o = 1 for the

example in figure 4.1), using weights Wjk between neurons Bj and Ck ( with Wok and Bo for the

bias). Each neuron value is subsequently passed through a transfer function, which may be linear

or nonlinear (Zurada, 1992). A common choice of nonlinear transfer function is a sigmoid, of the

general form:

1

uu 1 e

(4.4)

Where, u = Bj (or Ck ).

i j kA (i 1,2,....,m) B (j 1,2,.....,n) C (k 1,2,.....,O)

Figure 4.1. An (m x n x o) artificial neural network structure, showing a multilayer perceptron.

Nonlinearities are incorporated into the network via the activation and transfer functions in

each neuron. Complexities in the data are captured through the number of neurons in the hidden

layer. In adoption of Neural Network for practical purpose, it is desired to restrict the connections

between neurons. This is done by fixing some of the weights to zero, so that, they can dropped out

from the calculations, the working principal for subsequent adjustment of weight is in accordance

with the error propagation in the network. If increasing a given weight leads to more error, we

adjust the weight downwards and if increasing a weight leads to less error, we adjust the weight

upwards. Adjustment of all the up or down continues throughout this process until the weights and

error settled down.

To avoid over fitting to the data, a neural network is usually trained on a subset of inputs

and outputs to determine weights, and subsequently validated on the remaining (quasi-

independent) data to measure the accuracy of prediction.

4.3.4 Hybrid Approach

In recent times, the concept of combined model instead of single time series model is being

prepared for prediction purpose. Several researchers have used a hybrid principal component and

ANN approach to improve the accuracy of the prediction results of their long range forecasting

investigations. In the present investigation, the hybrid approach for weather forecasting is tried to

improve the accuracy of prediction. Several studies show that the techniques of combinations of

ANN with ARIMA offer a competitive edge over each of the individual model. Taskaya et al.

(2007) doughty the degrade performance of ARIMA neural network hybrids, if the relationship

between the linear and non-linear components is different from additive assumption. The

combination of MLR with ARIMA and MLR with ANN is proposed in the present study. The

hybrid of multiple linear regression with ARIMA and ANN techniques to analyze the weekly

weather parameters of all the seven parameter studied and included in the comparative study to

identify the best precise weather forecasting model.

Hybrid Model of Multiple Linear Regression and Autoregressive Integrated Moving Average

(MLR_ARIMA)

The composition of a multiple linear regression with autoregressive integrated moving

average model is proposed to develop a new hybrid model in this section. It is assumed that the

predictive performance improves by integrating two single models. For this purpose the selected

significantly contributed parameters obtained through stepwise regression analysis are used to

develop the MLR_ARIMA model and their performance is compared with all other models.

Hybrid Model of Multiple Linear Regression and Artificial Neural Network (MLR_ANN)

It has been observed in the current researches that a single model may not be sufficient to

identify all the characteristics of the time series data. The hybrid models decompose a time series

into linear and non-linear form and prove to be better approach in comparison to single model. In

this section the hybrid model of multiple linear regression with neural network approach is

proposed to yield more accurate results. Similar to previous model, the significantly contributed

parameters selected through stepwise regression analysis in multiple linear regression model are

used to develop the hybrid MLR_ANN model.

4.3.5 Performance evaluation criteria

Many analytical methods have been proposed for the evaluation and inter-comparison of

different models, which can be evaluated in terms of graphical representation and numerical

computations.

The graphical performance criteria involves: A linear scale plot of the predicted and observed

weather parameters for training and testing data sets for all the models.

The numerical performance criterion involves:

Mean error (BIAS): N

i i

i 1

1Y Y

N

(4.5)

Mean absolute error (MAE): N

i i

i 1

1Y Y

N

(4.6)

Root mean square error (RMSE):

2N

i i

i 1

1Y Y

N

(4.7)

Prediction error (PE):

i i

i

Y - Y

Y

(4.8)

Correlation Coefficient (r): This is obtained by performing a regression between the predicted

values and the actual values and is computed by

N

i ii i

i 1

2N N

2

i ii i

i 1 i 1

Y Y Y Y

Y Y Y Y

r

(4.9)

where, implies the average over the whole test set, N is the total number of forecast

outputs. iY and iY

are the actual and predicted values respectively for i = 1, 2, …..,N, iY and iY

are

the mean values of the actual and predicted values respectively.

For the best prediction, the BIAS, MAE and RMSE values should be small and PE should

be sufficiently small i.e., close to 0. But ‘r’ should be found closer to 1 (between 0 - 1) for

indicating better agreement between observed and predicted values. The recital of weather

forecasting models had been evaluated on the basis of Mat lab 7.0.1 version, students’ academic

SAS version and Microsoft Excel.

4.4 Development of forecasting model for weather parameters

4.4.1 Introduction

The most common problems, that a contemporary data analyst encounters, is pulling out of

significant conclusions about a intricate system using data from a solitary measured parameter.

The methodology presented above was applied to the 50 years monsoon weather data for the

months of June, July, August and September. The weather data was further classified into weeks

for further analysis. A total of 850 data sets (17 weeks x 50 years) of weekly parameters were

used. The most significantly contributed parameters were selected using stepwise regression

analysis based on training data set of 35 years i.e., 595 data sets and the remaining data sets (15

years) are used in testing of the developed models which are used for comparing the real and

predicted values.

4.4.2 Rainfall

The multiple linear regression model is fitted to predict the weekly rainfall as dependent

parameter taking the other weekly independent parameters as maximum temperature, minimum

temperature, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most

significantly contributed parameters are selected using stepwise regression analysis based on 595

data set of 35 years and the best fit multiple regression model is given below:

1 3 4 5 6Y = 391.90674 - 8.65592X -1.89389X + 2.46673X + 4.70662X - 8.50348X (4.10)

where, maximum temperature (X1), relative humidity at 7 am (X3), relative humidity at 2

pm (X4), pan evaporation (X5), bright sunshine (X6) are observed with coefficient of determination

as 37.66% as most contributing parameters for predicting the rainfall (Y), as dependent

parameter. The parameter minimum temperature (X2) seems to have least control over rainfall and

hence does not appear in the proposed multiple regression model. The stepwise regression

procedure for selecting the significant parameters for the rainfall parameter is mentioned in

Appendix A (a).

We next performed an ARIMA modeling of the data using all the weather parameters. The

rainfall parameter time series data set is stationary so we do not require any transformation in the

data set. Then we used the autocorrelation function (ACF) and partial autocorrelation function

(PACF) of the weekly rainfall time series (see figure 4.2) to estimate the values of ‘p’ and ‘q’ of

the ARIMA model. Note that while both the ACF and PACF have significant terms at lags 1and

17, they have maximum correlation coefficient (0.152) at lag 1. This possibly suggests that an

ARIMA of order 1 is the best fit for the rainfall weather data set. Using an iterative model building

process of identification, estimation, and diagnostic checking using all the weather parameters, we

finally selected an ARIMA (1, 0, 1) model as the most appropriate fit for the observed data.

Figure 4.2. Plots of autocorrelation and partial autocorrelation coefficients and time lags of

weakly Rainfall parameter.

Then an ANN model building process was performed using all the weather parameters. We

selected the best suited architecture of Feed Forward Neural Network Model for our weekly

rainfall data by comparing methods and changing the layer and number of neurons in each

network. This proposed model had an input environment with all the weather parameters, three

hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and 12

neurons in the third hidden layer) and one neuron in the output layer (see figure 4.3).

Figure 4.3. Artificial neural network structure for weekly Rainfall prediction parameter

Figure 4.4. Mapping of the number of epochs obtained for desired goal for ANN model for

Rainfall parameter

We used a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a

tan sigmoid activation function in the first, second and third hidden layer and a log sigmoid

activation function in the output layer. A set of random values distributed uniformly between -1 to

+1 are used to initialize the weight of the neural network model. 515 epochs are used to train the

neural network model with 0.003 goal (see figure 4.4). The program used for training the 595 data

sets using all the weather parameters for ANN model for predicting the rainfall parameter is

mentioned in Appendix B (a).

We next performed a hybrid MLR_ARIMA modeling of the data using the five weather

parameters, significantly selected through stepwise regression earlier, for predicting the rainfall

weather parameter. We used the same ACF and PACF of the rainfall weather parameter, having

maximum correlation coefficient (0.152) at lag 1 (see figure 4.2). Suggesting that an

MLR_ARIMA (1, 0, 1) model as the most appropriate fit using the significantly selected weather

parameters data set.

Finally we performed a hybrid MLR_ANN model building process using the same, more

significant parameters as suggested by the multiple regression model. We selected the same best

suited architecture of Feed Forward Neural Network Model as obtained while developing the

ANN model for weekly rainfall data set. This proposed hybrid MLR_ANN model had different

input environment, that is, only the significant parameters selected through stepwise regression

analysis are used, but the same three hidden layers (8 neurons in the first hidden layer, 10 neurons

in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in the output

layer as considered earlier while developing the ANN model (see figure 4.5). We used the same

Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid

activation function in the first, second and third hidden layer and a log sigmoid activation function

in the output layer. A new different set of random values distributed uniformly between -1 to +1

are used to initialize the weight of the MLR_ANN model. 415 epochs are used to train the neural

network model with the same 0.003 goal (see figure 4.6). The program used for training the 595

data sets using the significantly selected parameter for the hybrid MLR_ANN model of dependent

rainfall weather parameter is mentioned in Appendix C (a).

Figure 4.5. Hybrid MLR_ANN structure for weekly Rainfall prediction parameter.

Figure 4.6. Mapping of the number of epochs obtained for desired goal for hybrid MLR_ANN

model for Rainfall parameter.

4.4.3 Maximum temperature

The multiple regression model is fitted to predict the weekly maximum temperature as

dependent parameter taking the other weekly independent parameters as rainfall, minimum

temperature, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most



1 2 3 4 5 6Y = 34.9866 - 0.0025X + 0.3343X - 0.0778X - 0.0703X + 0.2465X + 0.0683X

(4.11)

where, rainfall (X1), minimum temperature (X2), relative humidity at 7 am (X3) and 2 pm

(X4), pan evaporation (X5), bright sunshine (X6) are observed as 84.32% contributing parameters

for predicting the maximum temperature (Y), as dependent parameter. The stepwise regression

procedure for selecting the significant parameters for the maximum temperature is mentioned in

Appendix A (b).


maximum temperature time series data set is found to be stationary data set. Then, we use the

autocorrelation function (ACF) and partial autocorrelation function (PACF) of the weekly

maximum temperature time series (see figure 4.7) to estimate the parameters (p and q) of the

ARIMA model. Note that while both the ACF and PACF have significant terms at lags 1, 16 and

17, they have maximum correlation coefficient (0.463) at lag 1. This possibly suggests that an

ARIMA of order 1 is best fit for the maximum temperature data. Using an iterative model building

process of identification, estimation, and diagnostic checking using all the weather parameters, we

finally selected an ARIMA (1, 0, 1) model as the most appropriate fit for the observed data.

Further, we performed an ANN model building process using all the same weather

parameters. We selected the best suited architecture of Feed Forward Neural Network Model for

our weekly maximum temperature data by comparing methods and changing the layer and number

of neurons in each network. This proposed model had an input environment with all the weather

parameters, three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second

hidden layer and 12 neurons in the third hidden layer) and one neuron in the output layer (see

figure 4.8). We used a Scaled Conjugate Gradient Algorithm for training this multilayer

perceptron, a tan sigmoid activation function in all the four layers, that is, the three hidden layers

and an output layer. A set of random values distributed uniformly between -1 to +1 are used to

initialize the weight of the neural network model. 386 epochs are used to train the neural network

model with 0.00000117083 goal (see figure 4.9). The program used for training the 595 data sets

using all the weather parameter for ANN model for predicting the maximum temperature

parameter is mentioned in Appendix B (b).

Since in the stepwise regression analysis, all the parameters are selected as significant

parameters, therefore, there is no possibility of developing the hybrid models for maximum

temperature weather parameter, that is, the hybrid model will be same as ARIMA and ANN

model.


weakly Maximum Temperature parameter.

Figure 4.8. Artificial neural network structure for weekly Maximum Temperature prediction

parameter.


Maximum Temperature.

4.4.4 Minimum temperature

The multiple regression model is fitted to predict the weekly minimum temperature as

dependent parameter taking the other weekly independent parameters as maximum temperature,

rainfall, relative humidity at 7 am and 2 pm, pan evaporation and bright sunshine. The most



1 3 4 6Y = 2.64644 + 0.52100X - 0.01740X + 0.09727X 0.09121X (4.12)

where, minimum temperature (Y) as the dependent parameter having 30.13% contribution

of the significant parameters maximum temperature (X1), relative humidity at 7 am (X3), relative

humidity at 2 pm (X4) and bright sunshine (X6). The parameters rainfall (X2) and pan evaporation

(X5) seems to have least control over minimum temperature and hence does not appear in the

proposed multiple regression model. The stepwise regression procedure for selecting the

significant parameters for the minimum temperature parameter is mentioned in Appendix A (c).


minimum temperature time series data set is stationary, so no transformation of the data set is

required. We then used the autocorrelation function (ACF) and partial autocorrelation function

(PACF) of the weekly minimum temperature time series (see figure 4.10) to estimate the values of

‘p’ and ‘q’ of the ARIMA model. Note that while both the ACF and PACF have significant terms

commonly at lags 1, 15, 16 and 17, they have maximum correlation coefficient (0.435) at lag 1.

This possibly suggests that an ARIMA of order 1 is the best fit for the minimum temperature data

set. Using an iterative model building process of identification, estimation, and diagnostic

checking using all the weather parameters, we finally selected an ARIMA (1, 0, 1) model as the

most appropriate fit for the observed data. But after applying ARIMA (1, 0, 1) it is observed that

there is lag 18 in the autocorrelation plots, so we applied ARIMA (1, 0, 18) model and obtained

the appropriate fit for the observed data.

Then an ANN model building process was performed using all the weather parameters. We

selected the best suited architecture of Feed Forward Neural Network Model for our weekly

minimum temperature data by comparing methods and changing the layer and number of neurons

in each network. This proposed model had an input environment with all the weather parameters,

three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and

12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.11). We used

a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid

activation function for all the hidden and output layers. A set of random values distributed

uniformly between -1 to +1 are used to initialize the weight of the neural network model. 594

epochs are used to train the neural network model with 0.000001717 goal (see figure 4.12). The

program used for training the 595 data sets using all the weather parameters for ANN model for

predicting the minimum temperature parameter is mentioned in Appendix B (c).


weakly Minimum Temperature parameter.

Figure 4.11. Artificial neural network structure for weekly Minimum Temperature prediction

parameter.


Minimum Temperature parameter.

We next performed a hybrid MLR_ARIMA modeling of the data using the four weather

parameters, significantly selected through stepwise regression earlier, for predicting the minimum

temperature weather parameter. We used the same ACF and PACF of the minimum temperature

weather parameter (see figure 4.10), we applied ARIMA (1, 0, 18) model and obtained the

appropriate fit using the significantly selected weather parameters data set as earlier applied for

ANN model.

Figure4.13. Hybrid MLR_ANN structure for weekly Minimum Temperature prediction

parameter.




ANN model for weekly minimum temperature data set. This proposed hybrid MLR_ANN model

had different input environment, that is, only the significant parameters selected through stepwise

regression analysis are used, but the same three hidden layers (8 neurons in the first hidden layer,

10 neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in

the output layer (see figure 4.13) are considered.

We used the same Scaled Conjugate Gradient Algorithm for training this multilayer

perceptron, a tan sigmoid activation function for all the hidden and output layers. A new different

set of random values distributed uniformly between -1 to +1 are used to initialize the weight of the

MLR_ANN model. 1028 epochs are used to train the neural network model with the same

0.000001717 goal (see figure 4.14). The program used for training the 595 data sets using the

significantly selected parameter for the hybrid MLR_ANN model of dependent Minimum

Temperature weather parameter is mentioned in Appendix C (b).

Figure 4.14. Mapping of the number of epochs obtained for desired goal for Hybrid MLR_ANN

model for Minimum Temperature parameter.

4.4.5. Relative Humidity at 7 AM

The multiple regression model is fitted to predict the weekly average relative humidity at 7

am as dependent parameter taking the other weekly independent parameters as maximum

temperature, minimum temperature, rainfall, relative humidity at 2 pm, pan evaporation and bright

sunshine. The most significantly contributed parameters are selected using stepwise regression

analysis based on 595 data set of 35 years and the best fit multiple regression model is given

below:

1 2 3 4 5 6Y = 116.06296 -1.31618X - 0.21386X - 0.01111X + 0.33298X - 0.60294X + 0.29476X

(4.13)

where, relative humidity at 7 am (Y) as the dependent parameter having 79.04%

contribution of the significant parameters maximum temperature (X1), minimum temperature (X2),

rainfall (X3), relative humidity at 2 pm (X4) pan evaporation (X5) and bright sunshine (X6). The

stepwise regression procedure for selecting the significant parameter for the relative humidity at 7

am parameter is mentioned in Appendix A (d).

We next performed an ARIMA modeling of the data using all the weather parameters.

Further, seeing the relative humidity at 7 AM time series data set it is clear that the data is

stationary and, therefore, does not require any transformation. We used the autocorrelation

function (ACF) and partial autocorrelation function (PACF) of the weekly average relative

humidity at 7 AM time series (see figure 4.15) to estimate the parameters (p and q) of the ARIMA

model. Here, while both the ACF and PACF have common significant terms at lags 1, 15, 16 and

17, but autocorrelation function (ACF) has maximum correlation coefficient (0.531) at lag 17 and

partial autocorrelation function (PACF) have maximum correlation coefficient (0.513) at lag 1.

Using an iterative model building process of identification, estimation, and diagnostic checking

using all the weather parameters, we finally selected an ARIMA (1, 0, 1) model as the most

appropriate fit for the observed data. But after applying ARIMA (1, 0, 1) it is observed that there

is lag 10 in the autocorrelation plots, so we applied ARIMA (1, 0, 10) model and obtained the

appropriate fit for the observed data without lags.

Next, we performed an ANN model building process using all the same weather


our weekly relative humidity at 7 am by comparing methods and changing the layer and number of

neurons in each network. This proposed model had an input environment with all the parameters,


12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.16). We used

a Scaled Conjugate Gradient Algorithm for training this multilayer perceptron, a tan sigmoid

activation function in the first, second and third hidden layer and a log sigmoid activation function

in the output layer.

A set of random values distributed uniformly between -1 to +1 are used to initialize the

weight of the neural network model. 747 epochs are used to train the neural network model with

0.0000166 goals (see figure 4.17). The program used for training the 595 data sets using all the

weather parameters for ANN model for predicting relative humidity at 7 AM parameter is

mentioned in Appendix B (d). Since in the stepwise regression analysis all the parameters are

selected as significant parameters, therefore, there is no possibility of developing the hybrid

models for relative humidity at 7 AM weather parameter, that is, the hybrid model will be same as

ARIMA and ANN model.


weakly average Relative Humidity at 7 AM parameter.

Figure 4.16. Artificial neural network structure for weekly average Relative Humidity at 7 AM



Relative Humidity at 7 AM parameter.

4.4.6. Relative Humidity at 2 PM

The multiple regression model is fitted to predict the weekly average relative humidity at 2

pm as dependent parameter taking the other weekly independent parameters as maximum

temperature, minimum temperature, relative humidity at 7 am, rainfall, pan evaporation and bright

sunshine. The most significantly contributed parameters are selected using stepwise regression

analysis based on 595 data set of 35 years and the best fit multiple regression model is given

below:

1 2 3 4 5 6Y = 46.85286 -1.85133X +1.74419X + 0.51835X + 0.02415X - 0.40761X -1.02706X

(4.14)

where, relative humidity at 2 pm (Y) as the dependent parameter having 84.45%

contribution of the significant parameters maximum temperature (X1), minimum temperature (X2),

relative humidity at 7 am (X3), rainfall (X4), pan evaporation (X5) and bright sunshine (X6). The

stepwise regression procedure for selecting the significant parameters for the relative humidity at 2

pm parameter is mentioned in Appendix A (e).


relative humidity at 2 pm time series data set required no transformation in the data since it is a

stationary data set. We then used the autocorrelation function (ACF) and partial autocorrelation

function (PACF) of the weekly average relative humidity at 2 pm time series (see figure 4.18) to

estimate the values of ‘p’ and ‘q’ of the ARIMA model. Note that while both the ACF and PACF

have common significant terms at lags 1, 15, 16 and 17, they have maximum correlation

coefficient (0.622) at lag 1. This possibly suggests that an ARIMA of order 1 is best fit for the

data. Using an iterative model building process of identification, estimation, and diagnostic

checking using all the weather parameters, we finally selected an ARIMA (1, 0, 0) model as the

most appropriate fit for the observed data.

Further we performed an ANN model building process using all the same weather


our weekly relative humidity at 2 PM by comparing methods and changing the layer and number

of neurons in each network. This proposed model had an input environment with all the weather

parameters, three hidden layers (8 neurons in the first hidden layer, 10 neurons in the second

hidden layer and 12 neurons in the third hidden layer) and one neuron in the output layer (see

figure 4.19). We used a Scaled Conjugate Gradient Algorithm for training this multilayer

perceptron, a tan sigmoid activation function in the three hidden layer and the output layer. A set

of random values distributed uniformly between -1 to +1 are used to initialize the weight of the

neural network model. 544 epochs are used to train the neural network model with 0.00003155

goals (see figure 4.20). The program used for training the 595 data sets using all the weather

parameter for ANN model for predicting relative humidity at 2 pm parameter is mentioned in

Appendix B (e).


weakly average Relative Humidity at 2 PM parameter.

Since in the stepwise regression analysis all the parameters are selected as significant

parameters, therefore, there is no possibility of developing the hybrid models for relative humidity

at 2 pm weather parameter, that is, the hybrid model will be same as ARIMA and ANN model.

Figure 4.19. Artificial neural network structure for weekly average Relative Humidity at 2 PM



Relative Humidity at 2 PM parameter.

4.4.7. Pan Evaporation

The multiple regression model is fitted to predict the weekly average pan evaporation as


minimum temperature, relative humidity at 7 am and 2 pm, rainfall, and bright sunshine. The most



1 3 4 5Y = -4.10879 + 0.53040X - 0.07685X - 0.02902X + 0.00351X (4.15)

where, maximum temperature (X1), relative humidity at 7 am (X3), relative humidity at 2

pm (X4) and rainfall (X5) are observed with coefficient of determination as 69.22% as most

contributing parameters for predicting the pan evaporation (Y), as dependent parameter. The

parameter minimum temperature (X2) and bright sunshine (X6) seems to have least control over

pan evaporation and hence does not appear in the proposed multiple regression model. The

stepwise regression procedure for selecting the significant parameters for the pan evaporation

parameter is mentioned in Appendix A (f).


pan evaporation time series data set is found to be stationary, so we did not require any

transformation of the data set. We then used the autocorrelation function (ACF) and partial

autocorrelation function (PACF) of the weekly average pan evaporation time series (see figure

4.21) to estimate the parameters (p and q) of the ARIMA model. Note that, both the ACF and

PACF have common significant terms at lags 1, 15, 16 and 17, but autocorrelation function (ACF)

has maximum correlation coefficient (0.605) at lag 17 and partial autocorrelation function (PACF)

have maximum correlation coefficient (0.548) at lag 1. Using an iterative model building process

of identification, estimation, and diagnostic checking, we finally selected an ARIMA (1, 0, 17)

model as the most appropriate fit for the observed data.

Further, we performed an ANN model building process using all the weather parameters.

We selected the best suited architecture of Feed Forward Neural Network Model for our weekly

pan evaporation by comparing methods and changing the layer and number of neurons in each

network. This proposed model had an input environment with all the weather parameters, three

hidden layers (8 neurons in the first hidden layer, 10 neurons in the second hidden layer and 12

neurons in the third hidden layer) and one neuron in the output layer (see figure 4.22).


weakly average Pan Evaporation parameter.

Figure 4.22. Artificial neural network structure for weekly average Pan Evaporation prediction

parameter.


tan sigmoid activation function in the first, second and third hidden layer and a log sigmoid

activation function in the output layer. A set of random values distributed uniformly between -1 to

+1 are used to initialize the weight of the neural network model. 117 epochs are used to train the

neural network model with 0.00000285 goals (see figure 4.23). The program used for training the

595 data sets using all the weather parameters for ANN model for predicting pan evaporation

parameter is mentioned in Appendix B (f).

We next performed a hybrid MLR_ARIMA modeling of the data using the four weather

parameters, significantly selected through stepwise regression earlier, for predicting the pan

evaporation weather parameter. We used the same ACF and PACF having maximum correlation

coefficient (0.605) at lag 17 and (0.548) at lag 1, respectively (see figure 4.21). Suggesting that an

MLR_ARIMA (1, 0, 17) model as the most appropriate fit using the significantly selected weather


Figure 4.23. Mapping of the number of epochs obtained for desired goal for ANN model for Pan

Evaporation parameter.




ANN model for weekly average pan evaporation data set. This proposed hybrid MLR_ANN

model had different input environment, that is, only the significant parameters selected through

stepwise regression analysis, but the same three hidden layers (8 neurons in the first hidden layer,

10 neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in

the output layer (see figure 4.24). We used the same Scaled Conjugate Gradient Algorithm for

training this multilayer perceptron, a tan sigmoid activation function in the first, second and third

hidden layer and a log sigmoid activation function in the output layer. A new different set of

random values distributed uniformly between -1 to +1 are used to initialize the weight of the

MLR_ANN model. 76 epochs are used to train the neural network model with 0.00000285 goals

(see figure 4.25). The program used for training the 595 data sets using the significantly selected

parameter for the hybrid MLR_ANN model of dependent pan evaporation weather parameter is

mentioned in Appendix C (c).

Figure 4.24. Hybrid MLR_ANN structure for weekly average Pan Evaporation prediction

parameter.


model for Pan Evaporation parameter.

4.4.8. Bright Sunshine

The multiple regression model is fitted to predict the weekly average bright sunshine as


minimum temperature, relative humidity at 7 am and 2 pm, pan evaporation and rainfall. The most



1 2 3 4 6Y = 7.86562 + 0.12606X - 0.13358X + 0.03621X - 0.07713X - 0.00597X (4.16)

where, bright sunshine (Y) as the dependent parameter having 46.68% contribution of the

significant parameters maximum temperature (X1), minimum temperature (X2) , relative humidity

at 7 am (X3), relative humidity at 2 pm (X4) and rainfall (X6). The parameter pan evaporation (X5)

seems to have least control over bright sunshine and hence does not appear in the proposed

multiple regression model. The stepwise regression procedure for selecting the significant

parameters for the bright sunshine parameter is mentioned in Appendix A (g).


bright sunshine time series data set is found to be stationary so we did not require any

transformation of the data set. We used the autocorrelation function (ACF) and partial

autocorrelation function (PACF) of the weekly average bright sunshine time series (see figure

4.26) to estimate the parameters (p and q) of the ARIMA model. Note that while both the ACF

and PACF have common significant terms at lags 1 and 17, they have maximum correlation

coefficient (0.356) at lag 1. This possibly suggests that an ARIMA of order 1 is best fit for the

bright sunshine data set. Using an iterative model building process of identification, estimation,

and diagnostic checking using all the weather parameters, we finally selected an ARIMA (1, 0, 0)

model as the most appropriate fit for the observed data.

Then we performed an ANN model building process using all the same weather


our weekly bright sunshine by comparing methods and changing the layer and number of neurons

in each network. This proposed model had an input environment with all the weather parameters,


12 neurons in the third hidden layer) and one neuron in the output layer (see figure 4.27).


tan sigmoid activation function in the first, second and third hidden layer and a purelin activation

function in the output layer. A set of random values distributed uniformly between -1 to +1 are

used to initialize the weight of the neural network model. 570 epochs are used to train the neural

network model with 0.00000219 goals (see figure 4.28). The program used for training the 595 data

sets using all the weather parameters for ANN model for predicting bright sunshine parameter is

mentioned in Appendix B (g).


weakly average Bright Sunshine parameter.

Figure 4.27. Artificial neural network structure for weekly average Bright Sunshine prediction

parameter.


Bright Sunshine parameter.

We next performed a hybrid MLR_ARIMA modeling of the data using the five weather

parameters, significantly selected through stepwise regression earlier, for predicting the bright

sunshine weather parameter. We used the same ACF and PACF of the bright sunshine weather

parameter, having maximum correlation coefficient (0.356) at lag 1 (see figure 4.26). Suggesting

that an ARIMA (1, 0, 0) model as the most appropriate fit using the significantly selected weather


Figure 4.29. Hybrid MLR_ANN structure for weekly average Bright Sunshine prediction

parameter.

Finally, we performed a hybrid MLR_ANN model building process using the same, more



ANN model for weekly average bright sunshine data set. This proposed hybrid MLR_ANN model

had different input environment, that is, only the significant parameters selected through stepwise

regression analysis, but the same three hidden layers (8 neurons in the first hidden layer, 10

neurons in the second hidden layer and 12 neurons in the third hidden layer) and one neuron in the

output layer (see figure 4.29) are considered.

We used the same Scaled Conjugate Gradient Algorithm for training this multilayer

perceptron, a tan sigmoid activation function in the first, second and third hidden layer and a

purelin activation function in the output layer as considered while developing the ANN model. A

new different set of random values distributed uniformly between -1 to +1 are used to initialize the

weight of the MLR_ANN model. 1257 epochs are used to train the neural network model with the

same 0.00000219 goal (see figure 4.30). The program used for training the 595 data sets using the

significantly selected parameter for the hybrid MLR_ANN model of dependent bright sunshine

weather parameter is mentioned in Appendix C (d).


Bright Sunshine parameter.

4.5 Comparison of prediction ability of forecasting models

4.5.1. Introduction

Recognition of pertinent model is an important errand, obtained through comparing the

predictive ability of forecasting models. The methodology presented above was applied to training

data set of 35 years, that is, 595 data sets, weekly weather data for the monsoon period. For the

training data sets, the comparison among the developed models is made for each weather

parameters by comparing the real and predicted values.

The comparison is made on the basis of the analytical methods, which can be evaluated in

terms of graphical representation in the form of linear scale plot and numerical computations

through mean error (BIAS), mean absolute error (MAE), root mean square error (RMSE),

prediction error (PE), and correlation coefficient (r). Here the comparison among the predicted and

observed values for all the seven parameters for training date set are discussed graphically in the

next sub sections.

4.5.2. Rainfall

The process developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and hybrid

MLR_ANN in section 4.4.2 are compared using the graphical method. The comparison among the

developed models is made to identify the finest model for the rainfall weather parameter by

comparing the real and predicted values. In figure 4.31, we provide a comparison of the actual

rainfall with the predicted rainfall using all the five developed models, for 595 data sets used for

training purposes, observing that predicted values by ANN and Hybrid MLR_ANN models are

tending more towards the actual values of weekly rainfall.

4.5.3. Maximum Temperature

The procedure developed through multiple linear regression, ARIMA and neural network in

section 4.4.3 are compared using the graphical method. The comparison among the developed models

is presented to identify the finest model for the maximum temperature weather parameter by

comparing the real and predicted values. In figure 4.32, we provide a comparison of the weekly actual

maximum temperature with the predicted maximum temperature using the three developed models, for

595 data sets used for training purposes. The values obtained through ANN model are tending more to

overlap the actual weekly maximum temperature values. A few points among them can be observed

viz. the last week of 1961, second week of 1965, 1973, first week of 1980 and seventh week of 1982,

1986.

4.5.4. Minimum Temperature

The course of action developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and

hybrid MLR_ANN in section 4.4.4 are compared using the graphical method. The comparison

among the developed models is discussed to identify the finest model for the minimum

temperature weather parameter by comparing the real and predicted values. In figure 4.33, we

provide a comparison of the weekly actual minimum temperature with the predicted minimum

temperature using all the developed models, for 595 data sets used for training purposes. For one

or two weeks of few years the predicted values obtained through the Hybrid MLR_ANN model

tended more towards the actual values of minimum temperature.

4.5.5. Relative Humidity at 7 AM

The guiding principle developed through multiple regression, ARIMA and neural network

in section 4.4.5 are compared using the graphical method. The comparison among the developed

models is fundamental, so as to identify the finest model for the relative humidity at 7 am, weather

parameter by comparing the real and predicted values. In figure 4.34, we provide a comparison of

the weekly actual relative humidity at 7 am with the predicted relative humidity at 7 am using the

three developed models, for 595 data sets used for training purposes. The predicted values

obtained through ANN model gives more precise results as can be seen graphically at few point’s

viz., first week of 1964, 1965, 1966, 1975, 1992; seventh week of 1969; second week of 1979,

1987 and twelfth week of 1982.

4.5.6. Relative Humidity at 2 PM

The procedure developed through multiple regression, ARIMA and neural network in

section 4.4.6 are compared using the graphical method. The comparison among the developed

models is presented, so as to identify the finest model for the relative humidity at 2 pm, weather

parameter by comparing the real and predicted values. In figure 4.35, we provide a comparison of

the weekly actual relative humidity at 2 pm with the predicted relative humidity at 2 pm using the

three developed models, for 595 data sets used for training purposes, indicating ANN model as

preferred model.

4.5.7. Pan Evaporation

The path developed through multiple regression, ARIMA, ANN, hybrid MLR_ARIMA and

hybrid MLR_ANN in section 4.4.7 are compared using the graphical method. The comparison

among the developed models are primary, so as to identify the premium model for the pan

evaporation weather parameter by comparing the real and predicted values. Figure 4.36, presents a

comparison of the weekly actual pan evaporation with the predicted pan evaporation using all the

developed models, for 595 data sets used for training purposes. The values obtained through ANN

and hybrid MLR_ANN models are tending to partly cover the actual weekly pan evaporation but we

can examine that hybrid MLR_ANN is leaning more towards the actual values of pan evaporation,

as observed in few point’s viz., first week of 1961, 1962, 1964, 1965, 1967, 1980, 1984; third week

of 1973, 1992, 1994; sixth week of 1972 and tenth week of 1972, 1975.

4.5.8. Bright Sunshine

The path developed through MLR, ARIMA, ANN, hybrid MLR_ARIMA and hybrid

MLR_ANN in section 4.4.8 are compared using the graphical method. The comparison among the

developed models are primary, so as to identify the premium model for the bright sunshine

weather parameter by comparing the real and predicted values. Figure 4.37, is presented with a

comparison of the weekly actual bright sunshine with the predicted bright sunshine using all the

developed models, for 595 data sets used for training purposes indicating hybrid MLR_ANN

model as a preferred model.

4.6 Conclusion

Complexity of the nature of weekly weather parameters record has been studied using the

Multiple Linear Regression, Autoregressive Integrated Moving Average, Artificial Neural

Network, Hybrid MLR_ARIMA and Hybrid MLR_ANN techniques. The weekly weather

parameters data for the months of June, July, August and September over a period of 35 years of

pantnagar region was used to develop and train the models. Since, the parameter selection (input

pattern) in the models is always a challenging task, so to reduce this complexicity we proposed the

hybrid model by introducing those parameters only which are found significant using stepwise

regression analysis to obtain valid non-bias results. The result showed that the variation during the

four months was among all the parameters and correlation between relative humidity at 7 am and

relative humidity at 2 pm was maximum (0.83736) and it was minimum between rainfall and

minimum temperature (-0.00666).

The above mentioned five models were developed for each weather parameter. In

developing the ANN and hybrid MLR_ANN models, the three hidden layers and scale conjugate

gradient algorithm for training were same for all the weather parameters, expect the input

parameters and the weight and biases considered. Observing the graphical presentation of each

weather parameter, it was concluded that ANN model is a preferred model in comparison to the

MLR and ARIMA models for all the weekly weather parameter. Finally, the study reveals that

hybrid MLR_ANN model can be used as an appropriate forecasting tool to estimate the weather

parameters, in contrast to the MLR, ARIMA, ANN and hybrid MLR_ARIMA models.

IDENTIFICATION OF PRECISE WEATHER FORECASTING MODEL

Introduction

Validation of weather forecasting model

Conclusion

CHAPTER 5

IDENTIFICATION OF PRECISE WEATHER FORECASTING MODEL

5.1 Introduction

The performance of the entire existing weather forecasting model was evaluated in the

previous chapter and the comparison of prediction ability of forecasting models indicates the

better performance of Artificial Neural Network model. All the seven weather parameters were

used to develop the MLR, ARIMA and ANN models. The most significantly contributed variables

selected for each weather parameter using stepwise regression analysis were used to develop the

hybrid MLR_ARIMA and hybrid MLR_ANN models. The comparison among the real and

predicted values, to identify the precise weather forecasting is made in this section and applied to

testing quasi-independent 255 data sets of 15 years weekly weather data for the

monsoon period which was not used while developing the model.

The comparison is made on the basis of the analytical methods, which can be evaluated in

terms of graphical representation in the form of linear scale plot and numerical computations

through mean error (BIAS), mean absolute error (MAE), root mean square error (RMSE),

prediction error (PE), and correlation coefficient (r). The details of the numerical computational

methodology are used as explained in section 4.3.5. The appropriateness effectiveness of these

models is demonstrated by comparing the actual value of all the seven weather parameters with

their predicted value and results are presented in next section.

5.2. Validation of weather forecasting model

5.2.1 Rainfall

A comparison of the performance of the actual weekly rainfall with its predicted value

using the proposed models are presented with those of other traditional forecasting models,

graphically in figure 5.1.

It is clearly perceptible that the predicted values of the 17 weeks of weekly monsoon

rainfall for the first three years 1996 to 1998 varied with slight fluctuation from the actual values.

Further, it identify that there is a large variation among few monsoon weeks of the two years, 1999

and 2000, since the predicted values are in a trend form but the actual value are having high

variation. Next, in the years 2001 and 2002 the actual values are slightly smaller than the predicted

values, but a large variation again due to the actual values different from the trend pragmatic

through previous years are observed for two to three monsoon weeks of the years 2003, 2004 and

2005.

Presently, it is also observed that in the last monsoon weeks of 2003 and beginning

monsoon weeks of 2004, very less variation is observed between the actual value and predicted

value through hybrid MLR_ANN model. Further, similar observation is identified for the few

monsoon weeks in the years 2007 and 2009. Thus indicating the hybrid MLR_ANN model is a

proficient predictor in comparison to the other considered models.

The estimates viz. mean error, mean absolute error, root mean square error, prediction

error and correlation coefficient are presented in table 5.1 for the testing data set. The bias for

testing data set is the least for hybrid MLR_ANN model than that obtained from hybrid

MLR_ARIMA, ANN, ARIMA and MLR models.

Table 5.1. Comparison of the performance of forecasting models for Rainfall parameter.

Techniques BIAS MAE RMSE PE CC

MLR 3.39615 53.91620 75.58739 0.66798 0.60251

ARIMA 1.57248 52.87119 74.90918 0.65503 0.61154

ANN -6.23647 49.74627 74.74657 0.61632 0.61441

MLR_ARIMA 3.59415 53.90974 75.50548 0.66790 0.60344

MLR_ANN -6.28196 49.33765 74.34915 0.61125 0.61894

The MAE further explains that the hybrid MLR_ANN model is more precise than ANN

model. The hybrid MLR_ANN model shows a smaller value for RMSE as compared to those of

the other models. The PE obtained for testing data from MLR, hybrid MLR_ARIMA, ARIMA and

ANN models is 0.66798, 0.66790, 0.65503 and 0.61632, respectively, while for hybrid

MLR_ANN is 0.61125 which is the least and indicating it as precise prediction model. Further, the

correlation coefficient is found to be highest for hybrid MLR_ANN model in comparison to other

models. Thus, the graphical representation as well as the numerical estimates both favored, the

hybrid multiple linear regression with artificial neural network (MLR_ANN) model as a preferred

performance in comparison to the other models, concluding that this hybrid technique can be used

as an effective rainfall forecasting tool in the Himalaya.

5.2.2 Maximum Temperature

Comparison of the performance of the weekly maximum temperature parameter is

presented graphically in figure 5.2. The weekly monsoon maximum temperature for the years

1996, 1999, 2000, 2001, 2002, 2005, 2006 2007 and 2008 clearly shows some variation among the

predicted value and actual values for most of the weeks of each year. But for the years 1997, 1998,

2003, 2004, 2009 and few starting weeks of 2010 very less variation is observed among the actual

values and predicted value through ANN model, indicating that ANN predicts comparatively

better than the other traditional models.


error and correlation coefficient are also presented in table 5.2 for the same data set. Since in the

stepwise regression analysis all the parameters are selected as significant parameters, therefore,

there is no possibility of developing the hybrid models for maximum temperature in the study

area. The bias obtained for testing data set is smaller for artificial neural network than the values

that are obtained from autoregressive integrated moving average and multiple linear regression.

Table 5.2. Comparison of the performance of forecasting models for Maximum Temperature.


MLR 0.87990 1.05235 1.45396 0.03228 0.88964

ARIMA 0.87050 1.04583 1.45192 0.03208 0.88881

ANN 0.79330 1.00275 1.40719 0.03076 0.88995

The MAE measure for testing data set for multiple linear regression model is 1.05235 and

for ARIMA model is 1.04583, while the same error measure is considerably lower at 1.00275 for

the artificial neural network model. The ANN model shows a lesser value for RMSE compared to

those of MLR and ARIMA models. The PE obtained for testing data from multiple linear

regression model is 0.03228 and ARIMA is 0.03208 and through artificial neural network model it

is 0.03076 which is smaller representing it as preferred prediction model. Moreover, the neural

network model had the highest correlation coefficient among all the models. These numerical

estimates hold up that the ANN model has a superior performance in comparison to ARIMA and

MLR model which coincides with the previous results.

5.2.3 Minimum Temperature

Comparison of the performance of the proposed model is presented with those of other

forecasting model of the weekly minimum temperature parameter graphically in figure 5.3. The

weekly minimum temperature for the monsoon season shows fluctuations among the actual and

predicted values. The predicted values developed through hybrid MLR_ANN models following a

trend based on previous years, showed least variation to the actual values for few weeks of the

years 1997, 1998, 1999, 2000, 2004 and 2009. Thus the hybrid MLR_ANN model proved to be

better performer and can be further analyzed more precisely by evaluating through numerical

performance of each model. The estimates viz. mean error, mean absolute error, root mean square

error, prediction error and correlation coefficient are presented in table 5.3 for the testing data set.

Table 5.3. Comparison of the performance of forecasting models for Minimum Temperature.

Techniques

BIAS MAE RMSE PE CC

MLR -0.79700 1.09592 1.38227 0.04416 0.44361

ARIMA -0.71713 1.08877 1.29726 0.04387 0.49334

ANN -0.77922 1.08745 1.33547 0.04382 0.49699

MLR_ARIMA -0.72705 1.08911 1.31870 0.04388 0.46572

MLR_ANN -0.72980 1.03726 1.29656 0.04179 0.50767

It can be seen from table 5.3 that the bias for testing data set is -0.72980 for hybrid

MLR_ANN model which is neither the lowest nor the highest among all the other compared

models. Since, the bias does not prove to be a perfect criterion for comparison among the

developed models, so we move forward to the other considered performance evaluation criteria’s.

Next, the MAE for testing data set is 1.09592, 1.08911, 1.08877 and 1.08745 for MLR, hybrid

MLR_ARIMA, ARIMA and ANN respectively; while the same error measure is lowest as

1.03726 for the hybrid MLR_ANN model. The RMSE is also found to be least as 1.29656 for the

hybrid MLR_ANN model in comparison to the other forecasting models. The PE obtained for

testing data from MLR model is 0.04416, hybrid MLR_ARIMA is 0.04388, ARIMA model is

0.04387 and ANN is 0.04382 and through hybrid MLR_ANN model is 0.04179 which is lesser

indicating it as preferred prediction model. Further, the correlation coefficient is observed highest

for hybrid MLR_ANN model. As the graphical representation as well as the numerical estimates

sustains that the hybrid MLR_ANN model has a preferred performance. It can be concluded that

this hybrid technique can be used as a reliable minimum temperature forecasting contrivance in

the Himalaya.

5.2.4 Relative Humidity at 7 AM

A comparison of the performance of the actual values with the predicted values of relative

humidity at 7 am is presented graphically in figure 5.4. The predicted values of the 17 weeks of

weekly average relative humidity at 7 am monsoon season, showed least variation of predicted

ANN values to the actual values for most of the weeks of all the 15 years, except for September

weeks of 1998, 2003 and 2007, and few weeks of 1999, 2004, 2008 and 2010. Thus, it can be

competently said that ANN model is an efficient performer than the other developed models.

In table 5.4, the estimates viz. mean error, mean absolute error, root mean square error,

prediction error and correlation coefficient for the testing data set are also given. Since in the

stepwise regression analysis, all the parameters are selected as significant parameters, therefore,

there is no possibility of developing the hybrid models for Relative Humidity at 7 AM weather

parameter in the present study area.

Table 5.4. Comparison of the performance of forecasting models for Relative Humidity at 7 AM .


MLR 1.13423 3.16400 4.11119 0.03663 0.89181

ARIMA 1.13182 3.08161 4.01749 0.03568 0.89820

ANN 1.06157 2.92431 3.87023 0.03385 0.90479

It can be observed from table 5.4 that the bias for testing data set is lesser for artificial

neural network than the values that are obtained from ARIMA and MLR. The MAE measure for

testing data set for multiple linear regression model is 3.16400 and for ARIMA model is 3.08161,

while the same error measure is considerably lower at 2.92431 for the artificial neural network

model. The artificial neural network model also shows a smaller value for RMSE judge against to

those of MLR and ARIMA models. The PE obtained for testing data from MLR, ARIMA and

ANN models is 0.03663, 0.03568 and 0.03385, indicating ANN model as the lowest among the

entire three prediction models. Further, the correlation coefficient is observed to be highest for

artificial neural network model as 0.90479 in comparison to 0.89181 and 0.09820 for multiple

linear regression and ARIMA models respectively. These numerical estimates maintain that the

ANN model has a preferred performance and can be used as a forecasting tool for relative

humidity at 7 am in the Himalayas range, which validate the previous findings.

5.2.5 Relative humidity at 2 PM

A comparison of the performance of the actual relative humidity at 2 pm with the predicted

relative humidity at 2 pm using the three forecasting models is presented graphically in figure 5.5.

The weekly average relative humidity at 2 pm of the monsoon season showed less variation

between the actual values and predicted values through ANN for most of the weeks of all years. A

reasonable variation of predicted values to the actual value can be observed in a week of 1996,

2005, 2006 and 2007. Thus, it can be said that ANN model is a better performer in comparison to

the other developed models.

The table 5.5, represents the estimates viz. mean error, mean absolute error, root mean

square error, prediction error and correlation coefficient for the testing data set. Since in the

stepwise regression analysis all the parameters are selected as significant parameters, therefore,

there is no possibility of developing the hybrid models for Relative Humidity at 2 PM weather

parameter in the present study area.

Table 5.5. Comparison of the performance of forecasting models for Relative Humidity at 2 PM.


MLR 1.71266 3.47877 4.93385 0.05245 0.93181

ARIMA 1.58552 3.38308 4.83330 0.05100 0.93333

ANN 0.45569 3.28471 4.55504 0.04952 0.93479

It can be seen from the above table that the bias for testing data set is the least for artificial

neural network model than the other considered models. The MAE measure for testing data set for

multiple linear regression model is 3.47877 and for ARIMA model is 3.38308, while the same

error measure is considerably lower at 3.28471 for the artificial neural network model. The

artificial neural network model shows a smaller value for RMSE compared to those of multiple

linear regression and ARIMA models. The PE obtained for testing data is 0.05245 and 0.05100

from multiple linear regression model and ARIMA, respectively, and through artificial neural

network model is 0.4952 which is smaller and can be preferred as predictive model.

Moreover, the correlation coefficient is observed to be highest for artificial neural network

model. Thus, these numerical estimates support the graphical presentation indicating that the

artificial neural network model has a preferred performance in comparison to multiple linear

regression and autoregressive integrated moving average models.

5.2.6 Pan Evaporation

Comparison on the performance of the proposed model is presented with those of other

forecasting model for the weekly average pan evaporation parameter graphically in figure 5.6. The

monsoon weekly average pan evaporation showed the least variation among the actual and

predicted values obtained through hybrid MLR_ANN for most of the weeks of almost all the

years, except for the two years 1999 and 2000, since these two years also followed a prediction

pattern developed from the previous years, are reasonably different, from the actually observed

values of weekly average pan evaporation. Thus, indicating that the hybrid MLR_ANN is a better

performer in comparison to the other models.



testing data set is the highest for hybrid MLR_ANN model than the values that are obtained from

hybrid MLR_ARIMA, ANN, ARIMA and MLR models.

Table 5.6. Comparison of the performance of forecasting models for Pan Evaporation.

Techniques

BIAS MAE RMSE PE CC

MLR -0.61433 1.12179 1.50834 0.20349 0.80829

ARIMA -0.55696 1.09292 1.46588 0.19826 0.80699

ANN -0.42118 1.07216 1.43977 0.19449 0.80355

MLR_ARIMA -0.58701 1.09695 1.48079 0.19899 0.80593

MLR_ANN -0.40118 1.07019 1.40800 0.19413 0.81262

The MAE for testing data set for multiple linear regression model is 1.12179 and for

hybrid MLR_ARIMA model is 1.09695, ARIMA model is 1.09292 and ANN is 1.07216 while the

same error measure is noticeably lower as 1.07019 for the hybrid MLR_ANN model. Moreover,

the hybrid MLR_ANN model shows a smaller value for RMSE as 1.40800 in comparison to the

other models. The PE obtained for testing data from multiple linear regression model is 0.20349,

hybrid MLR_ARIMA is 0.19899, ARIMA model is 0.19826 and ANN is 0.19449 and through

hybrid MLR_ANN model is 0.19413 which is the smallest, indicating it as preferred prediction

model. Further, the correlation coefficient is found to be the highest for neural network model.

These numerical estimates also support the graphical presentation, indicating that, the hybrid

multiple linear regression with artificial neural network (MLR_ANN) model has a favored

performance and can be used as a reliable pan evaporation forecasting tool in the Himalaya.

5.2.7 Bright Sunshine

Comparison of the performance of the proposed model is presented with those of other

forecasting model of the weekly average bright sunshine parameter graphically in figure 5.7. It can

be identified that the weekly average bright sunshine of the monsoon season reflects variation

between the actual and predicted values through all the five considered models, except for few

weeks among the years, a less variation is observed between the actual and predicted value

through hybrid MLR_ANN model. Indicating hybrid MLR_ANN model as a preferred performer

than the other methods considered and developed.



testing data set is the highest for hybrid MLR_ANN model than the values that are obtained from

hybrid MLR_ARIMA, ANN, ARIMA and MLR models.

Table 5.7. Comparison of the performance of forecasting models for Bright Sunshine.


MLR 0.11512 1.21750 1.48545 0.20045 0.79686

ARIMA 0.07957 1.19586 1.46565 0.19689 0.79746

ANN 0.24863 1.22118 1.46282 0.20106 0.81908

MLR_ARIMA 0.08059 1.19610 1.46593 0.19693 0.79742

MLR_ANN 0.30745 1.17726 1.43552 0.19383 0.81924

The MAE for testing data set for MLR model is 1.21750 and for hybrid MLR_ARIMA

model is 1.19610, ARIMA model is 1.19586 and ANN model is 1.22118 while the same error

measure is noticeably lower as 1.17726 for the hybrid MLR_ANN model. The hybrid MLR_ANN

model also shows a smaller value for RMSE as 1.43552 compared to those of MLR, ARIMA,

ANN and hybrid MLR_ARIMA models as 1.48545, 1.46565, 1.46282 and 1.46593 respectively,

for testing data set.

In general, a good forecast may have a relatively high BIAS value but a relatively low

MAE and RMSE values (if the predicted variable is well correlated with independent variables) or

low BIAS value and high MAE and RMSE (if the predicted variable is poorly correlated with

independent variables).

The PE obtained for testing data from MLR model is 0.20045, hybrid MLR_ARIMA

model is 0.19693, ARIMA model is 0.19689 and ANN model is 0.20106 and through hybrid

MLR_ANN model is 0.19383 which is the smallest indicating it as preferred prediction model.

Further, the correlation coefficient is found to be the highest for hybrid MLR_ANN model as

0.81924 than the other models. These numerical estimates, thus, support that the hybrid multiple

linear regression with artificial neural network (MLR_ANN) model has a better performance and

can be preferred as a reliable bright sunshine forecasting tool in the Himalaya.

5.3 CONCLUSION

Hybrid MLR_ANN, hybrid MLR_ARIMA, artificial neural network, autoregressive

moving average and multiple linear regression models are used to study the impediment of the

environment of weekly weather parameters. The weekly weather parameters data for the months

of June, July, August and September over a period of 15 years of pantnagar region was used to

identify the precise weather forecasting models. Since, the parameter assortment in the models is

always an exigent task, so we introduced those parameters for hybrid model building only which

are found significant using stepwise regression analysis during the training period.

The comparison among the five models shows the trend, based on previous years but the

actual values were fluctuating. The proposed hybrid MLR_ANN model was observed as precise

model in comparison to MLR, ARIMA, ANN and hybrid MLR_ARIMA models. In view of the

fact that, all the prediction models are consistent but the finest model is the lone having least mean

absolute error and root mean square error, prediction error and high correlation coefficient, as

observed in hybrid MLR_ANN model. It was observed that the ANN model is also precise

weather forecasting model as compared to MLR and ARIMA models which coincides with the

previous findings. At last, the study reveal that hybrid MLR_ANN model maintained can be used

as an appropriate forecasting interest to estimate the weather parameters, in contradiction to the

multiple linear regression, ARIMA, ANN and hybrid MLR_ARIMA models.

SUMMARY AND FUTURE SCOPE

Summary

Future scope

CHAPTER 6

SUMMARY AND FUTURE SCOPE

6.1. Summary

The role of statistical techniques for providing reliable predictions of weather parameters is

considered to be most important in the field of metrology all over the world. These predictions

influence the agricultural as well as the industrial strategies. In India, the month of June, July,

August and September are identified as the summer monsoon month. The summer monsoon in

early June marks the beginning of the principal raining season for the Himalaya. The monsoon

season in and around Indian Metrological Department (IMD) Pantnagar observatory which is

situated in the foot hills of Himalayas, ranges between 15 to 20 weeks. Accordingly our study is

based on a time series weather data set, collected at the IMD observatory at Pantnagar, India, over

a period of 50 years. Pantnagar, located at 29 N, 79.45 E approximately 293.89 meter above mean

sea level, in the tarai region of Uttarakhand. Assuming that the monsoon season in and around

Pantnagar ranges between 15 to 20 weeks: we consider a 17 weeks data set for our study during

1961-2010. The weekly data comprises of seven weather parameters, viz., Rainfall, Maximum and

Minimum Temperature, Relative Humidity at 7 AM and 2 PM, Bright Sunshine and Pan

Evaporation, collected during monsoon months June to September.

Thus providing reliable prediction and forecasting of weather parameters in the Himalayas

in particular and of India in general is an important challenges for planners and scientists. The

present study is planned with the following objectives:

1. To study, the distribution pattern of weather parameters.

2. Prediction of weather parameters using different forecasting model.

3. To compare the prediction ability of these model.

4. To identify the precise weather forecasting model.

5. To study the reliability of the developed model by comparing the forecast value with its

observed value.

In this research work we took up a study to identify the precise and reliable weather

forecasting model through comparison of several existing and proposed models. The thesis is

divided into seven chapters and the salient results obtained and the main significance of study is

summarized in the following paragraphs.

Probability Distribution: The descriptive statistics are computed for each weather parameters for

different study period. The best fit probability distribution was identified out of a large number of

commonly used probability distribution by using different goodness of fit tests. It was observed

that the best fit probability distribution obtained for the weather parameters data sets are different.

For seasonal weather parameters Normal distribution, Weibull (3P) distribution, Log-

Pearson 3 distribution, Gamma (3P) and Log-Gamma distribution represents the best fit

distribution for Rainfall, Maximum temperature, Relative humidity at 7AM, Pan Evaporation and

Bright Sunshine respectively, while Weibull (2P) was fitted for both Maximum Temperature and

Relative Humidity at PM. The best fit probability distributions for weekly weather parameters for

different study periods are different. In general, Generalized Extreme Value distribution, Weibull

(2P, 3P) distributions are most commonly best fit probability distribution for most of the weeks

among the different weather parameters.

Forecasting Models: The weekly weather parameters data for the months of June, July, August

and September over a period of 50 years of pantnagar region was worn in which 35 years data was

employed to develop and train the models and 15 years was used to test and validate the developed

models. The variation during the four months was among all the parameters and correlation

between relative humidity at 7 am and relative humidity at 2 pm was maximum (0.83736) and it

was minimum between rainfall and minimum temperature (-0.00666). Since, the parameter

selection in the models is always a difficult task, so we introduced those parameters for hybrid

model building only which are found significant using stepwise regression analysis during the

training period. Hybrid MLR_ANN, hybrid MLR_ARIMA, Artificial Neural Network, ARIMA

and Multiple Linear Regression models are used to study the environment of weekly weather

parameters. The models were developed and compared individually for each weather parameter

graphically and numerically through Mean Error, Mean Absolute Error, Root Mean Square Error,

Prediction Error and Correlation Coefficient. Finally, the study divulges that Hybrid MLR_ANN

model is an appropriate forecasting interest to estimate the weather parameters, in disparity to the

multiple linear regression, ARIMA, ANN and hybrid MLR_ARIMA models.

6.2. Future Scope

Weather forecasting is an art using astronomical and meteorological events to monitor

changes in the weather. Although with modern technology, particularly computers and weather

satellites and the availability of data provided by coordinated meteorological observing networks,

has resulted in enormous improvements in the accuracy of weather forecasting. But still with the

ever growing demand for more accurate and reliable weather forecasts, the field opens-up to

additional investigation. However there are some observations in this regard and issues related to

our research investigation which can be addressed in future.

In contrast to the traditional hybrid linear and non-linear methodologies, we can generally

say that the performance of the hybrid MLR_ANN model will not be worse than either of the

components used in isolation, so that it can be applied as an appropriate methodology for

combination linear and non-linear models for time series forecasting. Further, we can say that

more hybrid models can be developed using different ANN ensemble and by using more weather

parameters. Moreover, the inclusion of other seasonal factors may also improve the forecasting

accuracy. It is expected that instead of developing hybrid models based on two techniques, the

integrated hybrid model may be developed by combining more than two techniques together. This

integrated hybrid model may produce more precise model for future and may be an upcoming

topic for further studies.

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APPENDIX A(a)

Procedure followed for Stepwise Regression Analysis for

Dependent Variable: Rainfall

Stepwise Selection: Step 1

Variable Relative Humidity 2PM Entered: R-Square = 0.2996 and C (p) = 71.6676

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 963290 963290 253.60 <.0001

Error 593 2252448 3798.39389

Corrected Total 594 3215738

Parameter Standard

Variable Estimate Error Type II SS F Value Pr > F

Intercept -119.44984 12.16003 366524 96.49 <.0001

Relative Humidity 2PM 2.89509 0.18180 963290 253.60 <.0001

Bounds on condition number: 1, 1

--------------------------------------------------------------------------------------------------


Variable Bright Sunshine Entered: R-Square = 0.3444 and C (p) = 31.2555


Sum of Mean


Model 2 1107452 553726 155.48 <.0001

Error 592 2108286 3561.29453


Parameter Standard


Intercept 6.22513 22.99585 260.97892 0.07 0.7867


Bright Sunshine -9.84784 1.54782 144161 40.48 <.0001

Bounds on condition number: 1.7074, 6.8295

--------------------------------------------------------------------------------------------------


Variable Relative Humidity 7AM Entered: R-Square = 0.3545 and C (p) = 23.7044


Sum of Mean


Model 3 1139917 379972 108.18 <.0001

Error 591 2075821 3512.38808


Parameter Standard


Intercept 66.26725 30.19235 16920 4.82 0.0286

Relative Humidity 7AM -1.38703 0.45622 32465 9.24 0.0025


Bright Sunshine -9.17876 1.55283 122722 34.94 <.0001


--------------------------------------------------------------------------------------------------


Variable Maximum temperature Entered: R-Square = 0.3666 and C (p) = 14.2837


Sum of Mean


Model 4 1178736 294684 85.35 <.0001

Error 590 2037002 3452.54544


Parameter Standard


Intercept 379.40528 98.06595 51678 14.97 0.0001

Maximum Temperature -6.25287 1.86477 38820 11.24 0.0009

Relative Humidity 7AM -2.28785 0.52608 65295 18.91 <.0001


Bright Sunshine -8.74845 1.54488 110716 32.07 <.0001


--------------------------------------------------------------------------------------------------


Variable Pan Evaporation Entered: R-Square = 0.3766 and C (p) = 6.7459


Sum of Mean


Model 5 1211156 242231 71.17 <.0001

Error 589 2004582 3403.36528


Parameter Standard


Intercept 391.90674 97.44921 55045 16.17 <.0001

Maximum Temperature -8.65592 2.00849 63212 18.57 <.0001

Relative Humidity 7AM -1.89389 0.53770 42223 12.41 0.0005


Pan Evaporation 4.70662 1.52496 32420 9.53 0.0021

Bright Sunshine -8.50348 1.53589 104323 30.65 <.0001


--------------------------------------------------------------------------------------------------

All variables left in the model are significant at the 0.1500 level.

No other variable met the 0.1500 significance level for entry into the model.

Summary of Stepwise Selection

Variable Number Partial Model

Step Entered Variables In R-Square R-Square C (p) F Value Pr > F

1 Relative Humidity 2PM 1 0.2996 0.2996 71.6676 253.60 <.0001

2 Bright Sunshine 2 0.0448 0.3444 31.2555 40.48 <.0001

3 Relative Humidity 7AM 3 0.0101 0.3545 23.7044 9.24 0.0025

4 Maximum Temperature 4 0.0121 0.3666 14.2837 11.24 0.0009

5 Pan Evaporation 5 0.0101 0.3766 6.7459 9.53 0.0021

APPENDIX A(b)


Dependent Variable: Maximum Temperature


Variable Relative humidity 7AM Entered: R-Square = 0.7151 and C (p) = 477.2735


Sum of Mean


Model 1 3097.61259 3097.61259 1488.50 <.0001

Error 593 1234.05268 2.08103

Corrected Total 594 4331.66528

Parameter Standard


Intercept 53.93488 0.53853 20873 10030.3 <.0001

Relative humidity 7AM -0.23778 0.00616 3097.61259 1488.50 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 3343.92268 1671.96134 1002.08 <.0001

Error 592 987.74260 1.66848


Parameter Standard


Intercept 44.80651 0.89273 4203.00312 2519.05 <.0001


Pan Evaporation 0.36545 0.03008 246.31008 147.63 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 3480.07053 1160.02351 805.05 <.0001

Error 591 851.59475 1.44094


Parameter Standard


Intercept 44.14902 0.83238 4053.61208 2813.17 <.0001


Relative Humidity 2PM -0.06352 0.00653 136.14785 94.49 <.0001

Pan Evaporation 0.29327 0.02892 148.17111 102.83 <.0001


--------------------------------------------------------------------------------------------------


Variable Minimum Temperature Entered: R-Square = 0.8374 and C (p) = 24.8486


Sum of Mean


Model 4 3627.17774 906.79444 759.43 <.0001

Error 590 704.48753 1.19405


Parameter Standard


Intercept 35.78241 1.06880 1338.35354 1120.86 <.0001

Minimum Temperature 0.34132 0.03075 147.10722 123.20 <.0001



Pan Evaporation 0.23873 0.02678 94.87433 79.46 <.0001


--------------------------------------------------------------------------------------------------


Variable Rainfall Entered: R-Square = 0.8417 and C (p) = 10.5253


Sum of Mean


Model 5 3646.03416 729.20683 626.43 <.0001

Error 589 685.63112 1.16406


Parameter Standard


Intercept 35.85264 1.05543 1343.24493 1153.93 <.0001

Rainfall -0.00295 0.00073325 18.85641 16.20 <.0001




Pan Evaporation 0.24756 0.02653 101.32602 87.05 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 6 3652.41694 608.73616 526.96 <.0001

Error 588 679.24833 1.15518


Parameter Standard


Intercept 34.98664 1.11408 1139.25818 986.21 <.0001

Rainfall -0.00252 0.00075311 12.93483 11.20 0.0009




Pan Evaporation 0.24658 0.02644 100.50376 87.00 <.0001

Bright Sunshine 0.06837 0.02909 6.38279 5.53 0.0191


---------------------------------------------------------------------------------------------


All variables have been entered into the model.




1 Relative humidity 7AM 1 0.7151 0.7151 477.273 1488.50 <.0001

2 Pan Evaporation 2 0.0569 0.7720 266.052 147.63 <.0001


4 Minimum Temperature 4 0.0340 0.8374 24.8486 123.20 <.0001

5 Rainfall 5 0.0044 0.8417 10.5253 16.20 <.0001

6 Bright Sunshine 6 0.0015 0.8432 7.0000 5.53 0.0191

APPENDIX A(c)


Dependent Variable: Minimum Temperature


Variable Maximum Temperature Entered: R-Square = 0.0421 and C (p) = 217.5398


Sum of Mean


Model 1 62.20914 62.20914 26.06 <.0001

Error 593 1415.79260 2.38751


Parameter Standard


Intercept 20.25823 0.78396 1594.25897 667.75 <.0001

Maximum Temperature 0.11984 0.02348 62.20914 26.06 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 426.74307 213.37154 120.16 <.0001

Error 592 1051.25867 1.77577


Parameter Standard


Intercept -0.76196 1.61540 0.39509 0.22 0.6373


Relative Humidity 2PM 0.10114 0.00706 364.53393 205.28 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 441.48427 147.16142 83.91 <.0001

Error 591 1036.51748 1.75384


Parameter Standard


Intercept 0.51214 1.66446 0.16605 0.09 0.7584



Bright Sunshine -0.09958 0.03435 14.74120 8.41 0.0039


--------------------------------------------------------------------------------------------------


Variable Relative Humidity 7AM Entered: R-Square = 0.3013 and C (p) = 4.7836


Sum of Mean


Model 4 445.26193 111.31548 63.59 <.0001

Error 590 1032.73982 1.75041


Parameter Standard


Intercept 2.64644 2.20810 2.51435 1.44 0.2312


Relative humidity 7AM -0.01740 0.01185 3.77766 2.16 0.1423


Bright Sunshine -0.09121 0.03479 12.03446 6.88 0.0090


--------------------------------------------------------------------------------------------------






1 Maximum Temperature 1 0.0421 0.0421 217.540 26.06 <.0001


3 Bright Sunshine 3 0.0100 0.2987 4.9410 8.41 0.0039

4 Relative humidity 7AM 4 0.0026 0.3013 4.7836 2.16 0.1423

APPENDIX A(d)


Dependent Variable: Relative Humidity at 7 AM




Sum of Mean


Model 1 39178 39178 1488.50 <.0001

Error 593 15608 26.32032


Parameter Standard


Intercept 186.94704 2.60296 135767 5158.24 <.0001



--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 41967 20983 969.06 <.0001

Error 592 12819 21.65334


Parameter Standard


Intercept 128.80293 5.64091 11290 521.38 <.0001

Maximum Temperature -1.81043 0.12697 4402.23745 203.31 <.0001



--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 42755 14252 700.12 <.0001

Error 591 12030 20.35615


Parameter Standard


Intercept 118.78898 5.70115 8837.36012 434.14 <.0001



Pan Evaporation -0.71005 0.11410 788.29066 38.72 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 4 43117 10779 545.01 <.0001

Error 590 11669 19.77784


Parameter Standard


Intercept 119.67441 5.62340 8957.43103 452.90 <.0001


Rainfall -0.01290 0.00302 361.56017 18.28 <.0001


Pan Evaporation -0.62239 0.11432 586.18727 29.64 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 5 43257 8651.37901 441.99 <.0001

Error 589 11529 19.57350


Parameter Standard


Intercept 115.86488 5.77261 7885.47968 402.87 <.0001


Rainfall -0.01089 0.00309 242.83120 12.41 0.0005


Pan Evaporation -0.60811 0.11386 558.38604 28.53 <.0001

Bright Sunshine 0.31774 0.11875 140.13310 7.16 0.0077


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 6 43304 7217.36338 369.62 <.0001

Error 588 11482 19.52637


Parameter Standard


Intercept 116.06296 5.76706 7908.60919 405.02 <.0001


Minimum Temperature -0.21386 0.13743 47.28526 2.42 0.1202

Rainfall -0.01111 0.00309 252.18817 12.92 0.0004


Pan Evaporation -0.60294 0.11377 548.45500 28.09 <.0001

Bright Sunshine 0.29476 0.11952 118.75498 6.08 0.0139


--------------------------------------------------------------------------------------------------








3 Pan Evaporation 3 0.0144 0.7804 29.1147 38.72 <.0001

4 Rainfall 4 0.0066 0.7870 12.5982 18.28 <.0001

5 Bright Sunshine 5 0.0026 0.7896 7.4216 7.16 0.0077

6 Minimum Temperature 6 0.0009 0.7904 7.0000 2.42 0.1202

APPENDIX A(e)


Dependent Variable: Relative Humidity at 2 PM




Sum of Mean


Model 1 79294 79294 1319.49 <.0001

Error 593 35636 60.09429


Parameter Standard


Intercept 207.83164 3.93313 167795 2792.20 <.0001



--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 88469 44235 989.66 <.0001

Error 592 26460 44.69677


Parameter Standard


Intercept 156.25954 4.94592 44614 998.16 <.0001




--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 92813 30938 826.71 <.0001

Error 591 22117 37.42265


Parameter Standard


Intercept 61.95949 9.85360 1479.65519 39.54 <.0001



Relative humidity 7AM 0.53384 0.04955 4343.70133 116.07 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 4 95664 23916 732.41 <.0001

Error 590 19266 32.65388


Parameter Standard


Intercept 61.71801 9.20442 1468.13280 44.96 <.0001




Bright Sunshine -1.31787 0.14104 2850.99934 87.31 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 5 96814 19363 629.53 <.0001

Error 589 18116 30.75732


Parameter Standard


Intercept 49.13831 9.16702 883.75694 28.73 <.0001




Rainfall 0.02307 0.00377 1149.72761 37.38 <.0001

Bright Sunshine -1.03245 0.14462 1567.47160 50.96 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 6 97056 16176 532.15 <.0001

Error 588 17874 30.39725


Parameter Standard


Intercept 46.85286 9.14906 797.17553 26.23 <.0001




Rainfall 0.02415 0.00377 1246.78191 41.02 <.0001

Pan Evaporation -0.40761 0.14432 242.47563 7.98 0.0049

Bright Sunshine -1.02706 0.14379 1550.87910 51.02 <.0001


--------------------------------------------------------------------------------------------------







2 Minimum Temperature 2 0.0798 0.7698 281.489 205.28 <.0001


4 Bright Sunshine 4 0.0248 0.8324 48.8003 87.31 <.0001

5 Rainfall 5 0.0100 0.8424 12.9769 37.38 <.0001

6 Pan Evaporation 6 0.0021 0.8445 7.0000 7.98 0.0049

APPENDIX A(f)


Dependent Variable: Pan Evaporation




Sum of Mean


Model 1 3055.56776 3055.56776 1115.01 <.0001

Error 593 1625.05738 2.74040


Parameter Standard


Intercept -22.73697 0.83990 2008.26447 732.84 <.0001



--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 3204.44481 1602.22240 642.55 <.0001

Error 592 1476.18034 2.49355


Parameter Standard


Intercept -4.47870 2.49508 8.03440 3.22 0.0732




--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 3218.32405 1072.77468 433.57 <.0001

Error 591 1462.30110 2.47428


Parameter Standard


Intercept -5.95940 2.56285 13.37857 5.41 0.0204



Rainfall 0.00237 0.00100 13.87924 5.61 0.0182


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 4 3239.73488 809.93372 331.64 <.0001

Error 590 1440.89027 2.44219


Parameter Standard


Intercept -4.10879 2.62176 5.99821 2.46 0.1176



Relative Humidity 2PM -0.02902 0.00980 21.41083 8.77 0.0032

Rainfall 0.00351 0.00107 26.51822 10.86 0.0010


--------------------------------------------------------------------------------------------------








3 Rainfall 3 0.0030 0.6876 10.1554 5.61 0.0182

4 Relative Humidity 2PM 4 0.0046 0.6922 3.4119 8.77 0.0032

APPENDIX A(g)


Dependent Variable: Bright Sunshine




Sum of Mean


Model 1 1051.50791 1051.50791 419.47 <.0001

Error 593 1486.50536 2.50675


Parameter Standard


Intercept 12.76168 0.31239 4183.56363 1668.92 <.0001



--------------------------------------------------------------------------------------------------




Sum of Mean


Model 2 1146.64727 573.32364 243.94 <.0001

Error 592 1391.36601 2.35028


Parameter Standard


Intercept 11.98536 0.32616 3173.63613 1350.32 <.0001


Rainfall -0.00650 0.00102 95.13936 40.48 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 3 1164.58858 388.19619 167.05 <.0001

Error 591 1373.42470 2.32390


Parameter Standard


Intercept 14.57439 0.98662 507.10468 218.21 <.0001



Rainfall -0.00660 0.00102 97.97309 42.16 <.0001


-------------------------------------------------------------------------------------------------




Sum of Mean


Model 4 1172.11215 293.02804 126.57 <.0001

Error 590 1365.90113 2.31509


Parameter Standard


Intercept 12.90426 1.35205 210.88565 91.09 <.0001


Relative humidity 7AM 0.02245 0.01246 7.52357 3.25 0.0719


Rainfall -0.00627 0.00103 85.77070 37.05 <.0001


--------------------------------------------------------------------------------------------------




Sum of Mean


Model 5 1184.61776 236.92355 103.11 <.0001

Error 589 1353.39551 2.29779


Parameter Standard


Intercept 7.86562 2.54542 21.94101 9.55 0.0021

Maximum Temperature 0.12606 0.05404 12.50562 5.44 0.0200


Relative humidity 7AM 0.03621 0.01374 15.96364 6.95 0.0086


Rainfall -0.00597 0.00103 76.53762 33.31 <.0001


--------------------------------------------------------------------------------------------------







2 Rainfall 2 0.0375 0.4518 15.7398 40.48 <.0001

3 Minimum Temperature 3 0.0071 0.4589 9.9418 7.72 0.0056

4 Relative humidity 7AM 4 0.0030 0.4618 8.6718 3.25 0.0719

5 Maximum Temperature 5 0.0049 0.4668 5.2364 5.44 0.0200

APPENDIX B(a) Program for developing ANN model for Rainfall % Training for Rainfall

% Creating the feed forward network

net=newff([0.0236 0.0432;0.0172 0.0335;0.038 0.098; 0.016 0.092;0.0000 0.0185;

0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');

% Enter the pattern and target

p = [0.0381 0.0265 0.074 0.044 0.0122 0.0066

0.0347 0.0233 0.080 0.052 0.0065 0.0069

…………………………..…………………….

0.0328 0.024 0.093 0.065 0.0041 0.0088

0.032 0.0229 0.094 0.064 0.0034 0.0078];

p = p';

t = [0.0056

0.0738

………

0.0272

0.0112];

t = t';

% set initial weights&biases

net.IW{1,1} = [-0.8754 0.5691 -0.5960 -0.7580 -0.2941 -0.5808

0.6511 -0.2726 -0.0521 0.5720 -0.6031 0.5401

0.5375 0.3341 0.0824 0.2887 0.0589 -0.9999

0.3523 0.2958 0.9987 0.8578 0.1649 0.2789

-0.4838 -0.3115 0.6352 0.8444 0.6760 -0.9644

0.1908 -0.8439 0.2789 -0.5051 0.8061 0.1336

0.0303 0.3956 0.0722 -0.1205 0.0424 0.9686

-0.8503 0.9714 -0.9321 0.3208 -0.3348 0.1436];

net.LW{2,1} =[ 0.1669 0.6254 0.5631 0.0461 -0.0918 -0.2504 0.0376 -0.1060

0.3322 0.6177 -0.2275 -0.1279 0.0275 -0.0126 0.2725 0.0139

-0.2849 0.7109 -0.1916 -0.9661 0.9048 0.3261 0.6344 -0.6618

0.6657 -0.0168 0.7077 -0.1694 0.2635 0.2823 -0.1229 0.8654

0.5925 0.8494 -0.8233 0.2653 0.2465 0.1414 0.6385 0.1062

0.0052 -0.2024 -0.2001 0.9764 0.1687 -0.6857 0.8503 0.7989

0.9569 0.7751 0.4519 0.2634 0.2132 -0.5242 -0.2326 0.4896

-0.0316 0.0919 -0.7745 -0.4365 -0.0007 -0.4960 -0.0084 -0.9707

0.2303 0.5802 0.8999 -0.6458 0.7177 -0.5838 0.9549 -0.4680

0.2209 -0.2903 -0.6861 0.1895 -0.2585 0.2821 -0.6412 -0.0255];

net.LW{3,2} =

[-0.8486 -0.6747 0.3305 0.3452 -0.2362 0.7200 0.3338 0.3797 -0.2056 -0.3913

0.8301 -0.6520 0.0935 0.4716 -0.1065 -0.1097 0.3176 -0.6392 -0.8581 -0.0295

-0.0533 -0.0064 0.8642 0.6741 0.8208 -0.4858 -0.0578 0.1573 -0.2212 -0.5869

-0.2314 0.1757 0.7960 0.7165 0.8207 -0.0764 0.0396 0.3412 0.9951 -0.1821

-0.6669 -0.7202 -0.3719 -0.8201 -0.6210 0.4288 -0.6957 0.3734 -0.6647 -0.2793

0.4430 -0.1451 -0.1824 -0.2177 0.2350 -0.7160 -0.6020 -0.8999 -0.2222 0.6489

0.0563 -0.5688 0.6196 -0.6303 -0.5125 -0.5146 0.4519 0.8357 0.9454 0.8613

0.9196 0.4672 0.2423 -0.1273 0.8340 0.9841 0.6740 0.6118 -0.6070 0.3347

-0.1114 -0.2678 0.0344 0.9198 0.5722 -0.1578 0.4817 0.5853 0.4767 -0.6444

-0.5283 0.1231 -0.3697 0.2298 -0.9242 0.0088 -0.7207 0.0067 -0.2896 0.1226

-0.4075 0.6867 -0.3770 0.0153 0.0430 0.4637 -0.8477 -0.5013 0.2867 0.4182

-0.5730 -0.6748 0.1384 -0.6400 0.8806 0.6674 -0.4899 0.4453 0.5295 -0.4088];

net.LW{4,3} = [-0.2489 0.1706 -0.8142 0.9684 0.7488 -0.1751 0.8994 0.8689

0.5437 -0.0984 0.3068 -0.3575];

net.b{1} = [0.3660

-0.4437

-0.4755

-0.9644

0.9308

-0.6543

-0.2085

-0.3939];

net.b{2} = [ -0.0064

-0.4747

0.9436

-0.9568

-0.9078

0.6698

0.4679

0.2685

0.5845

-0.5981];

net.b{3} = [ 0.3816

-0.2013

0.2325

-0.6459

-0.3839

0.4836

-0.1686

0.1700

-0.3120

-0.0293

-0.6155

-0.0767];

net.b{4}=[0.0487];

% set the training parameters

net.trainParam.show=100;

net.trainParam.epochs=100000;

% net.trainParam.time=115;

net.trainParam.goal=0.003;

% train the network

net=train(net,p,t);

% simulate the trained network

y=sim(net,p)

APPENDIX B(b) Program for developing ANN model for Maximum Temperature

% Training for Maximum temperature


net = newff([0.0000 0.4432;0.0172 0.0340;0.0380 0.0980;0.0160 0.0920;0.0000

0.0185;0.0007 0.0116], [8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');


p = [0.0056 0.0265 0.0740 0.0440 0.0122 0.0066

0.0738 0.0233 0.0800 0.0520 0.0065 0.0069

…….…………………………………………..….

0.0272 0.0240 0.0930 0.0650 0.0041 0.0088

0.0112 0.0229 0.0940 0.0640 0.0034 0.0078];

p = p';

t = [0.0381

0.0347

..…….

0.0328

0.032];

t = t';


net.IW{1,1} = [0.1830 0.9827 -0.8075 0.9702 -0.2300 -0.1041

0.5743 0.6219 -0.7227 0.3095 -0.1905 0.4341

0.9856 -0.7181 0.2101 0.5207 -0.6333 0.0682

-0.9022 -0.7562 0.0774 -0.2469 -0.1813 -0.7190

-0.6815 -0.3828 -0.3076 0.6001 -0.4974 -0.1432

0.4774 -0.1611 -0.9719 0.5146 0.7935 0.6651

-0.6774 0.9253 -0.9237 0.3601 0.5258 0.8976

0.4031 0.8722 0.6227 0.5242 0.7912 0.1363];

net.LW{2,1} = [0.7155 -0.4739 0.8768 -0.2460 -0.1839 -0.0928 0.8877 0.8881

-0.7649 -0.3336 -0.7183 -0.0781 -0.9393 0.4054 -0.8248 0.7563

-0.0680 0.3087 0.3761 0.3995 0.4740 -0.4405 -0.8136 -0.7103

0.3111 0.8226 -0.5903 -0.3393 -0.5643 0.8982 0.7213 0.7331

0.2030 -0.8793 -0.6909 0.0901 -0.1530 0.3457 0.8256 0.0985

0.5692 0.9367 0.4832 -0.5163 -0.0804 -0.3912 -0.7015 0.8838

0.1234 -0.7528 -0.5495 -0.5984 -0.5395 -0.9752 -0.0063 0.9674

-0.3328 -0.3471 0.7951 0.4012 0.8578 -0.0749 -0.0243 0.8066

-0.1534 -0.7072 -0.8250 0.7536 -0.6620 -0.4152 0.7060 0.6676

0.0302 -0.0263 -0.6002 -0.6233 0.6482 -0.9846 -0.2759 0.4045];

net.LW{3,2} =

[-0.7892 0.7948 -0.5644 0.1354 -0.3702 0.8356 -0.5480 0.1078 0.4876 -0.4043

0.5195 -0.4049 0.3418 0.7638 0.5408 0.1434 0.3398 -0.6380 0.5458 0.7920

0.4372 -0.9754 -0.0865 0.4747 -0.0022 0.0400 -0.7902 0.3215 0.9271 -0.5078

0.8674 0.9659 0.0349 0.5260 -0.0902 0.7690 -0.3179 0.4917 -0.2053 0.4285

-0.6505 0.9109 0.1923 0.2575 -0.7420 -0.2751 0.8238 -0.5874 0.4226 -0.0624

0.0962 0.0151 0.3048 0.2957 -0.6096 0.2236 0.0455 0.8854 -0.4382 0.6825

0.6181 0.0326 -0.7992 -0.2536 0.4944 -0.3513 0.3241 0.8275 -0.3580 0.8405

-0.2734 0.4844 0.2711 -0.7802 -0.6452 -0.1765 -0.3122 -0.7835 -0.0471 -0.7535

0.8305 0.2894 -0.1686 0.3789 0.2623 -0.4946 -0.7184 -0.3682 -0.7482 0.6708

0.3940 -0.7313 0.0449 0.0082 0.3848 -0.4528 -0.6386 0.4990 -0.8370 -0.6484

0.4221 -0.4647 -0.9156 -0.8721 -0.3680 -0.4029 -0.6183 0.3187 -0.5224 -0.3911

0.8366 -0.5711 0.9745 0.3079 -0.1101 0.1189 0.9379 -0.2884 -0.1105 0.2908];

net.LW{4,3} = [-0.0692 0.1473 0.1255 0.0563 0.2041 0.0952 0.1070 0.1474

-0.0443 0.1328 -0.1395 0.1926];

net.b{1} = [-0.2063

-0.9503

0.0815

0.0754

0.5538

0.1737

-0.3515

-0.0912];

net.b{2} = [-0.9913

0.5159

0.1396

-0.5876

-0.1088

0.2714

0.7654

-0.9691

-0.4134

0.9016];

net.b{3} = [0.8224

-0.4396

-0.1421

-0.8167

0.5647

-0.0764

0.1147

-0.6510

0.8410

0.1498

0.4667

0.8328];

net.b{4} = [-0.0534];






% train the network

net = train(net,p,t);


y = sim(net,p)

APPENDIX B(c) Program for developing ANN model for Minimum Temperature % Training for Minimum temperature


net=newff([0.0236 0.0432;0.0000 0.4432;0.038 0.098; 0.016 0.092;0.0000 0.0185;

0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');


p = [ 0.0381 0.0056 0.074 0.044 0.0122 0.0066

0.0347 0.0738 0.08 0.052 0.0065 0.0069

………………………………

0.0328 0.0272 0.093 0.065 0.0041 0.0088

0.0320 0.0112 0.094 0.064 0.0034 0.0078];

p = p';

t = [0.0265

0.0233

………

0.024

0.0229];

t= t';

% set initial weights & biases

net.IW{1,1} = [0.5536 0.0430 0.2723 -0.3851 0.0014 0.0443

0.9695 0.4651 0.5682 0.2038 0.1346 0.5991

0.5100 -0.7832 0.7564 0.2457 -0.4487 -0.1632

0.9535 -0.7101 -0.6487 0.1373 -0.7912 -0.1160

-0.5104 -0.5715 -0.0948 0.2581 0.5745 -0.6275

-0.2897 -0.4698 -0.2563 0.1590 0.3275 -0.2306

0.6673 0.1785 -0.8307 0.7457 0.8579 0.9837

-0.0878 -0.5797 0.6688 -0.0553 0.0985 0.9830];

net.LW{2,1} = [-0.6010 0.0488 0.4981 0.1298 -0.4060 0.4664 0.1657 -0.6048

-0.1352 -0.8005 -0.9650 0.5459 -0.3057 -0.8117 -0.4227 0.2894

0.8353 0.0013 0.7041 0.7583 -0.4720 -0.9661 -0.3515 0.7300

-0.6360 -0.2446 -0.9679 -0.3216 0.7613 -0.4419 -0.9072 0.0408

0.9010 0.4517 0.0144 -0.1730 0.2443 -0.7843 0.7669 0.1533

0.3172 0.3161 -0.6449 0.0063 -0.4995 0.0023 0.8137 0.6739

0.9079 0.9721 0.3042 -0.7417 0.6953 0.5513 -0.6486 -0.4126

0.0550 -0.6494 -0.6131 -0.7381 0.3837 -0.7441 0.0888 0.9754

0.8614 -0.6895 0.8226 0.0267 -0.4730 -0.8976 -0.4635 -0.6496

-0.3869 0.2148 -0.2552 0.7695 0.5063 0.6647 0.3153 0.4653];

net.LW{3,2}=

[0.3018 -0.9096 -0.1047 0.6253 0.4674 -0.5164 0.1890 0.8921 -0.1206 0.8019

0.6692 -0.2640 -0.2445 -0.9090 0.3149 0.7342 0.4179 0.5778 -0.8540 -0.9360

-0.3564 -0.2943 0.0032 -0.1677 0.5503 0.4887 0.8478 0.2316 0.2608 0.0614

-0.9227 -0.6927 -0.5703 -0.8114 -0.0176 -0.8090 -0.7305 0.3228 0.3272 -0.6274

0.0439 0.8080 -0.0190 -0.4571 0.3789 0.0237 -0.9561 -0.3868 0.4882 -0.8610

0.3212 -0.7708 0.2412 0.7825 -0.0516 -0.1919 0.0745 -0.4586 0.1078 -0.7197

-0.4023 0.0042 -0.9565 0.5778 0.0245 0.5371 -0.9470 -0.9675 0.5835 -0.8114

-0.9942 -0.3228 0.7342 -0.2424 0.8829 -0.4090 -0.9720 0.2503 0.4133 -0.6446

0.5418 -0.5751 0.4136 0.5352 0.5687 -0.6623 -0.6044 0.4483 0.5935 -0.0937

0.1952 0.8292 0.2899 0.0719 -0.5043 -0.1085 0.5173 0.4314 0.2180 0.4337

-0.8530 -0.6065 -0.0009 0.0653 0.3079 0.0607 0.4455 0.0003 -0.1015 -0.5388

-0.7381 0.3235 -0.6587 0.1451 -0.5644 0.6793 -0.7501 0.5707 0.1169 -0.3810];

net.LW{4,3} = [0.5012 -0.1792 -0.1174 -0.0282 0.3103 0.0855 -0.0692 -0.3121 -

0.6469 0.0914 0.5573 -0.1817];

net.b{1} = [ -0.8705

-0.7203

-0.6073

0.1588

0.2553

0.9526

-0.4646

-0.0111];

net.b{2} = [ 0.9235

0.5283

-0.0836

0.6827

-0.3115

0.2728

0.4533

-0.1677

0.4202

-0.8918];

net.b{3} = [-0.8857

-0.5030

0.1551

0.8718

0.5743

-0.0888

-0.0141

-0.5285

0.8396

0.3056

-0.5666

-0.8621];

net.b{4} = [0.3720];






% train the network

net=train(net,p,t);


y=sim(net,p)

APPENDIX B(d) Program for developing ANN model for Relative Humidity at 7 AM

% Training for Relative Humidity at 7 AM


net = newff([0.0236 0.0432; 0.0172 0.0335;0.0000 0.4432;0.016 0.092; 0.0000 0.0185;

0.0007 0.0116], [8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');


p = [ 0.0381 0.0265 0.0056 0.044 0.0122 0.0066

0.0347 0.0233 0.0738 0.052 0.0065 0.0069

…………………………………………………..

0.0328 0.0240 0.0272 0.065 0.0041 0.0088

0.0320 0.0229 0.0112 0.064 0.0034 0.0078];

p = p';

t = [0.074

0.080

…..…

0.093

0.094];

t = t';


net.IW{1,1} = [0.3993 0.0559 -0.5671 -0.5619 0.1204 0.4636

0.6472 0.8795 -0.3328 0.0796 0.8470 -0.5849

0.1825 0.0479 0.6551 -0.0231 -0.9869 -0.0046

-0.0719 0.1821 -0.4123 -0.0032 0.1074 -0.3365

0.0785 -0.3795 0.4819 -0.1454 0.8002 0.3035

0.6397 0.5071 0.4646 0.1765 -0.2689 0.4831

0.0287 0.0059 0.0119 0.0260 -0.5120 0.8247

-0.2506 0.3626 0.9474 -0.0514 -0.4404 -0.4746];

net.LW{2,1} = [0.9260 0.0727 0.2154 0.8846 0.1803 0.1020 -0.6001 0.9212

-0.0342 -0.8027 0.5245 -0.2896 -0.0707 -0.0686 0.1523 0.4920

-0.3674 -0.6659 -0.5690 0.9120 -0.3458 -0.7657 0.7953 -0.6038

0.7335 -0.2760 0.2560 0.1025 -0.7507 -0.5497 0.4720 0.0999

0.1976 0.5876 0.1307 0.1320 0.0643 0.1058 0.7823 0.6258

-0.7387 -0.8299 -0.7319 0.7221 -0.9057 0.3150 0.5930 -0.7231

0.1501 -0.3159 -0.0717 0.5868 -0.2530 -0.6144 0.6100 -0.3937

0.7887 -0.7121 -0.5279 0.2268 -0.8491 -0.2602 -0.3849 0.7716

0.1960 -0.2286 -0.3955 0.8952 0.6934 0.4108 -0.0136 -0.2832

-0.0316 -0.1811 0.8871 -0.8321 -0.7273 -0.3724 -0.7060 -0.6157];

net.LW{3,2} =

[0.9303 -0.0139 0.0380 0.4166 0.7025 0.5978 -0.4936 -0.2215 -0.6515 0.8029

-0.1378 -0.6000 0.2000 -0.0268 0.6782 -0.2308 0.4191 0.0127 0.0514 0.5411

-0.1286 -0.7915 0.7275 -0.4092 -0.1055 0.6563 -0.1320 0.0308 -0.5607 0.4708

-0.0443 -0.2298 -0.4990 0.8830 0.8076 -0.3750 -0.1501 0.7050 0.8523 -0.6972

0.7379 0.3482 -0.5548 -0.5223 -0.1830 -0.6471 -0.4609 -0.8649 -0.5113 0.5506

-0.5910 0.3072 -0.2210 0.4513 -0.3957 0.8798 -0.2536 0.3912 -0.7545 0.2224

0.5711 -0.9355 0.1034 -0.2205 -0.8573 -0.5789 0.3808 0.5864 -0.4221 0.5794

0.3076 0.7616 -0.3087 0.4447 -0.7018 0.2237 0.3231 -0.0388 0.8255 0.7911

0.7320 -0.4140 0.2274 -0.4210 -0.6343 0.7362 0.7188 0.3095 -0.0006 0.6939

0.5271 0.7713 -0.4005 -0.3363 0.6316 -0.1129 -0.6021 0.7165 0.7476 -0.2067

0.8774 -0.1706 0.0936 0.2312 -0.2708 0.7386 0.3247 0.3825 -0.5177 0.5844

-0.6874 -0.3568 -0.0178 -0.2641 0.6061 -0.0261 -0.1514 0.5295 0.3511 -0.1898];

net.LW{4,3}= [0.9768 -0.8569 -0.9742 0.4239 0.5066 0.6002 -0.3358 0.1514

0.2937 0.0338 -0.5995 -0.4561];

net.b{1} = [-0.0096

-0.1848

-0.9375

0.2745

-0.1984

-0.9723

0.2074

0.6717];

net.b{2} = [-0.1418

0.2738

0.3458

-0.6403

-0.2777

0.0127

0.8046

0.0245

0.5307

0.0546];

net.b{3} = [-0.8557

0.5106

0.2926

0.8038

-0.5511

-0.1513

0.1840

-0.2164

0.7711

0.1041

0.7744

0.8390];

net.b{4} = [ -0.7024];






% train the network

net =train(net,p,t);


y = sim(net,p)

APPENDIX B(e) Program for developing ANN model for Relative Humidity at 2 PM

% Training for Relative Humidity at 2 PM


net = newff([0.0236 0.0432; 0.0172 0.0335;0.038 0.098;0.0000 0.4432;0.0000 0.0185;

0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','tansig'},'trainscg');


p = [0.0381 0.0265 0.0740 0.0056 0.0122 0.0066

0.0347 0.0233 0.0800 0.0738 0.0065 0.0069

…………………………………………………..

0.0328 0.0240 0.0930 0.0272 0.0041 0.0088

0.0320 0.0229 0.0940 0.0112 0.0034 0.0078];

p = p';

t = [0.044

0.052

……..

0.065

0.064];

t = t';


net.IW{1,1} = [0.7359 0.1843 0.9574 0.0541 -0.5302 0.1915

-0.1483 -0.3240 0.5386 -0.6799 -0.2885 0.4064

-0.0541 0.8281 -0.4575 -0.1265 -0.6093 0.3627

0.2008 0.0897 0.7760 -0.4324 -0.3566 0.3755

0.7916 -0.1845 0.6725 -0.5910 0.9761 -0.5051

0.1901 0.5141 -0.5460 0.3930 -0.2015 -0.4294

-0.3855 0.9105 -0.0842 0.0241 0.6178 -0.1599

0.7207 0.5913 0.8029 -0.2480 0.5532 -0.3286];

net.LW{2,1} = [ -0.1515 0.0597 0.3223 0.0325 0.4205 0.0592 0.0392 0.9734

-0.5860 0.8262 0.1230 -0.0356 0.2368 0.3968 -0.8404 -0.6009

0.4300 -0.7895 -0.4803 0.2303 -0.6325 -0.0268 0.9145 0.2492

0.2832 0.7708 0.6287 0.4624 -0.9252 -0.2994 -0.8239 0.5658

-0.0173 -0.0276 -0.5713 -0.7164 0.0090 0.1134 0.2793 0.6552

0.3992 -0.2867 0.8809 -0.5434 0.7010 -0.6220 -0.1316 0.8412

-0.1605 -0.0369 0.3897 -0.4651 0.7298 -0.2298 0.5777 -0.4485

-0.0646 -0.6122 0.1675 0.5054 0.1979 -0.4009 -0.0385 -0.0661

-0.2886 0.5084 0.1057 0.4251 -0.4550 0.7736 0.0845 -0.9929

0.6689 0.2743 -0.4505 0.5538 -0.4464 0.3179 0.4715 -0.5543];

net.LW{3,2} =

[0.6800 -0.3204 0.5834 -0.5676 -0.1674 0.6121 0.6261 -0.4545 0.6760 -0.7316

-0.1814 0.5979 0.6621 0.6532 0.6253 -0.5038 0.7562 -0.2141 0.1145 -0.5323

-0.0929 0.7605 0.7157 -0.8474 -0.5593 -0.6711 -0.0027 -0.3229 -0.1120 0.5892

0.1920 -0.6137 -0.3324 0.4786 -0.7747 0.7508 0.3353 -0.8811 -0.5460 -0.7564

0.1314 0.5747 -0.0391 0.6691 -0.7323 -0.4133 0.8818 -0.3188 0.3723 -0.8967

0.1848 -0.2698 0.4607 -0.8323 -0.0265 -0.7108 -0.8618 -0.5132 -0.2076 0.2117

-0.7576 0.6499 -0.8528 -0.6629 0.0937 0.6108 -0.0616 0.5420 -0.6451 -0.1679

0.8972 0.4395 -0.5422 -0.4070 0.4045 -0.8160 0.1586 -0.2606 0.6038 -0.1116

-0.8588 0.5257 0.5042 0.0594 -0.2184 -0.4312 -0.6882 -0.9036 0.6119 0.0779

0.0226 0.2483 -0.0519 0.7665 0.3992 -0.0655 0.4354 0.8971 0.5170 0.1730

0.5123 -0.5329 -0.4689 -0.3846 -0.1855 -0.8265 -0.4937 -0.4698 -0.1262 0.8056

0.4206 -0.5751 -0.8644 -0.0663 0.6990 -0.1432 0.2059 0.8181 0.6722 0.3270];

net.LW{4,3} = [-0.0766 -0.4366 -0.3554 -0.8038 -0.3609 0.6728 -0.8659 -0.8034

0.1600 -0.5794 -0.1973 0.0354];

net.b{1} = [-0.9832

0.0068

0.9386

-0.4354

-0.7773

0.4431

0.8719

-0.0728];

net.b{2}= [0.8723

0.5704

-0.0178

-0.3930

-0.0104

0.3372

-0.6547

-0.7064

-0.4553

0.5818];

net.b{3} = [-0.7931

0.4295

0.3078

-0.6710

-0.4580

-0.1164

0.0229

0.5187

-0.0030

0.2545

0.4883

0.7336];

net.b{4} = [-0.1122];






% train the network



y = sim(net,p)

APPENDIX B(f) Program for developing ANN model for Pan Evaporation % Training for Pan Evaporation


net=newff([0.0236 0.0432;0.0172 0.0335;0.0380 0.0980; 0.016 0.092; 0.0000 0.4432;

0.0007 0.0116],[8,10,12,1],{'tansig','tansig','tansig','logsig'},'trainscg');


p = [0.0381 0.0265 0.074 0.044 0.0056 0.0066

0.0347 0.0233 0.08 0.052 0.0738 0.0069

……………….………………………………..

0.0328 0.024 0.093 0.065 0.0272 0.0088

0.032 0.0229 0.094 0.064 0.0112 0.0078];

p = p';

t = [0.0122

0.0065

…….

0.0041

0.0034];

t = t';


net.IW{1,1} = [ -0.4092 -0.9839 -0.5397 0.2725 0.5089 -0.3987

-0.8912 -0.6087 0.7342 -0.2753 -0.9905 -0.8467

-0.4415 0.8863 0.6940 -0.7260 -0.2904 0.6507

0.7042 -0.4639 -0.2243 -0.8664 -0.7879 -0.1902

-0.4116 0.9310 -0.9683 -0.8323 0.0182 0.4231

0.0729 0.9679 -0.3609 -0.6115 0.0171 0.4061

-0.2921 -0.6636 0.8394 0.8251 -0.8558 0.0283

0.8973 -0.5499 0.6815 -0.1661 -0.9342 0.4265];

net.LW{2,1} = [-0.0233 -0.7616 0.3765 0.8237 -0.2273 -0.6645 -0.6830 0.0294

-0.2323 0.6478 -0.5732 0.0491 -0.8631 -0.6252 -0.1364 -0.0355

0.6712 0.7559 0.7828 -0.9838 -0.1100 -0.1012 0.7689 -0.2783

-0.2397 0.1351 0.9838 -0.7191 0.2285 -0.1063 -0.5203 0.0590

0.8100 -0.6106 -0.5614 -0.5072 0.4814 -0.7074 -0.7250 0.3408

0.5096 -0.5903 0.6014 -0.6320 -0.6054 0.1244 -0.8258 0.0195

-0.8097 -0.6649 0.6742 -0.9578 0.5613 0.7347 0.1129 -0.0748

-0.0109 -0.9295 -0.1454 -0.7372 -0.5355 -0.3679 0.8520 -0.9246

0.6237 -0.6560 0.0031 -0.5621 0.1688 -0.8662 -0.3380 -0.6366

-0.9506 0.0001 -0.6194 0.0373 0.0526 -0.7463 0.2886 -0.2908];

net.LW{3,2} =

[-0.0625 0.5352 0.3186 0.7122 -0.0415 0.5259 -0.7025 -0.0979 0.2204 -0.5865

0.7363 -0.2367 0.0692 0.8259 -0.2239 -0.3696 -0.6591 -0.2831 0.7458 -0.4855

0.1753 -0.6956 0.1714 -0.1048 0.4919 0.1439 -0.7313 0.8375 -0.6558 0.8587

0.1097 -0.2748 0.0113 0.0071 0.1185 -0.7124 0.4206 -0.5631 -0.9303 0.1635

0.3956 -0.8018 -0.5243 0.3055 -0.1133 0.6456 -0.3963 -0.9289 0.4450 0.4664

-0.0344 0.8625 0.4825 0.7005 -0.6820 0.4453 -0.5271 -0.4359 -0.3107 -0.2429

0.0025 -0.7645 0.2320 -0.4138 0.6134 -0.1513 0.4661 -0.1002 0.1569 -0.3581

-0.7840 0.0236 0.8105 0.5238 0.3795 0.0365 0.1130 -0.0853 -0.8366 0.9953

-0.7037 0.2739 -0.0718 0.6901 0.8057 -0.2848 -0.6999 -0.2539 -0.1903 0.9215

-0.8171 -0.1247 0.2110 -0.1482 0.1574 -0.1230 0.2671 -0.3619 0.2990 -0.3592

0.0832 -0.6319 -0.0688 -0.0815 0.7360 0.0706 -0.4168 0.4522 -0.7653 0.0038

-0.2731 0.0732 0.6095 0.2183 -0.1434 0.8660 0.2675 -0.1427 -0.7780 0.1498];

net.LW{4,3} = [0.4256 -0.0068 0.0028 0.0153 0.0247 -0.0619 -0.1291 -0.6879

0.0065 0.2692 0.0086 0.0468];

net.b{1}= [-0.9901

0.1540

-0.0596

0.9989

-0.8190

-0.4594

0.0302

0.3135];

net.b{2} = [-0.7062

0.5665

-0.2220

0.6866

-0.3450

0.4251

-0.7662

-0.0620

0.6925

-0.9648];

net.b{3} = [ 0.8081

-0.5466

-0.1421

-0.8709

-0.4226

0.2599

0.0083

-0.7717

-0.7710

-0.3026

0.4866

-0.8168];

net.b{4} = [ -0.8644];






% train the network

net=train(net,p,t);


y=sim(net,p)

APPENDIX B(g) Program for developing ANN model for Bright Sunshine % Training for Bright Sunshine


net=newff([0.0236 0.0432; 0.0172 0.0335; 0.038 0.098;0.016 0.0920;0.0000 0.0185;

0.0000 0.4432],[8,10,12,1],{'tansig','tansig','tansig','purelin'},'trainscg');


p = [0.0381 0.0265 0.074 0.044 0.0122 0.0056

0.0347 0.0233 0.08 0.052 0.0065 0.0738

…………………………………..

0.0328 0.024 0.093 0.065 0.0041 0.0272

0.032 0.0229 0.094 0.064 0.0034 0.0112];

p = p';

t = [0.0066

0.0069

……..

0.0088

0.0078];

t = t';


net.IW{1,1} = [-0.8535 0.5669 -0.1547 -0.5520 -0.7675 0.4131

0.1915 -0.6454 -0.0462 0.3010 0.5427 0.9058

-0.2946 -0.9411 -0.2110 -0.2481 0.9858 -0.8979

-0.3001 -0.2426 0.4265 0.8768 0.2576 0.6613

-0.5004 0.0477 0.9804 0.8947 -0.7140 0.5993

0.0643 -0.8281 0.9127 -0.5197 0.1732 -0.8671

-0.3453 -0.5308 0.6877 0.0677 -0.6298 -0.6339

0.5272 0.6893 -0.9391 -0.9730 0.9129 0.3312];

net.LW{2,1} = [-0.0165 -0.8781 0.1787 0.4896 0.6519 0.3350 0.0164 0.5407

0.8321 0.9109 -0.5751 0.0363 0.3604 -0.4380 -0.0497 0.1764

0.6101 -0.1766 -0.1908 0.8318 0.6970 -0.1683 0.0181 0.5219

0.5407 -0.1219 -0.5670 0.4181 0.3672 -0.4965 -0.9079 -0.9077

0.7888 -0.9364 0.6827 -0.2635 0.2550 -0.7381 0.7438 -0.3497

-0.8629 -0.9476 -0.8186 0.4906 -0.4659 0.3562 0.4617 -0.5893

-0.3644 -0.6823 0.2343 -0.2374 -0.7231 -0.8462 -0.0139 0.2752

0.0644 0.3444 -0.3105 -0.6914 0.4428 0.5727 0.6512 -0.7346

0.1995 0.2863 -0.8312 0.7299 0.7664 0.0098 -0.0303 0.1676

0.3587 -0.1578 0.7947 -0.7489 0.6266 0.9098 -0.5373 -0.7023];

net.LW{3,2} =

[-0.9401 -0.5537 -0.1126 -0.6935 -0.2994 -0.4936 0.0817 0.2900 0.8568 -0.3653

-0.7923 -0.0465 -0.5556 -0.2343 0.8661 0.2658 -0.9185 -0.0165 -0.3367 0.6481

-0.4383 0.5638 -0.2955 0.4648 -0.3946 -0.7700 0.5356 0.3340 -0.8727 -0.5542

0.0759 -0.3463 -0.1862 -0.9984 -0.2188 -0.5141 0.7060 0.2972 -0.9402 -0.2873

0.4266 0.4748 0.0738 0.1115 -0.1253 0.3665 -0.1424 -0.1122 -0.0345 0.3118

-0.0085 0.8326 0.5058 -0.6702 0.0791 0.6774 0.6242 -0.0511 0.9090 0.1599

0.1457 0.2216 0.7534 -0.1920 -0.7419 -0.8022 -0.3962 0.7158 0.6820 -0.3920

-0.3860 0.7718 0.3091 -0.1915 0.7806 -0.1262 0.3971 0.1017 0.8914 0.2116

0.1273 0.6336 0.4671 0.4048 -0.7848 0.4338 -0.6612 -0.6392 0.3986 -0.5327

-0.6735 0.9643 -0.0825 0.3088 0.1871 -0.6027 -0.2806 -0.4200 0.7341 -0.0280

0.0185 0.5422 -0.7114 0.1604 0.4994 -0.6282 0.0607 -0.1771 -0.4737 -0.7218

0.7677 -0.0460 -0.0155 0.2299 0.2946 0.3462 0.6229 0.1633 0.6460 0.4172];

net.LW{4,3} = [-0.2916 0.4238 -0.0932 0.1281 0.0077 0.0207 -0.0054 -0.0314 -

0.0180 0.1889 -0.0356 -0.7170];

net.b{1} = [0.2532

-0.6654

0.5751

0.6534

-0.5557

0.6703

-0.3919

-0.2879];

net.b{2} = [ -0.7839

-0.5423

-0.1824

-0.9420

-0.0624

-0.0237

-0.7544

0.0924

0.5790

0.9473];

net.b{3} = [0.9031

0.6027

0.0509

-0.9412

-0.4695

-0.1232

0.2131

-0.6329

0.8309

-0.1721

0.4098

0.0188];

net.b{4} = [ -0.7555];






% train the network

net=train(net,p,t);


y=sim(net,p)

APPENDIX C(a) Program for developing Hybrid MLR_ANN model for Rainfall % Training for Hybrid MLR_ANN for Rainfall


net = newff([0.0236 0.0432;0.0380 0.0980;0.0160 0.0920;0.0000 0.0185;

0.0007 0.0116],[8,10,12,1], {'tansig','tansig','tansig','logsig'},'trainscg');


P = [0.0381 0.0740 0.0440 0.0122 0.0066

0.0347 0.0800 0.0520 0.0065 0.0069

…………………………………………..

0.0328 0.0930 0.0650 0.0041 0.0088

0.0320 0.0940 0.0640 0.0034 0.0078];

P = P';

T = [0.0056

0.0738

………

0.0272

0.0112];

T = T';


net.IW{1,1} = [-0.0165 -0.5602 -0.9254 0.5428 -0.0347

0.9585 0.4071 0.9759 -0.2952 -0.4694

-0.5800 0.0012 0.6181 0.6248 -0.2739

-0.4363 0.9256 -0.5187 -0.6159 -0.3227

-0.6768 -0.9767 0.6415 0.5018 -0.7421

0.8588 -0.5151 -0.5123 0.3539 -0.2847

0.6738 -0.8447 -0.4710 -0.8824 0.6447

0.2823 0.0128 -0.0896 0.0977 0.0253];

net.LW{2,1} =

[ 0.4033 0.8710 0.0954 0.3346 0.0840 -0.4327 -0.8064 -0.4690

-0.2987 -0.0491 0.8702 0.1368 0.4376 0.8301 0.3531 -0.7863

0.8230 -0.9564 -0.8084 0.3470 -0.3655 -0.0281 -0.0353 0.1396

0.7490 -0.3869 -0.0854 0.8928 0.1007 0.2776 -0.1711 0.5790

0.5934 -0.5088 0.0043 0.6617 0.0539 0.8461 0.2708 -0.2750

0.7333 -0.7863 0.5775 0.8317 0.7859 -0.2347 -0.3770 -0.7176

-0.5908 0.8086 -0.0282 0.5866 -0.2306 0.9521 0.5944 0.0665

-0.4093 0.5681 0.5001 -0.0451 0.1344 0.9377 0.0803 -0.6780

-0.7031 -0.3047 -0.9889 0.5810 -0.1917 -0.4595 0.3169 -0.2686

0.3083 0.0619 0.2205 0.2461 -0.5910 -0.0581 0.4316 -0.4020];

net.LW{3,2} =

[-0.2992 0.0736 0.1809 -0.1987 0.0344 0.3314 0.7406 -0.3810 0.0014 -0.3096

0.0585 -0.8555 0.8627 -0.8211 -0.0035 0.7772 -0.5794 -0.5452 -0.9230 0.2821

-0.9921 0.5174 0.3439 -0.4363 0.1567 -0.5887 0.2269 0.1921 -0.7470 0.2161

-0.2624 -0.1125 -0.3643 -0.5726 -0.6557 -0.4331 0.3500 -0.8120 -0.3295 -0.8066

0.8511 -0.2858 0.8399 -0.4140 -0.8231 -0.9964 0.9448 -0.1758 -0.5992 0.5170

0.6006 0.1218 0.2986 0.0855 -0.1872 0.8996 0.2479 0.5829 -0.5938 -0.0209

0.1483 -0.1589 0.5353 -0.4556 0.0892 0.5796 -0.3841 -0.4574 -0.0395 -0.0366

-0.4478 0.2656 -0.0969 -0.3269 0.0018 -0.9472 -0.1770 0.7348 0.9646 0.4547

0.3409 0.3838 -0.9974 -0.7623 -0.5218 0.7049 0.3017 -0.2188 -0.1944 0.2978

-0.0821 0.6305 0.5099 0.2730 -0.7267 0.2925 0.1331 0.4866 -0.2774 0.6319

0.8429 0.9359 0.1330 0.1193 -0.1069 0.8959 -0.9942 0.5246 0.5098 0.3724

-0.2655 -0.3821 -0.4520 -0.0292 -0.7265 -0.2095 0.0033 0.6487 0.9488 0.0321];

net.LW{4,3} = [ 0.9979 0.5627 -0.5817 0.9911 -0.4803 -0.7350 0.0063 0.5415

0.4904 0.1651 0.5294 0.0905];

net.b{1} = [0.5913

-0.7489

-0.1915

0.0567

0.0964

0.5604

0.1472

-0.0661];

net.b{2} = [-0.0586

0.3628

-0.7196

0.0973

0.6545

-0.3077

0.1530

-0.3279

-0.1812

0.9241];

net.b{3} = [0.0867

0.1228

0.8659

0.0213

0.2901

0.7938

-0.1097

0.9280

-0.0195

0.2999

0.9186

-0.5942];

net.b{4} = [-0.6442];


net.trainParam.show = 100;

net.trainParam.epochs = 100000;

% net.trainParam.time = 115;

net.trainParam.goal = 0.003;

% train the network



y = sim(net,p)

APPENDIX C(b) Program for developing Hybrid ANN model for Minimum Temperature % Training for Hybrid MLR_ANN for Minimum temperature


net = newff([0.0236 0.0432;0.038 0.098; 0.016 0.092; 0.0007 0.0116],[8,10,12,1],

{'tansig','tansig','tansig','tansig'},'trainscg');


P = [0.0381 0.074 0.044 0.0066

0.0347 0.080 0.052 0.0069

………………………………..

0.0328 0.093 0.065 0.0088

0.0320 0.094 0.064 0.0078];

P = P';

T = [0.0265

0.0233

..…….

0.0240

0.0229];

T = T';


net.IW{1,1} = [0.1089 -0.6727 -0.7924 -0.5323

0.2135 0.5008 0.7948 0.4625

-0.1546 0.2521 0.3830 -0.5170

0.2101 0.2383 0.2221 0.0331

-0.3497 0.8230 -0.5889 -0.5643

-0.8754 0.4537 0.1024 -0.7260

0.0623 -0.2027 0.3841 0.5354

-0.4644 -0.5981 -0.9038 0.2432];

net.LW{2,1} = [-0.4277 0.0179 0.9611 -0.2847 0.1105 -0.1778 -0.3126 -0.8705

0.4252 -0.0500 -0.8309 0.2823 0.0191 -0.7340 0.2093 0.9023

-0.3982 0.3143 0.6544 0.6092 -0.6228 -0.3388 0.8481 0.8864

0.0745 -0.1689 0.8537 -0.1139 -0.7820 0.1127 -0.8242 0.9426

-0.4584 0.9795 -0.6219 -0.5359 0.6711 -0.8649 -0.2676 0.5663

-0.3195 -0.1308 -0.4303 -0.6655 -0.6043 -0.7324 0.2152 -0.2900

-0.1622 0.4071 0.0196 0.3368 -0.7206 0.9249 0.7834 -0.0955

-0.1597 -0.2529 -0.9409 0.0042 -0.2452 0.3254 -0.5294 0.7569

-0.5632 -0.8681 -0.0340 0.7844 0.2597 -0.6888 -0.5017 -0.5120

-0.3300 -0.7983 0.9042 -0.5364 -0.2758 -0.3137 0.0577 -0.0872];

net.LW{3,2} =

[0.0216 0.6615 -0.5375 0.0296 -0.0124 0.3568 -0.6221 -0.4996 0.2910 0.7199

0.0868 0.7066 -0.1379 0.0388 0.0877 0.4924 -0.6638 -0.5075 -0.6423 0.0039

0.0775 -0.4218 -0.6704 0.8214 0.1289 0.3250 -0.2822 0.2744 -0.7502 -0.5108

0.6607 0.0993 0.8013 0.4702 -0.4041 0.2922 -0.4714 0.2923 -0.2316 -0.6579

-0.8421 -0.3035 0.4866 0.4739 -0.3395 -0.1389 -0.4692 -0.2069 -0.2946 0.3367

0.4515 -0.1562 -0.5675 0.3154 0.6925 -0.8764 -0.1842 -0.4420 0.8363 -0.2085

-0.5962 -0.4865 -0.1099 -0.4632 0.4839 -0.0587 0.4288 0.5822 0.5679 -0.9318

-0.7712 -0.6244 -0.9410 0.9070 0.7567 -0.4974 0.2797 -0.0453 -0.9753 0.3295

-0.0588 -0.6184 0.8017 -0.4986 0.1131 -0.9043 -0.1239 -0.4225 -0.6122 -0.0001

0.5991 -0.9430 0.4101 -0.4603 -0.7504 0.6578 0.0244 -0.4394 -0.2440 0.0673

-0.3036 -0.8884 0.3107 0.5624 0.6066 0.5451 0.6421 0.7287 -0.6833 0.1758

0.6456 0.1542 0.0703 0.5252 0.8308 0.2982 0.7434 0.1186 0.8350 0.7662];

net.LW{4,3} = [-0.2025 0.2587 0.5378 0.1469 0.2400 0.4669 -0.3847 -0.0810

0.4419 0.0444 0.6404 0.4148];

net.b{1} = [ -0.6335

-0.9164

0.6704

-0.8871

0.2425

-0.0358

0.4393

0.6078];

net.b{2} = [ 0.8546

-0.5453

0.9412

-0.0433

0.2023

-0.1785

-0.6180

-0.5442

0.1806

-0.9093];

net.b{3} = [ 0.1424

0.8841

0.1328

0.6196

0.8312

-0.2198

0.8906

-0.2929

0.8971

0.8391

0.3129

0.2420];

net.b{4} = [0.6705];






% train the network



y = sim(net,p)

APPENDIX C(c) Program for developing Hybrid ANN model for Pan Evaporation

% Training for Hybrid MLR_ANN for Pan Evaporation


net = newff([0.0236 0.0432;0.0380 0.0980; 0.016 0.092; 0.0000 0.4432],[8,10,12,1],

{'tansig','tansig','tansig','logsig'},'trainscg');


P = [0.0381 0.074 0.044 0.0056

0.0347 0.080 0.052 0.0738

…………………………………

0.0328 0.093 0.065 0.0272

0.0320 0.094 0.064 0.0112];

P = P';

T = [0.0122

0.0065

.……..

0.0041

0.0034];

T = T';


net.IW{1,1} = [ -0.7272 0.9862 0.5016 -0.7559

0.3100 -0.3557 0.3844 0.3400

0.1637 0.4781 0.0540 0.2945

0.3590 0.2991 0.0643 -0.4392

-0.2047 0.3757 0.1413 -0.0442

0.8556 -0.1522 -0.6192 -0.2727

-0.2320 0.8968 -0.3116 -0.7358

0.5899 -0.6269 0.5045 0.3510];

net.LW{2,1} = [ -0.2135 0.4473 -0.7220 -0.1285 0.0845 -0.7815 -0.2210 -0.0254

0.3368 0.8847 0.4520 -0.7329 0.8688 -0.6782 0.7876 0.0363

0.3701 -0.1084 0.6084 0.1106 -0.3238 0.2478 -0.1009 0.7651

0.0292 0.2022 0.4127 -0.4327 0.0904 0.2583 0.0766 -0.2817

-0.8854 0.6838 0.4883 -0.0478 -0.6398 -0.9361 0.8232 -0.3119

-0.6100 0.8362 0.3270 0.8323 -0.0021 0.7533 0.9206 -0.4554

-0.2608 -0.8379 0.7985 -0.5701 -0.2513 -0.9704 -0.3494 0.8006

-0.6165 0.8619 -0.7470 0.1843 0.5553 0.1230 -0.0332 0.4142

0.6901 0.1444 -0.1737 0.3201 0.2916 -0.9144 -0.0566 -0.7507

-0.2038 -0.7971 -0.5943 0.7631 0.8431 0.4336 -0.7287 0.0105];

net.LW{3,2} =

[0.1557 0.4094 0.4130 -0.0939 -0.7884 -0.6147 -0.9166 0.1717 -0.8233 -0.5406

-0.5574 -0.0167 0.6716 -0.1286 0.8273 0.0231 -0.4696 0.3720 0.1806 -0.0803

0.5380 -0.7225 0.1275 0.2300 -0.0864 0.3762 0.0812 -0.0660 -0.0953 0.0520

0.2769 -0.2214 0.7927 -0.5422 -0.1949 -0.8049 0.5569 0.1738 0.6784 -0.8245

-0.6488 -0.7821 0.4875 -0.7928 -0.2820 0.5986 -0.2970 0.8247 -0.3944 -0.0402

-0.7540 0.6614 -0.5384 0.5072 -0.4650 0.5057 0.6111 0.5349 -0.7210 -0.4863

-0.4538 0.8376 0.8278 0.0409 0.9084 0.4262 -0.1011 -0.3536 -0.0742 0.7012

-0.8118 -0.5475 -0.6702 -0.6423 -0.3983 -0.0980 0.1542 0.1578 0.0552 0.2281

-0.6887 0.2576 0.3332 -0.5090 0.5357 -0.6282 0.2805 -0.5193 0.5750 -0.7378

0.7363 -0.7128 0.3516 -0.7168 -0.1879 0.6679 0.3366 0.5135 0.1297 0.8156

-0.7172 0.0069 0.7357 -0.4856 -0.4966 -0.9634 0.2556 0.6031 0.4909 0.6793

-0.0472 -0.4132 -0.1695 -0.2286 0.0003 -0.1881 0.4694 0.3752 0.5988 0.9887];

net.LW{4,3} = [0.3270 -0.6599 -0.6565 0.0350 0.1762 0.4022 0.4682 -0.1866

0.1506 -0.4282 0.8604 0.4187];

net.b{1} = [0.4896

-0.6720

-0.4822

-0.3595

0.0715

0.2632

0.8832

-0.1039];

net.b{2} = [0.7302

-0.4762

-0.9153

-0.6191

0.1788

-0.1960

-0.5998

-0.9473

0.4340

-0.7052];

net.b{3} = [-0.8028

0.5738

-0.0520

-0.8137

0.4453

0.1055

-0.1666

-0.4724

-0.9803

0.1444

-0.2031

-0.7824];

net.b{4} = [-0.3111];






% train the network



y = sim(net,p)

APPENDIX C(d) Program for developing Hybrid ANN model for Bright Sunshine

% Training for Hybrid MLR_ANN for Bright Sunshine


net = newff([0.0236 0.0432; 0.0172 0.0335; 0.038 0.098;0.016 0.0920;0.0000

0.4432],[8,10,12,1], {'tansig','tansig','tansig','purelin'},'trainscg');


P = [0.0381 0.0265 0.074 0.044 0.0056

0.0347 0.0233 0.080 0.052 0.0738

…………………………………………

0.0328 0.0240 0.093 0.065 0.0272

0.0320 0.0229 0.094 0.064 0.0112];

P = P';

T = [0.0066

0.0069

……….

0.0088

0.0078];

T = T';


net.IW{1,1} = [ 0.2535 0.8032 0.4855 0.0653 -0.2864

-0.2039 0.2511 -0.6987 0.6960 -0.7380

0.3083 -0.6499 -0.3556 -0.1896 -0.8882

-0.9468 -0.8670 -0.8440 0.9455 -0.1885

-0.3951 0.6232 0.2924 0.3258 0.3214

0.5504 0.1191 -0.2252 -0.9820 0.9634

-0.5582 0.1118 0.9788 0.1732 0.3051

0.5210 -0.3703 -0.7575 -0.5050 0.6340];

net.LW{2,1} = [ -0.8838 0.4402 -0.8666 0.9213 -0.1844 -0.3731 0.4570 -0.6265

0.8433 -0.8236 0.1431 0.8929 -0.3520 0.4636 0.3173 -0.9237

-0.4854 -0.1954 -0.7909 -0.0581 0.1324 -0.7104 0.0966 0.8494

-0.0146 0.5240 -0.4160 0.4259 0.3998 0.2870 0.6868 -0.7486

-0.2532 0.8772 0.5227 0.1875 0.7124 0.5905 0.8099 0.4093

-0.9875 -0.4068 -0.9776 0.8645 0.0461 0.2457 -0.7013 0.3713

-0.2534 0.2298 0.8699 0.5598 -0.1372 0.8524 -0.1342 -0.4537

-0.4086 -0.5370 0.5958 0.7701 -0.8375 0.8347 -0.3304 0.1210

0.4934 0.0580 -0.8741 0.3820 -0.9255 0.4894 -0.0820 0.3495

0.1691 -0.8404 -0.9516 0.0078 -0.4245 0.1088 -0.8149 -0.9201];

net.LW{3,2} =

[ 0.3296 0.7735 0.0756 0.8328 0.7670 -0.4081 0.3805 -0.5196 0.4570 0.2741

-0.4748 0.0588 0.7263 -0.5607 0.2643 0.7994 0.5495 -0.7471 -0.3801 -0.1198

0.8962 -0.4212 -0.1303 0.6565 0.3993 -0.3729 -0.2272 0.7454 0.6997 0.8395

-0.5805 0.1323 -0.7514 0.4964 0.6184 0.1572 0.6246 0.5602 -0.5959 0.5630

-0.4586 -0.5891 0.6003 -0.5022 -0.6595 0.2273 -0.6526 -0.5330 -0.0464 -0.6340

-0.8230 0.8332 -0.5941 0.3858 0.3046 0.7668 0.1651 -0.1934 -0.7275 -0.0343

0.5300 0.4139 0.9723 0.1301 0.4944 -0.5189 -0.3001 0.5363 0.7358 0.6723

-0.6276 -0.5261 -0.7997 -0.7174 -0.3420 -0.6313 -0.4725 -0.5147 -0.5239 0.1270

0.2037 0.8647 -0.5703 0.2161 -0.2602 0.1436 0.3466 -0.2824 0.3372 0.0417

0.4591 -0.6960 -0.3138 0.1939 -0.0357 -0.5145 0.2635 0.4528 -0.6164 0.1827

0.5458 -0.6874 -0.8361 0.7449 0.1117 0.6738 0.1509 -0.3140 -0.0875 -0.7798

-0.1765 -0.6446 0.8022 -0.8591 0.0230 -0.7721 0.0777 -0.1528 0.4186 0.7594];

net.LW{4,3} = [-0.0691 0.4847 0.0491 -0.4330 -0.4161 0.4828 0.0139 0.4180

-0.0993 0.2504 0.3449 -0.1747];

net.b{1} = [0.9807

0.6875

0.5157

0.5344

-0.2808

-0.1161

-0.8841

0.6626];

net.b{2} = [0.1246

-0.4557

0.1187

0.7682

0.4334

0.0097

-0.5773

-0.2063

0.5837

0.9203];

net.b{3} = [-0.8444

0.6469

-0.0153

0.7519

0.3512

0.1348

0.2617

-0.4950

0.8987

0.1456

0.4628

-0.8149];

net.b{4} = [-0.3443];






% train the network



y = sim(net,p)

COMPARATIVE STUDY OF FORECASTING MODELS BASED ON WEATHER …shodhganga.inflibnet.ac.in › bitstream...

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