Quantitative Methods in

Forecasting

Assoc. Prof. Christian Tanushev, Ph.D.

9 February 2015

Overview

• Review on Qualitative Methods of Foreceasting

• Types of Cross Sectional Study

• Statistical average: mode, median, mean

• Variance and Standard deviation

• Smoothing Forecasting Methods

– “Moving Averages”

– Exponential Smoothing

– Regression models

2

Forecasting - definitions

• Forecasting is the process of estimation in

unknown situations

• A forecast is a prediction based on knowledge

of past behavior.

• Normally, the prediction is expressed as a

probability. The prediction is an assertion of

likelihood that an outcome will take place.

3

Salvador Dali, Explosion Of Faith

In A Cathedral, 1974

4

Boian Donev, Pegasus, 1995

5

The Knowledge Spiral

6

Types of KnowledgeMode of

knowing

Knowing by

experience

Knowing how to

do, "know-how"

Conceptual (theoretical)

knowing

Example

"The benches in this

lecture-hall are not

comfortable."

"I know how to

design a good TV

couch."

"A suitable seat height for

British grown-ups is 44 cm."

Area of

validity

Pieces of knowledge

are detached and

valid only in one case

Knowledge can be

applied in several

instances

Knowledge can be applied

to all instances of the same

type. It contains mainly

general rules

Mode of

presentation

The essential sense of

"tacit" knowledge

cannot be explained

verbally

Skill of trade.Many

important points of

these cannot be

presented verbally

The knowledge can be

expressed in words and

exact models, and it can be

printed as a handbook

Method of

teaching the

knowledge

Cannot be taught. Can

be learned only by own

experience

The master shows how

the thing is done; the

student imitates the

master

Lectures and reading of text-

books

7

Cross Sectional Study

(No time perspective)One single case will be

studied, or a few similar

cases. Holistic view

Case study

- Exploratory Case Study (chemistry elements)

- Case Study Based on Earlier Theory (Periodic Table of

Elements of Mendeleev)

- Normative Case Study (nuclear decay)

A few different cases will be

studied. Holistic view

Comparative Study

- Descriptive Comparison

- Normative Comparison (sample)

A large number of different

cases will be studied

Classification

– Exploratory Classification

- Classification into Given Classes

- Normative Classification

Variables or measurements

from a large number of

cases are analyzed

Quantitative Analysis

- Analyzing Individual Variables

- Analyzing Relationships between Variables

- Normative Study of Variables

8

Case Study• The most usual target in case studies is to

describe the object or phenomenon - not only

its external appearance but also its internal

structure

9

Model

10

Exploratory Case Study• Exploratory study, in other words not basing

the study on any earlier model or theory, is

usually laborious, slow and uncertain, so

usually you will want to avoid such an

approach if you can. The normal method is to start with a thorough

search of literature.

• If you can distinguish any historical evolution that has taken place

around the object, it can help you to detect a dynamic invariance in

the phenomenon.

• In the case that your material consists of several similar objects or

cases your target normally becomes to find out what is common to

all the cases: what is the static invariance in them. When studying

artifacts it could be a typical form, pattern or proportion. When

studying people it could be their prevalent attitude, widespread taste

or a typical behavior.11

Case Study Based on Theory• It is common that in the beginning

of exploratory study you will take aholistic look at the objects. It means that you start by gathering as much information about the objects as possible, and postpone the task of cutting away unnecessary data until you get a better picture about what is necessary.

• Any object can be looked at from several different viewpoints, either from the angles of various established sciences or just from miscellaneous practical points of view.

12

Normative Case Study

• An important use of normative case study is to

guide the development of a new version of an

existing model of a product

13

Descriptive Comparison• In the initial phases of the study you only reach

descriptive answers to the question what the object is and what it is like, and from this basis you can then try to explain or answer the question why the object is as it is.

• In comparative analysis you can apply all the usual types of explanation: by earlier events, by later events, and contextual explanation. It can be useful to make a table

14

Normative Comparison

• In normative analysis one of the principal

criteria is evaluative like "satisfaction",

"usefulness" etc., and the aim of the study is to

point out the best (in this respect) among the

alternatives that are being studied.

• The final aim perhaps is not only to find the

best, but also to improve it or similar objects

later on

15

Exploratory Classification

• The goal in classification is always the same:

to reveal the systematic structure, invariance,

that exists in all the cases (population) that you

study.

16

Classification into Given

Classes• Fuzzy classification is a method which aims

at placing all the cases or specimens in one or other of the classes even if the "fit" is not perfect

– Cluster analysis, which is best suited when the classification is to be made on the basis of numerical data, and

– Typology which is appropriate for all other kinds of material. Typology is a method of classification where each class is formed around a "typical" or "pure" exemplar.

17

Linnean Classification System

KINGDOMSTRUCTURAL

ORGANIZATION

METHOD OF

NUTRITION

TYPES OF

ORGANISMS

NAMED

SPECIES

TOTAL

SPECIES

(estimate)

Monera

Small, simple single

prokaryotic cell (nucleus is

not enclosed by a membrane);

some form chains or mats

Absorb food

Bacteria, blue-green

algae, and

spirochetes

4,000 1,000,000

Protista

Large, single eukaryotic cell

(nucleus is enclosed by a

membrane); some form chains

or colonies

Absorb, ingest,

and/or

photosynthesize

food

Protozoans and algae

of various types80,000 600,000

Fungi

Multicellular filamentous

form with specialized

eukaryotic cells

Absorb food

Funguses, molds,

mushrooms, yeasts,

mildews, and smuts

72,000 1,500,000

Plantae

Multicellular form with

specialized eukaryotic cells;

do not have their own means

of locomotion

Photosynthesize

food

Mosses, ferns, woody

and non-woody

flowering plants

270,000 320,000

Animalia

Multicellular form with

specialized eukaryotic cells;

have their own means of

locomotion

Ingest food

Sponges, worms,

insects, fish,

amphibians, reptiles,

birds, and mammals

1,326,239 9,812,298

18

Normative Classification

• Normative is any cross tabulation where one of

the dimensions expresses an evaluation

19

Product A Product B Product C

Good 81 % 34 % 9 %

Average 4 % 36 % 60 %

Bad 15 % 30 % 31 %

Analyzing Individual Variables

• Often the preliminary exploration starts with a

single variable.

• Before you submit data to analysis, it will often

be useful to perform some preliminary

operations. These may include:

– Removal of data which are obviously erroneous or

irrelevant.

– Normalizing or reducing your data means that you

eliminate the influence of some well known but

uninteresting factor. 20

Types of Scales

• Nominal Scale. Any numbers used are mere labels

• Ordinal Scale. Numbers indicate the relative position of items

• Interval scale. Numbers indicate the magnitude of difference between items, no absolute zero point.

• Ratio scale. Numbers indicate magnitude of difference and there is a fixed zero point

21

Mode• An average is a statistic which characterizes the typical

value of your data and eliminates the random scattering of values

• A mode is the value that occurs the most frequently in a data set or a probability distribution. The mode of the sample: { 1, 1, 2, 2, 2, 4, 4, 5, 6 } is 2.– The sample: { -4, -2, -2, 0, 2, 2, 3} has two modes -2 & 2 –

bimodal distribution

22

Median

• A median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half.

• No algebraic positional average which takes the value of the unit that is at the middle of prearranged statistical row.– Median of the set: { 1, 1, 2, 2, 2, 4, 4, 5, 6 } There are 5

observations: ( 1, 2, 4, 5, 6 ) – uneven number, so we just pick the third observation and the median is 4.

– Median of the set: { 1, 2, 2, 2, 4, 4, 5 }. The observations are 4: ( 1, 2, 4, 5 ) – even number, so the median is half sum from the second and the third observations and median is (2+4)/2=3.

• The average price of housing is usually the median.

23

Median Graph

24

Statistical Arithmetic Mean

• Non weighted arithmetic mean (average)

• Weighted arithmetic mean

• The expected value of a random variable, which

is also called the population mean25

Arithmetic mean• Statistical set: 7, 12, 17, 24, 35 has a mean 19

• If we assume the probabilities relative to these

values as follows 0.1; 0.4; 0.2; 0.2; 0.1, then

the weighted average will be 17.2

26

195

95

5

3524171271

n

x

x

n

i

i

2.171

2.17

1.02.02.04.01.0

1.0.352.0*242.0*174.0*121.0*7*

1

1

n

i

i

n

i

i

w

wx

x

Normal Distribution• If your studies involve people, your measurements

quite often turn out to be distributed according to a certain curve, the so called Gauss curve

• One of its properties is that 68% of all measurements will differ from the mean by no more than the standard deviation and 95% by two standard deviations

27

Averages in Economic Forecasting

28

Weighted Average

(11,98=5*6+6*18+7*20+...+32*1)

Median (The value of the 100-th

from 199 observations)

Dispersion (Statistical Variation)

• Once you have calculated the average value, it

would sometimes be interesting to describe how

far the singular values are scattered around the

average. To this end, you may choose between a

variety of statistics

• In connection with the arithmetic mean you will

often want to calculate the standard deviation

29

Variance Formulas

• Dispersion is a measure in variability or spread

in a variable or a probability distribution

• The formulas for a population and a sample

will differ

30

n

xxn

i

i

1

2

2

1

1

2

2

n

xxn

i

i

Variance of a Random Variable

• The variance of a random variable is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean). Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out.

31

2

1

2 *Pr

n

iii rEr

n

iii rrE

1

*Pr

Standard Deviation

• The positive square root of the variance, called the standard deviation, has the same units as the original variable and can be easier to interpret for this reason

• The standard deviation is a measure of risk, it evaluates the probable deviation of the real values from the expected values.

32

2

n

xxn

i

i

1

2

2

Diachronic, or historical, study

Holistic study of

the evolution of

individuals or

specimens

Analyzing Development

- Describing Development

- Explaining Development

- Normative Study of

Development

Study of the

development of

variables

Study of Time Series

33

Describing Development

• Individual development. Temporal view of an

industrial product

• Development of a class.

• A time series is a line of variable values collected

under a period of time, usually at even intervals.

– The curve is the most usual presentation for time

series.

– Time is normally presented on the horizontal x-axis.

34

Explaining Development

• Often a mere description of the changes in the object

of study does not suffice, and the researcher is asked

to uncover also the reasons and/or effects of the

changes.

• The reasons can be taken either from:

– the past (causal explanation).

– the concurrent context

– the future (i.e. from the intentions of people)

35

Normative Study of Development

• The focal point is to evaluate specific criteria –utility, effectiveness, functionality, safety, beauty, ecofriendliness, price

36

Dominant goal in the design

theory of architecture:

Style of

architecture:

BeautyDoric, Ionian and Corinthian

styles

Religious salvation The Gothic style

Beauty (the classical goal

restored)

Renaissance, baroque, rococo,

neo-classical style

Individualisml'Art Nouveau and other personal

styles like Gaudi's

Utility Functionalism

Beauty – Doric Order

37

Beauty – Ionic Order

38

Beauty – Corinthian Order

39

Religion Salvation – Gothic Style

40

Beauty - Renaissance

41

Individualism – Gaudi Personal Style

42

Utility – Functionalism

43

Time Series

44

Jefferson Memorial

45

Lincoln Memorial

46

Time Series

• If we take a closer look at the variation of the time series, it often reveals components, all of which have their specific regularities which can be analyzed. The most usual of these components are:

– A trend is a linear direction of development over a period of time.

– A periodic variation is a cyclical variation recurring in a similar form all over again.

– Conjuncture variation occurs repeatedly in the same way as a periodic variation, but its length and form vary

– Random variation is usually eliminated by means of the flexible average method.

47

Time Series - Graph

48

Extrapolation• Extrapolation is the most usual method of forecasting. It is

based on the assumption that present development will continue in the same direction and with unvarying speed (or alternatively, with steadily growing or diminishing speed, i.e. a logarithmic extrapolation).

49

d

d

1 2 3 4 5 t

d

Smoothing Methods

• Smoothing Techniques apply to a dynamic set

of historical (observed in practice) data to

calculate the forecast the future value for some

future event of the series.

• The basic notion inherent in smoothing

methods is that there is some pattern to the

series of data and that this pattern will continue

in the future; consequently, past events are

drawn upon to predict or forecast future events

50

Three Basic Methods

• The accuracy of the forecast with smoothing

techniques depends primarily on

– the cohesiveness of the series of historical data and

– how far in the future the forecast is made.

• Smoothing Techniques

– “Moving Averages”

– Exponential Smoothing

– Regression models

51

Moving Averages

The method of moving averages uses the average

of some specified historical period to forecast

the value of a future period.

Moving average for 3, 5, 9, 21 periods...

52

n

AAAAF ntttt

t121

1

...

averagesmovingthecalculatetousedperiodsofnumbern

periodtheforvaluesobservedrealyAдоA

periodnexttheforvalueforecastedaisFwhere

ntt

t

1

1

A Double Moving Average

• A Double Moving Average – we calculate it

from already calculated moving averages for

diferent periods of time

53

n

MAMAMAMAF ntttt

t

'

1

'

2

'

1

''' ...

averagesmovingthecalculatetousedperiodsofnumbern

periodtheforaveragesmovingcalculatedMAдоMA

valueforecastedaisFwhere

ntt

t

'

1

'

''

Example of a “moving average” (MA)

54

Year Value 3-years MA Double 3-years MA 5-years MA

2007 31.6

2008 30.5

2009 31.8

2010 34.2 31.3

2011 36.3 32.2

2012 39.3 34.1 32.9

2013 41.7 36.6 32.5 34.4

2014 50.0 39.1 34.3 36.7

2015 46.8 43.7 36.6 40.3

2016 43.7 46.2 39.8 42.8

2017 52.1 46.8 43.0 44.3

2018 63.3 47.5 45.6 46.9

Model of “A Double Moving

Average ”

55

forecastwewhichfuturetheintimeofperiodsofnumbert

averagemovingthecalculatetousedyearsofnumbern

averagesmovingsecondandfirsttheofevaluationMAиMA

tscoefficien

valueforecastedaisFwhere

tt

t

,

,

''

1

'

1

1

)(1

2

2

*

''

1

'

1

''

1

'

1

1

tt

tt

t

MAMAn

MAMA

tF

Model’s Forecast

56

Year Value 3-yrs. MADouble 3-yrs.

MA

α =

2MA’t+1-MA”t+1

Level

β = 2/(n-1) *

(MA’t+1-MA”t+1)

Trend

Ft+1 =

α + β*P

Forecast

2007 28.2

2008 31.6

2009 30.5

2010 31.8 30.10

2011 34.2 31.30

2012 36.3 32.17

2013 39.3 34.10 31.19 37.01 2.91 39.92

2014 41.7 36.60 32.52 40.68 4.08 44.76

2015 50.0 39.10 34.29 43.91 4.81 48.72

2016 46.8 43.67 36.60 50.73 7.07 57.80

2017 43.7 46.17 39.79 52.54 6.38 58.92

2018 52.1 46.83 42.98 50.69 3.86 54.54

Exponential Moving Average

57

11 *1 ttt FAF

periodstimeofnumbern

tcoefficienweightingn

tperiodforvalueactualtheA

tperiodforMAlexponentiaforecastofvaluetheF

where

t

t

;1

2

1

)(

1

111 tttt FAFF

Exponential Moving Average

Forecast Year Value Forecast

α = 0.2

Forecast

α = 0.5

2007 28.2

2008 31.6 28.20 28.20

2009 30.5 28.88 29.90

2010 31.8 29.20 30.20

2011 34.2 29.72 31.00

2012 36.3 30.62 32.60

2013 39.3 31.75 34.45

2014 41.7 33.26 36.88

2015 50.0 34.95 39.29

2016 46.8 37.96 44.64

2017 43.7 39.73 45.72

2018 52.1 40.52 44.7158

EMA – a Trigger in Technical

Analysis

59

26-days EMA

longer and slower

12-days EMA

shorter and

faster

Sell

Signal

Buy

Signal

Regression Analysis

• Explore if a certain variable is causally

dependent on one or more other variables

60

errorforecast

linetheofslopetheofanglethe

axisуwithpointcross

variabletindependenx

variabledependenty

i

i

i

iii xy *

Linear Trend in Time:

Calculating α and β

61

22

..

ttn

ytytn

n

ty

.

ii ty *

Linear Trend Equation Example

62

t y

Week t2 Sales ty

1 1 150 150

2 4 157 314

3 9 162 486

4 16 166 664

5 25 177 885

S t = 15 S t2 = 55 S y = 812 S ty = 2499

(S t)2 = 225

Linear Trend Calculation

63

3.6225275

1218012495

22555*5

812*152499*5

5.1435

15*3.6812

ty *3.65.143

Linear Regression

64

2009 4

2010 8

2011 10

2012 12

2013 13

2014 14

2015 14

2016 ?

2017 ?

2009 4

2010 8

2011 10

2012 12

2013 13

2014 14

2015 14

2016 17.14

2017 18.75

y = 1.607x + 4.2861

0

2

4

6

8

10

12

14

16

18

20

2009 2010 2011 2012 2013 2014 2015 2016 2017

Forecast Accuracy

• Forecast Error - difference between the actual value and predicted value for a given time period

et = At – Ft• Mean Absolute Deviation (MAD)

– Average absolute error

• Mean Squared Error (MSE)

– Average of squared error

• Mean Absolute Percent Error (MAPE)

– Average absolute percent error

65

MAD, MSE, and MAPE

66

n

FA

MAD

n

t

tt

1

n

FA

MSE

n

t

tt

1

2

n

A

FA

MAPE

n

t t

tt

1

100*

Calculation of MAD, MSE, MAPE

67

Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/A)*100

1 217 215 2 2 4 0.92

2 213 216 -3 3 9 1.41

3 216 215 1 1 1 0.46

4 210 214 -4 4 16 1.90

5 213 211 2 2 4 0.94

6 219 214 5 5 25 2.28

7 216 217 -1 1 1 0.46

8 212 216 -4 4 16 1.89

-2 22 76 10.26

MAD= 2.75

MSE= 9.50

MAPE= 1.28

Regression Log Function

68

2009 4 4

2010 8 8

2011 10 10

2012 12 12

2013 13 13

2014 14 14

2015 14 14

2016 15.32017 16.0

y = 5.3577ln(x) + 4.1892

0

2

4

6

8

10

12

14

16

18

2009 2010 2011 2012 2013 2014 2015 2016 2017

Summary

• Quantitative methods of forecasting provide us

with an expected value of a certain indicator.

• These measures helps us to obtain a clear

picture of the future

• Excellent knowledge of statistics helps us to be

more precise in forecasting economic variables

69