Meteorological Forecasting Using Type 2 Fuzzy Logic System

7/31/2019 Meteorological Forecasting Using Type 2 Fuzzy Logic System

1/27

Prepared By Selver Arslan

May 2012


2/27

Time series represents a powerful mathemetical tool to model

the temporal evolution of phenomenon in physical , economic

and biological environment.In this light of this meteorological

forecasting can be rendered as a complex time series

prediction problem. To overcome the problem of the residual non-stationary of the

random components of data and to consider other sources of

uncertainity , such as the model uncertainity related to tuning

of the predictor parameters in this work we extend theapproach on very recent class of fuzzy logic systems.


3/27

Most time series patterns can be described in terms of 2

basic components: Trends and seasonality.Trends represents a

general systematic linear or nonlinear component that changes

overtime.Seasonality may have a formally analogous nature.

But it repeats itself in systematic intervals over time.

We consider the measurements of temperature and relative

humidity collected 2004 in about one year in Torino Italy at a

sample rate 900 second .


4/27


5/27


6/27

Missing data is a frequent occurring phenomenon in many

practical contexts.For example , the measurements can not be

acquired directly. Or the measured values are unreliable or

missing or interrupted data collection occurs.All these casesfrequently happen.

In these cases a few missed data can be extrapolated by a data

fitting prosedure.Unfortunately ,in our application many data

are missed as shown below.Thus , we only consider the valuesof T and RH collected during the first 7 months of the year.


7/27


8/27

There is no automatic technique that can identify trend

components in the time series data; However as long as the

trend is monotonous (increasing or decreasing).The analysis is

typically not very difficult.When random components arepresent , the first step is smoothing.

Lastly a trend can be exracted by performing a data

fitting.Many monotonous time series data can be adequatelyapproximated by a linear function.


9/27


10/27

Seasonal dependency is another general component of the time

series pattern.It is formally defined as correlational dependency

of the order k between each ith element of the series and the

(i-k)th element is measured by autocorrelation ; k is usually

called the lag.Seasonality can be visually identified in the seriesas a pattern that repeats every k elements

The seasonal component may be extracted by filtering or by a

time frequency analysis.In our application the maximum

frequency of the seasonal component is so slow that very highorder filters should be used.


11/27


12/27

Once seasonality and trend have been identified,and a random

component has been reduced, then a residual component still

represents the core of the data.This is called the chaoticcomponent which depends on some initial state that does not

repeat over time and that can not be reduced or eliminated by

averaging.


13/27


14/27

All the steps described above aim to exract a stationary time

series from the collected data.This issue is crucial in many

forecasting methods.Meteorological time series are quite

complex as shown.And many different components contribute

to the temporal evolution of the data.The non-stationary of theresidual component is clear and it can be also shown by

performing a local variance estimation of the time series, by

considering 100 samples at a time .In doing so we obtain the

plot shown below.


15/27


16/27

STEP 1:First in a Type-1 FLS we expect a crisp value for the

input value , while in a type 2 FLS we can consider a

membership function for the input value x.This issue is crucial

when data are affected by random contributios or non-stationarynormal random component with a time dependent variance 2xsuch that x is between |1,2| We can easily build the MF as

shown below.


17/27


18/27


19/27

Analogously , if the dominant uncertainity contribution relates

to the central value of the MF as it occurs in the fuzzy sets

pattitioning the input domain.Then an MF with uncertain

central value and fixed width is built as shown below.

Once the input domain of each input (data of the time series in

our context) has been partitioned by means of type 2 MFs such

as depicted in figure below , consider the generic jth fuzzy ruleof the defined predictor as :

If (Xn-p+1 is A1) and (Xn-p+2 is A2) and (Xn is Ap+1) then (Xn+1 is Ap+1)


20/27

STEP 2 : At this point , the evaluation of each of eachantecedent is performed by implementing the intersection of

the MF of the input variable and the considered MF of the

input domain.The intersection can be accomplished by several

operators.Here , we implement the minimum operator.Each

antecedent has an interval of membership degrees associated

figure below and denoted as [lx1,ux1] and [lx2,ux2] where l and u

stand for lower and upper.

STEP 3 :To implement the and between two antecedents , weapply again the minimum operator to the 2 intervals achieved

above obtaining the interval [l,u] as shown figure below.This

interval represents the degree of the truth of antecedents in a

certain rule.


21/27


22/27

STEP 4: Implication operators are needed to apply the degreeof truth of each rule to the consequent .In our context, with theoutput and the input domains the same, the MF of the output

variable considered in a rule are weighted by the interval [l,u]resulting in a clipping of the MFs of the output variable.Thesituation is depicted in figure below in the case of 2 firedconsequents.

STEP 5:This step involves fuzzy rules aggregation.In particular,

an orbetween different rules is implemented by the maximumoperator among the lower and upper degrees of each firedconsequent respectively.The result are two MFs a lower andupper one as shown in figure 12 where ,for instance weaggregate the two rules that have been activated in the previous

steps considering the uncertain region bounded by the lowerand upper MFs.This is footprint of uncertainity.


23/27


24/27

STEP 6:

This step shows the important difference in using type-2 FLS with

respect to type 1 FLS. A type reducer has to be applied before

defuzzifying the output.Type reduction extracts an interval for the

output value from the uncertain region .Among different methods

proposed in the literature , here we apply the centroid type reduction.

The procedure works as follows.Define 3 vectors.Y vector represents

sampling points.C vector represents centers.S vector represents

spreads.The output value is between ylow and yhigh which is achived atmost Ns iteration.Center of the centroid is calculated according to

standart centroid center formula.


25/27


26/27


27/27

This paper presents a novel approach to forecast Meteorological

phenomena. It is based on the signal decomposition and on an

architecture that uses a Type-2 FLS in order to predict the

chaotic components.Signal decomposition permits to focus the

prediction only on chaotic components. This procedure allows toobtain longterm prediction bounded errors. Using Type-2 FLS is

also veryimportant because this class of systems has shown

small sensitivity to measurements uncertainty and noise. The

whole system has been designed and simulated on historical

meteorological data previously acquired. First results show

promising performance with respect to other methodologies

Meteorological Forecasting Using Type 2 Fuzzy Logic System

Documents

Transcript of Meteorological Forecasting Using Type 2 Fuzzy Logic System