to i-th rule. r is the number of rules in the rule base.

Given the output of the individual consequents yi , the global output y of the

Takagi-Sugeno model is computed by using the fuzzy mean formula:

r

i=1 βi (x)yi

y= (6.19)

r

i=1 βi (x)

6.3 Fuzzy Model Identi¬cation 115

where βi (x) = µAi (x) is the degree of membership or belonging of X into the

fuzzy set Ai .

The conjunctive form of the antecedent 6.17 can be also formulated as follows,

IF x1 is Ai,1 AND ... AND xp is Ai,p THEN yi = ai x + bi

with degree of membership or ful¬llment

βi (x) = µAi,1 (x1 ) § µAi,2 (x2 ) § ... § µAi,p (xp ).

By normalizing this membership degree of rule antecedent with

βi (x)

φi (x) = (6.20)

r

j=1 βj (x)

then we can interpret the a¬ne Takagi-Sugeno model as the quasilinear model

with a dependent input parameter (Wolkenhauer, 2001):

r r

y= φi (x)ai x+ φi (x)bi = a (x) + b(x). (6.21)

i=1 i=1

Usually it is di¬cult to implement multidimensional fuzzy sets, therefore the

antecedent 6.17 is commonly interpreted as a combination of equations with a

one-dimensional fuzzy set for each variable x.

6.3.3 Model Identi¬cation

The basic principle of model identi¬cation by product space clustering is to

approximate a non linear regression problem by decomposing it to several local

linear sub-problems described by IF-THEN rules. A comprehensive discussion

can be found in Giles and Draeseke (2001).

The identi¬cation and estimation of the fuzzy model used for cases of multi-

variate data as follows, suppose

y = f (x1 , x2 , ..., xp ) +

where is assumed independent, identical and normally distributed.

116 6 Money Demand Modelling

If the error has a mean of zero, then the fuzzy function interprets the conditional

mean of variable output y. Therefore, the assumption now is the use of linear

least square as the basis for analysis.

Step 1: For each xr and y, separately partition n observations of the sample n

into fuzzy clusters cr by using fuzzy clustering (where r = 1, ..., p).

Step 2: Consider all possible combinations of c fuzzy cluster towards the num-

ber of variable input p, where:

k

c= cr (6.22)

r=1

Step 3: Make a model by using data taken from each fuzzy cluster.

yij = βi0 + βi1 x1ij + βi2 x2ij + ... + βip xpij + (6.23)

ij

where j = 1, ..., n; i = 1, ..., c.

Step 4: Predict the conditional mean of x by using:

c

i=1 (bi0 + bi1 x1k + ... + bip xpk )wik

yk =

ˆ ; k = 1, ..., n. (6.24)

c

wik

i=1

where

p

wik = δij urjk ; i = 1, ..., c

r=1

δij is a selector that selects a membership value for each fuzzy of j cluster if

the cluster is associated with the i-th (default δij = 1).

The fuzzy predictor of the conditional mean y is a weighted average of linear

predictor based on the fuzzy partition of explanatory variables, with a member-

ship value varying continuously through the sample observations. The e¬ect of

this condition is that the non-linear system can be e¬ectively modelled.

Furthermore, a separately modelling of each fuzzy cluster, including the use of

fuzzy logic having the ”IF” form, input data are found in this region. ”THEN”

is likely to be a predictor of the response variables.

The modelling technique based on fuzzy sets can be clustered in a local mod-

elling method, because it uses partitions of a domain process on a number of

6.3 Fuzzy Model Identi¬cation 117

Indonesian Money Demand

4.4

Fuzzy TS

Log(Money Demand)

4.3

True Value

4.2

4.1

0 10 20 30 40 50

Time: 1990:I-2002:III

Two dimensional plot of Indonesian Money Demand with Fuzzy TS vs Time

fuzzy region. In each region of the input space, a rule is de¬ned to specify

models from its output. The rules here are described as a local sub-model of a

system. TakagiSugeno is one of the models used here.

6.3.4 Modelling Indonesian Money Demand

In this section we prepare a model for M2 money demand in Indonesia us-

ing the approach of fuzzy model identi¬cation. This constructing model was

based on the three-monthly data taken from 1990:1-2002:III which consisted of

three variables, Real Money Demand (M2), Real GNP (GNP), and Long Term

Interest Rate (r).

The result of fuzzy clustering with the M2 and GNP variables shows the data

formed of three clusters. Of the three clusters, it™s di¬cult to determine their

real clusters so that we arrange them as the fourth cluster.

On the other hand, a clustering using M2 and r variables perform of 2 clusters.

The intersection of these two clustering stages would result in 4 di¬erent clusters

as shown in Table 6.2.

The ¬rst and the second clusters, that is the period of 1990:1-1994:3 and 1994:4-

1998:3, show that GNP has a positive e¬ect of money demand, while r is neg-

ative. In the second period, however, the e¬ect of r is not signi¬cant. This

phenomenon is consistent with the result gained using the standard economet-