Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 184

Table 4.2(a). Cluster centers (V) generated by Gustafson-Kessel algorithm

v 1 for input u , or
X 1

v 2 for input y ,or
X 2

v 3 for output g

-0.4099 1.0024 0.2146

-0.8820
1.1066

-0.8681
-0.1326

-0.3636
-0.0055

Table 4.2(b). Variance parameters of GMFs determined from fitting the projected data

Serial number of
antecedent GMFs

For input u , or X 1 For input y , or X 2

First GMF
Second GMF

2.1398
1.0178

1.5698
1.6112

Third GMF 0.9319 2.4221

Table 4.2(c). Consequents' parameters of Takagi-Sugeno rules

Theta0 Theta1 Theta2

-0.4706
0.5056
-0.1839

0.0750
0.1282
0.4057

-0.0765
0.1685
0.3783

Here, we apply the Gustafson-Kessel clustering algorithm to construct the
desired fuzzy model using the first two columns of the XIO = [ u , y , g ] matrix as the
input data and the third column as desired output data, i.e. the data (pattern) matrix
here is constructed as Z = [ XIO ] T . The first 200 (training) samples (rows of XIO
matrix) were used for fuzzy rules generation by applying the Gustafson-Kessel
clustering algorithm using the following parameter settings: number of clusters c =
3, fuzziness exponent m = 2 and termination tolerance = 0.001. Accordingly, three
clusters with cluster centers V = [ v 1 , v 2 , v 3 ] and partition matrix U of size 3 u

were obtained. Projecting the first two rows of the U matrix on to the input
dimension and, thereafter, by fitting the Gaussian function of the form

200

^ `

three antecedent fuzzy membership functions for each
input were obtained (Figure 4.6(a)).

2 2

y exp 4 log(2) ( x v ) / V ,

i