AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

280
HYBRID INTELLIGENT SYSTEMS
How does an ANFIS learn?
An ANFIS uses a hybrid learning algorithm that combines the least-squares
estimator and the gradient descent method (Jang, 1993). First, initial activation
functions are assigned to each membership neuron. The function centres of the
neurons connected to input x i are set so that the domain of x i is divided equally,
and the widths and slopes are set to allow sufficient overlapping of the respective
functions.
In the ANFIS training algorithm, each epoch is composed from a forward pass
and a backward pass. In the forward pass, a training set of input patterns (an
input vector) is presented to the ANFIS, neuron outputs are calculated on the
layer-by-layer basis, and rule consequent parameters are identified by the leastsquares
estimator. In the Sugeno-style fuzzy inference, an output, y, is a linear
function. Thus, given the values of the membership parameters and a training
set of P input-output patterns, we can form P linear equations in terms of the
consequent parameters as:
8
y d ð1Þ ¼ 1 ð1Þf 1 ð1Þþ 2 ð1Þf 2 ð1Þþ...þ n ð1Þf n ð1Þ
y d ð2Þ ¼ 1 ð2Þf 1 ð2Þþ 2 ð2Þf 2 ð2Þþ...þ n ð2Þf n ð2Þ
><
.
y d ðpÞ ¼ 1 ðpÞf 1 ðpÞþ 2 ðpÞf 2 ðpÞþ...þ n ðpÞf n ðpÞ
.
.
>:
y d ðPÞ ¼ 1 ðPÞf 1 ðPÞþ 2 ðPÞf 2 ðPÞþ...þ n ðPÞf n ðPÞ
ð8:14Þ
or
8
y d ð1Þ ¼ 1 ð1Þ½ k 10 þ k 11 x 1 ð1Þþk 12 x 2 ð1Þþ...þ k 1m x m ð1ÞŠ
þ 2 ð1Þ½ k 20 þ k 21 x 1 ð1Þþk 22 x 2 ð1Þþ...þ k 2m x m ð1ÞŠ þ ...
þ n ð1Þ½ k n0 þ k n1 x 1 ð1Þþk n2 x 2 ð1Þþ...þ k nm x m ð1ÞŠ
y d ð2Þ ¼ 1 ð2Þ½ k 10 þ k 11 x 1 ð2Þþk 12 x 2 ð2Þþ...þ k 1m x m ð2ÞŠ
þ 2 ð2Þ½ k 20 þ k 21 x 1 ð2Þþk 22 x 2 ð2Þþ...þ k 2m x m ð2ÞŠ þ ...
þ n ð2Þ½ k n0 þ k n1 x 1 ð2Þþk n2 x 2 ð2Þþ...þ k nm x m ð2ÞŠ
><
.
y d ðpÞ ¼ 1 ðpÞ½ k 10 þ k 11 x 1 ðpÞþk 12 x 2 ðpÞþ...þ k 1m x m ðpÞŠ
þ 2 ðpÞ½ k 20 þ k 21 x 1 ðpÞþk 22 x 2 ðpÞþ...þ k 2m x m ðpÞŠ þ ...
þ n ðpÞ½ k n0 þ k n1 x 1 ðpÞþk n2 x 2 ðpÞþ...þ k nm x m ðpÞŠ
.
y d ðPÞ ¼ 1 ðPÞ½ k 10 þ k 11 x 1 ðPÞþk 12 x 2 ðPÞþ...þ k 1m x m ðPÞŠ
þ 2 ðPÞ½ k 20 þ k 21 x 1 ðPÞþk 22 x 2 ðPÞþ...þ k 2m x m ðPÞŠ þ ...
>:
þ n ðPÞ½ k n0 þ k n1 x 1 ðPÞþk n2 x 2 ðPÞþ...þ k nm x m ðPÞŠ
ð8:15Þ