Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 161

Algorithm 4.1. Automated rules generation algorithm for time series prediction
Given the time series X = {X 1 , X 2 , X 3 ,…, X q } for t = {1, 2, 3, …., q}, the
Mamdani-type fuzzy rules are generated as follows:
x Step 1. Partition the time series data into MISO form
XI(t) = [X{t-(D-1)d}, X{t-(D-2)d}, …., X{t-d}, X{t}]
XO(t) = [X(t+L)],
For four-inputs system D = 4, and select sampling interval d = 6, and
lead time of forecast L = 6.
x Step 2. Divide the domain interval [X lo , X hi ]
into (n-1) = 2N overlapping fuzzy regions.
X lo = min(X), X hi = max(X).
Assign to each region a GMF and
denote them as G 1 , G 2 , …., G n .
x Step 3. Compute S = ( X hi - X lo ) /2N, so that the
mean parameters of GMFs are:
C 1 = X lo , C 2 = C 1 + S, ..., C r = C 1 + (r-1)S, ...,
C n = C 1 + (n-1)S = X hi .
and variance parameters of GMFs are:
Sigma_ G1 = Sigma_ Gn = Sigma 1 ,
Sigma_ G2 = Sigma_ G3 =, ..., = Sigma_ G(n-1) =Sigma 2 .
Select two suitable values for Sigma 1 , Sigma 2 , so that
two adjacent fuzzy regions partially overlap.
x Step 4. Fuzzify all the crisp inputs and output.
For any input X ki or output Y k compute the degree of membership in
all Gaussian regions.
0 < µ G j (X ki ) = f j (X ki ) = exp(-0.5.(X ki -C j ) 2 /(Sigma _Gj ) 2 ) 1.
Say, for i = 1, k = 1, and for j = 1, 2, 3, ..., n.
x Step 5. Arrange all degrees of membership in an (n×1) column
vector.
Similarly, compute the degree of membership for i= 2, 3, 4 and
for Y k , etc., when k = 1, and j =1,2, 3, …,n; etc.
Arrange them all in a column vector form of size (n×1).
When all such column vectors each of size (n×1) are arranged side by
side sequentially they result in a Mu-matrix of size {n×(i max +1)}. For
four-inputs and one-output system the Mu-matrix is of size (n×5).
x Step 6. Select the maximum value of degree of membership from each
column and record the corresponding row number i.e. integer value of
r, such that 0 <µ G r ( X ki ) = max ( µ G j ( X ki )) 1, j = 1, 2, 3, …, n and 1 r
n.
Note the value of r from each column of the Mu-matrix, such that
1 r = integer n. This is the key step of the automated rule
generation algorithm. Now create the fuzzy rule based on the r values
and the corresponding degree of membership.
x Step 7 . Create the rule list and solve the rule conflict problems (if
any) using the degree of rule. Also remove the redundant rule from the
rule list (if any).