Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 158

4.5.2 Modifications Proposed for Automated Rules Generation

The described rules generation algorithm requires much manual effort in handling
the numerous data sets (say 500) and the large number of triangular membership
functions (say 20 to 50). To reduce this, the modification of some of the
operational steps and the use of Gaussian membership functions of the following
form are proposed (Palit and Popovic, 1999):

^ `

2

(; , ) exp ( 2

2

(4.2)

f )

xc V x c V

i j j i j j

j

for simplification of computation of degree of membership for all values of X i .
Dividing the domain interval [ X lo , X hi ] into ( n -1) overlapping fuzzy regions,
with nN , N is some integer value, and assigning to each region a Gaussian

membership function, the mean C and the variance V will also have

2 1

nN
values, such as C 1 to C n and V 1 to V n respectively. For ease of computer program
implementation, the domain interval is divided into ( n -1) equal regions such that

2 1

CX , CC X X 2, N CC X X 2, N ..., CX and

1 lo 2 1 hi lo 3 2 hi lo n hi

" For forecasting of Mackey-Glass chaotic
time series, for example, n = 17, V a = 0.08, V b = 0.04, and the domain interval [0.4,
1.4] were selected. The fuzzy regions in this case are denoted by G 1 , G 2 , G 3 , ..., G n ,
etc. for convenience and G indicates the Gaussian membership functions (GMFs).
For any input X i within the domain interval the degree of membership

VVV , VV V V .

1 n a 2 3 n 1 b

P X fX will be within [0, 1], for j = 1, 2, 3, ..., n . If

X C

then

G i j i i j

j

1,

whereas the degree of membership

P will be zero
only if X rf and for other values of X i in the domain interval the degree of
membership can assume any value between 0 to 1. The fuzzy rules can now be
generated in the usual way.
With the above modifications and after preprocessing the time series data, the
automated fuzzy rules generation continues with fixing the domain interval as
>

P X X fX

G i G i j i

j j

@

X , X { ª min X , max X º

¼ and with dividing the domain interval into ( n -1)

¬

lo hi

equal regions, where nN 2 , and N is any suitable integer value such that each 1

1

segment is of length SX X n , on which the accuracy of the forecast

hi lo

depends.
Now, the total number

nN 2 of GMFs G 1 to G n , over the entire domain 1

with CX , CCS , CCSC ..., 31 , S

r CC r S

..., 1,

1 lo 2 1 3 2 1 1

,
are assigned with suitably selected values of V a and V b . The integer n and, hence,
the C 2 to C n -1 and V values are chosen such that two consecutive membership
functions partially overlap. The forecasting accuracy also depends greatly on the
extent of overlapping. It has been observed that the overlaps that are too large or
too narrow may deteriorate the forecasting accuracy of the time series.

CC n SX and 1 , VV ! V V , whereas VVV

n 1 hi 2 3 n 1 b 1 n a