Evolutionary Computation - Computational Intelligence in Time Series Forecasting - page 203

p d \ and with p = 1, 2, ..., N , the objective is to determine the fuzzy logic
system described above such that the performance function of the network, i.e. sum
square error (SSE) is minimized by optimal settings of the network's free
parameters

c V and y l . The SSE of the network is defined as

l

,

l

i i

2

N

p T

S 0.5 e 0.5 E E .

¦

p 1

, represents the approximation error of the network and y p
represents the output y of the network due to the presentation of p th input pattern
X p . We further assume that M , which corresponds to the number of implemented
GMFs for the partitioning of the input domain and also the number of implemented
fuzzy rules, is already given. Therefore, in order to train the network, i.e. for the
optimal settings of the network's free parameters, the binary-coded GA can be
applied.
For this purpose all the free parameters of the network are encoded in a binary
bit string or chromosome. For M fuzzy rules and n inputs to the system the total
number of mean parameters plus variance parameters of the GMFs along with
singleton rules' consequents will be of size

where, e

p

y

p

d

p

u u. Therefore, for a
network with n = 2 inputs and with M = 5 rules, each chromosome must encode

2 Mn M 1

uu parameter values.
Now if each parameter (say mean parameter of the GMF) of the network is
represented by N p bits, which include the first one bit as a sign bit, followed by N c
characteristic bits and N m mantissa bits, then N p = (1 + N c + N m ) and in this case N p
= 12 bits is selected.
Therefore, the entire bit length of each chromosome will be

25 2 5 1 25

^ ` p

u u = 300 bits. For example, if a parameter i c assumes a
decimal value -2.4256, then the first digit (2) before the decimal point is known as
the characteristic part and the remaining four digits (4256) after the decimal point
represent the mantissa part. Therefore, in order to represent any decimal number
within +3.99 to -3.99 we can use a 12-bit binary number, where the first bit, say 0,
will represent the “+ve” sign and 1 will represent the “-ve” sign, followed by the
next two bits, which can represent only four decimal numbers 0, 1, 2 or 3, and the
remaining nine bits represent the mantissa part.
For instance, the 12-bit number (1111 1111 1111) can represent the parameter
value (1*2 1 + 1*2 0 + 1*2 -1 + 1*2 -2 + ... + 1*2 -9 ) = -3.9980, whereas the 12-bit
number (0111 1111 1111) represents the decimal number +3.9980. Similarly, any
other combination of such 12 bits will represent any number between -3.9980 and
+3.9980.
Alternatively, the parameters within the above range can be encoded as
equivalent binary numbers as follows. Suppose the number -2.55 or +2.55 has to be
encoded into the equivalent 12-bit binary number, then just represent the -255 as
(1000 1111 1111) and similarly +255 as (0000 1111 1111), neglecting the
position of the decimal point during the encoding. However, during decoding

L 2 Mn MN 1