Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 165

> @

T u u u u. 741 NN 1 741 1 351

and the resulting column vector

>@

T

T ª

11

, , !" ! . ,

1

;; ,

MM

, ,

M

º

TT T TT T

¬ ¼

01 n 0 1 n

So, it is to be noted that once the parameters of antecedent fuzzy sets (using the
Wang-Mendel's approach, or its modification, or by fuzzy clustering) are
determined, which are required for computation of degree of fulfilment of each
rule for a given set of ( N training samples) inputs, the linear TS rule's consequent
parameters can be determined easily by LSE technique as described above.

4.6 Forecasting Time Series Using the Fuzzy Logic Approach

In the forecasting examples described below, the shifted values X ( t + L ) are the
predicted values based on sampled past values of a time series up to the point t , i.e .
[ X { t -( D -1) d }, X { t -( D -2) d }, ..., X ( t - d ), X ( t )]. Therefore, the predictor to be
implemented should map from D sample data points, sampled at every d time
units, as its input values to the predicted value as its output. Depending on the
availability of the time series data and on its complexity, the D , d , and L values are
selected. In our case D = 4 and d = L = 6 have been selected, corresponding to a
four-inputs system, and to a sampling interval of six time units.
Hence, for each t > 18, the input data represent a four dimensional vector and
the output data a scalar value

XI ( t ) = [ X ( t -18), X ( t -12), X ( t -6), X ( t) ]

XO ( t ) = [ X ( t +6)].

Supposing that there are m input-output data sets, we generally use the first m /2
input-output data values (training samples) for fuzzy rule generation and the
remaining m /2 input data sets for verification of forecasting accuracy with the
fuzzy logic approach.

4.6.1 Forecasting Chaotic Time Series: An Example

As an example, forecasting of a chaotic time series is considered in this section.
The chaotic series are generated from deterministic nonlinear systems that are
sufficiently complicated as they appear to be random, however, because of the
underlying nonlinear deterministic maps that generate the series, chaotic time
series are not random time series (Wang and Mendel , 1992). The chaotic series, for
our experiment, is obtained by solving the Mackey-Glass differential equation
(Junhong, 1997; M ATLAB , 1998). Lapedes and Farber (1987) also used
feedforward neural networks for the prediction of the same chaotic time series and
reported that the neural network gave the best predictions in comparison with