Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 187

essential that it is a nonlinear combination of various forecasts of a given time
series. The latter has been reconfirmed by many studies, which have revealed that
only the nonlinearity provides the combination with the guarantee to produce better
forecasts than either of the combination components separately. This is mainly
because, here, we have a kind of synergic effect.
In this section we describe an application example where a fuzzy model has
been used as a nonlinear forecasts combiner. For this purpose, we consider once
again the temperature series discussed in Chapter 3, along with its two forecasted
series. The temperature series selected is a non-stationary, non-seasonal time
series. Moreover, the original temperature series with 226 observations was
obtained from a chemical process by temporarily disconnecting the controllers
from the pilot plant involved and recording the subsequent temperature fluctuation
every minute (Box and Jenkins, 1976).
The two separate forecasts of the selected temperature time series were made,
one by applying the Box-Jenkins ARMA/ARIMA method (Box and Jenkins, 1976)
and the other by applying Holt's exponential smoothing technique (Chatfield,
1980). In order to utilize the fuzzy model as a nonlinear forecasts combiner, here,
we used both the forecasted series as two inputs to the fuzzy model to be
developed, and the original temperature series as the desired output from the fuzzy
model. The two forecasted series and the original time series have been rearranged
as the first, second and the third columns respectively of a HBXIO matrix.
Thereafter, the first 150 rows from the HBXIO matrix were used as training data
and the remaining rows, i.e. 151 to 224 rows of HBXIO matrix were used as test
samples to evaluate the efficiency of the combination approach described (Palit
and Popovic, 2000). It is to be noted that by applying conventional forecasting
methods on the original temperature series we obtained only 224 data points in
both cases.
Using the modified and automated rule-generation algorithm, Mamdani-type
fuzzy rules were generated from the training data based on the implemented n = 21
GMFs, and fixing X lo = 18, X hi = 28,

V Care has been taken
to make the rule base somewhat compact by eliminating the conflicting rules and
unnecessary redundant rules. Thereafter, a nonlinear combination of forecasts with
the fuzzy model was generated, based on the above rule base and utilizing only the
input data from the validation data sets (see Figure 4.7(b)). Finally, the
performance of the approach was measured by computing performance indices,
such as SSE, RMSE etc. , for the validation data set as illustrated in Table 4.3.
From Table 4.3 it can be seen that the SSE and RMSE achieved with the proposed
fuzzy model is much better than the individual forecast generated either by the
Box-Jenkins method or by Holt's exponential smoothing technique. The reported
result obviously confirms the high suitability of the fuzzy logic approach as a
nonlinear forecasts combiner.

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0.4, and 0.2.