Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 169

(validation data) are used for verification of forecasting accuracy, and for all four
inputs and for the output the domain interval [ X lo , X hi ] { [0.4, 1.4] have been
selected. Four different fuzzy logic systems with 17, 27, 37, and 51 Gaussian
membership functions (GMFs) have been investigated.
For the fuzzy predictor, the Mamdani-type fuzzy rules were initially generated
using the first 500 rows (training data) of the XIO matrix; thereafter, redundant and
conflicting rules were removed from the rule list. For that purpose, only c 1 = 0.4
and c n = 1.4 were selected and the values c 2 , c 3 ,..., c n -1 were calculated for equal
divisions of all ( n -1) intervals. For the first system, i.e. with n = 17 GMFs, V a =
0.08 and V b = 0.04 were selected. Similarly, V a = 0.08 and V b = 0.02 were selected
for the second and third systems (with n = 27 and 37 GMFs), whereas V a = V b =
0.02 were selected for the fourth (with n = 51 GMFs) fuzzy predictor. Figure 4.4(a)
shows the partitioning of universes of discourse for the first input and output of the
predictor with the n = 17 GMFs, and Figure 4.4(b) through Figure 4.4(e) show the
results of forecasting, along with the forecasting errors, for the investigated
systems. Note that, because of good prediction accuracy, forecasted series can
hardly be distinguished from the original chaotic series except for Figure 4.4(b).
The performance functions like SSE (0.5 E T .E ), with E as a column vector of
errors and T indicating transposition of the E vector, and RMSE indicating the
efficiency of the individual fuzzy system investigated, are also computed and listed
in Table 4.1 for mutual comparison. The results from the Table 4.1 confirm the
high suitability of the proposed approach, based on automatically generated fuzzy
rules for forecasting of Mackey-Glass chaotic time series. From Table 4.1 it also
follows that, when the number of GMFs is increased from 17 to 51, the forecasting
accuracy is significantly increased.
4.7 Rules Generation by Clustering
Automated data driven rule generation, as described above, works considerably
well for nonlinear time series modelling and forecasting. However, the fuzzy rule
base generated in this way is generally very large, because each set of input-output
pair generates a fuzzy rule. This is true even after the removal of redundant and
conflicting rules from the rule base generated. For instance, using the first 500
input-output data sets of Mackey-Glass chaotic time series and using 27 number of
GMFs, which are used for partitioning of input and output universes of discourse,
the generated fuzzy rules, after the removal of conflicting and redundant rules, are
still of the order of 350. This definitely imposes a large amount of computational
load for fuzzy inferencing. To avoid this, an alternative approach, based on a fuzzy
clustering algorithm was proposed that, for instance, uses only a few fuzzy rules
for nonlinear time series modelling and forecasting.
4.7.1 Fuzzy Clustering Algorithms for Rules Generation
Clustering algorithms are mathematical tools useful in identifying the natural
groupings of data, based on common similarities, from a large data set to produce a