Transparent Fuzzy/Neuro-fuzzy Modelling - Computational Intelligence in Time Series Forecasting - page 272

where it can be seen that the first two triangular fuzzy sets after adaptation become
highly overlapping and they approximately represent the same concept.
Consequently, assigning them any meaningful label, such as low or medium, is no
longer appropriate. Furthermore, some of the fuzzy sets extracted from numerical
data may be similar to the universal set U . Such fuzzy sets are irrelevant because,
for all the elements within the universe of discourse, they have degree of
membership approximately equal to 1, which fails to categorize the data. The
opposite effect is similarity to a singleton fuzzy set (see Figure 7.1 (right)). In this
case, a particular data point has degree of membership equal to 1 and for all other
data points it gives zero degrees of membership. If a rule has one or more such
fuzzy sets in it's premise, then it may never fire, and thus the rule does not
contribute to the output model. However, it should be noted that such a rule may
represent an exception in the overall model behaviour and, therefore, deserves a
special care as it's removal may force one to neglect the exceptionality in the
model behaviour.

7.3.2 Compact and Transparent Modelling Scheme

We will now turn our attention to the application problem of similarity-driven
simplification to enhance the transparency and compactness of a fuzzy rule base. In
order to reduce the redundancy of fuzzy models obtained from data, this
simplification can naturally be combined with a data-driven modelling tool, which
results in a transparent fuzzy model scheme. This is the approach followed in
nonlinear time series modelling for the purpose of forecasting it's future values. As
such, a data-driven modelling tool, either of the fuzzy clustering or the neuro-fuzzy
method, can be considered. However, other methods, such as Wang and Mendel's
(1992a) approach, or it's modification by Palit and Popovic (1999a) for rule base
generation or fuzzy modelling, can also be considered. Setnes et al. (1998a)
considered a similarity-driven simplification in combination with fuzzy-neural
networks, and Setnes and Roubos (2000), and Roubos and Setnes (2001)
considered the genetic-fuzzy approach for second-order nonlinear plant modelling
using Wang data (Wang and Yen, 1999), the principal steps of which for a
transparent modelling scheme are described below.

Step 1: Model Structure Selection

x The relevant input and output variables that are used for fuzzy model
building are determined. Here, the structure selection for dynamic systems
means translation of the identification problem into the equivalent
regression problem that can be solved in a static manner (Babuška, 1996).
Frequently, a reasonable choice of model structure can be made by the
user, based on prior knowledge about the process. For the time series
forecasting problem considered in this chapter, four input variables and one
output variable are considered, so that the input data is a vector of size

1u and output is a scalar.

Step 2: Data Clustering