Neuro-fuzzy Approach - Computational Intelligence in Time Series Forecasting - page 225

degree of fulfilment

zb of the corresponding rule, which, after multiplication
with the corresponding Takagi-Sugeno rule consequent

l

l
m y , is used as input to the
summation block (+) at the final output layer. The output of this summation node is
the final defuzzified output value, which, being crisp in nature, is directly
compatible with the real world data. Once the fuzzy system of the above choice is
represented as a feedforward network, the algorithm used for its training is less
relevant.
A similar fuzzy model with singleton rule consequents, trained with standard
backpropagation algorithm, was used by Wang and Mendel (1992b) for
identification of various nonlinear plants.
Forecasting of time series is primarily based on numerical input-output data. To
demonstrate this for neuro-fuzzy networks, a Takagi-Sugeno-type model, i.e . with
linear rules consequent (and also a singleton model as a special case), is selected
(Palit and Popovic, 1999; Palit and Babuška, 2001). Here, the number of
membership functions to be implemented for fuzzy partitioning of input universes
of discourse happens to be equal to the number of a priori selected fuzzy rules. To
accelerate the convergence speed of the training algorithm and to avoid other
inconveniences, the Levenberg-Marquardt training algorithm (described in the
Section 6.4.2.3) or the adaptive genetic algorithm (AGA) can also be used.
In Chapter 4 it was shown that in forecasting of various nonlinear time series
the fuzzy logic approach with automatically generated fuzzy rules (Wang and
Mendel, 1992c; Palit and Popovic, 1999) works reasonably well. However, it was
emphasized that the performance of fuzzy logic systems depends greatly on a set of
well-consistent fuzzy rules and on the number of fuzzy membership functions
implemented, along with their extent of overlapping. Therefore, determination of
the optimum overlapping values of adjacent membership functions is very
important in the sense that overlapping values too large or too small may
deteriorate the forecasting accuracy. In the absence of firm guiding rules for
optimum selection of overlapping, this selection mechanism was rather seen more
as an art than as a science, mainly relying on a trial-and-error approach.
Alternatively, very time-consuming heuristic approaches, such as the evolutionary
computation or the genetic algorithms (Setnes and Roubos, 2000), can be used for
this purpose.
Fuzzy logic systems encode numerical crisp values using linguistic labels, so it
is difficult and time consuming to design and fine tune the membership functions
related to such labels. However, neural networks' learning ability can automate this
process. The combination of both fuzzy logic and neural network implementations
can thus facilitate development of hybrid forecasters.
As an example we will consider the neural-networks-like architecture of the
neuro-fuzzy system (Figure 6.6) and the training algorithm selected will fine tune
the randomly generated system parameters. The great advantage of this scheme is
that, apart from the user-selected number of fuzzy rules to be implemented, all
other fuzzy parameters are automatically set by the training algorithm, so that the
user does not need to bother about the optimal settings of fuzzy region
overlappings and the like. Therefore, the approach to be described here is often