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Evolving Fuzzy Systems

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Abbreviations

Evolving system :

In the context of this article the term ‘evolving’ is used in the sense of the self‐development of a system (in terms of both its structure and parameters) based on the stream of data coming to the system on‐line and in real‐time from the environment and the system itself. The system is assumed to be mathematically described by a set of fuzzy rules of the form:

$$ \begin{aligned} &\textit{Rule}^i\colon\\ &\textit{IF\enskip}(\textit{Input$_1$ is close to prototype$_1^i$})\\ &\textit{\enskip AND\enskip\dots\enskip AND\enskip}(\textit{Input$_n$ is close to prototype$_n^i$})\textit{\enskip}\\ &\textit{THEN\enskip}(\textit{Output$^i$}=\overline{\textit{Inputs}}^\textit{T}\enskip\textit{ConseqPara ms}) \\ \end{aligned} $$
(1)

In this sense, this definition strictly follows the meaning of the English word “evolving” as described in [34], p. 294, namely “unfolding; developing; being developed, naturally and gradually”. Contrast this to the definition of “evolutionary ” in the same source, which is “development of more complicated forms of life (plants, animals) from earlier and simpler forms”. The terms evolutionary or genetic are also associated with such phenomena (respectively operators that mimic these) as chromosome crossover, mutation,selection and reproduction, parents andoff‐springs [32]. Evolving (fuzzy and neuro‐fuzzy ) systems do not deal with such phenomena. They rather consider a  gradual development of the underlying (fuzzy or neuro‐fuzzy) system structure.

Fuzzy system structure :

Structure of a fuzzy (or neuro‐fuzzy) system is constituted of a set of fuzzy rules (1). Each fuzzy rule is composed of antecedent (IF) and consequents (THEN) parts. They are linguistically expressed. The antecedent part consists of a number of fuzzy sets that are linked with fuzzy logic aggregators such as conjunction, disjunction, more rarely, negation [43]. In the above example, a conjunction (logical AND) is used. It can be mathematically described by so‐called t‐norms or t‑conorms between membership functions. The most popular membership functions are Gaussian, triangular, trapezoidal [73]. The consequent part of the fuzzy rules in the so‐called Takagi–Sugeno (TS) form is represented by mathematical functions (usually linear). The structure of the TS fuzzy system can also be represented as a neural network with a specific (five layer) composition (Fig. 1). Therefore, these systems are also called neuro‐fuzzy (NF).

Figure 1
figure 1_192

Structure of the (neuro‐fuzzy) system of TS type

Full size image

The number of fuzzy rules and inputs (which in case of classification problems are also called features or attributes) is also a part of the structure.

The first layer consists of neurons corresponding to the membership functions of a wparticular fuzzy set. This layer takes the inputs, x and gives as output the degree, µ to which these fuzzy descriptors are satisfied. The second layer represents the antecedent parts of the fuzzy rules. It takes as inputs the membership function values and gives as output the firing level of the ith rule, τ i . The third layer of the network takes as inputs the firing levels of the respective rule, τ i and gives as output the normalized firing level, λ i as “center of gravity” [43] of τ i . As an alternative one can use the “winner takes all” operator. This operator is used usually in classification, while the “center of gravity” is preferred for time‐series prediction and general system modeling and control. The fourth layer aggregates the antecedent and the consequent part that represents the local sub‐systems (singletons or hyper planes). Finally, the last 5th layer forms the total output of the NF system. It performs a weighed summation of local sub‐systems.

Fuzzy system parameters:

Parameters of the NF system of TS type include the center, c and spread, σ of the Gaussians or parameters of the triangular (or trapezoidal) membership functions. An example of a Gaussian type membership function can be given as:

$$ \mu =\text{e}^{-\frac{1}{2}\left( {\frac{d}{r}} \right)^2} $$
(2)

where d denotes distance between a data sample (point in the data space) and a prototype/cluster center (focal point of a fuzzy set); r is the radius of the cluster (spread of the membership function).

Note that the distance can be represented by Euclidean (the most typical example), Mahalonobis [33], cosine etc. forms.

These parameters are associated with the antecedent part of the system. Consequent part parameters are coefficients of the (usually) linear functions, singleton coefficients or coefficients of more complex functions (e. g. exponential) if such ones are used.

$$ y_{i}= a_{i0}+a_{i1}x_{1}+\dots+a_{in}x_{n} $$
(3)

where a denotes parameters of the consequent part; x denote the inputs (features); i is the index of the ith fuzzy rule; n is the number (dimensionality) of the inputs (features).

Potential :

Potential is a mathematical measure of the data density. It is calculated at a data point, z and represents numerically the accumulated proximity (density) of the data surrounding this data point. It resembles the probability distribution used in so‐called Parzen windows [33] and is described in [26,72] by a Gaussian‐like function:

$$ P(z)=\text{e}^{-\frac{1}{2r}\overline \sigma^{2}} $$
(4)

where \( { z=[x,y] } \) denotes the joint (input/output) vector;

\( { \overline \sigma_k^2 =\frac{1}{k-1}\sum\limits_{i=1}^{k-1} {d^2(z_k ,z_i)} } \) is the variance of the data in terms of the cluster center.

In [3,9] the Cauchy function is used which has the same properties as the Gaussian but is suitable for recursive calculations.

$$ P(z)=\frac{1}{1+\overline \sigma^{2}}\:. $$
(5)
Age of a cluster or fuzzy rule :

The age of the (evolving) cluster is defined as the accumulated time of appearance of the samples that form the cluster which support that fuzzy rule.

$$ A^i=k-\frac{\sum\limits_{l=1}^{S_k^i} {k_l} } {S_k^i} $$
(6)

where k denotes the current time instant; \( { S_k^i } \) denotes the support of the cluster that is the number of data samples (points) that are in the zone of influence of the cluster (formed by its radius). It is derived by simple counting of data samples (points) at the moment of their arrival (when they are first read) and assigned to the nearest cluster [10].

The values of A vary from 0 to k and the derivative of A in respect to time is always less or equal to 1 [17]. An “old” cluster (fuzzy rule) has not been updated recently. A “young” cluster (fuzzy rule) is one that has predominantly new samples or recent ones. The (first and second) derivatives of the age are very informative and useful for detection of data “ shift ” and “drift” [17].

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Acknowledgments

The author would like to thank Mr. Xiaowei Zhou for his assistance in producing the illustrative material, Dr. Jose Macias Hernandez forkindly providing real data from the oil refinery CEPSA, Santa Cruz, Tenerife, Spain, Dr. Richard Buswell, Loughborough University and ASHRAE(RP‐1020) for the real air conditioning data, and Dr. Edwin Lughofer from Johannes Kepler University of Linz, Austria for providing real data fromcar engines.

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    Plamen Angelov

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Angelov, P. (2009). Evolving Fuzzy Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_192

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