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Let us discuss several special cases of the filtering fuzzy sets used in the compression algorithm.
The difference between the original fuzzy relation R and its decompressed version (where ||.|| stands for a suitable distance function). For fixed values of p and r (that induce a required compression rate) the minimization of this distance is accomplished by changing the family of the fuzzy filters 8.5.2. GA-optimized data miningThe fundamental objective of data mining is to reveal easily interpretable, stable and meaningful patterns in a abundant flood of data. Data mining subsumes classic modeling by focussing on patterns rather than precise functional relationships (functions) with a significant number of almost impossible to interpret parameters. It is appropriate to proceed with a relatively straightforward example, that is, however, quite representative for a broad class of systems. Let us discuss a series of triples of data k = 1, 2, , N originating from a certain dynamic system. The usual modeling activities (as commonly envisioned in system identification) start from a certain form of the model, say The flexibility of the model stems from a vector of parameters (param); to cope with the available experimental data the model is optimized in a parametric fashion. In essence, data mining assumes no particular form of the functions but searches for patterns that could be eventually materialized into the number of functional dependences. The experimental data can be arranged into a two-dimensional array indexed by x(k) and u(k) whose entries are just the corresponding values in the next time instant x(k+1). An interpretation of the table calls for a suitable information granularity; this comes in the form of an information granule defined for control (u(k)) and state variable (x(k), x(k+1)). To concentrate on the generic form of the method, we start with the codebooks
The same table can be interpreted as a collection of patterns. For example, for the (i, j)-th entry we come up with the summarization of some experimental data The multiplicity of some of the patterns for the entries result in nondeterministic behavior of the model; note also that so far we have not confined ourselves to any particular class of functions (say, linear, polynomials, trigonometric, etc.). As an example of this form of model representation, Fig. 8.9 summarizes a well-known Box-Jenkins data set for several delay times τ. Here the individual data elements are organized as triples (u(k-τ), x(k), x(k+ 1)).
The nondeterministic nature of the model can be expressed by a Petri net; below we illustrate the patterns as the firing of the transition (tr) is caused by Ai and Bj and results in a transfer of tokens that now become located at several output places (here A2, A3, and A7), Fig. 8.10.
The leading optimization criterion is straightforward: our intent is to reduce a nondeterministic behavior of the model by making the multiple entries of the relational tableau as homogeneous as possible. A simple index reflecting this would be a variance of the indexes of the landmarks that become associated with the given entry of the table. Comprehensively, we consider an additive format of the performance index to be minimized where σ2ij denotes the variance of the (i, j)-th entry of the table. The minimization of Q is achieved by moving around the endpoints of the landmarks (sets) defined in the control and state space. If instead of sets, the landmarks are expressed as fuzzy sets then an accumulation of evidence involves the degrees of activation (membership values) combined AND-wise. This leads to the formula Its value represents a degree of activation of A1 as it appears at the (i, j)-th entry of the table and becomes implied by the triple ( x(k), u(k), x(k+1)).
Copyright © CRC Press LLC
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