EarthWeb   
HomeAccount InfoLoginSearchMy ITKnowledgeFAQSitemapContact Us
     

   
  All ITKnowledge
  Source Code

  Search Tips
  Advanced Search
   
  

  

[an error occurred while processing this directive]
Previous Table of Contents Next


Let us discuss several special cases of the filtering fuzzy sets used in the compression algorithm.

(i)  Ai and Bi are disjoint sets covering X and Y. The solution (decompressed image) is computed based upon the supports of the filtering sets. Note that

Here the convolution formula becomes simplified

and gij is just a maximal membership value derived over the Cartesian product of Ai and Bj. Note that the larger the filtering sets, the more extended the area over which a maximum of R is determined. Lower granularity of Ai and Bi delivers a higher compression rate but at the expense of quality of the reconstructed image.

(ii)  Ai and Bi are singletons, that is fuzzy sets with a single nonzero membership value.

Then the convolution returns the original image; obviously no compression is achieved.

The difference between the original fuzzy relation R and its decompressed version is regarded as the performance measure; our intent is to minimize the distance

(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 and . Their parametric as well as structural optimization is completed using evolutionary computation.

8.5.2. GA-optimized data mining

The 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 and treated as sets. Let us now scan data, triple after triple, and convert it into the corresponding granule. The entries of the resulting linguistic table defined over . can be multiple; the same combination (Cartesian product) of Ai and Bj may lead to different A1s, Fig. 8.8.


Figure 8.8  Linguistic table with multiplicity of entries.

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)).


Figure 8.9  Box - Jenkins data set for τ = 0, 2, 4, and 8

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.


Figure 8.10  Petri net as a model of the linguistic table with multiple inputs

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)).


Previous Table of Contents Next

Copyright © CRC Press LLC

HomeAccount InfoSubscribeLoginSearchMy ITKnowledgeFAQSitemapContact Us
Products |  Contact Us |  About Us |  Privacy  |  Ad Info  |  Home

Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction in whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement.