Data engineering : fuzzy mathematics in systems theory and by Olaf Wolkenhauer

By Olaf Wolkenhauer

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The ‘squares criterion’ is fundamental to systems and data analysis. The in least-squarescriterion is commonly used to describe the objective function system identification. It will be introduced in the following section and will play a prominent role throughout this book. 1 THE LEAST-SQUARES CRITERION You’ve got to draw the line somewhere. as they say. In this section, we introduce a technique which allows us to identify system models, as discussed in Section 1, from sampled data. The most commonly used criterion to quantify the quality of the model fitting the data is called ‘least-squares’ criterion.

If data are considered to be random, the uncertainty of outcomes in Y is characterized by someprobability distribution or density p(y) which quantifies t,he ‘likelihood’ of whether any particular value in Y will, on average, occur or not. In a probabilistic setting, it is common practice to associate the outcomes in Y with a random variable, denoted y. For some event, represented by function 5 specifying subset A c Y, the expectation of the characteristic of event A: subset A, is defined as the probability +m E[b] = s-co CA(Y) P(y>dY where b(Y) = -J AP(Y) G Pr(A) 1 ifyEA, (2 .

8 Example of partial linkage between c and <‘. The aim of a product space representation is to obtain for a given J: E X a unique representation in form of ‘coordinates’: The expectation is therefore that two observables provide a more comprehensive description of elements of X. Considering a pair of observables (<, c’), as if they describe one, we would say that two abstract states are equivalent if neither e nor <’ can distinguish between them. That is, if the pair (<, {‘) imposes a single equivalence relation Et<’ on X, where Etc/ holds if and only if c(xi) = c(xz) and <‘(xl) = t/(x2)* Th e equivalence classes of this new relation Ett 1 are formed from the intersections of the equivalence classes of EE with Et/.

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