By Vladimir S. Cherkassky, Filip Mulier
An interdisciplinary framework for studying methodologies—covering facts, neural networks, and fuzzy common sense, this ebook offers a unified remedy of the rules and strategies for studying dependencies from facts. It establishes a normal conceptual framework during which a number of studying tools from facts, neural networks, and fuzzy common sense will be applied—showing few primary rules underlie so much new equipment being proposed at the present time in information, engineering, and machine technology. whole with over 100 illustrations, case experiences, and examples making this a useful textual content.
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Additional info for Learning from Data: Concepts, Theory, and Methods, Second Edition
For two known multivariate normal distributions, the optimal decision rule is a polynomial of degree 2 (Fukunaga 1990): f ðxÞ ¼ I n 1 2ðx T À1 1 À m0 ÞT ÆÀ1 0 ðx À m0 Þ À 2ðx À m1 Þ Æ1 ðx À m1 Þ þ c > 0g; ð2:44Þ where P detð 0 Þ Pðy ¼ 0Þ P À ln : c ¼ ln detð 1 Þ Pðy ¼ 1Þ ð2:45Þ The boundary of this decision rule is a paraboloid. To produce a good decision rule, we must estimate the two d Â d covariance matrices accurately because it is their inverses that are used in the decision rule. In practical problems, there are often not enough data to provide accurate estimates, and this leads to a poor decision rule.
These properties hold for both the frequentist and Bayesian views of probability. This view of uncertainty is applicable if an observer is capable of unambiguously recognizing occurrence of an event. For example, an ‘‘interest rate cut’’ is an unambiguous event. However, in many situations the events themselves occur to a certain subjective degree, and (useful) characterization of uncertainty amounts to specifying a degree of such partial occurrence. For example, consider a feature weight whose values light, medium, and heavy correspond to overlapping intervals as shown in Fig.
Note that the linear decision rule, which assumes equal covariances, does not match the underlying class distributions. However, the first-order model provides the lowest classification error (Fig. 2). 5 Nonparametric Methods The development of nonparametric methods was an attempt to deal with the main shortcoming of classical techniques: that of having to specify the parametric form of the unknown distributions and dependencies. Nonparametric techniques require few assumptions for developing estimates; however, this is at the expense of requiring a large number of samples.