By Yurij Kharin (auth.)
This booklet is worried with very important difficulties of strong (stable) statistical pat tern acceptance while hypothetical version assumptions approximately experimental facts are violated (disturbed). development attractiveness thought is the sector of utilized arithmetic within which prin ciples and strategies are developed for type and identity of gadgets, phenomena, tactics, occasions, and indications, i. e. , of items that may be laid out in a finite set of beneficial properties, or houses characterizing the gadgets (Mathematical Encyclopedia (1984)). levels in improvement of the mathematical concept of trend acceptance should be saw. on the first degree, until eventually the center of the Nineteen Seventies, trend recogni tion thought used to be replenished typically from adjoining mathematical disciplines: mathe matical records, sensible research, discrete arithmetic, and knowledge thought. This improvement level is characterised by way of profitable answer of trend reputation difficulties of alternative actual nature, yet of the easiest shape within the feel of used mathematical versions. one of many major techniques to resolve trend acceptance difficulties is the statisti cal strategy, which makes use of stochastic types of characteristic variables. below the statistical process, the 1st level of trend reputation idea improvement is characterised via the belief that the chance info version is understood precisely or it's esti mated from a consultant pattern of enormous dimension with negligible estimation error (Das Gupta, 1973, 1977), (Rey, 1978), (Vasiljev, 1983)).
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This publication is anxious with vital difficulties of strong (stable) statistical pat tern attractiveness while hypothetical version assumptions approximately experimental facts are violated (disturbed). trend popularity concept is the sphere of utilized arithmetic within which prin ciples and strategies are built for type and id of gadgets, phenomena, methods, occasions, and indications, i.
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Extra info for Robustness in Statistical Pattern Recognition
J 1988) an approach to asymptotic analysis of classification error probability for the Gaussian model and for the model with block-independent feature variables under the special Kolmogorov- Deev asymptotics of increasing dimenSIOn: n -+ 00, m -+ 00, min -+ A, 0 < A < 00 was developed. A considerable part of investigations of small-sample effects was performed under the Gaussian model assumption. 2-type distortions) are mostly investigated by computer experiments (Raudis, 1976). In Chapters 3, 4 we develop the method of asymptotic expansions of classification risk for the analysis of small-sample effects in the cases of both parametric and nonparametric families of probability distributions (Kharin, 1981~1983).
L)~2r1 (/12 - /1d) = X o. 56). 66) is inapplicable in the pattern recognition problems where the classes are distinguished by the covariance matrices ~1' ~2 only and mathematical expectations /11, /12 have equal or close values. 5 Probability Models of Data and Optimal Decision Rules 1. Optimal Decision Rules in Discrete Space of Feature Variables Until now in constructing and investigating optimal pattern recognition algorithms, we assumed the continuous observation model (when the observed feature vector x = (Xj) E RN may be any point of the N-dimensional Euclidean space).
Misclassification of Training Sample In the training sample A; of size only (1 - Ei) . i( i E S), and the remaining Ei . 1j; 0 :::; Ei :::; E+i. This situation is often appears in practice, for example, in biomedical data processing (Kox, 1974), (Launer, 1979), (Kharin, 1983). , 1976), (Lachenbruch, 1979), (Chittineni, 1980), (Krishnan, 1988). ni 2. 46 Violations of Model Assumptions and ... 4. Outliers in Training Sample The training sample Ai( i E S) from the i- th class is contaminated by outliers (Aivazyan et al.