Combined approach to pattern classification in parametric case
Michał Woźniak · 2007
Abstract: This paper is devoted to the methods for combining heterogeneous sets of learning data: set of training examples and set of IF-THEN rules with unprecisely formulated weights. Adopting the probabilistic (Bayes) model of recognition task and assuming known form of class conditional probability density functions (CPDFs) with unknown parameters, the recognition algorithm via fusion of both sets of data is presented. Proposed concept of combining of input data consists in treating of both sets as sources of information about unknown parameters of CPDFs, which leads to the modified maximum likelihood (ML) method of parametr estimation. In proposed procedure the likelihood function is maximized taking into account constraints provided by the set of rules. A series of numerical examples with computer generated data for several cases which differ in form and number of rules is considered. To find feasible solution of ML problem two approaches were employed. The first method uses the Kuhn-Tucker conditions for the nonlinear problem with inequality constraints, the second one however is approximated procedure which does not guarantee the optimality of result. For each solution the estimation error, i.e. distance between values of estimator and parameter is calculated as a measure of its quality, which allows us to rank procedures and to imply some practical conclusions.