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Granularity transformation of qualitative criterion, orthogonality of qualitative mapping system, and pattern recognition

A kind of granularity transformation of qualitative criterion, which called subdivision of qualitative criterion and its inducing orthogonality of qualitative mapping family are presented in this paper. The inner product between a decomposable property and qualitative mapping and the orthogonal expansion of property whose orthogonal bases is the qualitative mapping family are discussed. It is shown that in qualitative mapping orthogonal system the recognition of a pattern can be converted into recognition of a vector whose components are coefficients of the orthogonal expansion of the pattern respectively. proposition p(o) of object o with quantity x of attribute a(o) whose truth value varies with the qualitative criterion (α,β) can be represented by the Qualitative Mapping τp(o)(x,(α,β)), and definitions of qualitative mappings whose qualitative criteria are an interval, a vector of intervals or a grid of n-dimension parallelepiped respectively have been given. The relation between inner transformation of qualitative criterion and Artificial Neuron has been discussed. In spite the truth value of the property p(x,o) of object o with quantity x of attribute a(o) can be judged by a qualitative criterion (α,β), when the granularity of (α,β) is big, the set Sp={o|p(o,x)} classifying by p(o,x) may be too rough to be generally adopted in a real application, therefore, we usually divide the criterion in to smaller segments. For example, native place is one of the best properties for classifying people. Usually the criterion of native place are countries which divide the earth into more then one hundred portions, but in one country, for better distinction, we need more exact criterion such as provinces, cities, regions or even small towns and villages that divide a country into smaller