The Method Of Fuzzy Discriminant Risk Analysis

It was Lofti Zadeh, a Professor at the University of California at Berkeley, who had the insight as well as the courage to initiate in 1965 the radically new paradigm by introducing the concept of a fuzzy set. The reaction to this proposal was fairly typical of a paradigm shift. The concept itself has emerged as fundamental and fresh idea. Fuzzy sets, in which the membership is a matter of degree rather than a matter of either affirmation or denial, involve capturing, representing, and working with linguistic notions-objects with unclear boundaries.Lately fuzzy set theory has become a subject of considerable attention. The growing visibility of the theory is a result of its highly successful practical applications in many areas, such as decision making, clustering and pattern recognition, processing and understanding of images, diagnosis (medical, engineering, cognitive), risk analysis and reliability theory, engineering design, database and information retrieval systems, expert systems, business and management, economic theory, processing of natural language, and numerous other application areas.Discriminant analysis deals with a special type of multivariate analysis problem. Its greatest attribute is that based on the known historical data information of some samples in every population, it evolves classified laws of objective things, creates discriminant functions and decision rules, and then allocate the units not readily known to belong to a particular population into one of those populations according to the evolved discriminant functions and decision rules. But in order to distinct the populations from each other, significant testing should be done first. Besides, none of the decision rules are expected to be perfect, there will always be certain probabilities of misclassifications. So, finally the misclassification rates must be estimated.This paper introduces the fuzzy theory into the discriminant analysis, and discusses the problem of discriminant analysis with two populations in a three- step process. In order to invoke the usual method of real-valued data, we consider theδ-cut(orδ-levle) set of fuzzy data because the end points of theδ-cut set (closed interval) of a fuzzy number are real numbers.Firstly, this paper tests whether the fuzzy means of the two populations(G_v :Σ), v = 1,2) have significant differences. Because the observed data are all fuzzy, null or alternative hypotheses shouldn't be accepted at a significance level in a precise sense as usual. In this case, the notion of degree of confidence which is connected with the result is used. i.e.If the common covariance matrixΣis known, thenIf the common covariance matrixΣis unknown, thenLetIn this paper, let l(K_{H|^<sub>0,δ} = 1 -δ_H0^min denote the belief degree under which H|^<sub>0 is accepted at the level of significance a; And let l(K_{H|^<sub>1,δ}) = 1 -δ_H|^<sub>1^min denote the belief degree under which H|^<sub>1 is accepted at the level of significanceα. Moreover, a real numberγ∈[0,1] would be given in advance as a matter of experience. If l₀>γ) thenμ|^<sub>1 andμ|^<sub>2 have significant differences at the significance levelα. Now, it is reasonable to classify A|^. If l₀≤γ, thenμ|^<sub>1 andμ|^<sub>2 don't have significant differences. Now, it does not make sense to classify A|^. Secondly, this paper does discriminant analysis. When we distinct A| is from G₁ or G₂, the conception of fuzzy Mahalanobis distance is introduced. And we will obtaining the following fuzzy discriminant functions;If the common covariance matrixΣis known, thenIf the common covariance matrixΣis unknown, thenThe classification rule in this paper is(i) When coreW| < 0, there exists a minimumδ₁∈[0,1], such that D|^<sub>1 (?) D|^<sub>2. So we can say that the simple A| is assigned to the group G₂ at the levelδ₁.(ii) When coreW| > 0, there exists a minimumδ₂∈[0,1], such that D|^<sub>1 < D|^<sub>2. So we can say that the simple A| is assigned to the group G₁ at the levelδ₂.(iii) When coreW| = 0, coreA| is coincident with care. Since the probability of misclassification would be too large, it's difficult to assign A| into one of the two groups. Thus other rules need to be applied to do further classification.In (i) (or (ii)) if D_<sup><sub>1 (?) D|^<sub>2 (or D|^<sub>1 < D|^<sub>2) for allδ∈[0,1], we can positively state that the simple A| is classified to the group G₂ (or G₁). Ifδ₁ (orδ₂) is too close to 1, the probabilities of misclassifications would be both too large whichever A is classified into so that it is meaningless.Finally, this paper estimates the risk of misclassifications and obtains fuzzyprobabilities of misclassifications: where And gettingandThereby the purpose that is to get fuzzy probabilities of misclassifications from the crisp normal distribution table is brought about.Besides, the method is illustrate in one example.