Power Functions Of Statistical Criteria Defined By Bans

The basic problem for developers of monitoring systems for technological processes is to exclude the false alarms. False alarms generate the interruption of technological process and lead to the manual analysis of the reasons of the wrong system behavior. In the paper it is offered to use the statistical techniques with probabilities of false alarms equal to zero. This class of statistical decisions is based on concept of bans of probability measures in a finite space. Conditions under which powers of statistical criteria accept value 1 on a finite step are found. These conditions are formulated in terms of supports of probability measures. INTRODUCTION The paper deals with the mathematical model of monitoring of a technological system behavior with finite set of states. Suppose that such monitoring systems solve the task with the help of statistical techniques. In the mathematical models the trajectories of functioning of such system are represented by infinite sequences in which each coordinate accepts value in the finite fixed alphabet. Application of statistical techniques on a set of infinite sequences demands a probability measure P which describes the correct behavior of analyzable system. The wrong system behavior is described by a probability distribution Q. Different wrong behaviors of the technological system can be described by different distributions of probabilities on space of the infinite sequences. However in the elementary case it is possible to assume that distribution of the wrong behavior of technological system is unique and known. In practice the monitoring system of technological process observes initial sections of trajectories and for each step n it tests the hypothesis H0, n that the distribution of the observed section of trajectory is defined by probablity distribution measure Pn which is the projection of measure P on the first n coordinates. The alternative hypothesis H1, n in the elementary case is defined by measure Qn which is projection of measure Q on the first n coordinates. Criteria of testing of hypotheses H0, n against alternatives H1, n allow to make the decision about the wrong behavior of technological system. The basic problem for developers of such monitoring systems is the false alarms appearance when the correct behavior of technological process is perceived as wrong (Axelson, 1999). False alarms generate interruption of technological process, and that the worst, they lead to necessity of the manual analysis of the reasons of the wrong system behavior. For this purpose in the paper it is offered to use the statistical techniques for monitoring with probabilities of false alarms equal to zero. This class of statistical decisions is based on concept of the ban (Grusho and Timonina, 2011; Grusho et al., 2013). The ban of a probability measure in the considered scheme is a vector for which probability of its appearance is equal to 0 in a finite projection of measure. Any statistical criterion for testing H0, n against H1, n is defined by a critical set Sn of vectors of length n. When the observed vector is in Sn then it leads to the acceptance of alternative H1, n. If all vectors in Sn are bans of a measure Pn, say that the criterion is defined by bans of a measure P . Existence and properties of the criteria determined by bans were researched in papers (Grusho and Timonina, 2011; Grusho et al., 2013, 2014). In particular, the behavior of power function of criteria was researched in case of n → ∞. Conditions of consistency of sequence of the statistical criteria determined by bans, i.e. conditions when powers of criteria tend to 1 in case of n→∞ are found. Specialists believed that all properties of power functions for finite n were defined by numerical values of probability distributions P and Q. However in this paper conditions under which power functions of criteria accept value 1 on a finite step are found. These conditions are formulated in terms of supports of probability measures for the main measure P on space of the infinite sequences and for alternatives. Information about supports of measures is known not always. Therefore in the paper we built the constructive check of conditions for existence of criteria with the power function equals to 1 on a finite step N . The article is structured as follows. Section 2 introduces definitions and previous results. In Section 3 the Proceedings 29th European Conference on Modelling and Simulation ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978-0-9932440-0-1 / ISBN: 978-0-9932440-1-8 (CD) main results are proved. In Conclusion we shortly analyze applications of constructed sequences of tests. MATHEMATICAL MODEL. BASIC DEFINITIONS AND PREVIOUS RESULTS Let’s consider mathematical model of some technological process. Let X = {x1, ..., xm} be a finite set, X be a Cartesian product of X, X∞ be a set of all sequences when i-th element belongs to X. Define A be a σ-algebra on X∞, generated by cylindrical sets. A is also Borel σ-algebra in Tychonoff product X∞, where X has a discrete topology (Bourbaki, 1968; Prokhorov and Rozanov, 1993). On (X∞, A) a probability measure P is defined. For any n = 1, 2, ..., assume that probability distribution Pn is a projection of measure P on the first n coordinates of random sequences from X∞. It is clear that for every Bn ⊆ X Pn(Bn) = P (Bn ×X∞). (1) LetDn(P ) be the support of a measure Pn (Prokhorov and Rozanov, 1993): Dn(P ) = {xn ∈ X, Pn(xn) > 0}. Define cylindrical set ∆n(P ) as follows: ∆n(P ) = Dn(P )×X∞. The sequence of cylindrical sets ∆n(P ), n=1,2,..., is not increasing and ∆(P ) = lim n→∞ ∆n(P ) = ∞ ⋂