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The screening of data sets is essential to modern technology. The use of classical group testing to isolate objects that are individually positive has become the standard experimental procedure in many applied settings. Work is just beginning in applying group testing techniques to the identification of subsets of objects that are collectively positive. This paper addresses the development of probabilistic group testing methods that lead to the identification of positive combinations of objects with specific applications to data mining. Group testing for complexesThe screening of data sets is essential to modern technology. Whenever the objective is to find "positive objects" in a data set, a test indicating whether at least one positive is in a specific part of the data set can greatly facilitate their isolation. Such tests are called binary group tests and the general mathematical method behind the identification of the positives using such tests is known as classical group testing [4]. The use of classical group testing to isolate objects that are individually positive has become the standard experimental procedure in many applied settings [2][3][4]7]. This paper addresses the development of efficient group testing methods that lead to the identification of positive combinations of objects. Computationally feasible methods that can identify positive combinations of objects have applications to medical genetics, biomolecular computing, computer security, software testing, data mining, coding theory, and marketing. In this paper, we specifically address possible applications to data mining [1,[8][9][10].Throughout this paper, all simple lower case italic variables are non-negative integers. Given set S, |S| denotes its cardinality. Let [t] = {1, 2, . . . , t} represent a finite set with t elements. Suppose we have an unknown collection of subsets . . . , S , . . . , S d } of [t], where an antichain. We refer to as the set of positive complexes and we simply call a subset in a complex. In thegroup testing for complexes (GTC) problem, (or a portion of ) must be identified by performing certain 0,1 tests on subsets (groups) from [t] [6,[11][12][13][14][15]. These tests are called complex group tests. A subset is said to be positive if it completely contains a complex; otherwise the subset is said to be negative. In short, a subset P ⊂ [t] is positive if and only if E-mail addresses: macula@geneseo.edu (A.J. Macula), leonard.popyack@rl.af.mil (L.J. Popyack).

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System diagnostics facilitates reliability enhancement and condition-based maintenance of technical systems. Diagnostic problems can be formulated and solved only on the basis of mathematical models reflecting the complex stochastic nature of the failure development process. This process is affected by numerous continuous factors, but its outcome constitutes a random event. These specifics limit the applications of traditional regression models. The concept of a "continuous process with discrete-event output" is introduced, and cluster analysis is employed as a modeling approach facilitating the solution of various diagnostic problems. Important aspects of cluster analysis are the definition of the informativity criterion, selection of "informative subspaces," definition of separating rules, and finally, formulation of a cluster model are presented. Various enhancements of cluster analysis are proposed. A cluster model is utilized for the definition of so-called "probabilistic space" that, in conjunction with Bayes' technique, facilitates the solution of failure prediction problems. Application of cluster models for failure analysis and prediction is illustrated by numerical examples.

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