This chapter summarizes some methods of inferring approximate diagnostic tests. Considering the sets of approximately minimal diagnostic tests as “characteristic portraits” of object classes we have developed a model of commonsense reasoning by analogy. The system DEFINE of analogical inference with some results of its application is described. Mining approximate functional, implicative dependencies and association rules is based on the same criteria and on applying the same algorithm realized in the Diagnostic Test Machine described shortly in this chapter. Some results of inferring “crisp” and approximate tests with the use of Diagnostic Test Machine are give in Appendix to this chapter.
TopIntroduction
Real-world problems require very often the necessity of approximate reasoning. Several theories have been advanced to model human rough reasoning and to deal with the uncertainty and imprecision of data. The most known among them are the fuzzy set theory (Zade, 1965), the probability theory (Pearl, 1988), and the rough set theory (Pawlak, 1982). The fuzzy set theory is used more than the others. It serves as a “transformer” of numerical scales into symbolic scales the values of which are usually linguistic terms or concepts. Note that the interpretation (meaning) of linguistic terms depends on the context of reasoning. Linguistic terms are relativistic: a man of average height will be high in a “Lilliputians” country and low in a country of giants. The fuzzy approach serves as an “interface between numerical and conceptual scales” (Dubois et al., 2006). So, if we obtained a fuzzy rules or scales as a result of learning, then we have to use them for pattern recognition or decision making in the same context. By this reason, diagnostic psychological tests are reconstructed in time.
Machine learning is inherently inductive process and, naturally, even “crisp” rules have a certain degree of confidence. We shall distinguish some kinds of uncertainty:
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Rule (dependency) is not satisfied in a given dataset, but it could be satisfied if to eliminate some examples from this dataset. It is said that this rule has the degree of confidence and support calculated with respect to the dataset;
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Rule (dependency) is satisfied in a given dataset, but the dataset imperfect or insufficient for reliable conclusions. In this case, the calculation of statistical significance for the rule and its confidence limits has to be estimated (if the dataset has probabilistic nature). If it is not the case, then it is necessary to check the steadiness of the rule under changing the conditions of observations or under appearing new examples;
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The task “to find all the rules satisfying a given restriction” is replaced by the task “to find only a part of rules satisfying a given restriction”;
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The request “to find minimal (maximal) or optimal rule with respect to a given criterion” is replaced by the request “to find quasi–minimal (maximal) or quasi – optimal rule, i.e. rule approximating a given criterion.
In this chapter, we concentrate on some fast heuristics for inferring approximately minimal diagnostic tests and their application in the tasks of forest aerial image interpretation (Naidenova, 1981; Naidenova, & Polegaeva, 1983; 1985).
TopSome Fast Heuristics For Inferring Approximately Minimal Diagnostic Tests (Amdts)
We begin with the definitions and terminology.
Let {A1, ….An } be a set of multi-valued attributes. Combinations of attributes’ values are object descriptions. A set of object descriptions (objects, for short) is denoted by Q. Let P(Q) = {Q1, Qk} be a partition of the set Q unto k disjoint classes.
The problem is to construct for all pairs of sets Qi, Qj ∈ P(Q), i, j = 1, ….., k, a collection (of hopefully small size) of diagnostic tests Tij distinguishing the examples of Qi (viewed as the set of positive examples) from the examples of Qj (viewed as the set of negative examples) (i.e., these tests correctly classify all the examples Qi and reject all the examples of Qj).