 # The Concept of Good Classification (Diagnostic) Test

Xenia Naidenova (Military Medical Academy, Russia)
DOI: 10.4018/978-1-60566-810-9.ch007
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## Abstract

In this chapter, the definition of good diagnostic test and the characterization of good tests are introduced and the concepts of good maximally redundant and good irredundant tests are given. The algorithms for inferring all kinds of good diagnostic tests are described in detail.
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## Definition Of Good Diagnostic Test

In chapter 6, we considered the set L(I(T)) of partitions produced by closing atomic partitions of I(T) with the use of operations + and * on partitions. This set is the algebraic lattice with constants over U. Now we consider the table of examples T(UК) with partition interpretation I(TK) over U’= UК and the algebraic lattice L(I(U’)) with constants over U’, where K is a given goal attribute.

Let XU is a test for a goal attribute К, and partitions P(X), P(K) are the interpretations of X and K, respectively.

Denote by Q(K) the set of all diagnostic tests for K: Q(K) ={X: XU: P(X) ≤ P(K))}.

Sub-lattice LK(I(U’)) of L(I(U’)): LK(I(U’)) = {Р: РP(K)} is the principal ideal generated by P(K) in L(I(U’)) (see, please, definition 6.3 of principal ideal in Chapter 6).

For definition of good test, we use partition dependency: P(X) ⊆ P(К) (ХКX*K = XX + K = K).

If P(X) = P(K), then X is the ideal approximation of classification K.

Of all tests X, XU, P(X) ⊂ P(K), the good test will be Х such that P(X) is the closest to P(K) element of LK(I(U’)), i. e., for all P(Y), YU condition (P(X) ⊆ P(Y) ⊆ P(K)) implies P(X) = P(Y). Thus, we come to the following definition of a good diagnostic test.

Definition 7.1. A collection XU is a good test or a good approximation of К of T if the following conditions are satisfied

• а) XQ(K);

• b)

there does not exist a collection of attributes Z, ZU, XZ such that ZQ(K) and P(X) < P(Z) ≤ P(K).

We introduce the concept of the best diagnostic test as follows.

Definition 7.2. A good test X, XU is the best one for a given classification K of T if the number of classes in partition P(X) is the smallest for all tests of Q(K).

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