The Concept of Good Classification (Diagnostic) Test

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(U∪К) with partition interpretation I(T∪K) over U’= U∪К and the algebraic lattice L(I(U’)) with constants over U’, where K is a given goal attribute.

Let X ⊆ U 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: X ⊆ U: P(X) ≤ P(K))}.

Sub-lattice LK(I(U’)) of L(I(U’)): LK(I(U’)) = {Р: Р ≤ P(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) (Х ≤ К ≡ X*K = X ≡ X + K = K).

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

Of all tests X, X ⊆ U, 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), Y ⊆ U 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 X ⊆ U is a good test or a good approximation of К of T if the following conditions are satisfied

• а) X ∈ Q(K);

  • b)

    there does not exist a collection of attributes Z, Z ⊆ U, X ≠ Z such that Z ∈ Q(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, X ⊆ U 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).