Determinantal divisors for the degree of independence of a contingency matrix

A contingency table summarizes the conditional frequencies of two attributes and shows how these two attributes are dependent on each other. Thus, this table is a fundamental tool for pattern discovery with conditional probabilities, such as rule discovery. This paper proposes formal analysis of a contingency table based on linear algebra. The analysis shows that the rank of a contingency table plays a very important role in evaluating the degree of statistical independence. Especially, from the viewpoint of the degree of independence, we have three classes: (complete) dependence, partial dependence, and independence. Also, determinantal divisors provides information on the degree of dependencies between the matrix of the whole elements and its submatrices.