Meaning of Marginal Distributions and Residuals in a Contingency Table

Abstract

A contingency table summarizes the conditional frequencies of two attributes and shows how these two attributes are dependent on each other with the information on a partition of universe generated by these attributes. % Thus, this table can be viewed as a relation between two attributes with respect to information granularity. % This paper focuses on decomposition of a contingency matrix by using a matrix of expected values based on marginal distribution (expected matrix). Especially when the rank of a matrix is full, say, $r$, the difference between a original matrix and the expected matrix will become $r-1$ at most. Moreover, the sum of rows or columns will become zero, which means that the information of one rank correponds to information on the frequency of a contingency matrix.