QUANTIFYING MONOTONE CORRELATION IN VISUAL PATTERN ANALYSIS WITH MATHEMATICAL STATISTICS

Abstract [English]

A two-way frequency table that categorizes all corporate bonds rated by Standard & Poor's and Moody's, or the well-known father-son social mobility data matrix that is frequently cited as an example of a Markov chain in human resources, are two examples to consider. Given that bond ratings and social class have a similar natural ordering, both are instances of contingency tables with rows and columns that represent ordinal categorical variables. The degree of association between such row and column variables is measured and quantified in this study. When it comes to cardinal variables, correlation provides a clear indicator of linkage that doesn't require category scaling. A measure of connection could be calculated using sup correlation, a scheme similar to eigen analysis, if the order relationships involved do not need to be respected. Methods for ordinal categorical data, like Spearman's or Kendall's tau, assign numerical values to the categories. We introduce a few novel methods for assessing the correlation between ordinal variables. The idea of monotone correlation is used to develop four new statistical measures of monotone relationships. The particular circumstance that produced the data determines whether each of these metrics is appropriate. An iterative process is the only way to obtain these metrics of association. These metrics are evaluated and the corresponding monotone scalings are obtained using a nonlinear optimization approach.