Detection of higher order covariate relations embedded in multi-dimensional data

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This research aims to develop new methods applicable to detect monotone and locally monotone higher order covariate relations embedded in multidimensional symbolic data. We obtained the following three major results.(1) Detection of locally monotonic chain structures embedded in multidimensional symbolic data : We developed a method that is able to detect higher order covariate relations. By this method we can detect higher order polynomial structures, sinusoidal structures, and others in multidimensional symbolic data table without functional identification process.(2) A generalized correlation coefficient : By applying a well known correlation coefficient to local regions associated with each data sample and by aggregating the local correlations, we have a generalized correlation coefficient that is able to evaluate higher order covariate relations between two feature variables.(3) The characterization of monotone structures by the nesting property and its application to symbolic data analysis : We frequently use histogram representations in order to reduce given huge data tables. By the virtue of monotone property of the cumulative distribution functions, we developed the quantile method to the principal component analysis for histogram valued data tables. The quantile method may also be able to treat other research problems in symbolic data analysis.