Expectation-based and Quantile-based Probabilistic Support Vector Machine Classification for Histogram-Valued Data

A histogram-valued random variable represents its value by a list of pairs of bins and their corresponding probabilities or relative frequencies. This type of data is a part of the symbolic data. There are many cases such as colors in image learning where histogram-valued data are naturally found. This study focuses on classification of the histogram-valued data by extending two approaches of support vector machine (SVM), namely, the expected-based and quantile-based probabilistic SVM on histogram-valued data. In both approaches, the cases of linear and nonlinear problems as well as the least-square classification are discussed. In addition, the extension to multi-class classification is also discussed. To compare the performance of the proposed procedures a simulation study has been done based on some generated data sets. The data are generated from various distributions with various parameters to represent different cases of classification, including binary and multi-class classification. Further, the methods are applied on two different real data sets. From the results, it can be concluded that our proposed methods perform well on wide range of classification problems.