Change-point detection in multinomial data with a large number of categories

Abstract

We consider a sequence of multinomial data for which the probabilities associated with the categories are subject to abrupt changes of unknown magnitudes at unknown locations. When the number of categories is comparable to or even larger than the number of subjects allocated to these categories, conventional methods such as the classical Pearson's chi-squared test and the deviance test may not work well. Motivated by high-dimensional homogeneity tests, we propose a novel change-point detection procedure that allows the number of categories to tend to infinity. The null distribution of our test statistic is asymptotically normal and the test performs well with finite samples. The number of change-points is determined by minimizing a penalized objective function based on segmentation, and the locations of the change-points are estimated by minimizing the objective function with the dynamic programming algorithm. Under some mild conditions, the consistency of the estimators of multiple change-points is established. Simulation studies show that the proposed method performs satisfactorily for identifying change-points in terms of power and estimation accuracy, and it is illustrated with an analysis of a real data set.

Graph-based multiple change-point detection

“…Lavielle & Teyssiere (2006) introduced a set of methods based on penalized Gaussian log-likelihood to detect changes in covariance structure. Wang et al (2018) improved Pearson's chi-squared test for multinomial data, and added a penalty term to allow for multiple change-point selection.…”

Structural change detection in ordinal time series

“…Because of the importance of parameter stability, it is necessary to detect the structural change. Studies of structural change detection has been a popular research subject in statistics, see Csörgö and Horváth [ 2 ], Bai and Perron [ 3 ], Lee et al [ 4 ], Perron [ 5 ], Gombay [ 6 ], Wang et al [ 7 ], Chen et al [ 8 ], Baranowski et al [ 9 ], Wang et al [ 10 ], Chen [ 11 ] and Liu et al [ 12 ] for reviews of the field.…”

“…Structural changes detection in categorical data have been considered as well. Höhle [ 13 ] proposed a prospective CUSUM change-point detection procedure to detect a structural change in categorical time series; Wang et al [ 10 ] described a procedure based on high-dimensional homogeneity test to detect and estimate multiple change-point in multinomial data; Plasse and Adams [ 14 ] illustrated a multiple change-point detection method for categorical data streams, which could adaptively monitor the category probabilities. As generalized linear regression models for categorical time series allow for parsimonious modeling and incorporation of random time-dependent covariates, Fokianos and Kedem [ 15 ] suggested the generalized linear model for categorical time series modeling.…”

Sequential change-point detection in a multinomial logistic regression model

“…Hudecová [18] studied the change-point problem within the framework of autoregressive models for binary time series. Wang et al [19] proposed a novel change-point detection procedure motivated by high-dimensional homogeneity tests to estimate the locations of multiple change-points in multinomial data with a large number of categories.…”