A fuzzy classification approach to piecewise regression models

Piecewise regression models have been widely applied to various areas in which break-point detection is important and break locations in piecewise regression models are essential to know when and how the pattern of data structure changes. One major difficulty with likelihood approaches to break-point location for regression models is the non-smoothness of the likelihood function with respect to break-points regarded as parameters. Although several non-standard methods have been proposed to overcome this problem, their applications may be limited due to additional required techniques or restrictions. Other shortcomings of existing methods are such as only a single break-point allowed, an unchanged variance or model continuity demanded, and heavy calculations involved. Locating break-points is similar to classifying data into groups of similar items. Fuzzy clustering is powerful and widely applied to substantial areas but less used in break-point detection. In this paper, we propose a new fuzzy classification maximum likelihood break-point (FCM-BP) method to simultaneously estimate break-points and regression parameters free of non-differentiability. We transform a piecewise regression model into a fuzzy class model and then employ a fuzzy classification maximum likelihood procedure to derive estimates. The non-differentiability problem is bypassed through mixture modelling with break-points regarded as class variables. The proposed FCM-BP is robust to initial values and it can detect the location and the magnitude of changes in mean, in variance, and in model coefficients simultaneously, allowing multiple break-points in all of these. FCM-BP is feasible for models with continuous or discontinuous break-points, whereas several of the well-known statistical methods are available for continuous models only. The proposed method is demonstrated and compared with other existing methods through extensive experiments using numerical and real datasets. Experimental results demonstrate the effectiveness and the superiority of FCM-BP, and real data applications reflect its broad practicability.