Stepwise possibilistic c-regressions

In 1993, Hathaway and Bezdek combined switching regressions with fuzzy c-means (FCM) to create fuzzy c-regressions (FCR). The FCR algorithm had been widely studied and applied in various areas. However, membership of the FCR does not always correspond to the degree of belonging and it can be inaccurate in a noisy environment. Krishnapuram and Keller (1993) proposed a possibilistic c-means (PCM) whereby membership gives a much better explanation of the degree of belonging and it is more robust to noise and outliers than the FCM. Therefore, we incorporate possibilistic clustering into switching regression models and term it possibilistic c-regressions (PCR). Although a PCR ameliorates the problem of outliers and noisy points, a PCR still depends heavily on initial values. In this paper, we propose a schema for a nested stepwise procedure for a PCR, called a stepwise PCR (SPCR) method, which repeats the PCR on a series of nested subsets using the clustering results of the previous subset as good initial values for the PCR for the succeeding subset. When the smallest subset, D1, is determined at the location where the c regression models separate sufficiently, the number of clusters and the cluster centers in D1 are determined using a modified mountain method, so data in D1 is properly partitioned, which provides good initial values for the following PCR. The proposed SPCR is unsupervised, does not require initialization and is unaffected by noise and outliers. Several experiments and real examples demonstrate the superiority and effectiveness of the proposed SPCR method.