Fuzzy Blocking Regression Models
Regression analysis is a well known and a widely used technique in multivariate data analysis. The efficiency of it is extensively recognized. Recently, several proposed regression models have exploited the spatial classification structure of data. The pu
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Abstract. Regression analysis is a well known and a widely used technique in multivariate data analysis. The efficiency of it is extensively recognized. Recently, several proposed regression models have exploited the spatial classification structure of data. The purpose of this inclusion of the spatial classification structure is to set a heterogeneous data structure to homogeneous structure in order to adjust the heterogeneous data structure to a single regression model. One such method is a blocking regression model. However, the ordinal blocking regression model can not reflect the complex classification structure satisfactorily. Therefore, the fuzzy blocking regression models are offered to represent the classification structure by using fuzzy clustering methods. This chapter’s focus is on the methods of the fuzzy clustering based blocking regression models. They are extensions of the conventional blocking regression model.
1 Introduction Fuzzy clustering techniques are widely used in a large number of areas, since this clustering can obtain a result describing real-world phenomenon that is substantiated by robust solutions and low calculation cost when compared with hard clustering. Hard clustering means classifying the given observation into exclusive clusters. This way, we can discriminate clearly whether an object belongs to a cluster or not. However, such a partition is not sufficient to represent many real situations. So, a fuzzy clustering method is offered to allow clusters to have uncertainty boundaries, this method allows one object to belong to some overlapping clusters with some grades [2], [9], [13]. By exploiting such advantages of fuzzy clustering, several hybrid methods have been proposed in the area of multivariate data analysis [14]. In particular, the regression methods use the result of fuzzy clustering as spatial weights of data structure and represent the nonlinearity of data by using the weights. Fuzzy c-regression model [8], geographically weighted regression model [3], and fuzzy cluster loading models [14] are typical examples. These models use weights which show the degree of contribution of objects to clusters and represent the distribution of the weights by using fuzzy membership function or probabilistic density function. The estimates of the regression coefficients are obtained by the weighted least squares method. That is, these models are implemented by L.C. Jain et al. (Eds.): Comp. Intel. Para.: Innov. Applications, SCI 137, pp. 195–217, 2008. c Springer-Verlag Berlin Heidelberg 2008 springerlink.com
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multiplying the weights to the data and attempts to set a heterogeneous data structure to homogeneous structure. On the other hand, a regression model which attempts to solve this problem by adding different intercepts according to different clusters (or blocks) [4]. In this model, identification of clusters (or blocks) are represented by dummy variables and the values of the dummy variables are given in advance by using external information to identify the blocks. Consider
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