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Abstract: In this study, we introduce the concept of collaborative fuzzy clustering—a conceptual and algorithmic machinery for the collective discovery of a common structure (relationships) within a finite family of data residing at individual data sites. There are two fundamental features of the proposed optimization environment. First, given existing constraints which prevent individual sites from exchanging detailed numeric data, any communication has to be realized at the level of information granules. The specificity of these granules impacts the effectiveness of ensuing collaborative activities. Second, the fuzzy clustering realized at the level of the individual data site has to constructively consider the findings communicated by other sites and act upon them while running the optimization confined to the particular data site. Adhering to these two general guidelines, we develop a comprehensive optimization scheme and discuss its two-phase character in which the communication phase of the granular findings intertwines with the local optimization being realized at the level of the individual site and exploits the evidence collected from other sites. The proposed augmented form of the objective function is essential in the navigation of the overall optimization that has to be completed on a basis of the data and available information granules. The intensity of collaboration is optimized by choosing a suitable tradeoff between the two components of the objective function. The objective function based clustering used here concerns the well-known Fuzzy C-Means (FCM) algorithm. Experimental studies presented include some synthetic data, selected data sets coming from the machine learning repository and the weather data coming from Environment Canada. [Copyright &y& Elsevier]

In this paper, we propose an iterative algorithm for multiple regression with fuzzy variables. While using the standard least-squares criterion as a performance index, we pose the regression problem as a gradient-descent optimisation. The separation of the evaluation of the gradient and the update of the regression variables makes it possible to avoid undue complication of analytical formulae for multiple regression with fuzzy data. The origins of fuzzy input data are traced back to the fundamental concept of information granulation and an example FCM-based granulation method is proposed and illustrated by some numerical examples. The proposed multiple regression algorithm is applied to one-, three-and nine-dimensional synthetic data sets as well as the 13-dimensional Boston Housing dataset from the machine learning repository. The algorithm''s performance is illustrated by the corresponding plots of convergence of regression parameters and the values of the prediction error of the resulting regression model. General comments on the numerical complexity of the proposed algorithm are also provided. [Copyright &y& Elsevier]