Hard and fuzzy c-regression models for data with tolerance in independent and dependent variables

C-regression models are known as very useful tools in many fields. Since now, many trials to construct c-regression models for data with uncertainty in independent and dependent variables have been done. However, there are few c-regression models for data with uncertainty in independent variables in comparison with dependent variables now. The reason is as follows. The models are constructed using optimal solutions which is derived by solving an optimization problem “analytically”. The problem for data with uncertainty in dependent variables can be easily solved but it is very difficult to solve the problem for data with uncertainty in independent variables “analytically”. Therefore, most of the models for data with uncertainty in independent variables are constructed in which the solutions are calculated “numerically”. By the way, we have proposed “tolerance” of a convenient tool to handle data with uncertainty and applied it to some of clustering algorithms. This concept of tolerance is very useful. The reason is that we can handle data with uncertainty in the framework of optimization to use the concept, without introducing some particular measure between intervals. Especially when we handle the data with missing values of its attributes in the framework of optimization like as fuzzy c-means clustering, this tool is effective. Besides, we think that the tolerance is also available when we consider to construct a regression model for data with uncertainty in independent and dependent variables. In this paper, we first derive the optimal solutions for c-regression models for data with uncertainty in independent and dependent variables “analytically” by using the concept of tolerance. Second, we construct hard and fuzzy c-regression models for data with tolerance in independent and dependent variables. Moreover, we estimate effectiveness of the algorithms through some numerical examples.