Regression Analysis Based On Fuzzy Points Data

For classical regression analysis, all training data are assumed to be independent identically distributed random samples and are treated uniformly in the construction of regression equation. However, in many real-world applications, the effects of the training data are different. It is often that some training data may be more important than others, so the different training data should make different contributions to the fitting curve. We require that important training data can make much contributions to the fitting curve. In this paper, we apply a confidence weight to each training data and reformulate the classical linear regression methods—the least squares estimator, ridge estimator, principal component estimator and so on. Meanwhile, we analyze its statistical characteristics which similar to the properties of parameters estimator of the classical linear regression. Four numerical examples are used to demonstrate our proposed method. The bigger the circles are, the more important they are to the fitting curve, so they can make much contributions to the fitting curve. The fitting curve is apparently closer to the data which have bigger weight. When all fuzzy weights degenerate to one point, our proposed method becomes the classical linear regression method. Therefore, the classical linear regression is special form of our method.In addition, we discuss discriminant analysis which the linear regression model based on fuzzy points data are used and Logistic regression model. For the problem of discriminant analysis, each class has different effect on the researchful problem, some classes data may be more important than others. We apply a confidence weight to each class data such that the important data must be classified correctly and may not care about some classes data whether or not they are misclassified. Three numerical examples further show the properties of our method.