One characteristic of computational statistics is the processing of enormous amounts of data. It is now possible to analyze large amounts of highdimensional data through the use of high-performance contemporary computers. In general, however, several problems occur when the number of dimensions becomes high. The first problem is an explosion in execution time. For example, the number of combinations of subsets taken from p variables is 2p; when p exceeds 20, calculation becomes difficult pointing terms of computation time. When p exceeds 25, calculation becomes an impossible no matter what type of computer is used. This is a fundamental situation that arises in the selection of explanatory variables during regression analysis. The second problem is the sheer cost of surveys or experiments. When questionnaire surveys are conducted, burden is placed on the respondent because there are many questions. And since there are few inspection items to a patient, there are few the burdens on the body or on cost. The third problem is the essential restriction of methods. When the number of explanatory variables is greater than the data size, most methods are incapable of directly dealing with the data; microarray data are typical examples of this type of data. For these reasons, methods for dimension reduction without loss of statistical information are important techniques for data analysis. In this chapter, we will explain linear and nonlinear methods for dimension reduction; linear methods reduce dimension through the use of linear combinations of variables, and nonlinear methods do so with nonlinear functions of variables. We will also discuss the reduction of explanatory variables in regression analysis. Explanatory variables can be reduced with several linear combinations of explanatory variables.
