On The Mathematical Model Of Correspondence Analysis

In this paper, we discussed the model of correspondence analysis(or dual scaling) based on the criterion of internal consistency and by means of matrix analysis in detail and clearly. The weighting coefficients of vaviables and individuals can be obtained from the following two dual equations (x'f^-1x-(1/t)gg')a=λGa(x'G^-1x-(1/t)ff')b=λFbEspecially, we indicated the connection between the correspondence analysis and variance analysis based on the relation between the correlation ratio and the F-statistic. Based on the Nishisato's book,we discussed 4 equivalent models of corrspondence analysis and indicate the relationship between it and regression analysis, canonical correlation analysis and so on. Especially, we proved the equivalance between our model and Banzécri's model which is too complex to be understanded. The connection between correspondence analysis and principal component analysis was studied in the paper. We indicated that these two methods can be unified under homogeneity criterion. Refering to the paper written by Xia and Yang, a new method, stepwise correspondence analysis, was proposed. When the multivariate analysis is applied to solving practical problem, people usually attempt to collect the variables as many as possible, whereas, because of the correlation between variables, some variables can be replaced by others. The superfluous variables usually cause some trouble to our computation, so they should be eliminated. An important problem in multivariate analysis is about variable selection. As to the questions with dependent variable, using stepwise regression or stepwise discrimination, we can eliminate some variables, or equivalently select only a few important variables easily from many original variables. In view of the question without dependent variable, in this paper, we will introduce the stepwise correspondence analysis. Using this method, we can also select a few important variables from many original variables based on the configuration of all individuals. The criteria is based upon the following viewpoint: Admit that n individuals can be described quite well by original m variables. The scores V give the configuration of n individuals in a r dimensional Euclidean space, and describe the relationship between n individuals. Our aim is to eliminate some variables, or equivalently select l(