The Ridge Estimation And Principal Components Estimation Of Factor Score In Factor Analysis

Factor analysis is a kind of important tool of scientific research, applied widely in the many variables data processing of such fields as industry, agriculture, economy, biology and medicine, etc.. Factor analysis synthesizes to be quantity less factors with intricate relation variables (or samples ), in order to reproduce the inter-reaction between primitive variable and factor , and can also classify the variable according to different factors at the same time . It also is belong to in the multivariate statistical analysis method ,which is a process of dimension reduction .As the last step in the factor analysis, As the factor analysis last the step, the factor scores expresses the public factor to the linear combination of the variable (or sample) . Because in the factor score function the number of equation is smaller than the number of variable, therefore, we can not precisely calculate to the factor score, only can carry on the estimate to it. Utilized the Thomson regression law , the calculation formula of the factor scores is:F=A(X'X)^-1XIn the having type: (X'X)^-1, Multicollinearity relation may come up in the column vector. In this time the determinant | XX'| approximately equals 0, and λp≈0, 1/λp is very big. And causes to the very large deviation of the estimated F value, therefore it is bad to the stability of the observation error. This article studies the Multicollinearity question in the formula of the factor score,proposes the improvement method, and analyses the special function that ridgeestimation and the principal components estimation reacts on the factor scores. In order tosolute the bad stability above mentioned, in general we insert the positive numbersalong the principal diagonal in the morbid state (XX) artificially , thus causesλp slightly big somewhat. The formula of the factor score can be Obtains. Then the definition of the ridge estimate is: F(k) = A1 (X'X + kIp )-1 X 0< k < +∞In the formula, at that time of k = 0, it was the regression estimate of Thomson; and When k→ +∞, then.F(k)→0. Thereupon this article has produced the method of the best parameter k definition. Here some influential means and principle will be introduced:1) ridge trace stable2) make the error of mean square littleDepended on principal components estimation, the Multicollinearity variable in the formula of the factor score can be improved and we can gain a satisfactional result.Using the Matlabe software and the statistical software, we obtain the result that has been revised by the ridge estimation and the principal components estimation. And the stability of the result is better than that depended on the conventional factor score.