البحث الأول

Predicting  Multivariate  Responses  in  Multiple  Linear Regression: Case of One Different Independent variable.

 

 El-Houssainy A. Rady*, Sayad M.El Sayad*,MohamedI.El-Hor**


Abstract

 

While studying the problem of predicting several response variables from the same set of independent variables, the use of correlations between the response variables to improve predictive accuracy is considered and is compared with the usual procedure of doing individual regressions of each response variable on the common set of predictor variables. Breiman and Friedman (1997) introduced a new procedure called the curds and whey method. Its use can substantially reduce prediction errors when there are correlations between responses while maintaining accuracy even if the responses are uncorrelated. They applied this procedure when each response  variables depend on all the same set of explanatory variables. In this paper we will apply this method when each response variables depend on one different independent variable.

 

 

Keywords: CURDS AND WHEY METHOD; CANONICAL REGRESSION; CROSS-VALIDATION.

 

1. Introduction

 

The idea of predicting several quantities using a common set of predictor variables have so many applications and have been studied by so many authors(van der Merwe, A. and Zidek, J. V. ,1980,1989; Breiman and Friedman ,1997 ;S. Srivastavaand Tumulesh K. S. Solanky ,2003). A major paper by Breiman and Friedman (1997) is considered on of the basic papers in this  area concentrating on taking the advantage of correlations between response variables to improve predictive accuracy compared with the usual procedures used in that matter.

(5.2) The conclusion

 

The result of the simulation study are summarized by the respective means of the performance measure values (11) and (12) for each method over the 10 replications for each situation. Figure (1) and Figure (2) display box plots of the mean values of equations (11) and (12) respectively over all the 120 situations covered by the simulation study, i.e. each box plot summarizes the distribution of 120 mean values; and summarizes the distribution of the average overall mean-square error ratio and show comparable performance the Curds and Whey method of one different independent variable in the three cases of the (signal) variance responses, C&W-GCV1 represent the (signal) variance for each response equal to 1.0, C&W-CGV2 represent the average (signal) variance for responses equal to 1.0, C&W-CGV3 represent the sum of (signal) variances for the responses equal to 1.0 based on the two performance measures A (m) (11) and I (m) (12) respectively.


From the simulation results, we find that in the  cases of the (signal) variance for each response equal to 1.0 and the average (signal) variance for responses equal to 1.0, the estimation of the mean square error from the ordinary least square method is less than the estimation of the mean square error from Curds and Whey method in all models that we study in our simulation. But in case the sum of (signal) variances for the responses equal to 1.0, the estimation of the mean square error from the Curds and Whey method is less than the estimation of the mean square error from ordinary least square method in 95% from the all models.


REFERENCES


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البريد الالكتروني [email protected]

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