USDA FONeST service RESEARCH NOTE NE-236 



1977 



^^rthea8tern 

 Txperiment lit 



FOREST SERVICE, U.S. DEPT. OF AGRICULTURE, 6816 MARKET STREET, UPPER DARBY, PA. 19082 



5 RIDGE: A COMPUTER PROGRAM FOR 



J CALCULATING RIDGE REGRESSION ESTIMATES 



3 by DONALD E. HILT 



^ Research Forester 



S and 

 u DONALD W. SEEGRIST 



g Mathematical Statistician 



USDA Forest Service 

 l^ortheastern Forest Experiment Station 



.4 6.s//7/(:-^— Least-squares coefficients for multiple-regression models may 

 be unstable when the independent variables are highly correlated. Ridge 

 regression is a biased estimation procedure that produces stable estimates 

 of the coefficients. Ridge regi-ession is discussed, and a computer program 

 for calculating the ridge coefficients is presented. 



KEYWORDS: Regression, computer progi'am, correlated variables. 



Multiple-regression models are widely used in 

 forestry. In some studies, the independent 

 variables are highly correlated. In this case the 

 least-squares coefficients may be too large in 

 absolute value, and the signs may reverse with 

 small changes in the data. With highly cor- 

 related data, one should consider estimation 

 methods that reduce the effects of the correla- 

 tion and produce stable regression coefficients 

 (M(trqK(irdf and Snee 1975). 



The purpose of this note is to discuss ridge- 

 regression methods and present a computer 

 program for ridge regression. A list of 

 references is also given. 



Ridge Regression 



The observational equations for a multiple- 

 regression model can be written as 



in which Y is the nxl vector of observations, X is 

 the nxp matrix of independent variables, is 

 a pxl vector of parameters unknown, and e is 

 the nxl vector of errors. It is assumed tfiat 

 E(^) = 0,and E( ^^'^^ )= ^ % 

 The least-squares estimate of id is 



£=(X'X)-'X'Y. [1] 



V 



