Ill-conditioned Matrix and Its Improvement in Regression
Mo Hui · ACTA AGRONOMICA SINICA · 2006
In regression analysis,the information matrix X′X is an important factor because of b=(X′X)~(-1)X′Y.If the determinant value of X′X,det(X′X),is close to zero,the inverse of the X′XX′X)~(-1),will extremely inflate,the error mean square for regression coefficient will largely increase,and in consequence the regression fitting will be poor robustness and low precision.Thus the matrix X′X of det(X′X)≈0 is called matrix.In this paper the determinant value of correlative matrix R of X variables,det(R), is used as a synthetic index for ill-conditioning,i.e.if the det(R) lies in the intervals[0.01,0.01] and[0.0001,0.0001] but nonzero,the corresponding matrix X′X can be regarded as ill-conditioned and seriously ill-conditioned,respectively.The ill-conditioned X′X results from the linear dependency among columns in matrix.Three diagnostic criteria,including linear correlation coefficient,multiple determination coefficient and condition index,can measure the degree of the column dependency.In order to improve or eliminate the ill-conditioning of X′X,four methods,i.e.(1) to reduce the original regression model,(2) to collect the new data,(3) to add the restrictive condition for regression coefficients and(4) to adopt the non-customary regression procedure such as the ridge regression and the generalized inverse[WTHX]M~- regression,are suggested.The importance of diagnosing the ill-conditioning and the evaluation for improved ill-conditioning are also discussed briefly.