Local influence in principal component analysis: relationship between the local influence and influence function approaches revisited

Abstract

Influence analysis is developed on the basis of Cook's local influence in its original form, i.e., using the likelihood displacement as the criterion function for studying jointly as well as singly influential observations in principal component analysis (PCA). It is found that the derived influential directions are equivalent to the standardized vectors of the principal component scores obtained by PCA with metric V − of the influence functions of PCA parameters θ ̄ ̂ , where V is a consistent estimate for the asymptotic covariance matrix of θ ̄ ̂ and superscript ( −) indicates a g-inverse. It is a special case of the equivalence discussed by Tanaka and Zhang (Comp. Statist. Data Anal. 32 (1999) 197) in general statistical modeling. This special case broadens the range where the general theory holds. A numerical example is given to illustrate the performance of the proposed influence analysis and to compare the results with those obtained in previous studies.