Factor Analysis and Principal Components

https://www.jstage.jst.go.jp/article/jjscs1988/7/1/7_1_1/_pdf

Recent advance in sensitivity analysis in multivariate statistical methods

. Sliced inverse regression (SIR) is a dimension reduction technique that is both efficient and simple to implement. The procedure itself relies heavily on estimates that are known to be highly non-robust and, as such, the issue of robustness is often raised. This paper looks at the robustness of SIR by deriving and plotting the influence function for a variety of contamination structures. The sample influence function is also considered and used to highlight that common outlier detection and deletion methods may not be entirely useful to SIR. The asymptotic variance of the estimates is also derived for the single index model when the explanatory variable is known to be normally distributed. The asymptotic variance is then compared for varying choices of the number of slices for a simple model example.

Influence functions for sliced inverse regression.

Sliced inverse regression, sliced inverse regression II and sliced average variance estimation are three related dimension-reduction methods that require relatively mild model assumptions. As an approximation for the relative influence of single observations from large samples, the influence function is used to compare the sensitivity of the three methods to particular observational types. The analysis carried out here helps to explain why there is a lack of agreement concerning the preferability of these dimension-reduction procedures in general. An efficient sample version of the influence function is also developed and evaluated.

Implications of influence function analysis for sliced inverse regression and sliced average variance estimation

A 50-year personal journey through time with principal component analysis

In the context of sensitivity analysis in principal component analysis, Tanaka (1988) tackles the problem of the stability of the subspace spanned by dominant principal components. He derives the influence functions related to the projection operator on this subspace and to the spectral decomposition of the covariance or correlation matrix as sensitivity indicators. We suggest here a different approach based on some measures of closeness between the subspaces spanned by the initial eigenvectors and their corresponding version derived from an infinitesimal perturbation of the data distribution.

Jacques Bénasséni

Papers

Sensitivity coefficients for the subspaces spanned by principal components