Abstract
In a recent paper on sensitivity of subspaces spanned by principal components, [5] introduces an influence measure based on second order expansion of the RV and GCD coefficients which are commonly used as measures of similarity between two matrices. The goal of this short note is to point out that the paper of [2] is based on a similar approach. However this work seems unknown to Prendergast since it is missing in his references. A comparison of the two papers is provided together with a brief review of some related works.
Highlights
MSC 2010 subject classifications: Primary 62F35; secondary 62H12
In a recent paper on sensitivity of subspaces spanned by principal components, Prendergast [5] introduces an influence measure based on second order expansion of the RV and GCD coefficients which are commonly used as measures of similarity between two matrices
The goal of this short note is to point out that the paper of Castaño-Tostado and Tanaka [2] is based on a similar approach
Summary
Tanaka [7] considers the projection operator P = VVt onto the column space of V. When F is modified to Fǫ, P is modified to Pǫ = VǫVǫt which can be expressed in a convergent power series: Pǫ. Letting S′ denote the complement of S and yk = vkt (x − μ), Tanaka derives the influence function P(1) of P as: P(1). That P(1) can be used as sensitivity measure. In practice, F is generally unknown and must be estimated by the empirical c.d.f. Fbased on a sample. In his numerical study, following Critchley [3], Tanaka constructs three sample versions of this influence function. The reader is referred to these two papers for further details
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