Asymptotic and bootstrap tests for subspace dimension

TY - JOUR

T1 - Asymptotic and bootstrap tests for subspace dimension

AU - Nordhausen, Klaus

AU - Oja, Hannu

AU - Tyler, David E.

N1 - Funding Information: David E. Tyler’s research was partially supported by the National Science Foundation, United States Grant No. DMS-1407751 . Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation. The authors wish to thank the editors and reviewers for their valuable comments. Publisher Copyright: © 2021 The Author(s)

PY - 2022/3

Y1 - 2022/3

N2 - Many linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices. The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are illustrated in real data examples.

AB - Many linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices. The eigen-decomposition of one scatter matrix with respect to another is then often used to determine the dimension of the signal subspace and to separate signal and noise parts of the data. Three popular dimension reduction methods, namely principal component analysis (PCA), fourth order blind identification (FOBI) and sliced inverse regression (SIR) are considered in detail and the first two moments of subsets of the eigenvalues are used to test for the dimension of the signal space. The limiting null distributions of the test statistics are discussed and novel bootstrap strategies are suggested for the small sample cases. In all three cases, consistent test-based estimates of the signal subspace dimension are introduced as well. The asymptotic and bootstrap tests are illustrated in real data examples.

KW - Independent component analysis

KW - Order determination

KW - Principal component analysis

KW - Sliced inverse regression

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U2 - https://doi.org/10.1016/j.jmva.2021.104830

DO - https://doi.org/10.1016/j.jmva.2021.104830

M3 - Article

SN - 0047-259X

VL - 188

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

M1 - 104830

ER -