On the Incommensurability Phenomenon

Donniell E. Fishkind, Cencheng Shen, Youngser Park and Carey E. Priebe ()
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Donniell E. Fishkind: Johns Hopkins University
Cencheng Shen: Johns Hopkins University
Youngser Park: Johns Hopkins University
Carey E. Priebe: Johns Hopkins University

Journal of Classification, 2016, vol. 33, issue 2, No 3, 185-209

Abstract: Abstract Suppose that two large, multi-dimensional data sets are each noisy measurements of the same underlying random process, and principal components analysis is performed separately on the data sets to reduce their dimensionality. In some circumstances it may happen that the two lower-dimensional data sets have an inordinately large Procrustean fitting-error between them. The purpose of this manuscript is to quantify this “incommensurability phenomenon”. In particular, under specified conditions, the square Procrustean fitting-error of the two normalized lower-dimensional data sets is (asymptotically) a convex combination (via a correlation parameter) of the Hausdorff distance between the projection subspaces and the maximum possible value of the square Procrustean fitting-error for normalized data. We show how this gives rise to the incommensurability phenomenon, and we employ illustrative simulations and also use real data to explore how the incommensurability phenomenon may have an appreciable impact.

Keywords: Incommensurability phenomenon; Procrustes fitting; Principal components analysis; Grassmannian; Hausdorff distance (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s00357-016-9203-9

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