Some Practical Aspects and Variants of Normal Theory-Based Discriminant Rules

2005 ◽  
pp. 129-167
Keyword(s):  
Normal Theory

The distribution of volume reductions induced by isotropic random projections

1999 ◽  
Vol 31 (04) ◽  
pp. 985-994 ◽  
Author(s):  
Jørgen Nielsen
Keyword(s):  
Simple Proof ◽  
Stochastic Geometry ◽  
Random Variables ◽  
Random Projections ◽  
Regularity Conditions ◽  
Affine Subspace ◽  
Multivariate Normal ◽  
Normal Theory ◽  
Wilks Lambda ◽  
Isotropic Random

In this paper, isotropic random projections of d-sets in ℝ n are studied, where a d-set is a subset of a d-dimensional affine subspace which satisfies certain regularity conditions. The squared volume reduction induced by the projection of a d-set onto an isotropic random p-subspace is shown to be distributed as a product of independent beta-distributed random variables, for d ≤ p. One of the proofs of this result uses Wilks' lambda distribution from multivariate normal theory. The result is related to Cauchy's and Crofton's formulae in stochastic geometry. In particular, it can be used to give a new and quite simple proof of one of the classical Crofton intersection formulae.

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