The noncentral Bartlett decompositions and shape densities

Colin Goodall and Kanti V. Mardia

Journal of Multivariate Analysis, 1992, vol. 40, issue 1, 94-108

Abstract: In shape analysis, it is usually assumed that the matrix X:N-K of the co-ordinates of landmarks in K is isotropic Gaussian. Let Y:(N-1)-K be the centered matrix of landmarks from X so that Y ~ N([mu], [sigma]2I). Let Y=TT be the Bartlett decomposition of Y into lower triangular, T, and orthogonal, [Gamma], components. The matrix T denotes the size-and-shape of X. For N-1>=K (the usual case in multivariate analysis is N-1 =2 the distribution of T is related to the noncentral Wishart distribution, an integral over the orthogonal group, [Gamma]=±1. To derive the distribution of T when [Gamma]=+1, so that [Gamma] is a rotation, we investigate extending the method of random orthogonal transformations, especially when rank [mu]=K>=2. The case K=2 is tractable, but the case K=3 is not. However, by a direct method we obtain the shape density when rank [mu]=K=3 and [Gamma]=1 as a computable double-series of trigonometric integrals. However, for K>3, the density is not tractable which is not surprising in view of the same problem for the standard non-central Wishart distribution.

Keywords: Bartlett; decompositions; integral; over; SO(3); lower; triangular; matrix; QR-decomposition; random; orthogonal; transformation; shape; densities; size; shape; distribution; special; orthogonal; group; Wishart; distribution (search for similar items in EconPapers)
Date: 1992
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Citations: View citations in EconPapers (4)

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