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The noncentral Bartlett decompositions and shape densities

Author

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  • Goodall, Colin
  • Mardia, Kanti V.

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.

Suggested Citation

  • Goodall, Colin & Mardia, Kanti V., 1992. "The noncentral Bartlett decompositions and shape densities," Journal of Multivariate Analysis, Elsevier, vol. 40(1), pages 94-108, January.
  • Handle: RePEc:eee:jmvana:v:40:y:1992:i:1:p:94-108
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    Citations

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    Cited by:

    1. Frank G. Ball & Ian L. Dryden & Mousa Golalizadeh, 2008. "Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 1-22, March.
    2. Díaz-García, José A. & González-Farías, Graciela, 2005. "Singular random matrix decompositions: distributions," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 109-122, May.
    3. Díaz-García, José A. & Jáimez, Ramón Gutierrez & Mardia, Kanti V., 1997. "Wishart and Pseudo-Wishart Distributions and Some Applications to Shape Theory," Journal of Multivariate Analysis, Elsevier, vol. 63(1), pages 73-87, October.
    4. Díaz García, José A. & González Farías, Graciela, 2002. "Singular random matrix decompositions: distributions," DES - Working Papers. Statistics and Econometrics. WS ws024211, Universidad Carlos III de Madrid. Departamento de Estadística.

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