The Annals of Statistics

Shape changes in the plane for landmark data

Michael J. Prentice and Kanti V. Mardia

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This paper deals with the statistical analysis of matched pairs of shapes of configurations of landmarks in the plane. We provide inference procedures on the complex projective plane for a basic measure of shape change in the plane, on observing that shapes of configurations of $(k + 1)$ landmarks in the plane may be represented as points on $\mathbb{C} P^{k-1}$ and that complex rotations are the only maps on $\mathbb{C} S^{k-1}$ which preserve the usual Hermitian inner product. Specifically, if $u_1, \dots, u_n$ are fixed points on $\mathbb{C} P^{k-1}$ represented as $\mathbb{C} S^{k-1}/U(1)$ and $v_1, \dots, v_n$ are random points on $\mathbb{C} P^{k-1}$ such that the distribution of $v_j$ depends only on $||v_j^* Au_j||^2$ for some unknown complex rotation matrix A, then this paper provides asymptotic inference procedures for A. It is demonstrated that shape changes of a kind not detectable as location shifts by standard Euclidean analysis can be found by this frequency domain method. A numerical example is given.

Article information

Ann. Statist., Volume 23, Number 6 (1995), 1960-1974.

First available in Project Euclid: 15 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H10: Distribution of statistics
Secondary: 62H11: Directional data; spatial statistics

Shape unitary matrices spherical regression configuration


Prentice, Michael J.; Mardia, Kanti V. Shape changes in the plane for landmark data. Ann. Statist. 23 (1995), no. 6, 1960--1974. doi:10.1214/aos/1034713642.

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