The Annals of Statistics

Shape changes in the plane for landmark data

Michael J. Prentice and Kanti V. Mardia

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713642

Digital Object Identifier
doi:10.1214/aos/1034713642

Mathematical Reviews number (MathSciNet)
MR1389860

Zentralblatt MATH identifier
0858.62039

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

Keywords
Shape unitary matrices spherical regression configuration

Citation

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. https://projecteuclid.org/euclid.aos/1034713642


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