×

Spatial statistics. (English) Zbl 1100.62574

Summary: When the distribution of \(\mathbf {X} \in \mathbb R^p\) depends only on its distance to some \(\theta_0 \in \mathbb R^p\), we discuss results from Hössjer and Croux and Neeman and Chang on rank score statistics. Similar results from Neeman and Chang are also given when \(\mathbf {X}\) and \(\theta_0\) are constrained to lie on the sphere in \(\mathbb R^p\). Results from Ko and Chang on \(M\) estimation for spatial models in Euclidean space and the sphere are also discussed. Finally we discuss a regression type model: the image registration problem. We have landmarks \(U_i\) on one image and corresponding landmarks \(V_i\) on a second image. It is desired to bring the two images into closest coincidence through a translation, rotation and scale change. The techniques and principles of this paper are summarized through extensive discussion of an example in three-dimensional image registration and a comparison of the \(L_1\) and \(L_2\) registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of \(\mathbf {X}\) depends only on its distance to \(\theta_0\), introduce symmetries to spatial models which, if properly used, greatly simplify statistical calculations. These symmetries can be expressed in a more general setting by using the notion of statistical group models. (2) When working with a non-Euclidean parameter space \(\Theta\) such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress \(\Theta\) in Euclidean parameters.

MSC:

62H11 Directional data; spatial statistics
62H35 Image analysis in multivariate analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Brown, B. M. (1983). Statistical uses of the spatial median. J. Roy. Statist. Soc. Ser. B 45 25–30. · Zbl 0508.62046
[2] Brown, B. M. (1985). Multi-parameter linearization theorems. J. Roy. Statist. Soc. Ser. B 47 323–331. · Zbl 0569.62046
[3] Chang, T. (1986). Spherical regression. Ann. Statist. 14 907–924. JSTOR: · Zbl 0605.62079
[4] Chang, T. (1993). Spherical regression and the statistics of tectonic plate reconstructions. Internat. Statist. Rev. 61 299–316.
[5] Chang, T. and Ko, D. (1995). \(M\)-estimates of rigid body motion on the sphere and in Euclidean space. Ann. Statist. 23 1823–1847. JSTOR: · Zbl 0843.62064
[6] Chang, T. and Rivest, L.-P. (2001). \(M\)-estimation for location and regression parameters in group models: A case study using Stiefel manifolds. Ann. Statist. 29 784–814. · Zbl 1012.62022
[7] Chang, T. and Tsai, M.-T. (2003). Asymptotic relative Pitman efficiency in group models. J. Multivariate Anal. 85 395–415. · Zbl 1051.62004
[8] Downs, T. D. (1972). Orientation statistics. Biometrika 59 665–676. · Zbl 0269.62027
[9] Fisher, N. I. (1985). Spherical medians. J. Roy. Statist. Soc. Ser. B 47 342–348. · Zbl 0605.62055
[10] Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data. Cambridge Univ. Press. · Zbl 0651.62045
[11] Hettmansperger, T. P. and McKean, J. W. (1998). Robust Nonparametric Statistical Methods . Wiley, New York. · Zbl 0887.62056
[12] Hössjer, O. and Croux, C. (1995). Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter. J. Nonparametr. Statist. 4 293–308. · Zbl 1381.62113
[13] Jupp, P. E. and Mardia, K. V. (1989). A unified view of the theory of directional statistics, 1975–1988. Internat. Statist. Rev. 57 261–294. · Zbl 0707.62095
[14] Kirkwood, B., Royer, J.-Y., Chang, T. and Gordon, R. (1999). Statistical tools for estimating and combining finite rotations and their uncertainties. Geophysical J. International 137 408–428.
[15] Ko, D. and Chang, T. (1993). Robust \(M\)-estimators on spheres. J. Multivariate Anal. 45 104–136. · Zbl 0777.62056
[16] Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Statist. 5 201–213. · Zbl 0857.62056
[17] Neeman, T. M. (1995). Rank statistics for spherical data. Ph.D. dissertation, Univ. Virginia.
[18] Neeman, T. M. and Chang, T. (2001). Rank score statistics for spherical data. In Algebraic Methods in Statistics and Probability (M. A. G. Viana and D. St. P. Richards, eds.) 241–254. Amer. Math. Soc., Providence, RI. · Zbl 1012.62060
[19] Rancourt, D., Rivest, L.-P. and Asselin, J. (2000). Using orientation statistics to investigate variations in human kinematics. Appl. Statist. 49 81–94. · Zbl 0974.62107
[20] Reeds, J. (1975). Discussion of “Defining the curvature of a statistical problem (with applications to second order efficiency),” by B. Efron. Ann. Statist. 3 1234–1238. · Zbl 0321.62013
[21] Watson, G. S. (1983). Statistics on Spheres . Wiley, New York. · Zbl 0646.62045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.