Book contents
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
2 - Examples
Published online by Cambridge University Press: 05 May 2012
- Frontmatter
- Contents
- Commonly used notation
- Perface
- 1 Introduction
- 2 Examples
- 3 Location and spread on metric spaces
- 4 Extrinsic analysis on manifolds
- 5 Intrinsic analysis on manifolds
- 6 Landmark-based shape spaces
- 7 Kendall's similarity shape spaces ??km
- 8 The planar shape space ??k2
- 9 Reflection similarity shape spaces R??km
- 10 Stiefel manifolds Vk,m
- 11 Affine shape spaces A??km
- 12 Real projective spaces and projective shape spaces
- 13 Nonparametric Bayes inference on manifolds
- 14 Nonparametric Bayes regression, classification and hypothesis testing on manifolds
- Appendix A Differentiable manifolds
- Appendix B Riemannian manifolds
- Appendix C Dirichlet processes
- Appendix D Parametric models on Sd and ??k2
- References
- Index
Summary
This chapter collects together, and describes in a simple manner, a number of applications of the theory presented in this book. The examples are based on real data, and, where possible, results of parametric inference in the literature are cited for comparison with the new nonparametric inference theory.
Data example on S1: wind and ozone
The wind direction and ozone concentration were observed at a weather station for 19 days. Table 2.1 shows the wind directions in degrees. The data are taken from Johnson and Wehrly (1977). The data viewed on the unit circle S1 are plotted in Figure 3.1. We compute the sample extrinsic and intrinsic mean directions, which come out to be 16.71 and 5.68 degrees, respectively. They are displayed in the figure. We use angular coordinates for the data in degrees lying between [0°, 360°) as in Table 2.1. An asymptotic 95% confidence region for the intrinsic mean as obtained in Section 3.7, Chapter 3, turns out to be
{(cos θ, sin θ) : - 0.434 ≤ θ ≤ 0.6324}.
The corresponding end points of this arc are also displayed in Figure 3.1.
Johnson and Wehrly (1977) computed the so-called angular–linear correlation ρAL = maxα{ρ(cos(θ - α), X)}, where X is the ozone concentration when the direction of wind is θ. Here ρ denotes the true coefficient of correlation. Based on the sample counterpart rAL, the 95% confidence interval for ρAL was found to be (0.32, 1.00).
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- Nonparametric Inference on ManifoldsWith Applications to Shape Spaces, pp. 8 - 20Publisher: Cambridge University PressPrint publication year: 2012