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).

The real projective space ℝPm is the manifold of all lines through the origin in ℝm+1. It may be identified with the sphere Sm modulo the identification of a point p with its antipodal point -p. For m = 1 or 2, this is the axial space. Extrinsic and intrinsic inferences on ℝPm are developed first in this chapter. This aids in the nonparametric analysis on the space of projective shapes, identified here, subject to a registration, with (ℝPm)k-m-2.

Introduction

Consider a k-ad picked on a planar image of an object or a scene in three dimensions. If one thinks of images or photographs obtained through a central projection (a pinhole camera is an example of this), a ray is received as a landmark on the image plane (e.g., the film of the camera). Because axes in three dimensions comprise the projective space ℝP2, the k-ad in this view is valued in ℝP2. For a k-ad in three dimensions to represent a k-ad in ℝP2, the corresponding axes must all be distinct. To have invariance with regard to camera angles, one may first look at the original noncollinear three-dimensional k-ad u and achieve affine invariance by its affine shape (i.e., by the equivalence class Au, A ∈ GL(3, ℝ)), and finally take the corresponding equivalence class of axes in ℝP2 to define the projective shape of the k-ad as the equivalence class, or orbit, with respect to projective transformations on ℝP2.

In this chapter we adapt and extend the nonparametric density estimation procedures on Euclidean spaces to general (Riemannian) manifolds.

Introduction

So far in this book we have used notions of center and spread of distributions on manifolds to identify them or to distinguish between two or more distributions. However, in certain applications, other aspects of the distribution may also be important. The reader is referred to the data in Section 14.5.3 for such an example. Also, our inference method so far has been frequentist.

In this chapter and the next, we pursue different goals and a different route. Our approach here and in the next chapter will be nonparametric Bayesian, which involves modeling the full data distribution in a flexible way that is easy to work with. The basic idea will be to represent the unknown distribution as an infinite mixture of some known parametric distribution on the manifold of interest and then set a full support prior on the mixing distribution. Hence the parameters defining the distribution are no longer finite-dimensional but reside in the infinite-dimensional space of all probabilities. By making the parameter space infinite-dimensional, we ensure a flexible model for the unknown distribution and consistency of its estimate under mild assumptions. All these will be made rigorous through the various theorems we will encounter in the subsequent sections.

For a prior on the mixing distribution, a common choice can be the Dirichlet process prior (see Ferguson, 1973, 1974). We present a simple algorithm for posterior computations in Section 13.4.

Manifolds of greatest interest in this book are spaces of shapes of k-ads in ℝm, with a k-ad being a set of k labeled points, or landmarks, on an object in ℝm. This chapter introduces these shape spaces.

Introduction

The statistical analysis of shape distributions based on random samples is important in many areas such as morphometrics, medical diagnostics, machine vision, and robotics. In this chapter and the chapters that follow, we will be interested mainly in the analysis of shapes of landmark-based data, in which each observation consists of k > m points in m dimensions, representing k landmarks on an object, called a k-ad. The choice of landmarks is generally made with expert help in the particular field of application. Depending on the way the data are collected and recorded, the appropriate shape of a k-ad is the maximal invariant specified by its orbit under a group of transformations.

For example, one may look at k-ads modulo size and Euclidean rigid body motions of translation and rotation. The analysis of this invariance class of shapes was pioneered by Kendall (1977, 1984) and Bookstein (1978). Bookstein's approach is primarily registration-based, requiring two or three landmarks to be brought into a standard position by translating, rotating and scaling the k-ad. We would prefer Kendall's more invariant view of a shape identified with the orbit under rotation (in m dimensions) of the k-ad centered at the origin and scaled to have a unit size.

Digital images today play a vital role in science and technology, and also in many aspects of our daily life. This book seeks to advance the analysis of images, especially digitized ones, through the statistical analysis of shapes. Its focus is on the analysis of landmark-based shapes in which a k-ad, that is, a set of k labeled points or landmarks on an object or a scene, is observed in two or three dimensions, usually with expert help, for purposes of identification, discrimination, and diagnostics.

In general, consider the k-ad to lie in ℝm (usually, m = 2 or 3) and assume that not all the k points are the same. Then the appropriate shape of the object is taken to be the k-ad modulo a group of transformations.

For example, one may first center the k-ad, by subtracting the mean of the k-ad from each of the k landmarks, to remove the effect of location. The centered k-ad then lies in a hyperplane of dimension mk - m, because the sum of each of the m coordinates of the centered k points is zero. Next one may scale the centered k-ad to unit size to remove the effect of scale or size. The scaled, centered k-ad now lies on the unit sphere Sm(k-1)-1 in a Euclidean space (the hyperplane) of dimension m(k - 1) and is now called the preshape of the k-ad.

This chapter develops nonparametric Bayes procedures for classification, hypothesis testing and regression. The classification of a random observation to one of several groups is an important problem in statistics. This is the objective in medical diagnostics, the classification of subspecies, and, more generally, the target of most problems in image analysis. Equally important is the estimation of the regression function of Y given X and the prediction of Y given a random observation X. Here Y and X are, in general, manifold-valued, and we use nonparametric Bayes procedures to estimate the regression function.

Introduction

Consider the general problem of predicting a response Y ∈ Y based on predictors X ∈ X, where Y and X are initially considered to be arbitrary metric spaces. The spaces can be discrete, Euclidean, or even non-Euclidean manifolds. In the context of this book, such data arise in many chapters. For example, for each study subject, we may obtain information on an unordered categorical response variable such as the presence/absence of a particular feature as well as predictors having different supports including categorical, Euclidean, spherical, or on a shape space. In this chapter we extend the methods of Chapter 13 to define a very general nonparametric Bayes modeling framework for the conditional distribution of Y given X = x through joint modeling of Z = (X, Y). The flexibility of our modelling approach will be justified theoretically through Theorems, Propositions, and Corollaries 14.1, 14.2, 14.3, 14.4, and 14.5.