Extreme shape analysis

Summary.  We consider the analysis of extreme shapes rather than the more usual mean- and variance-based shape analysis. In particular, we consider extreme shape analysis in two applications: human muscle fibre images, where we compare healthy and diseased muscles, and temporal sequences of DNA shapes from molecular dynamics simulations. One feature of the shape space is that it is bounded, so we consider estimators which use prior knowledge of the upper bound when present. Peaks-over-threshold methods and maximum-likelihood-based inference are used. We introduce fixed end point and constrained maximum likelihood estimators, and we discuss their asymptotic properties for large samples. It is shown that in some cases the constrained estimators have half the mean-square error of the unconstrained maximum likelihood estimators. The new estimators are applied to the muscle and DNA data, and practical conclusions are given.     

NEW LARGE SAMPLE AND BOOTSTRAP METHODS ON SHAPE SPACES IN HIGH LEVEL ANALYSIS OF NATURAL IMAGES

This paper, dedicated to the 80th birthday of Professor C. R. Rao, deals with asymptotic distributions of Fréchet sample means and Fréchet total sample variance that are used in particular for data on projective shape spaces or on 3D shape spaces. One considers the intrinsic means associated with Riemannian metrics that are locally flat in a geodesically convex neighborhood around the support of a probability measure on a shape space or on a projective shape space. Such methods are needed to derive tests concerning variability of planar projective shapes in natural images or large sample and bootstrap confidence intervals for 3D mean shape coordinates of an ordered set of landmarks from laser images.     

Dihedral angles principal geodesic analysis using nonlinear statistics

Statistics, as one of the applied sciences, has great impacts in vast area of other sciences. Prediction of protein structures with great emphasize on their geometrical features using dihedral angles has invoked the new branch of statistics, known as directional statistics. One of the available biological techniques to predict is molecular dynamics simulations producing high-dimensional molecular structure data. Hence, it is expected that the principal component analysis (PCA) can response some related statistical problems particulary to reduce dimensions of the involved variables. Since the dihedral angles are variables on non-Euclidean space (their locus is the torus), it is expected that direct implementation of PCA does not provide great information in this case. The principal geodesic analysis is one of the recent methods to reduce the dimensions in the non-Euclidean case. A procedure to utilize this technique for reducing the dimension of a set of dihedral angles is highlighted in this paper. We further propose an extension of this tool, implemented in such way the torus is approximated by the product of two unit circle and evaluate its application in studying a real data set. A comparison of this technique with some previous methods is also undertaken.     

A Common Situation Conducive to Bizarre Distribution Shapes

Conditions conducive to bizarre-shaped distributions are fairly common in certain areas of research where, for perfectly valid reasons, an important causal variable is left uncontrolled. Theoretical rationale and actual examples are given to show how such distributions are generated, to exemplify their shapes, and to indicate their prevalence in practice. An L-shaped type tends to occur when time scores are recorded for a task subject to infrequent but time-consuming errors; other types occur when measuring behavior influenced by social conformity, or under other circumstances. The L shape appears to be far more conducive to nonrobustness than are previously investigated shapes.     

Functional Analysis of Variance,Discriminant Analysis,and Clustering in a Manifold of Elastic Curves

We develop functional data analysis techniques using the differential geometry of a manifold of smooth elastic functions on an interval in which the functions are represented by a log-speed function and an angle function. The manifold's geometry provides a method for computing a sample mean function and principal components on tangent spaces. Using tangent principal component analysis, we estimate probability models for functional data and apply them to functional analysis of variance, discriminant analysis, and clustering. We demonstrate these tasks using a collection of growth curves from children from ages 1–18.     

A unified approach to marginal equivalence in the general framework of group invariance

ABSTRACT

Two Bayesian models with different sampling densities are said to be marginally equivalent if the joint distribution of observables and the parameter of interest is the same for both models. We discuss marginal equivalence in the general framework of group invariance. We introduce a class of sampling models and derive marginal equivalence when the prior for the nuisance parameter is relatively invariant. We also obtain some robustness properties of invariant statistics under our sampling models. Besides the prototypical example of v-spherical distributions, we apply our general results to two examples—analysis of affine shapes and principal component analysis.     

Exploring the variability of DNA molecules via principal geodesic analysis on the shape space

Most of the linear statistics deal with data lying in a Euclidean space. However, there are many examples, such as DNA molecule topological structures, in which the initial or the transformed data lie in a non-Euclidean space. To get a measure of variability in these situations, the principal component analysis (PCA) is usually performed on a Euclidean tangent space as it cannot be directly implemented on a non-Euclidean space. Instead, principal geodesic analysis (PGA) is a new tool that provides a measure of variability for nonlinear statistics. In this paper, the performance of this new tool is compared with that of the PCA using a real data set representing a DNA molecular structure. It is shown that due to the nonlinearity of space, the PGA explains more variability of the data than the PCA.     

A superiority problem in misspecified restricted linear models

Abstract

In this article we study the relationship between principal component analysis and a multivariate dependency measure. It is shown, via simulated examples and real data, that the information provided by principal components is compatible with that obtained via the dependency measure δ. Furthermore, we show that in some instances in which principal component analysis fails to give reasonable results due to nonlinearity among the random variables, the dependency statistic δ still provides good results. Finally, we give some ideas about using the statistic δ in order to reduce the dimensionality of a given data set.     

Between-group comparison of principal components — some sampling results

Sampling distributions arc Investigated for the critical angles proposed by Krzanowski (1979) as a means of comparing principal component analyses done on two sets of individuals, each of which has had the same p responses measured on it. A summary of the distributions is presented, by giving mean angles and 95% points. The emphasis is on the null case, when the two sets of individuals are assumed to be independent samples from the same population, but a limited study is also made of the case where the two samples come from different populations.     

Semiparametric mixture models and repeated measures: the multinomial cut point model

Summary.  Suppose that we have m repeated measures on each subject, and we model the observation vectors with a finite mixture model.  We further assume that the repeated measures are conditionally independent. We present methods to estimate the shape of the component distributions along with various features of the component distributions such as the medians, means and variances. We make no distributional assumptions on the components; indeed, we allow different shapes for different components.     

Bayesian principal component regression with data-driven component selection

Principal component regression (PCR) has two steps: estimating the principal components and performing the regression using these components. These steps generally are performed sequentially. In PCR, a crucial issue is the selection of the principal components to be included in regression. In this paper, we build a hierarchical probabilistic PCR model with a dynamic component selection procedure. A latent variable is introduced to select promising subsets of components based upon the significance of the relationship between the response variable and principal components in the regression step. We illustrate this model using real and simulated examples. The simulations demonstrate that our approach outperforms some existing methods in terms of root mean squared error of the regression coefficient.     

Testing high-dimensional normality based on classical skewness and Kurtosis with a possible small sample size

Abstract

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples.     

Quadratic perturbation expansions of certain functions of eigenvalues and eigenvectors and their application to sensitivity analysis in multivariate methods

Tanaka (1988) lias derived the influence functions, which are equivalent to the perturbation expansions up to linear terms, of two functions of eigenvalues and eigenvectors of a real symmetric matrix, and applied them to principal component analysis. The present paper deals with the perturbation expansions up to quadratic terms of the same functions and discusses their application to sensitivity analysis in multivariate methods, in particular, principal component analysis and principal factor analysis. Numerical examples are given to show how the approximation improves with the quadratic terms.     

Estimation for distributions with monotone likelihood ratio:case of vector valued parameter

An extension of a result about the estimation in Karlin and Rubin is given for the following case:The sample space, the parameter space and the decision space are subsets of a multi-dimensional Euclidean space, there is defined a suitable partial ordering in each of spaces, and a probability distribution has monotone likelihood ratio with respect to the partial orderings (see Ishii, 1976). In the special case when the loss function is quadratic a simple proof of a result in Karlin and Rubin is given. Stein's estimators are discussed as examples.     

A Note on Estimation in Hilbertian Linear Models

We study estimation and prediction in linear models where the response and the regressor variable both take values in some Hilbert space. Our main objective is to obtain consistency of a principal component‐based estimator for the regression operator under minimal assumptions. In particular, we avoid some inconvenient technical restrictions that have been used throughout the literature. We develop our theory in a time‐dependent setup that comprises as important special case the autoregressive Hilbertian model.     

Membranes,mitochondria and amoebae: shape models

Most real-world shapes and images are characterized by high variability- they are not rigid, like crystals, for example—but they are strongly structured. Therefore, a fundamental task in the understanding and analysis of such image ensembles is the construction of models that incorporate both variability and structure in a mathematically precise way. The global shape models introduced in Grenander's general pattern theory are intended to do this. In this paper, we describe the representation of two-dimensional mitochondria and membranes in electron microscope photographs, and three-dimensional amoebae in optical sectioning microscopy. There are three kinds of variability to all of these patterns, which these representations accommodate. The first is the variability in shape and viewing orientation. For this, the typical structure is represented via linear, circular and spherical templates, with the variability accomodated via the application of transformations applied to the templates. The transformations form groups: scale, rotation and translation. They are locally applied throughout the continuum and of high dimension. The second is the textural variability; the inside and outside of these basic shapes are subject to random variation, as well as sensor noise. For this, statistical sensor models and Markov random field texture models are used to connect the constituent structures of the shapes to the measured data. The third variability type is associated with the fact that each scene is made up of a variable number of shapes; this number is not assumed to be known a priori. Each scene has a variable number of parameters encoding the transformations of the templates appropriate for that scene. For this, a single posterior distribution is defined over the countable union of spaces representing models of varying numbers of shapes. Bayesian inference is performed via computation of the conditional expectation of the parametrically defined shapes under the posterior. These conditional mean estimates are generated using jump-diffusion processes. Results for membranes, mitochondria and amoebae are shown.     

Influential observation in complex normal data for problems in allometry

Abstract

The aim of this paper is to investigate how some results related to the complex normal distribution are relevant in size and shape analysis. Our main focus is on the derivation of influential measures. In particular, Cook and Kullback–Leibler distances are combined with their respective asymptotic results as well as to an alternative process of defining cut-off points. Some numerical examples illustrate how these measures are used in practice. We perform an application to simulated and actual data. Results provide evidence that the methodology based on Kullback–Leibler distance outperforms one in terms of the Cook classic distance.     

Membranes, mitochondria and amoebae: shape models

Most real-world shapes and images are characterized by high variability- they are not rigid, like crystals, for example—but they are strongly structured. Therefore, a fundamental task in the understanding and analysis of such image ensembles is the construction of models that incorporate both variability and structure in a mathematically precise way. The global shape models introduced in Grenander's general pattern theory are intended to do this. In this paper, we describe the representation of two-dimensional mitochondria and membranes in electron microscope photographs, and three-dimensional amoebae in optical sectioning microscopy. There are three kinds of variability to all of these patterns, which these representations accommodate. The first is the variability in shape and viewing orientation. For this, the typical structure is represented via linear, circular and spherical templates, with the variability accomodated via the application of transformations applied to the templates. The transformations form groups: scale, rotation and translation. They are locally applied throughout the continuum and of high dimension. The second is the textural variability; the inside and outside of these basic shapes are subject to random variation, as well as sensor noise. For this, statistical sensor models and Markov random field texture models are used to connect the constituent structures of the shapes to the measured data. The third variability type is associated with the fact that each scene is made up of a variable number of shapes; this number is not assumed to be known a priori. Each scene has a variable number of parameters encoding the transformations of the templates appropriate for that scene. For this, a single posterior distribution is defined over the countable union of spaces representing models of varying numbers of shapes. Bayesian inference is performed via computation of the conditional expectation of the parametrically defined shapes under the posterior. These conditional mean estimates are generated using jump-diffusion processes. Results for membranes, mitochondria and amoebae are shown.     

Sensitivity Coefficient in Principal Component Analysis: Robust Case

In the classical principal component analysis (PCA), the empirical influence function for the sensitivity coefficient ρ is used to detect influential observations on the subspace spanned by the dominants principal components. In this article, we derive the influence function of ρ in the case where the reweighted minimum covariance determinant (MCD¹) is used as estimator of multivariate location and scatter. Our aim is to confirm the reliability in terms of robustness of the MCD¹ via the approach based on the influence function of the sensitivity coefficient.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.