A nonparametric approach to 3D shape analysis from digital camera images — I

This article for the first time develops a nonparametric methodology for the analysis of projective shapes of configurations of landmarks on real 3D objects from their regular camera pictures. A fundamental result in computer vision, emulating the principle of human vision in space, claims that, generically, a finite 3D configuration of points can be retrieved from corresponding configurations in a pair of camera images, up to a projective transformation. Consequently, the projective shape of a 3D configuration can be retrieved from two of its planar views, and a projective shape analysis can be pursued from a sample of images. Projective shapes are here regarded as points on projective shape manifolds. Using large sample and nonparametric bootstrap methodology for extrinsic means on manifolds, one gives confidence regions and tests for the mean projective shape of a 3D configuration from its 2D camera images. Two examples are given: an example of testing for accuracy of a simple manufactured object using mean projective shape analysis, and a face identification example. Both examples are data driven based on landmark registration in digital images.     

A statistical method for estimating the proportion of differentially expressed genes

Microarrays have been widely used to identify differentially expressed genes. One related problem is to estimate the proportion of differentially expressed genes. For some complex diseases, the amount of differentially expressed genes may be relatively small and these genes may only have subtly differential expressions. For these microarray data, it is generally difficult to efficiently estimate the proportion of differentially expressed genes. In this study, I propose a likelihood-based method coupled with an expectation-maximization (E-M) algorithm for estimating the proportion of differentially expressed genes. The proposed method has favorable performances if either (i) the P values of differentially expressed genes are homogeneously distributed or (ii) the proportion of differentially expressed genes is relatively small. In both of these situations, I showed through simulations that the proposed method gave satisfactory performances when it was compared to other existing methods. As applications, these methods were applied to two microarray gene expression data sets generated from different platforms.     

Nonparametric time series regression

Consider ak-times differentiable unknown regression function(·) of ad-dimensional measurement variable. LetT() denote a derivative of(·) of orderm and setr=(k–m)/(2k+d). Given a bivariate stationary time series of lengthn, under some appropriate conditions, a sequence of local polynomial estimators of the functionT() can be chosen to achieve the optimal rate of convergencen ^–r inL ₂ norms restricted to compacts; and the optimal rate (n ^–1 logn) ^r in theL norms on compacts. These results generalize those by Stone (1982,Ann. Statist.,10, 1040–1053) which deals with nonparametric regression estimation for random (i.i.d.) samples. Applications of these results to nonlinear time series problems will also be discussed.This work was completed while the author was visiting Mathematical Sciences Research Institute at Berkeley, California. Research was supported in part by NSF Grant DMS-8505550, NC Board of Science and Technology Development Award 90SE06 and UNC Research Council.     

Nonparametric inference for sequential <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> systems

The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.     

Tests of significance using selected sample quantiles

Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.     

Nonparametric density estimation for nonmixing approximable Stochastic Processes

The author deals with nonparametric density estimation for stochastic processes which satisfy the L _∞-approximability property. He considers a Parzen–Rosenblatt estimator of the density for general stationary L _∞-approximable processes. He states conditions under which it is consistent and investigates its rate of convergence. Finally, he applies his results to general nonmixing linear processes and nonmixing nonlinear autoregressive processes.     

Bandwidth choice for local polynomial estimation of smooth boundaries

Local polynomial methods hold considerable promise for boundary estimation, where they offer unmatched flexibility and adaptivity. Most rival techniques provide only a single order of approximation; local polynomial approaches allow any order desired. Their more conventional rivals, for example high-order kernel methods in the context of regression, do not have attractive versions in the case of boundary estimation. However, the adoption of local polynomial methods for boundary estimation is inhibited by lack of knowledge about their properties, in particular about the manner in which they are influenced by bandwidth; and by the absence of techniques for empirical bandwidth choice. In the present paper we detail the way in which bandwidth selection determines mean squared error of local polynomial boundary estimators, showing that it is substantially more complex than in regression settings. For example, asymptotic formulae for bias and variance contributions to mean squared error no longer decompose into monotone functions of bandwidth. Nevertheless, once these properties are understood, relatively simple empirical bandwidth selection methods can be developed. We suggest a new approach to both local and global bandwidth choice, and describe its properties.     

Sequential change-point detection for mixing random sequences under composite hypotheses

The problem of sequential detection of a change-point in the density function of one-dimensional distribution of observations from a mixing random sequence is considered when both before and after a change-point this density function belongs to a certain family of distributions, i.e. in the situation of composite hypotheses. A new quality criterion for change-point detection is proposed. The asymptotic a priori lower bound for this criterion is proved for wide class of methods of change-point detection. An asymptotically optimal method of change-point detection is proposed for which this lower bound is attained asymptotically. In particular, for the case of a simple hypothesis before a change-point, this method coincides with the generalized cumulative sums (CUSUM) method.      

Structural Change Models with an Application in Cryogenic Thermometry

In a linear regression framework, structural change models are proposed for the detection of abrupt changes in parameter values. Two models are discussed: 1) the pure structural change model, where all the components of the parameter are allowed to change and are tested all together, and 2) the partial structural change model, where only some of the parameter β components might change. For an on-line implementation, a sliding window algorithm is introduced. The procedure was successfully applied to phase transition identification in cryogenic thermometry. *Work partially funded under EU SofTools_MetroNet Contact N. G6RT-CT-2001-05061.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.