Assessing spinal shape

Idiopathic scoliosis is the most common spinal deformity, affecting perhaps as many as 5% of children. Early recognition of the condition is essential for optimal treatment. A widely used technique for identification is based on a somewhat crude angle measurement from a frontal spinal X-ray. Here, we provide a technique and new summary statistical measures for classifying spinal shape, and present results obtained from clinical X-rays.     

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.     

Familial spinal shape

It has been proposed that there is a familial relationship in the shape of the spine. This paper describes a pilot study investigating familial shape in the sagittal plane (side view), using three data sets of normal Leeds schoolchildren. The study is exploratory in nature, because only small samples were available. Data acquisition was by means of the Quantec system, which obtains surface shape measurements and extracts a line representing the spinal curve. The coordinates of the spine line in the sagittal plane are then used to investigate familial correlations of spinal shape. The spine lines first undergo some preprocessing, including Procrustes rotations to remove location, rotation and size effects. Smoothed principal component analysis of the curves provides suitable shape variables, and familial correlations between curves are then investigated. The covariates of sex and height are also investigated in the analysis. It does appear that there could be some evidence for familial correlations in sagittal spinal shape, although a further large-scale study is required. Finally, a discussion of the approach and other alternatives is considered.     

On clustering shape data

ABSTRACT

Among the statistical methods to model stochastic behaviours of objects, clustering is a preliminary technique to recognize similar patterns within a group of observations in a data set. Various distances to measure differences among objects could be invoked to cluster data through numerous clustering methods. When variables in hand contain geometrical information of objects, such metrics should be adequately adapted. In fact, statistical methods for these typical data are endowed with a geometrical paradigm in a multivariate sense. In this paper, a procedure for clustering shape data is suggested employing appropriate metrics. Then, the best shape distance candidate as well as a suitable agglomerative method for clustering the simulated shape data are provided by considering cluster validation measures. The results are implemented in a real life application.     

Assessing shape differences in populations of shapes using the complex watson shape distribution

This paper presents a novel Bayesian method based on the complex Watson shape distribution that is used in detecting shape differences between the second thoracic vertebrae for two groups of mice, small and large, categorized according to their body weight. Considering the data provided in Johnson et al. (1988), we provide Bayesian methods of estimation as well as highest posterior density (HPD) estimates for modal vertebrae shapes within each group. Finally, we present a classification procedure that can be used in any shape classification experiment, and apply it for categorizing new vertebrae shapes in small or large groups.     

Directional statistics and shape analysis

This paper highlights distributional connections between directional statistics and shape analysis. In particular, we provide a test of uniformity for highly dispersed shapes, using the standard techniques of directional statistics. We exploit the isometric transformation from triangular shapes to a sphere in three dimensions, to provide a rich class of shape distributions. A link between the Fisher distribution and the complex Bingham distribution is re-examined. Some extensions to higher-dimensional shapes are outlined.     

Estimation of the pareto shape parameter

The generalized jackknife is used to reduce the bias of an estimator, based on frequency moments, of the Pareto shape parameter. Computations of amount of bias reduction and simulations of mean-squared errors are presented.     

Bias-correction for Weibull common shape estimation

A general method for correcting the bias of the maximum likelihood estimator (MLE) of the common shape parameter of Weibull populations, allowing a general right censorship, is proposed in this paper. Extensive simulation results show that the new method is very effective in correcting the bias of the MLE, regardless of censoring mechanism, sample size, censoring proportion and number of populations involved. The method can be extended to more complicated Weibull models.     

Image search using trained flexible shape models

This paper describes a technique for building compact models of the shape and appearance of flexible objects seen in two-dimensional images. The models are derived from the statistics of sets of images of example objects with 'landmark' points labelled on each object. Each model consists of a flexible shape template, describing how the landmark points can vary, and a statistical model of the expected grey levels in regions around each point. Such models have proved useful in a wide variety of applications. We describe how the models can be used in local image search and give examples of their application.     

Image search using trained flexible shape models

This paper describes a technique for building compact models of the shape and appearance of flexible objects seen in two-dimensional images. The models are derived from the statistics of sets of images of example objects with ‘landmark’ points labelled on each object. Each model consists of a flexible shape template, describing how the landmark points can vary, and a statistical model of the expected grey levels in regions around each point. Such models have proved useful in a wide variety of applications. We describe how the models can be used in local image search and give examples of their application.     

The complex Watson distribution and shape analysis

The complex Watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis. We describe the density, obtain the integrating constant and provide large sample approximations. Maximum likelihood estimation and hypothesis testing procedures for one and two samples are described. The particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains. We make some observations about Bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions.     

Robust estimation of multivariate location and shape

In this paper, we describe an overall strategy for robust estimation of multivariate location and shape, and the consequent identification of outliers and leverage points. Parts of this strategy have been described in a series of previous papers (Rocke, Ann. Statist., in press; Rocke and Woodruff, Statist. Neerlandica 47 (1993), 27–42, J. Amer. Statist. Assoc., in press; Woodruff and Rocke, J. Comput. Graphical Statist. 2 (1993), 69–95; J. Amer. Statist. Assoc. 89 (1994), 888–896) but the overall structure is presented here for the first time. After describing the first-level architecture of a class of algorithms for this problem, we review available information about possible tactics for each major step in the process. The major steps that we have found to be necessary are as follows: (1) partition the data into groups of perhaps five times the dimension; (2) for each group, search for the best available solution to a combinatorial estimator such as the Minimum Covariance Determinant (MCD) — these are the preliminary estimates; (3) for each preliminary estimate, iterate to the solution of a smooth estimator chosen for robustness and outlier resistance; and (4) choose among the final iterates based on a robust criterion, such as minimum volume. Use of this algorithm architecture can enable reliable, fast, robust estimation of heavily contaminated multivariate data in high (> 20) dimension even with large quantities of data. A computer program implementing the algorithm is available from the authors.     

Inference in affine shape theory under elliptical models

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

Complex elliptical distributions with application to shape analysis

We introduce a general class of complex elliptical distributions on a complex sphere that includes many of the most commonly used distributions, like the complex Watson, Bingham, angular central Gaussian and several others. We study properties of this family of distributions and apply the distribution theory for modeling shapes in two dimensions. We develop maximum likelihood and Bayesian methods of estimation to describe shape and obtain confidence bounds and credible regions for shapes. The methodology is illustrated through an example where estimation of shape of mouse vertebrae is desired.     

Statistical models foe geometric components of shape change

Data for studies of biological shape often consist of the locations of individually named pointslandmarks considered to be homologous' (to correspond biologically) from form to form. In 1917 D'Arcy Thompson introduced an elegant model of homology as deformation: the configuration of landmark locations for any one form is viewed as a finite sample from a smooth mapping representing its biological relationship to any other form of the data set. For data in two dimensions, multivariate statistical analysis of landmark locations may proceed unambiguously in terms of complex-valued shape coordinates (e,v) = (C?A)/(B?A) for sets of landmark triangles ABC. These are the coordinates of one vertex/landmark after scaling so that the remaining two vertices are at (0,0) and (1,0). Expressed in this fashion, the biological interpretation of the statistical analysis as a homology mapping would appear to depend on the triangulation. This paper introduces an analysis of landmark data and homology mappings using a hierarchy of geometric components of shape difference or shape change. Each component is a smooth deformation taking the form of a bivariate polynomial in the shape coordinates and is estimated in a manner nearly invariant with respect to the choice of a triangulation.     

Size and shape analysis of palmar interdigital areas

As pointed out by Hauser & Abraham, dermatoglyphic methodology investigates either shape or size rather than both variables simultaneously. Hauser & Abraham have given a method to analyse simultaneously the shape and size of the third palmar interdigital area. We describe analytically their selection of the shape variables and indicate how their method can fit into a general framework of size and shape analysis. Their data are reanalysed and we describe some advantages of their choice of variables. However, their size and shape variables are highly correlated.