Ian L. Dryden

NON-EUCLIDEAN STATISTICS FOR COVARIANCE MATRICES, WITH APPLICATIONS TO DIFFUSION TENSOR IMAGING

The statistical analysis of covariance matrices occurs in m any important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy and to carry out principal geodesic analysis of covariance matrices. In medical image analysis a particular type of covariance matrix arises in diffusion weighted imaging called a diffusion tensor. The diffusion tensor is a 3 × 3 covariance matrix which is estimated at each voxel in the brain, and is obtained by fittin g a physically-motivated model on measurements from the Fourier transform of the molecule displacement density (Basser et al., 1994). A strongly anisotropic diffusion tensor indicates a strong direction of white matter fibre tracts, and plots of measures of anisotropy are very useful t o neurologists. A measure that is very commonly used in diffusion tensor imaging is Fractional Anisotropy

Serum Proteomic Fingerprinting Discriminates Between Clinical Stages and Predicts Disease Progression in Melanoma Patients

PURPOSE Currently known serum biomarkers do not predict clinical outcome in melanoma. S100-beta is widely established as a reliable prognostic indicator in patients with advanced metastatic disease but is of limited predictive value in tumor-free patients. This study was aimed to determine whether molecular profiling of the serum proteome could discriminate between early- and late-stage melanoma and predict disease progression. PATIENTS AND METHODS Two hundred five serum samples from 101 early-stage (American Joint Committee on Cancer [AJCC] stage I) and 104 advanced stage (AJCC stage IV) melanoma patients were analyzed by matrix-assisted laser desorption/ionisation (MALDI) time-of-flight (ToF; MALDI-ToF) mass spectrometry utilizing protein chip technology and artificial neural networks (ANN). Serum samples from 55 additional patients after complete dissection of regional lymph node metastases (AJCC stage III), with 28 of 55 patients relapsing within the first year of follow-up, were analyzed in an attempt to predict disease recurrence. Serum S100-beta was measured using a sandwich immunoluminometric assay. RESULTS Analysis of 205 stage I/IV serum samples, utilizing a training set of 94 of 205 and a test set of 15 of 205 samples for 32 different ANN models, revealed correct stage assignment in 84 (88%) of 96 of a blind set of 96 of 205 serum samples. Forty-four (80%) of 55 stage III serum samples could be correctly assigned as progressors or nonprogressors using random sample cross-validation statistical methodologies. Twenty-three (82%) of 28 stage III progressors were correctly identified by MALDI-ToF combined with ANN, whereas only six (21%) of 28 could be detected by S100-beta. CONCLUSION Validation of these findings may enable proteomic profiling to become a valuable tool for identifying high-risk melanoma patients eligible for adjuvant therapeutic interventions.

SHAPE DISTRIBUTIONS FOR LANDMARK DATA

The paper obtains the exact distribution of Booksteins shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendalls uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Booksteins result for three landmarks. The adequacy of these approximations is also studied.

GENERAL SHAPE DISTRIBUTIONS IN A PLANE

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendalls or Booksteins shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

Periods of Rest in fMRI Contain Individual Spontaneous Events which are Related to Slowly Fluctuating Spontaneous Activity

fMRI studies of brain activity at rest study slow (<0.1 Hz) intrinsic fluctuations in the blood‐oxygenation‐level‐dependent (BOLD) signal that are observed in a temporal scale of several minutes. The origin of these fluctuations is not clear but has previously been associated with slow changes in rhythmic neuronal activity resulting from changes in cortical excitability or neuronal synchronization. In this work, we show that individual spontaneous BOLD events occur during rest, in addition to slow fluctuations. Individual spontaneous BOLD events were identified by deconvolving the hemodynamic impulse response function for each time point in the fMRI time series, thus requiring no information on timing or a‐priori spatial information of events. The patterns of activation detected were related to the motor, visual, default‐mode, and dorsal attention networks. The correspondence between spontaneous events and slow fluctuations in these networks was assessed using a sliding window, seed‐correlation analysis, where seed regions were selected based on the individual spontaneous event BOLD activity maps. We showed that the correlation varied considerably over time, peaking at the time of spontaneous events in these networks. By regressing spontaneous events out of the fMRI signal, we showed that both the correlation strength and the power in spectral frequencies <0.1 Hz decreased, indicating that spontaneous activation events contribute to low‐frequency fluctuations observed in resting state networks with fMRI. This work provides new insights into the origin of signals detected in fMRI studies of functional connectivity. Hum Brain Mapp, 2013.

Statistical analysis on high-dimensional spheres and shape spaces

We consider the statistical analysis of data on high-dimensional spheres and shape spaces. The work is of particular relevance to applications where high-dimensional data are available—a commonly encountered situation in many disciplines. First the uniform measure on the infinite-dimensional sphere is reviewed, together with connections with Wiener measure. We then discuss densities of Gaussian measures with respect to Wiener measure. Some nonuniform distributions on infinite-dimensional spheres and shape spaces are introduced, and special cases which have important practical consequences are considered. We focus on the high-dimensional real and complex Bingham, uniform, von Mises–Fisher, Fisher–Bingham and the real and complex Watson distributions. Asymptotic distributions in the cases where dimension and sample size are large are discussed. Approximations for practical maximum likelihood based inference are considered, and in particular we discuss an application to brain shape modeling.

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.

Pivotal Bootstrap Methods for k-Sample Problems in Directional Statistics and Shape Analysis

We propose a novel bootstrap hypothesis testing approach for the problem of testing a null hypothesis of a common mean direction, mean polar axis, or mean shape across several populations of real unit vectors (the directional case) or complex unit vectors (the two-dimensional shape case). Multisample testing problems of this type arise frequently in directional statistics and shape analysis (as in other areas of statistics), but to date there has been relatively little discussion of nonparametric bootstrap approaches to this problem. The bootstrap approach described here is based on a statistic that can be expressed as the smallest eigenvalue of a certain positive definite matrix. We prove that this statistic has a limiting chi-squared distribution under the null hypothesis of equality of means across populations. Although we focus mainly on the version of the statistic in which neither isotropy within populations nor constant dispersion structure across populations is assumed, we explain how to modify the statistic so that either or both of these assumptions can be incorporated. Our numerical results indicate that the bootstrap approach proposed here may be expected to perform well in practice.

Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendalls shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].

Landmark-based morphometrics of the normal adult brain using MRI.

We describe the application of statistical shape analysis to homologous landmarks on the cortical surface of the adult human brain. Statistical shape analysis has a sound theoretical basis. Landmarks are identified on the surface of a 3-D reconstruction of the segmented cortical surface from magnetic resonance image (MRI) data. Using publicly available software (morphologika) the location and size dependence of the landmarks are removed and the differences in landmark distribution across subjects are analysed using principal component analysis. These differences, representing shape differences between subjects, can be visually assessed using wireframe models and transformation grids. The MRI data of 58 adult brains (27 female and 15 left handed) were examined. Shape differences in the whole brain are described which concern the relative orientation of frontal lobe sulci. Analysis of all 116 hemispheres revealed a statistically significant difference (P < 0.001) between left and right hemispheres. This finding was significant for right- but not left-handed subjects alone. No other significant age, gender, handedness, or brain-size correlations with shape differences were found.