Kriging prediction for manifold-valued random fields

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jmva.2015.12.00

Rejoinder for “A spatial modeling approach for linguistic object data : analysing dialect sound variations across Great Britain”

"Rejoinder for “A Spatial Modeling Approach for Linguistic Object Data: Analyzing Dialect Sound Variations Across Great Britain”." Journal of the American Statistical Association, 114(527), pp. 1103–110

A spatial modeling approach for linguistic object data : analysing dialect sound variations across Great Britain

Dialect variation is of considerable interest in linguistics and other social sciences. However, traditionally it has been studied using proxies (transcriptions) rather than acoustic recordings directly. We introduce novel statistical techniques to analyze geolocalized speech recordings and to explore the spatial variation of pronunciations continuously over the region of interest, as opposed to traditional isoglosses, which provide a discrete partition of the region. Data of this type require an explicit modeling of the variation in the mean and the covariance. Usual Euclidean metrics are not appropriate, and we therefore introduce the concept of d-covariance, which allows consistent estimation both in space and at individual locations. We then propose spatial smoothing for these objects which accounts for the possibly nonconvex geometry of the domain of interest. We apply the proposed method to data from the spoken part of the British National Corpus, deposited at the British Library, London, and we produce maps of the dialect variation over Great Britain. In addition, the methods allow for acoustic reconstruction across the domain of interest, allowing researchers to listen to the statistical analysis. Supplementary materials for this article are available online

Tests for separability in nonparametric covariance operators of random surfaces

The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional {data analysis} settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full non-separable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.Research Supported by EPSRC grant EP/K021672/2.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by the Institute of Mathematical Statistics

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II

Let $L=\Delta-\nabla\varphi\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $d\mu=e^{-\varphi}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla \varphi|\leq\theta$ and $\infty$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975) 289-297).Comment: Final version. The original proof of Theorem 2.1 using Li-Yau gradient estimate method has been moved to the appendix. The new proof is simple and direc

Micro-CT imaging of onchocerca infection of simulium damnosum s.l. blackflies and comparison of the peritrophic membrane thickness of forest and savannah flies

Onchocerciasis is a neglected tropical disease (NTD) caused by Onchocerca Diesing 1841 (Spirurida: Onchocercidae) nematodes transmitted by blackflies. It is associated with poverty and imposes a significant health, welfare and economic burden on many tropical countries. Current methods to visualize infections within the vectors rely on invasive methods. However, using micro-computed tomography techniques, without interference from physical tissue manipulation, we visualized in three dimensions for the first time an L1 larva of an Onchocerca species within the thoracic musculature of a blackfly, Simulium damnosum s.l. Theobald 1903 (Diptera: Simuliidae), naturally infected in Ghana. The possibility that thicker peritrophic membranes in savannah flies could account for their lower parasite loads was not supported, but there were limits to our analysis. While there were no statistically significant differences between the mean thicknesses of the peritrophic membranes, in the anterior, dorsal and ventral regions, of forest and savannah blackflies killed 34–48min after a blood-meal, the thickness of the peritrophicmembrane in the posterior region could not be measured. Micro-computed tomography has the potential to provide novel information on many other parasite/vector systems and impactful images for public engagement in health education

Distances and inference for covariance operators

A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance operator itself is an object of interest for statistical analysis. Distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable for functional data. In particular, an infinite dimensional analogue of the Procrustes size and shape distance is developed. The convergence of the finite dimensional approximations to the infinite dimensional distance metrics is also shown. For inference, a Fréchet estimator of both the covariance operator itself, and the average covariance operator is introduced. A permutation procedure to test the equality of the covariance operator between two groups is also considered. Additionally, the issue of using such distances for extrapolation to make predictions is explored. As an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross-linguistic dependence as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the Romance languages. 2