Davide Pigoli

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

Kriging prediction for manifold-valued random fields

K

Estimation of the mean for spatially dependent data belonging to a Riemannian manifold

-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.

Permutation tests for the equality of covariance operators of functional data with applications to evolutionary biology

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.

Research Data Supporting "Tests for separability in nonparametric covariance operators of random surfaces"

The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications. The aim of this work is to introduce models for spatial dependence among Riemannian data, with a special focus on the case of positive definite symmetric matrices. First, the Riemannian semivariogram of a field of positive definite symmetric matrices is defined. Then, we propose an estimator for the mean which considers both the non Euclidean nature of the data and their spatial correlation. Simulated data are used to evaluate the performance of the proposed estimator: taking into account spatial dependence leads to better estimates when observations are irregularly spaced in the region of interest. Finally, we address a meteorological problem, namely, the estimation of the covariance matrix between temperature and precipitation for the province of Quebec in Canada.

Distances and inference for covariance operators

A. Cabassi was supported by the MRC (project reference MC_UP_0801/1) and by a “Tesi all’estero” scholarship from Politecnico di Milano, Italy. The authors also wish to thank The Washington State University College of Arts and Sciences, Office of International Programs and Office of Research for travel grants to P.A. Carter.

The analysis of Acoustic Phonetic Data: exploring differences in the spoken Romance languages

#### Software - covsep-1.0.1.tar.gz : R package implementing the methodology of the paper. #### Data for reproducing the Numerical simulations: - reproduce-sims.R: R script to reproduce the simulation studies; WARNING: the simulations take a lot of time. - SIM2016a-SIM12feb.RData: results of the simulation studies, and parameters to reproduce them. - grid-sims2.RData: results of the simulation studies (Figure 5 and Figures S2 & S3 of the supplement), and parameters to reproduce them. - sim_function.R: function performing the simulation studies #### Data application: - Acoustic_Data_part1.RData: R workspace containing the preprocessed log-spectrograms considered in the phonetic application. (part 1) - Acoustic_Data_part2.RData: R workspace containing the preprocessed log-spectrograms considered in the phonetic application. (part 2) - Info_Acoustic_Data.txt: text file with information about the phonetic sounds. This includes the language, the word being pronounced and the gender of the speaker. - separability_Acoustic_Data.R: script to replicate the test for separability for the phonetic data as described in the manuscript.