1,287 research outputs found
Permutation tests for the equality of covariance operators of functional data with applications to evolutionary biology
Get PDFIn this paper, we generalize the metric-based permutation test for the equality of covariance operators proposed by Pigoli et al. (2014) to the case of multiple samples of functional data. To this end, the non-parametric combination methodology of Pesarin and Salmaso (2010) is used to combine all the pairwise comparisons between samples into a global test. Different combining functions and permutation strategies are reviewed and analyzed in detail. The resulting test allows to make inference on the equality of the covariance operators of multiple groups and, if there is evidence to reject the null hypothesis, to identify the pairs of groups having different covariances. It is shown that, for some combining functions, step-down adjusting procedures are available to control for the multiple testing problem in this setting. The empirical power of this new test is then explored via simulations and compared with those of existing alternative approaches in different scenarios. Finally, the proposed methodology is applied to data from wheel running activity experiments, that used selective breeding to study the evolution of locomotor behavior in mice
Estimation of the mean for spatially dependent data belonging to a Riemannian manifold
No full textThe 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
Simulation and modeling of spatially correlated positive definite symmetric matrices
No full textThe statistical analysis of positive definite symmetric matrices is becoming increasingly important in many applications. The focus of this work is on spatially correlated random positive definite symmetric matrices. First, the empirical Riemannian semivariogram of a field of positive definite symmetric matrices is defined, using the Riemannian distance on the manifold of positive definite symmetric matrices. Then, we propose an estimator of the mean matrix which takes into account both the non Euclidean nature of the data and the spatial correlation. Simulated data are then used to evaluate
the performance of the proposed estimator
Kriging prediction for manifold-valued random fields
Get PDFThe 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
A Ticklish Problem
No full textAbstract
The book Mr. Tickle is a childhood favourite for many. But this fantastical story of a man with “extraordinary long arms” is helping statisticians derive a more realistic understanding of the differences in regional accents. By Marius A. Tirlea, Shahin Tavakoli, Davide Pigoli and John A. D. Aston</jats:p
Kriging Riemannian Data via Random Domain Decompositions
Get PDFData taking value on a Riemannian manifold and observed over a complex spatial domain are becoming more frequent in applications, for example, in environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the nonstationarity of the generating random process, the nonlinearity of the manifold, and the complex topology of the domain. In this article, we propose to use a random domain decomposition approach to estimate an ensemble of local models and then to aggregate the predictions of the local models through Fréchet averaging. The algorithm is introduced in complete generality and is valid for data belonging to any smooth Riemannian manifold but it is then described in details for the case of the manifold of positive definite matrices, the hypersphere and the Cholesky manifold. The predictive performances of the method are explored via simulation studies for covariance matrices and correlation matrices, where the Cholesky manifold geometry is used. Finally, the method is illustrated on an environmental dataset observed over the Chesapeake Bay (USA). Supplementary materials for this article are available online
Wavelets in functional data analysis : estimation of multidimensional curves and their derivatives
No full textA wavelet-based method is proposed to obtain accurate estimates of curves in more than one dimension and of their derivatives. By means of simulation studies, this novel method is compared to another locally-adaptive estimation technique for multidimensional functional data, based on free-knot regression splines. This comparison shows that the proposed method is particularly attractive when the curves to be estimated present strongly localized features. The multidimensional wavelet estimation method is thus applied to multi-lead electrocardiogram records, where strongly localized features are indeed expected
Mathematical foundations of functional Kriging in Hilbert spaces and Riemannian manifolds
Get PDFIn this chapter, we review the mathematical framework for spatial prediction (kriging) for complex data. We focus here on the approach developed within the area of Object-Oriented Spatial Statistics which grounds on the foundational idea that the atom of the geostatistical analysis is the entire data point, regardless of its complexity. This is seen as an indivisible unit rather than a collection of features, and accordingly embedded as a point within a space of objects, called feature space. We illustrate here the kriging methods when data belong to Hilbert space and Riemannian manifolds, in stationary or nonstationary settings and discuss the estimators that can be used for the mean and the covariance structure
Wavelet smoothing for curves in more than one dimension
No full textThe estimation of smooth functions from their noisy and discrete observation is the first step in Functional Data Analysis. The choice of the basis functions is crucial in the process, since its properties influence the subsequent analysis.
Wavelets offer functional basis systems with the property of localization both in space and frequency. We developed a procedure to use wavelet bases for the estimation of curves in more than one dimension, with a particular focus on the possibility of obtaining also estimates of curves derivatives from wavelet expansion. This method is tested on the estimation of centreline and radius of the Internal Carotid Artery (ICA) for patients suspected to be affected by cerebral aneuris
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