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12
Matrix positivity preservers in fixed dimension
, 2015
"... A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomi ..."
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A classical theorem of I.J. Schoenberg characterizes functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size. Obtaining similar characterizations in fixed dimension is intricate. In this note, we provide a solution to this problem in the polynomial case. As consequences, we derive tight linear matrix inequalities for Hadamard powers of positive semidefinite matrices, and a sharp asymptotic bound for the matrix cube problem involving Hadamard powers.
Validity of covariance models for the analysis of geographical variation
"... 1. Due to the availability of large molecular datasets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parametrised biological models. 2. We focus here on a class of parametric covariance models that received sustained attent ..."
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1. Due to the availability of large molecular datasets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parametrised biological models. 2. We focus here on a class of parametric covariance models that received sustained attention lately and show that the conditions under which they are valid mathematical models have been overlooked so far. 3. We provide rigorous results for the construction of valid covariance models in this family. 4. We also outline how to construct alternative covariance models for the analysis of geographical variation that are both mathematically well behaved and easily implementable. 5. The full R code to reproduce the numerical analysis is available from
Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres
"... Abstract. Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an L2positive definite and zonal kernel on the unit sphere of Cq in order that the kernel can be recovered as a generalized convolution root of an equa ..."
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Abstract. Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an L2positive definite and zonal kernel on the unit sphere of Cq in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel. Key words: positive definiteness; zonal kernels; recovery formula; convolution roots; Zernike or disc polynomials 2010 Mathematics Subject Classification: 33C55; 41A35; 41A63; 42A82; 42A85 1
Covariance functions for mean square differentiable processes on spheres
"... Many applications in spatial statistics involve data observed over large regions on the Earth’s surface. There is a large statistical literature devoted to covariance functions capable of modeling the degree of smoothness in data on Euclidean spaces. We adapt some of this work to covariance functio ..."
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Many applications in spatial statistics involve data observed over large regions on the Earth’s surface. There is a large statistical literature devoted to covariance functions capable of modeling the degree of smoothness in data on Euclidean spaces. We adapt some of this work to covariance functions for processes on spheres, where the natural distance is great circle distance. In doing so, we define the notion of mean square differentiable processes on spheres and give necessary and sufficient conditions for an isotropic covariance function on a sphere to correspond to an m times mean square differentiable process. These conditions imply that if a process on a Euclidean space is restricted to a sphere of lower dimension, the process will retain its mean square differentiability properties. The restriction requires the covariance function to take Euclidean distance as its argument. To address the issue of whether covariance functions using Euclidean distance result in poorly fitting models, we introduce an analog to the Matérn covariance function that is valid on spheres with great circle distance metric, as the usual Matérn that is only generally valid with Euclidean distance. These covariance functions are compared with several others in applications involving satellite and climate model data. 1