Searcharxiv⌕ Search

arXiv subjects

Tasmin L. Symons

Publications and source records attributed to Tasmin L. Symons.

8 recordsLinked to original sources
arXiv2021

Gaussian random fields: with and without covariances

We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration matrices).

math.PR↗
arXiv2021

Integral Representations of Ultraspherical Polynomials II

In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from a theoretical point of view, it is not numerically satisfactory as it involves polynomials of high degree, which are numerically unstable. Here we sum this series to obtain an integral, which is numerically tractable.

math.CA↗
arXiv2020

Spatiotemporal mapping of malaria prevalence in Madagascar using routine surveillance and health survey data

Malaria transmission in Madagascar is highly heterogeneous, exhibiting spatial, seasonal and long-term trends. Previous efforts to map malaria risk in Madagascar used prevalence data from Malaria Indicator Surveys. These cross-sectional surveys, conducted during the high transmission season most recently in 2013 and 2016, provide nationally representative prevalence data but cover relatively short time frames. Conversely, monthly case data are collected at health facilities but suffer from biases, including incomplete reporting. We combined survey and case data to make monthly maps of prevalence between 2013 and 2016. Health facility catchments were estimated and incidence surfaces, environmental and socioeconomic covariates, and survey data informed a Bayesian prevalence model. Prevalence estimates were consistently high in the coastal regions and low in the highlands. Prevalence was lowest in 2014 and peaked in 2015, highlighting the importance of estimates between survey years. Seasonality was widely observed. Similar multi-metric approaches may be applicable across sub-Saharan Africa.

stat.AP↗
arXiv2019

Gaussian random fields on the sphere and sphere cross line

We review the Dudley integral for the Belyaev dichotomy applied to Gaussian processes on spheres, and discuss the approximate (or restricted) continuity of paths in the discontinuous case. We discuss also the spatio-temporal case, of sphere cross line. In the continuous case, we investigate the link between the smoothness of paths and the decay rate of the angular power spectrum, following Tauberian work of the first author, Malyarenko, and Lang and Schwab.

math.PR↗
arXiv2017

Probability, Statistics and Planet Earth, II:The Bochner-Godement theorem for symmetric spaces

The Bochner-Godement theorem extends the classical Bochner and Bochner-Schoenberg theorems from the context of Euclidean spaces and spheres to general symmetric spaces. We show how it also includes recent results on products of symmetric spaces: the Berg-Porcu theorem (sphere cross line), and the Guella-Menegatto-Peron theorem (products of spheres), and discuss related results and applications.

math.CA↗
arXiv2017

Probability, Statistics and Planet Earth, I: Geotemporal covariances

The study of covariances (or positive definite functions) on the sphere (the Earth, in our motivation) goes back to Bochner and Schoenberg (1940--42) and to the first author (1969, 1973), among others. Extending to the geotemporal case (sphere cross line, for position and time) was for a long time an obstacle to geostatistical modelling. The characterisation question here was raised by the authors and Mijatović in 2016, and answered by Berg and Porcu in 2017. Extensions to multiple products (of spheres and lines) follows similarly (Guella, Menegatto and Peron, 2016). We survey results of this type, and related applications e.g. in numerical weather prediction.

stat.ME↗
arXiv2016

Brownian Manifolds, Negative Type and Geo-temporal Covariances

We survey Brownian manifolds -- manifolds that can parametrise Brownian motion -- and those that cannot. We consider covariances of space-time processes, particularly those when space is the sphere -- geo-temporal processes. There are connections with functions of negative type.

math.PR↗