arXiv subjects

Jake P. Grainger

Publications and source records attributed to Jake P. Grainger.

6 recordsLinked to original sources

arXiv2025

The partial K function

The K function and its related statistics have been an enduring tool in the analysis of spatial point processes, providing an easy to compute and interpret summary statistic for characterising the interactions between points of one type, or between two different types of points. In this paper, we introduce a partial K function, enabling us to account for some of the effects of the other point types when analysing point-point interactions. The partial K function we introduce reduces to the usual K function when the other points are independent of the points of interest and has a similar interpretation. Using examples, we demonstrate how the partial K function can unpick dependence between point types that would otherwise be hidden in the usual K function. We also discuss important bias correction steps and hyperparameter selection. In addition, we introduce an extension to account for other spatial covariates, and demonstrate the methodology on the Lansing Woods dataset.

stat.ME↗

arXiv2023

Spectral estimation for spatial point processes and random fields

Spatial variables can be observed in many different forms, such as regularly sampled random fields (lattice data), point processes, and randomly sampled spatial processes. Joint analysis of such collections of observations is clearly desirable, but complicated by the lack of an easily implementable analysis framework. We fill this gap by providing a multitaper analysis framework using coupled discrete and continuous data tapers, combined with the discrete Fourier transform for inference. Using this set of tools is important, as it forms the backbone for practical spectral analysis. In higher dimensions it is important not to be constrained to Cartesian product domains, and so we develop the methodology for spectral analysis using irregular domain data tapers, and the tapered discrete Fourier transform. We discuss its fast implementation, and the asymptotic as well as large finite domain properties. Estimators of partial association between different spatial processes are provided as are principled methods to determine their significance, and we demonstrate their practical utility on a large-scale ecological dataset.

stat.ME↗

arXiv2023

Visualizing the Wavenumber Content of a Point Pattern

Spatial point patterns are a commonly recorded form of data in ecology, medicine, astronomy, criminology, epidemiology and many other application fields. One way to understand their second order dependence structure is via their spectral density function. However, unlike time series analysis, for point patterns such approaches are currently underutilized. In part, this is because the interpretation of the spectral representation of the underlying point processes is challenging. In this paper, we demonstrate how to band-pass filter point patterns, thus enabling us to explore the spectral representation of point patterns in space by isolating the signal corresponding to certain sets of wavenumbers.

stat.AP↗

arXiv2022

What is the Fourier Transform of a Spatial Point Process?

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fourth order moments of the Fourier transform using Campbell's theorem. Third we determine how to implement tapering, an important component for spectral analysis of other stochastic processes. Fourth we answer the question of how to produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process.

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arXiv2022

A multivariate pseudo-likelihood approach to estimating directional ocean wave models

Ocean buoy data in the form of high frequency multivariate time series are routinely recorded at many locations in the world's oceans. Such data can be used to characterise the ocean wavefield, which is important for numerous socio-economic and scientific reasons. This characterisation is typically achieved by modelling the frequency-direction spectrum, which decomposes spatiotemporal variability by both frequency and direction. State-of-the-art methods for estimating the parameters of such models do not make use of the full spatiotemporal content of the buoy observations due to unnecessary assumptions and smoothing steps. We explain how the multivariate debiased Whittle likelihood can be used to jointly estimate all parameters of such frequency-direction spectra directly from the recorded time series. When applied to North Sea buoy data, debiased Whittle likelihood inference reveals smooth evolution of spectral parameters over time. We discuss challenging practical issues including model misspecification, and provide guidelines for future application of the method.

stat.AP↗

arXiv2021

Estimating the parameters of ocean wave spectra

Wind-generated waves are often treated as stochastic processes. There is particular interest in their spectral density functions, which are often expressed in some parametric form. Such spectral density functions are used as inputs when modelling structural response or other engineering concerns. Therefore, accurate and precise recovery of the parameters of such a form, from observed wave records, is important. Current techniques are known to struggle with recovering certain parameters, especially the peak enhancement factor and spectral tail decay. We introduce an approach from the statistical literature, known as the de-biased Whittle likelihood, and address some practical concerns regarding its implementation in the context of wind-generated waves. We demonstrate, through numerical simulation, that the de-biased Whittle likelihood outperforms current techniques, such as least squares fitting, both in terms of accuracy and precision of the recovered parameters. We also provide a method for estimating the uncertainty of parameter estimates. We perform an example analysis on a data-set recorded off the coast of New Zealand, to illustrate some of the extra practical concerns that arise when estimating the parameters of spectra from observed data.

stat.AP↗