Isotropic covariance functions on graphs and their edges

Anderes, Ethan; Møller, Jesper; Rasmussen, Jakob Gulddahl

doi:10.1214/19-aos1896

Cited by 27 publications

(52 citation statements)

References 27 publications

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“…We expect that the same techniques could be applied on a linear network. We have not attempted this because there are some unresolved difficulties in constructing Cox models on a linear network (Baddeley et al 2017;Anderes et al 2020). Gibbs point process models on a network are in the early stages of development (van Lieshout 2018).…”

Section: Discussionmentioning

confidence: 99%

Variable selection using penalised likelihoods for point patterns on a linear network

Rakshit

McSwiggan

Nair

et al. 2021

Aus NZ J of Statistics

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Motivated by the analysis of a comprehensive database of road traffic accidents, we investigate methods of variable selection for spatial point process models on a linear network. The original data may include explanatory spatial covariates, such as road curvature, and 'mark' variables attributed to individual accidents, such as accident severity. The treatment of mark variables is new. Variable selection is applied to the canonical covariates, which may include spatial covariate effects, mark effects and mark-covariate interactions. We approximate the likelihood of the point process model by that of a generalised linear model, in such a way that spatial covariates and marks are both associated with canonical covariates. We impose a convex penalty on the log likelihood, principally the elastic-net penalty, and maximise the penalised loglikelihood by cyclic coordinate ascent. A simulation study compares the performances of the lasso, ridge regression and elastic-net methods of variable selection on their ability to select variables correctly, and on their bias and standard error. Standard techniques for selecting the regularisation parameter often yielded unsatisfactory results. We propose two new rules for selecting which are designed to have better performance. The methods are tested on a small dataset on crimes in a Chicago neighbourhood, and applied to a large dataset of road traffic accidents in Western Australia.

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Section: Discussionmentioning

confidence: 99%

Variable selection using penalised likelihoods for point patterns on a linear network

Rakshit

McSwiggan

Nair

et al. 2021

Aus NZ J of Statistics

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“…This highlights the fact that dealing with DPPs in non-Euclidean spaces S can be quite challenging (cf. Anderes et al, 2020).…”

Section: Determinantal Point Processesmentioning

confidence: 99%

Statistical learning and cross-validation for point processes

Cronie¹,

Moradi²,

Biscio³

2021

Preprint

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This paper presents the first general (supervised) statistical learning framework for point processes in general spaces. Our approach is based on the combination of two new concepts, which we define in the paper: i) bivariate innovations, which are measures of discrepancy/prediction-accuracy between two point processes, and ii) point process cross-validation (CV), which we here define through point process thinning. The general idea is to carry out the fitting by predicting CV-generated validation sets using the corresponding training sets; the prediction error, which we minimise, is measured by means of bivariate innovations. Having established various theoretical properties of our bivariate innovations, we study in detail the case where the CV procedure is obtained through independent thinning and we apply our statistical learning methodology to three typical spatial statistical settings, namely parametric intensity estimation, nonparametric intensity estimation and Papangelou conditional intensity fitting. Aside from deriving theoretical properties related to these cases, in each of them we numerically show that our statistical learning approach outperforms the state of the art in terms of mean (integrated) squared error.

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“…Moreover, in Euclidean space, a covariance function is called isotropic if it is a function of the Euclidean norm of the difference between locations. Unlike its counterpart in Euclidean space, the definition of stationarity is less clear on networks (Anderes et al, 2020). Nevertheless, a space-time covariance function is said to be isotropic within components if C(s 1 , s…”

Section: Parametric Space-time Covariance Modelsmentioning

confidence: 99%

“…Covariance functions receive special attention due to the fact that a Gaussian random process is completely determined by its first-and second-order moments (Porcu et al, 2020). Although a broad range of classes of space-time covariance models are available in Euclidean space (De Iaco et al, 2013) and a thorough review has recently been given by Porcu et al (2020), corresponding results for pure spatial linear networks are few and far between -a recent exception being Anderes et al (2020) -and space-time results on networks are practically non-existent. To fill this gap, in this article we develop Fourier-free space-time covariance functions on generalized linear networks, using space embedding and scale mixture approaches.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Space-Time Covariance Models on Networks with An Application on Streams

Tang¹,

Zimmerman²

2020

Preprint

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The second-order, small-scale dependence structure of a stochastic process defined in the space-time domain is key to prediction (or kriging). While great efforts have been dedicated to developing models for cases in which the spatial domain is either a finite-dimensional Euclidean space or a unit sphere, counterpart developments on a generalized linear network are practically non-existent. To fill this gap, we develop a broad range of parametric, non-separable space-time covariance models on generalized linear networks and then an important subgroup -Euclidean trees by the space embedding technique -in concert with the generalized Gneiting class of models and 1-symmetric characteristic functions in the literature, and the scale mixture approach.We give examples from each class of models and investigate the geometric features of these covariance functions near the origin and at infinity. We also show the linkage between different classes of space-time covariance models on Euclidean trees. We illustrate the use of models constructed by different methodologies on a daily stream temperature data set and compare model predictive performance by cross validation.

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Isotropic covariance functions on graphs and their edges

Cited by 27 publications

References 27 publications

Variable selection using penalised likelihoods for point patterns on a linear network

Variable selection using penalised likelihoods for point patterns on a linear network

Statistical learning and cross-validation for point processes

Space-Time Covariance Models on Networks with An Application on Streams

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