Searcharxiv⌕ Search

arXiv subjects

Finn Lindgren

Publications and source records attributed to Finn Lindgren.

At least 37 records · Page 2Linked to original sources
arXiv2022

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.

stat.ME↗
arXiv2021

Ranking earthquake forecasts using proper scoring rules: Binary events in a low probability environment

Operational earthquake forecasting for risk management and communication during seismic sequences depends on our ability to select an optimal forecasting model. To do this, we need to compare the performance of competing models with each other in prospective forecasting mode, and to rank their performance using a fair, reproducible and reliable method. The Collaboratory for the Study of Earthquake Predictability (CSEP) conducts such prospective earthquake forecasting experiments around the globe. One metric that has been proposed to rank competing models is the Parimutuel Gambling score, which has the advantage of allowing alarm-based (categorical) forecasts to be compared with probabilistic ones. Here we examine the suitability of this score for ranking competing earthquake forecasts. First, we prove analytically that this score is in general improper, meaning that, on average, it does not prefer the model that generated the data. Even in the special case where it is proper, we show it can still be used in an improper way. Then, we compare its performance with two commonly-used proper scores (the Brier and logarithmic scores), taking into account the uncertainty around the observed average score. We estimate the confidence intervals for the expected score difference which allows us to define if and when a model can be preferred. Our findings suggest the Parimutuel Gambling score should not be used to distinguishing between multiple competing forecasts. They also enable a more rigorous approach to distinguish between the predictive skills of candidate forecasts in addition to their rankings.

stat.AP↗
arXiv2020

Spatial 3D Matérn priors for fast whole-brain fMRI analysis

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the proposed spatial priors have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.

stat.ME↗
arXiv2018

Spatial modelling with R-INLA: A review

Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically-sized datasets from scratch is time-consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R-INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R-INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matérn Gaussian random fields. In this review, we discuss the large success of spatial modelling with R-INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight-forward approaches for non-stationary spatial models and non-separable space-time models.

stat.ME↗
arXiv2018

A Case Study Competition Among Methods for Analyzing Large Spatial Data

The Gaussian process is an indispensable tool for spatial data analysts. The onset of the "big data" era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various alternatives to the full Gaussian process that are more amenable to handling big spatial data have been proposed. These modern methods often exploit low rank structures and/or multi-core and multi-threaded computing environments to facilitate computation. This study provides, first, an introductory overview of several methods for analyzing large spatial data. Second, this study describes the results of a predictive competition among the described methods as implemented by different groups with strong expertise in the methodology. Specifically, each research group was provided with two training datasets (one simulated and one observed) along with a set of prediction locations. Each group then wrote their own implementation of their method to produce predictions at the given location and each which was subsequently run on a common computing environment. The methods were then compared in terms of various predictive diagnostics. Supplementary materials regarding implementation details of the methods and code are available for this article online.

stat.ME↗
arXiv2017

Efficient Covariance Approximations for Large Sparse Precision Matrices

The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the covariance matrix, such as the marginal variances, which may be non-trivial to obtain when the dimension is large. This paper introduces a fast Rao-Blackwellized Monte Carlo sampling based method for efficiently approximating selected elements of the covariance matrix. The variance and confidence bounds of the approximations can be precisely estimated without additional computational costs. Furthermore, a method that iterates over subdomains is introduced, and is shown to additionally reduce the approximation errors to practically negligible levels in an application on functional magnetic resonance imaging data. Both methods have low memory requirements, which is typically the bottleneck for competing direct methods.

stat.CO↗
arXiv2017

Constructing Priors that Penalize the Complexity of Gaussian Random Fields

Priors are important for achieving proper posteriors with physically meaningful covariance structures for Gaussian random fields (GRFs) since the likelihood typically only provides limited information about the covariance structure under in-fill asymptotics. We extend the recent Penalised Complexity prior framework and develop a principled joint prior for the range and the marginal variance of one-dimensional, two-dimensional and three-dimensional Matérn GRFs with fixed smoothness. The prior is weakly informative and penalises complexity by shrinking the range towards infinity and the marginal variance towards zero. We propose guidelines for selecting the hyperparameters, and a simulation study shows that the new prior provides a principled alternative to reference priors that can leverage prior knowledge to achieve shorter credible intervals while maintaining good coverage. We extend the prior to a non-stationary GRF parametrized through local ranges and marginal standard deviations, and introduce a scheme for selecting the hyperparameters based on the coverage of the parameters when fitting simulated stationary data. The approach is applied to a dataset of annual precipitation in southern Norway and the scheme for selecting the hyperparameters leads to concervative estimates of non-stationarity and improved predictive performance over the stationary model.

stat.ME↗
arXiv2017

Calculating probabilistic excursion sets and related quantities using excursions

The R software package excursions contains methods for calculating probabilistic excursion sets, contour credible regions, and simultaneous confidence bands for latent Gaussian stochastic processes and fields. It also contains methods for uncertainty quantification of contour maps and computation of Gaussian integrals. This article describes the theoretical and computational methods used in the package. The main functions of the package are introduced and two examples illustrate how the package can be used.

stat.CO↗
arXiv2016

Quantifying the uncertainty of contour maps

Contour maps are widely used to display estimates of spatial fields. Instead of showing the estimated field, a contour map only shows a fixed number of contour lines for different levels. However, despite the ubiquitous use of these maps, the uncertainty associated with them has been given a surprisingly small amount of attention. We derive measures of the statistical uncertainty, or quality, of contour maps, and use these to decide an appropriate number of contour lines, that relates to the uncertainty in the estimated spatial field. For practical use in geostatistics and medical imaging, computational methods are constructed, that can be applied to Gaussian Markov random fields, and in particular be used in combination with integrated nested Laplace approximations for latent Gaussian models. The methods are demonstrated on simulated data and an application to temperature estimation is presented.

stat.ME↗
arXiv2015

Going off grid: Computationally efficient inference for log-Gaussian Cox processes

This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making novel use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, while an approximation based on a counting process on a partition of the domain only achieves first-order convergence. The given results improve on the general theory of convergence of the stochastic partial differential equation models, introduced by Lindgren et al. (2011). The new method is demonstrated on a standard point pattern data set and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of Chakraborty et al. (2011). The second extension constructs a log-Gaussian Cox process on the world's oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

stat.CO↗
arXiv2015

Does non-stationary spatial data always require non-stationary random fields?

A stationary spatial model is an idealization and we expect that the true dependence structures of physical phenomena are spatially varying, but how should we handle this non-stationarity in practice? We study the challenges involved in applying a flexible non-stationary model to a dataset of annual precipitation in the conterminous US, where exploratory data analysis shows strong evidence of a non-stationary covariance structure. The aim of this paper is to investigate the modelling pipeline once non-stationarity has been detected in spatial data. We show that there is a real danger of over-fitting the model and that careful modelling is necessary in order to properly account for varying second-order structure. In fact, the example shows that sometimes non-stationary Gaussian random fields are not necessary to model non-stationary spatial data.

stat.ME↗
arXiv2015

Estimation of a non-stationary model for annual precipitation in southern Norway using replicates of the spatial field

Estimation of stationary dependence structure parameters using only a single realisation of the spatial process, typically leads to inaccurate estimates and poorly identified parameters. A common way to handle this is to fix some of the parameters, or within the Bayesian framework, impose prior knowledge. In many applied settings, stationary models are not flexible enough to model the process of interest, thus non-stationary spatial models are used. However, more flexible models usually means more parameters, and the identifiability problem becomes even more challenging. We investigate aspects of estimation of a Bayesian non-stationary spatial model for annual precipitation using observations from multiple years. The model contains replicates of the spatial field, which increases precision of the estimates and makes them less prior sensitive. Using R-INLA, we analyse precipitation data from southern Norway, and investigate statistical properties of the replicate model in a simulation study. The non-stationary spatial model we explore belongs to a recently introduced class of stochastic partial differential equation (SPDE) based spatial models. This model class allows for non-stationary models with explanatory variables in the dependence structure. We derive conditions to facilitate prior specification for these types of non-stationary spatial models.

stat.ME↗
arXiv2014

Exploring a New Class of Non-stationary Spatial Gaussian Random Fields with Varying Local Anisotropy

Gaussian random fields (GRFs) constitute an important part of spatial modelling, but can be computationally infeasible for general covariance structures. An efficient approach is to specify GRFs via stochastic partial differential equations (SPDEs) and derive Gaussian Markov random field (GMRF) approximations of the solutions. We consider the construction of a class of non-stationary GRFs with varying local anisotropy, where the local anisotropy is introduced by allowing the coefficients in the SPDE to vary with position. This is done by using a form of diffusion equation driven by Gaussian white noise with a spatially varying diffusion matrix. This allows for the introduction of parameters that control the GRF by parametrizing the diffusion matrix. These parameters and the GRF may be considered to be part of a hierarchical model and the parameters estimated in a Bayesian framework. The results show that the use of an SPDE with non-constant coefficients is a promising way of creating non-stationary spatial GMRFs that allow for physical interpretability of the parameters, although there are several remaining challenges that would need to be solved before these models can be put to general practical use.

stat.ME↗
arXiv2013

A skew Gaussian decomposable graphical model

This paper propose a novel decomposable graphical model to accommodate skew Gaussian graphical models. We encode conditional independence structure among the components of the multivariate closed skew normal random vector by means of a decomposable graph and so that the pattern of zero off-diagonal elements in the precision matrix corresponds to the missing edges of the given graph. We present conditions that guarantee the propriety of the posterior distributions under the standard noninformative priors for mean vector and precision matrix, and a proper prior for skewness parameter. The identifiability of the parameters is investigated by a simulation study. Finally, we apply our methodology to two data sets.

stat.ME↗
arXiv2013

Multivariate Gaussian Random Fields Using Systems of Stochastic Partial Differential Equations

In this paper a new approach for constructing \emph{multivariate} Gaussian random fields (GRFs) using systems of stochastic partial differential equations (SPDEs) has been introduced and applied to simulated data and real data. By solving a system of SPDEs, we can construct multivariate GRFs. On the theoretical side, the notorious requirement of non-negative definiteness for the covariance matrix of the GRF is satisfied since the constructed covariance matrices with this approach are automatically symmetric positive definite. Using the approximate stochastic weak solutions to the systems of SPDEs, multivariate GRFs are represented by multivariate Gaussian \emph{Markov} random fields (GMRFs) with sparse precision matrices. Therefore, on the computational side, the sparse structures make it possible to use numerical algorithms for sparse matrices to do fast sampling from the random fields and statistical inference. Therefore, the \emph{big-n} problem can also be partially resolved for these models. These models out-preform existing multivariate GRF models on a commonly used real dataset.

stat.ME↗
arXiv2013

Multivariate Gaussian Random Fields with Oscillating Covariance Functions using Systems of Stochastic Partial Differential Equations

In this paper we propose a new approach for constructing \emph{multivariate} Gaussian random fields (GRFs) with oscillating covariance functions through systems of stochastic partial differential equations (SPDEs). We discuss how to build systems of SPDEs that introduces oscillation characteristics in the covariance functions of the multivariate GRFs. By choosing different parametrization of the equations, some GRFs can be made with oscillating covariance functions but other fields can have Matérn covariance functions or close to Matérn covariance functions. The multivariate GRFs constructed by solving the systems of SPDEs automatically fulfill the hard requirement of nonnegative definiteness for the covariance functions. The approximate weak solutions to the systems of SPDEs are used to represent the multivariate GRFs by multivariate Gaussian \emph{Markov} random fields (GMRFs). Since the multivariate GMRFs have sparse precision matrices (inverse of the covariance matrices), numerical algorithms for sparse matrices can be applied to the precision matrices for sampling and inference. Thus from a computational point of view, the \emph{big-n} problem can be partially solved with these types of models. Another advantage of the method is that the oscillation in the covariance function can be controlled directly by the parameters in the system of SPDEs. We show how to use this proposed approach with simulated data and real data examples.

stat.ME↗
arXiv2013

Non-stationary Spatial Modelling with Applications to Spatial Prediction of Precipitation

A non-stationary spatial Gaussian random field (GRF) is described as the solution of an inhomogeneous stochastic partial differential equation (SPDE), where the covariance structure of the GRF is controlled by the coefficients in the SPDE. This allows for a flexible way to vary the covariance structure, where intuition about the resulting structure can be gained from the local behaviour of the differential equation. Additionally, computations can be done with computationally convenient Gaussian Markov random fields which approximate the true GRFs. The model is applied to a dataset of annual precipitation in the conterminous US. The non-stationary model performs better than a stationary model measured with both CRPS and the logarithmic scoring rule.

stat.ME↗