Finn Lindgren

Multivariate spatio‐temporal modelling for assessing Antarctica's present‐day contribution to sea‐level rise

Antarctica is the worlds largest fresh-water reservoir, with the potential to raise sea levels by about 60 m. An ice sheet contributes to sea-level rise (SLR) when its rate of ice discharge and/or surface melting exceeds accumulation through snowfall. Constraining the contribution of the ice sheets to present-day SLR is vital both for coastal development and planning, and climate projections. Information on various ice sheet processes is available from several remote sensing data sets, as well as in situ data such as global positioning system data. These data have differing coverage, spatial support, temporal sampling and sensing characteristics, and thus, it is advantageous to combine them all in a single framework for estimation of the SLR contribution and the assessment of processes controlling mass exchange with the ocean. In this paper, we predict the rate of height change due to salient geophysical processes in Antarctica and use these to provide estimates of SLR contribution with associated uncertainties. We employ a multivariate spatio-temporal model, approximated as a Gaussian Markov random field, to take advantage of differing spatio-temporal properties of the processes to separate the causes of the observed change. The process parameters are estimated from geophysical models, while the remaining parameters are estimated using a Markov chain Monte Carlo scheme, designed to operate in a high-performance computing environment across multiple nodes. We validate our methods against a separate data set and compare the results to those from studies that invariably employ numerical model outputs directly. We conclude that it is possible, and insightful, to assess Antarcticas contribution without explicit use of numerical models. Further, the results obtained here can be used to test the geophysical numerical models for which in situ data are hard to obtain.

Constructing Priors that Penalize the Complexity of Gaussian Random Fields

ABSTRACT 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 penalized 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 penalizes complexity by shrinking the range toward infinity and the marginal variance toward 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 nonstationary GRF parameterized 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 conservative estimates of nonstationarity and improved predictive performance over the stationary model. Supplementary materials for this article are available online.

Point process models for spatio-temporal distance sampling data from a large-scale survey of blue whales

Distance sampling is a widely used method for estimating wildlife population abundance. The fact that conventional distance sampling methods are partly design-based constrains the spatial resolution at which animal density can be estimated using these methods. Estimates are usually obtained at survey stratum level. For an endangered species such as the blue whale, it is desirable to estimate density and abundance at a finer spatial scale than stratum. Temporal variation in the spatial structure is also important. We formulate the process generating distance sampling data as a thinned spatial point process and propose model-based inference using a spatial log-Gaussian Cox process. The method adopts a flexible stochastic partial differential equation (SPDE) approach to model spatial structure in density that is not accounted for by explanatory variables, and integrated nested Laplace approximation (INLA) for Bayesian inference. It allows simultaneous fitting of detection and density models and permits prediction of density at an arbitrarily fine scale. We estimate blue whale density in the Eastern Tropical Pacific Ocean from thirteen shipboard surveys conducted over 22 years. We find that higher blue whale density is associated with colder sea surface temperatures in space, and although there is some positive association between density and mean annual temperature, our estimates are consistent with no trend in density across years. Our analysis also indicates that there is substantial spatially structured variation in density that is not explained by available covariates.

Bayesian adaptive smoothing splines using stochastic differential equations

The smoothing spline is one of the most popular curve-fitting methods, partly because of empirical evidence supporting its effectiveness and partly because of its elegant mathematical formulation. However, there are two obstacles that restrict the use of smoothing spline in practical statistical work. Firstly, it becomes computationally prohibitive for large data sets because the number of basis functions roughly equals the sample size. Secondly, its global smoothing parameter can only provide constant amount of smoothing, which often results in poor performances when estimating inhomogeneous functions. In this work, we introduce a class of adaptive smoothing spline models that is derived by solving certain stochastic differential equations with finite element methods. The solution extends the smoothing parameter to a continuous data-driven function, which is able to capture the change of the smoothness of underlying process. The new model is Markovian, which makes Bayesian computation fast. A simulation study and real data example are presented to demonstrate the effectiveness of our method.

Quantifying the Uncertainty of Contour Maps

ABSTRACT 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, which 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.

Efficient Covariance Approximations for Large Sparse Precision Matrices

ABSTRACT 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 nontrivial to obtain when the dimension is large. This article 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.

Latent Gaussian random field mixture models

For many problems in geostatistics, land cover classification, and brain imaging the classical Gaussian process models are unsuitable due to sudden, discontinuous, changes in the data. To handle data of this type, we introduce a new model class that combines discrete Markov random fields (MRFs) with Gaussian Markov random fields. The model is defined as a mixture of several, possibly multivariate, Gaussian Markov random fields. For each spatial location, the discrete MRF determines which of the Gaussian fields in the mixture that is observed. This allows for the desired discontinuous changes of the latent processes, and also gives a probabilistic representation of where the changes occur spatially. By combining stochastic gradient minimization with sparse matrix techniques we obtain computationally efficient methods for both likelihood-based parameter estimation and spatial interpolation. The model is compared to Gaussian models and standard MRF models using simulated data and in application to upscaling of soil permeability data.

A skew Gaussian decomposable graphical model

This paper proposes 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 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.