arXiv subjects

David Bolin

Publications and source records attributed to David Bolin.

At least 73 records · Page 4Linked to original sources

arXiv2019

Joint spatial modeling of significant wave height and wave period using the SPDE approach

The ocean wave distribution in a specific region of space and time is described by its sea state. Knowledge about the sea states a ship encounters on a journey can be used to assess various parameters of risk and wear associated with the journey. Two important characteristics of the sea state are the significant wave height and mean wave period. We propose a joint spatial model of these two quantities on the north Atlantic ocean. The model describes the distribution of the logarithm of the two quantities as a bivariate Gaussian random field. This random field is modeled as a solution to a system of coupled stochastic partial differential equations. The bivariate random field can model a wide variety of non-stationary anisotropy and allows for arbitrary, and different, differentiability for the two marginal fields. The parameters of the model are estimated on data of the north Atlantic using a stepwise maximum likelihood method. The fitted model is used to derive the distribution of accumulated fatigue damage for a ship sailing a transatlantic route. Also, a method for estimating the risk of capsizing due to broaching-to, based on the joint distribution of the two sea state characteristics, is investigated. The risks are calculated for a transatlantic route between America and Europe using both data and the fitted model. The results show that the model compares well with observed data. Also, it shows that the bivariate model is needed and cannot simply be approximated by a model of significant wave height alone.

stat.AP↗

arXiv2018

Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford-Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.

math.NA↗

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

Linear Mixed-Effects Models for Non-Gaussian Repeated Measurement Data

We consider the analysis of continuous repeated measurement outcomes that are collected through time, also known as longitudinal data. A standard framework for analysing data of this kind is a linear Gaussian mixed-effects model within which the outcome variable can be decomposed into fixed-effects, time-invariant and time-varying random-effects, and measurement noise. We develop methodology that, for the first time, allows any combination of these stochastic components to be non-Gaussian, using multivariate Normal variance-mean mixtures. We estimate parameters by max- imum likelihood, implemented with a novel, computationally efficient stochastic gradient algorithm. We obtain standard error estimates by inverting the observed Fisher-information matrix, and obtain the predictive distributions for the random-effects in both filtering (conditioning on past and current data) and smoothing (conditioning on all data) contexts. To implement these procedures, we intro- duce an R package, ngme. We re-analyse two data-sets, from cystic fibrosis and nephrology research, that were previously analysed using Gaussian linear mixed effects models.

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arXiv2018

Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in $\mathbb{R}^d$ is considered. The differential operator is given by the fractional power $L^β$, $β\in(0,1)$, of an integer order elliptic differential operator $L$ and is therefore non-local. Its inverse $L^{-β}$ is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator $L^{-β}$ is approximated by a weighted sum of non-fractional resolvents $(I + t_j^2 L)^{-1}$ at certain quadrature nodes $t_j>0$. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for $L=κ^2-Δ$, $κ> 0$, with homogeneous Dirichlet boundary conditions on the unit cube $(0,1)^d$ in $d=1,2,3$ spatial dimensions for varying $β\in(0,1)$ attest the theoretical results.

math.NA↗

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

Level set Cox processes

The log-Gaussian Cox process (LGCP) is a popular point process for modeling non-interacting spatial point patterns. This paper extends the LGCP model to handle data exhibiting fundamentally different behaviors in different subregions of the spatial domain. The aim of the analyst might be either to identify and classify these regions, to perform kriging, or to derive some properties of the parameters driving the random field in one or several of the subregions. The extension is based on replacing the latent Gaussian random field in the LGCP by a latent spatial mixture model. The mixture model is specified using a latent, categorically valued, random field induced by level set operations on a Gaussian random field. Conditional on the classification, the intensity surface for each class is modeled by a set of independent Gaussian random fields. This allows for standard stationary covariance structures, such as the Matérn family, to be used to model Gaussian random fields with some degree of general smoothness but also occasional and structured sharp discontinuities. A computationally efficient MCMC method is proposed for Bayesian inference and we show consistency of finite dimensional approximations of the model. Finally, the model is fitted to point pattern data derived from a tropical rainforest on Barro Colorado island, Panama. We show that the proposed model is able to capture behavior for which inference based on the standard LGCP is biased.

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arXiv2017

A three-dimensional statistical model for imaged microstructures of porous polymer films

A thresholded Gaussian random field model is developed for the microstructure of porous materials. Defining the random field as a solution to stochastic partial differential equation allows for flexible modelling of non-stationarities in the material and facilitates computationally efficient methods for simulation and model fitting. A Markov Chain Monte Carlo algorithm is developed and used to fit the model to three-dimensional confocal laser scanning microscopy images. The methods are applied to study a porous ethylcellulose/hydroxypropylcellulose polymer blend that is used as a coating to control drug release from pharmaceutical tablets. The aim is to investigate how mass transport through the material depends on the microstructure. We derive a number of goodness-of-fit measures based on numerically calculated diffusion through the material. These are used in combination with measures that characterize the geometry of the pore structure to assess model fit. The model is found to fit stationary parts of the material well.

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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↗

arXiv2017

A Bayesian General Linear Modeling Approach to Cortical Surface fMRI Data Analysis

Cortical surface fMRI (cs-fMRI) has recently grown in popularity versus traditional volumetric fMRI, as it allows for more meaningful spatial smoothing and is more compatible with the common assumptions of isotropy and stationarity in Bayesian spatial models. However, as no Bayesian spatial model has been proposed for cs-fMRI data, most analyses continue to employ the classical, voxel-wise general linear model (GLM) (Worsley and Friston 1995). Here, we propose a Bayesian GLM for cs-fMRI, which employs a class of sophisticated spatial processes to flexibly model latent activation fields. We use integrated nested Laplacian approximation (INLA), a highly accurate and efficient Bayesian computation technique (Rue et al. 2009). To identify regions of activation, we propose an excursions set method based on the joint posterior distribution of the latent fields, which eliminates the need for multiple comparisons correction. Finally, we address a gap in the existing literature by proposing a novel Bayesian approach for multi-subject analysis. The methods are validated and compared to the classical GLM through simulation studies and a motor task fMRI study from the Human Connectome Project. The proposed Bayesian approach results in smoother activation estimates, more accurate false positive control, and increased power to detect truly active regions.

stat.AP↗

arXiv2017

Comparison of hidden Markov chain models and hidden Markov random field models in estimation of computed tomography images

There is an interest to replace computed tomography (CT) images with magnetic resonance (MR) images for a number of diagnostic and therapeutic workflows. In this article, predicting CT images from a number of magnetic resonance imaging (MRI) sequences using regression approach is explored. Two principal areas of application for estimated CT images are dose calculations in MRI-based radiotherapy treatment planning and attenuation correction for positron emission tomography (PET)/MRI. The main purpose of this work is to investigate the performance of hidden Markov (chain) models (HMMs) in comparison to hidden Markov random field (HMRF) models when predicting CT images of head. Our study shows that HMMs have clear advantages over HMRF models in this particular application. Obtained results suggest that HMMs deserve a further study for investigating their potential in modelling applications where the most natural theoretical choice would be the class of HMRF models.

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arXiv2016

Fast Bayesian whole-brain fMRI analysis with spatial 3D priors

Spatial whole-brain Bayesian modeling of task-related functional magnetic resonance imaging (fMRI) is a great computational challenge. Most of the currently proposed methods therefore do inference in subregions of the brain separately or do approximate inference without comparison to the true posterior distribution. A popular such method, which is now the standard method for Bayesian single subject analysis in the SPM software, is introduced in Penny et al. (2005b). The method processes the data slice-by-slice and uses an approximate variational Bayes (VB) estimation algorithm that enforces posterior independence between activity coefficients in different voxels. We introduce a fast and practical Markov chain Monte Carlo (MCMC) scheme for exact inference in the same model, both slice-wise and for the whole brain using a 3D prior on activity coefficients. The algorithm exploits sparsity and uses modern techniques for efficient sampling from high-dimensional Gaussian distributions, leading to speed-ups without which MCMC would not be a practical option. Using MCMC, we are for the first time able to evaluate the approximate VB posterior against the exact MCMC posterior, and show that VB can lead to spurious activation. In addition, we develop an improved VB method that drops the assumption of independent voxels a posteriori. This algorithm is shown to be much faster than both MCMC and the original VB for large datasets, with negligible error compared to the MCMC posterior.

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arXiv2016

Whole-brain substitute CT generation using Markov random field mixture models

Computed tomography (CT) equivalent information is needed for attenuation correction in PET imaging and for dose planning in radiotherapy. Prior work has shown that Gaussian mixture models can be used to generate a substitute CT (s-CT) image from a specific set of MRI modalities. This work introduces a more flexible class of mixture models for s-CT generation, that incorporates spatial dependency in the data through a Markov random field prior on the latent field of class memberships associated with a mixture model. Furthermore, the mixture distributions are extended from Gaussian to normal inverse Gaussian (NIG), allowing heavier tails and skewness. The amount of data needed to train a model for s-CT generation is of the order of 100 million voxels. The computational efficiency of the parameter estimation and prediction methods are hence paramount, especially when spatial dependency is included in the models. A stochastic Expectation Maximization (EM) gradient algorithm is proposed in order to tackle this challenge. The advantages of the spatial model and NIG distributions are evaluated with a cross-validation study based on data from 14 patients. The study show that the proposed model enhances the predictive quality of the s-CT images by reducing the mean absolute error with 17.9%. Also, the distribution of CT values conditioned on the MR images are better explained by the proposed model as evaluated using continuous ranked probability scores.

stat.AP↗

arXiv2016

Statistical Modeling and Estimation of Censored Pathloss Data

Pathloss is typically modeled using a log-distance power law with a large-scale fading term that is log-normal. However, the received signal is affected by the dynamic range and noise floor of the measurement system used to sound the channel, which can cause measurement samples to be truncated or censored. If the information about the censored samples are not included in the estimation method, as in ordinary least squares estimation, it can result in biased estimation of both the pathloss exponent and the large scale fading. This can be solved by applying a Tobit maximum-likelihood estimator, which provides consistent estimates for the pathloss parameters. This letter provides information about the Tobit maximum-likelihood estimator and its asymptotic variance under certain conditions.

cs.IT↗

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↗

arXiv2016

Spatially adaptive covariance tapering

Covariance tapering is a popular approach for reducing the computational cost of spatial prediction and parameter estimation for Gaussian process models. However, tapering can have poor performance when the process is sampled at spatially irregular locations or when non-stationary covariance models are used. This work introduces an adaptive tapering method in order to improve the performance of tapering in these problematic cases. This is achieved by introducing a computationally convenient class of compactly supported non-stationary covariance functions, combined with a new method for choosing spatially varying taper ranges. Numerical experiments are used to show that the performance of both kriging prediction and parameter estimation can be improved by allowing for spatially varying taper ranges. However, although adaptive tapering outperforms regular tapering, simply dividing the data into blocks and ignoring the dependence between the blocks is often a better method for parameter estimation.

stat.CO↗

arXiv2016

Efficient adaptive MCMC through precision estimation

A novel adaptive Markov chain Monte Carlo algorithm is presented. The algorithm utilizes sparsity in the partial correlation structure of a density to efficiently estimate the covariance matrix through the Cholesky factor of the precision matrix. The algorithm also utilizes the sparsity to sample efficiently from both MALA and Metropolis Hasting random walk proposals. Further, an algorithm that estimates the partial correlation structure of a density is proposed. Combining this with the Cholesky factor estimation algorithm results in an efficient black-box AMCMC method that can be used for general densities with unknown dependency structure. The method is compared with regular empirical covariance adaption for two examples. In both examples, the proposed method's covariance estimates converge faster to the true covariance matrix and the computational cost for each iteration is lower.

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arXiv2013

Non-Gaussian Matérn fields with an application to precipitation modeling

The recently proposed non-Gaussian Matérn random field models, generated through Stochastic Partial differential equations (SPDEs), are extended by considering the class of Generalized Hyperbolic processes as noise forcings. The models are also extended to the standard geostatistical setting where irregularly spaced observations are modeled using measurement errors and covariates. A maximum likelihood estimation technique based on the Monte Carlo Expectation Maximization (MCEM) algorithm is presented, and it is shown how the model can be used to do predictions at unobserved locations. Finally, an application to precipitation data over the United States for two month in 1997 is presented, and the performance of the non-Gaussian models is compared with standard Gaussian and transformed Gaussian models through cross-validation.

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