David Bolin

Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping

A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the Gaussian Matern fields and a wide family of fields with oscillating covariance functions. Nonstationary covariance models are obtained by spatially varying the parameters in the SPDEs, and the model parameters are estimated using direct numerical optimization, which is more efficient than standard Markov Chain Monte Carlo procedures. The model class is used to estimate daily ozone maps using a large data set of spatially irregular global total column ozone data.

Fast estimation of spatially dependent temporal vegetation trends using Gaussian Markov random fields

There is a need for efficient methods for estimating trends in spatio-temporal Earth Observation data. A suitable model for such data is a space-varying regression model, where the regression coefficients for the spatial locations are dependent. A second order intrinsic Gaussian Markov Random Field prior is used to specify the spatial covariance structure. Model parameters are estimated using the Expectation Maximisation (EM) algorithm, which allows for feasible computation times for relatively large data sets. Results are illustrated with simulated data sets and real vegetation data from the Sahel area in northern Africa. The results indicate a substantial gain in accuracy compared with methods based on independent ordinary least squares regressions for the individual pixels in the data set. Use of the EM algorithm also gives a substantial performance gain over Markov Chain Monte Carlo-based estimation approaches.

A comparison between Markov approximations and other methods for large spatial data sets

The Matern covariance function is a popular choice for modeling dependence in spatial environmental data. Standard Matern covariance models are, however, often computationally infeasible for large data sets. Recent results for Markov approximations of Gaussian Matern fields based on Hilbert space approximations are extended using wavelet basis functions. Using a simulation-based study, these Markov approximations are compared with two of the most popular methods for computationally efficient model approximations, covariance tapering and the process convolution method. The methods are compared with respect to their computational properties when used for spatial prediction (kriging), and the results show that, for a given computational cost, the Markov methods have a substantial gain in accuracy compared with the other methods.

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

Abstract 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. HighlightsA fast method for Bayesian inference in task‐fMRI with spatial 3D priors is proposed.Sparse techniques for high‐dimensional Gaussian sampling give great speed‐ups.Using exact inference shows that SPMs variational Bayes can lead to false activity.An improved variational Bayesian method shows increased speed and accuracy.

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 is 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.

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 nonstationarities 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.

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.

Modeling the Polarimetric mm-Wave Propagation Channel Using Censored Measurements

This paper presents results based on polarimetric radio channel measurements at 60 GHz in a small room and in an empty, unfurnished, medium-sized room. The measurements in the small meeting room were performed using dual-polarized virtual antenna array elements at both the transmitter and receiver sides and includes LOS and NLOS scenarios. In the unfurnished room, a directional horn antenna was scanned in the azimuth plane and the Rx antenna was an omnidirectional antenna. Based on these measurements, the paper presents experimental parameter estimates for a novel model of the polarimetric propagation paths, that includes the power-decay as well as a correlated fading for the different polarization components. This work also includes an estimation method that takes measurement limitations caused by the noise floor and the limited cross-polarization discrimination of the antennas into account using censored samples.

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

ABSTRACT Two principal areas of application for estimated computed tomography (CT) images are dose calculations in magnetic resonance imaging (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. Obtained results suggest that HMMs deserve a further study for investigating their potential in modeling applications, where the most natural theoretical choice would be the class of HMRF models.