The Log-Gaussian Cox Process is a commonly used model for the analysis of spatial point pattern data. Fitting this model is difficult because of its doubly-stochastic property, i.e., it is an hierarchical combination of a Poisson process at the first level and a Gaussian Process at the second level. Various methods have been proposed to estimate such a process, including traditional likelihood-based approaches as well as Bayesian methods. We focus here on Bayesian methods and several approaches that have been considered for model fitting within this framework, including Hamiltonian Monte Carlo, the Integrated nested Laplace approximation, and Variational Bayes. We consider these approaches and make comparisons with respect to statistical and computational efficiency. These comparisons are made through several simulation studies as well as through two applications, the first examining ecological data and the second involving neuroimaging data.
Summary Statistical modelling of functional magnetic resonance imaging data is challenging as the data are both spatially and temporally correlated. Spatially, measurements are taken at thousands of contiguous regions, called voxels, and temporally measurements are taken at hundreds of time points at each voxel. Recent advances in Bayesian hierarchical modelling have addressed the challenges of spatiotemporal structure in functional magnetic resonance imaging data with models incorporating both spatial and temporal priors for signal and noise. Whereas there has been extensive research on modelling the functional magnetic resonance imaging signal (i.e. the convolution of the experimental design with the functional choice for the haemodynamic response function) and its spatial variability, less attention has been paid to realistic modelling of the temporal dependence that typically exists within the functional magnetic resonance imaging noise, where a low order auto‐regressive process is typically adopted. Furthermore, the auto‐regressive order is held constant across voxels (e.g. AR(1) at each voxel). Motivated by an event‐related functional magnetic resonance imaging experiment, we propose a novel hierarchical Bayesian model with automatic selection of the auto‐regressive orders of the noise process that vary spatially over the brain. With simulation studies we show that our model is more statistically efficient and we apply it to our motivating example.
Time series analysis of fMRI data is an important area of medical statistics for neuroimaging data. Spatial models and Bayesian approaches for inference in such models have advantages over more traditional mass univariate approaches; however, a major challenge for such analyses is the required computation. As a result, the neuroimaging community has embraced approximate Bayesian inference based on mean-field variational Bayes (VB) approximations. These approximations are implemented in standard software packages such as the popular statistical parametric mapping software. While computationally efficient, the quality of VB approximations remains unclear even though they are commonly used in the analysis of neuroimaging data. For reliable statistical inference, it is important that these approximations be accurate and that users understand the scenarios under which they may not be accurate. We consider this issue for a particular model that includes spatially varying coefficients. To examine the accuracy of the VB approximation, we derive Hamiltonian Monte Carlo (HMC) for this model and conduct simulation studies to compare its performance with VB in terms of estimation accuracy, posterior variability, the spatial smoothness of estimated images, and computation time. As expected, we find that the computation time required for VB is considerably less than that for HMC. In settings involving a high or moderate signal-to-noise ratio (SNR), we find that the 2 approaches produce very similar results suggesting that the VB approximation is useful in this setting. On the other hand, when one considers a low SNR, substantial differences are found, suggesting that the approximation may not be accurate in such cases and we demonstrate that VB produces Bayes estimators with larger mean squared error. A comparison of the 2 computational approaches in an application examining the hemodynamic response to face perception in addition to a comparison with the traditional mass univariate approach in this application is also considered. Overall, our work clarifies the usefulness of VB for the spatiotemporal analysis of fMRI data, while also pointing out the limitation of VB when the SNR is low and the utility of HMC in this case.
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