Does a Gibbs sampler approach to spatial Poisson regression models outperform a single site MH sampler?

In this paper we present and evaluate a Gibbs sampler for a Poisson regression model including spatial e ects. The approach is based on Frühwirth-Schnatter and Wagner (2004b) who show that by data augmentation using the introduction of two sequences of latent variables a Poisson regression model can be transformed into an approximate normal linear model. We show how this methodology can be extended to spatial Poisson regression models and give details of the resulting Gibbs sampler. In particular, the influence of model parameterisation and di erent update strategies on the mixing of the MCMC chains is discussed. The developed Gibbs samplers are analysed in two simulation studies and applied to model the expected number of claims for policyholders of a German car insurance company. The mixing of the Gibbs samplers depends crucially on the model parameterisation and the update schemes. The best mixing is achieved when collapsed algorithms are used, reasonable low autocorrelations for the spatial e ects are obtained in this case. For the regression e ects however, autocorrelations are rather high, especially for data with very low heterogeneity. For comparison a single component Metropolis Hastings algorithms is applied which displays very good mixing for all components. Although the Metropolis Hastings sampler requires a higher computational e ort, it outperforms the Gibbs samplers which would have to be run considerably longer in order to obtain the same precision of the parameters

A new spatial model for predicting multivariate counts : anticipating pedestrian crashes across neighborhoods and firm births across counties

textTransportation research regularly relies on data exhibiting both space and time dimensions. Thanks to the rise of smartphones, Bluetooth, and other devices, geo-referenced data collection enables application of more behaviorally realistic -- but complex -- models that account for spatial autocorrelation, temporal correlation, and possible time-space interactions (e.g., time-lagged effects from a neighboring unit's response). One promising area is crash count prediction, where crash frequencies (and severities) at zones, intersections, and along roadways will generally exhibit some spatial relationships, due to missing variables, causal mechanisms, and other ties. This dissertation work proposes and estimates a spatial multivariate count model and provides two case studies to implement such model. One case study is in the context of pedestrian-vehicle crash counts across zones in Austin, Texas, while accounting for network features (e.g., lane-miles and intersection density), land use factors (such as land use entropy and residential accessibility to commercial activities), population and job densities, and school access. The other case study pertains to new firm births by industries across U.S. counties while controlling for population density, agglomeration economies (e.g., percentage of firms with more than 100 people), wealth, and median age. The new model specification captures region-wide heterogeneity (thanks to extra variation introduced by the lognormal component in the mean crash-rate specification), correlations across two (or more) count types (in the same zone), and spatial autocorrelation among unobserved components. This new approach and associated application allow analysts to distinguish covariates' effects on multivariate crash and other counts from spatial spillover effects and cross-response correlations. This work adds to the literature by providing guidance on what types of specifications best reflect spatial count data while facilitating estimation (using large data sets) and illuminating the level and nature of spatial autocorrelation, multivariate correlation, and region-wide (latent) heterogeneity that exists in crash data after controlling for a host of observable factors.Civil, Architectural, and Environmental Engineerin

Topics in statistics of spatial-temporal disease modelling

This thesis is concerned with providing further statistical development in the area of space-time modelling with particular application to disease data. We briefly consider the non-Bayesian approaches of empirical mode decomposition and generalised linear modelling for analysing space-time data, but our main focus is on the increasingly popular Bayesian hierarchical approach and topics surrounding that. We begin by introducing the hierarchical Poisson regression model of Mugglin et al. [36] and a data set provided by NHS Direct which will be used to illustrate our results through-out the remainder of the thesis. We provide details of how a Bayesian analysis can be performed using Markov chain Monte Carlo (MCMC) via the software LinBUGS then go on to consider two particular issues associated with such analyses. Firstly, a problem with the efficiency of MCMC for the Poisson regression model is likely to be due to the presence of non-standard conditional distributions. We develop and test the 'improved auxiliary mixture sampling' method which introduces auxiliary variables to the conditional distribution in such a way that it becomes multivariate Normal and an efficient block Gibbs sampling scheme can be used to simulate from it. Secondly, since MCMC allows modelling of such complexity, inputs such as priors can only be elicited in a casual way thereby increasing the need to check how sensitive our output is to changes to the prior. We therefore develop and test the 'marginal sensitivity' method which, using only one MCMC output sample, quantifies how sensitive the marginal posterior distributions are to changes to prior parameter