Bayesian modeling with spatial curvature processes

Spatial process models are widely used for modeling point-referenced variables arising from diverse scientific domains. Analyzing the resulting random surface provides deeper insights into the nature of latent dependence within the studied response. We develop Bayesian modeling and inference for rapid changes on the response surface to assess directional curvature along a given trajectory. Such trajectories or curves of rapid change, often referred to as \emph{wombling} boundaries, occur in geographic space in the form of rivers in a flood plain, roads, mountains or plateaus or other topographic features leading to high gradients on the response surface. We demonstrate fully model based Bayesian inference on directional curvature processes to analyze differential behavior in responses along wombling boundaries. We illustrate our methodology with a number of simulated experiments followed by multiple applications featuring the Boston Housing data; Meuse river data; and temperature data from the Northeastern United States

Spatial risk estimation in Tweedie compound Poisson double generalized linear models

Tweedie exponential dispersion family constitutes a fairly rich sub-class of the celebrated exponential family. In particular, a member, compound Poisson gamma (CP-g) model has seen extensive use over the past decade for modeling mixed response featuring exact zeros with a continuous response from a gamma distribution. This paper proposes a framework to perform residual analysis on CP-g double generalized linear models for spatial uncertainty quantification. Approximations are introduced to proposed framework making the procedure scalable, without compromise in accuracy of estimation and model complexity; accompanied by sensitivity analysis to model mis-specification. Proposed framework is applied to modeling spatial uncertainty in insurance loss costs arising from automobile collision coverage. Scalability is demonstrated by choosing sizable spatial reference domains comprised of groups of states within the United States of America.Comment: 34 pages, 10 figures and 12 table

Spatial Tweedie exponential dispersion models

This paper proposes a general modeling framework that allows for uncertainty quantification at the individual covariate level and spatial referencing, operating withing a double generalized linear model (DGLM). DGLMs provide a general modeling framework allowing dispersion to depend in a link-linear fashion on chosen covariates. We focus on working with Tweedie exponential dispersion models while considering DGLMs, the reason being their recent wide-spread use for modeling mixed response types. Adopting a regularization based approach, we suggest a class of flexible convex penalties derived from an un-directed graph that facilitates estimation of the unobserved spatial effect. Developments are concisely showcased by proposing a co-ordinate descent algorithm that jointly explains variation from covariates in mean and dispersion through estimation of respective model coefficients while estimating the unobserved spatial effect. Simulations performed show that proposed approach is superior to competitors like the ridge and un-penalized versions. Finally, a real data application is considered while modeling insurance losses arising from automobile collisions in the state of Connecticut, USA for the year 2008.Comment: 26 pages, 3 figures and 7 table

Wombling for Spatial and Spatiotemporal Processes: Applications to Insurance Data

In this dissertation we focus on developing a framework for detecting zones of rapid change within a surface that is spatially or spatiotemporally indexed, which is termed as wombling. This is considered under two different approaches featuring a semi-parametric model and a model based parametric setup. The second approach starts with a Gaussian process specification on the spatial or spatiotemporal component. We focus on applying the developed framework to build models for analyzing spatiotemporally referenced insurance data arising from losses due to automobile collision in the state of Connecticut, in the year 2008. For the semi-parametric models we adopt a penalization based approach which imposes a penalty on the graph Laplacian that promotes spatial clustering. A coordinate descent algorithm is developed to estimate model parameters. The approach is extended further to encompass various members of the exponential dispersion family. We operate under a double generalized linear model framework featuring joint modeling of the mean and dispersion, coupled with spatial uncertainty quantification. A geographic boundary analysis framework is then developed to detect zones characterizing significant spatial signal in the response. For the parametric setup we resort to Bayesian hierarchical modeling of the spatial and spatiotemporally referenced insurance data. We develop a model based inferential wombling or boundary analysis framework for performing curvilinear wombling on open and closed curves within the spatial surface, while quantifying their evolution in time with respect to the spatial signal. This is facilitated by developing gradient and curvature processes arising from a Gaussian process specification on the response surface. Necessary and sufficient conditions are derived that ensure existence of such derivative processes. We operate within a hierarchical modeling framework that allows us to specify this dependence of the response on policy level covariates within the data, while also allowing us to infer about topological characteristics of the residual spatial surface. These are supplemented with synthetic illustrations under both Gaussian and latent Gaussian specifications and real data applications that demonstrate the inferential capability of our proposed framework

Wombling for Spatial and Spatiotemporal Processes: Applications to Insurance Data

In this dissertation we focus on developing a framework for detecting zones of rapid change within a surface that is spatially or spatiotemporally indexed, which is termed as wombling. This is considered under two different approaches featuring a semi-parametric model and a model based parametric setup. The second approach starts with a Gaussian process specification on the spatial or spatiotemporal component. We focus on applying the developed framework to build models for analyzing spatiotemporally referenced insurance data arising from losses due to automobile collision in the state of Connecticut, in the year 2008. For the semi-parametric models we adopt a penalization based approach which imposes a penalty on the graph Laplacian that promotes spatial clustering. A coordinate descent algorithm is developed to estimate model parameters. The approach is extended further to encompass various members of the exponential dispersion family. We operate under a double generalized linear model framework featuring joint modeling of the mean and dispersion, coupled with spatial uncertainty quantification. A geographic boundary analysis framework is then developed to detect zones characterizing significant spatial signal in the response. For the parametric setup we resort to Bayesian hierarchical modeling of the spatial and spatiotemporally referenced insurance data. We develop a model based inferential wombling or boundary analysis framework for performing curvilinear wombling on open and closed curves within the spatial surface, while quantifying their evolution in time with respect to the spatial signal. This is facilitated by developing gradient and curvature processes arising from a Gaussian process specification on the response surface. Necessary and sufficient conditions are derived that ensure existence of such derivative processes. We operate within a hierarchical modeling framework that allows us to specify this dependence of the response on policy level covariates within the data, while also allowing us to infer about topological characteristics of the residual spatial surface. These are supplemented with synthetic illustrations under both Gaussian and latent Gaussian specifications and real data applications that demonstrate the inferential capability of our proposed framework