Kernel Averaged Predictors for Space and Space-Time Processes
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In many spatio-temporal applications a vector of covariates is measured alongside a spatio-temporal response. In such cases, the purpose of the statistical model is to quantify the change, in expectation or otherwise, in the response due to a change in the predictors while adequately accounting for the spatio-temporal structure of the response, the predictors, or both. The most common approach for building such a model is to confine the relationship between the response and the predictors to a single spatio-temporal coordinate. For spatio-temporal problems, however, the relationship between the response and predictors may not be so confined. For example, spatial models are often used to quantify the effect of pollution exposure on mortality. Yet, an unknown lag exists between time of exposure to pollutants and mortality. Furthermore, due to mobility and atmospheric movement, a spatial lag between pollution concentration and mortality may also exist (e.g. subjects may live in the suburbs where pollution levels are low but work in the city where pollution levels are high).
The contribution of this thesis is to propose a hierarchical modeling framework which captures complex spatio-temporal relationships between responses and covariates. Specifically, the models proposed here use kernels to capture spatial and/or temporal lagged effects. Several forms of kernels are proposed with varying degrees of complexity. In each case, however, the kernels are assumed to be parametric with parameters that are easily interpretable and estimable from the data. Full distributional results are given for the Gaussian setting along with consequences of model misspecification. The methods are shown to be effective in understanding the complex relationship between responses and covariates through various simulated examples and analyses of physical data sets.
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