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Bayesian spatial and spatiotemporal models based on multiscale factorizations

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Abstract We review the literature on spatial and spatiotemporal models based on spatial multiscale factorizations. Specifically, we review models based on wavelets and Kolaczyk–Huang factorizations for Gaussian and Poisson data. These multiscale models decompose spatial and spatiotemporal datasets into many small components, called multiscale coefficients, at multiple levels of spatial resolution. Then analysis proceeds independently for each multiscale coefficient. After that, aggregation equations are used to coherently combine the analyses from the multiple multiscale coefficients to obtain a statistical analysis at the original resolution level. The computational cost of such analysis grows linearly with sample size. Furthermore, computations for these models are scalable, parallelizable, and fast. Therefore, these multiscale models are tremendously useful for the analysis of massive spatial and spatiotemporal datasets. This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical Models > Bayesian Models Data: Types and Structure > Image and Spatial Data
(a) Realization of a one‐dimensional nonobserved Gaussian random field (solid line) and 1,024 noisy observations (circles); (b) empirical wavelet coefficients; (c) thresholded wavelet coefficients; (d) true field (solid line), observations (circles), wavelets‐estimated random field (dashed line), and Gaussian process (GP)‐estimated random field (dotted line)
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Graphical representation of dynamic multiscale spatiotemporal models. At each time t, t = 1, …, T, the spatiotemporal data ytL are decomposed into a set of empirical multiscale coefficients. These empirical multiscale coefficients are assumed to be noisy measurements of latent multiscale coefficients. Finally, the latent multiscale coefficients evolve through time according to a dynamic process indexed by a vector of hyperpameters
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Density of sample correlations of empirical wavelet coefficients (solid line) for a Gaussian process with Matérn correlation function; and density of sample correlations of white noise (dashed line). For most practical purposes, the two densities are indistinguishable
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Computational time in seconds of wavelet analysis (Panel (a)) and Gaussian process analysis (Panel (b)) versus sample size. For direct comparison, Panel (b) includes the computational time for restricted maximum likelihood (REML) Gaussian process (GP) analysis (solid line) and wavelets analysis (dashed line), with the wavelet line visually appearing to be a horizontal line. Because of computational time constraints, here the maximum sample size considered for REML GP analysis was 214 = 16, 384 which took about 15.7 hr. In contrast, the maximum sample size considered for wavelet analysis was 220 = 1, 048, 576 which took about 0.99 s
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Data: Types and Structure > Image and Spatial Data
Statistical Models > Bayesian Models
Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods

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