Browsing by Subject "Poisson process"
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Item Open Access A Mean Field Approach to Watershed Hydrology(2016) Bartlett, Mark Stephan, JrSociety-induced changes to the environment are altering the effectiveness of existing management strategies for sustaining natural and agricultural ecosystem productivity. At the watershed scale, natural and agro-ecosystems represent complex spatiotemporal stochastic processes. In time, they respond to random rainfall events, evapotranspiration and other losses that are spatially variable because of heterogeneities in soil properties, root distributions, topography, and other factors. To quantify the environmental impact of anthropogenic activities, it is essential that we characterize the evolution of space and time patterns of ecosystem fluxes (e.g., energy, water, and nutrients). Such a characterization then provides a basis for assessing and managing future anthropogenic risks to the sustainability of ecosystem productivity.
To characterize the space and time evolution of watershed scale processes, this dissertation introduces a mean field approach to watershed hydrology. Mean field theory (also known as self-consistent field theory) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The mean field theory approximates a complex multi-component system by considering a lumped (or average) effect of all individual components acting on a single component. Thus, the many body problem is reduced to a one body problem. For watershed hydrology, a mean field theory reduces the numerous point component effects to more tractable watershed averages resulting in a consistent method for linking the average watershed fluxes (evapotranspiration, runoff, etc.) to the local fluxes at each point.
The starting point for this work is a general point description of the soil moisture, rainfall, and runoff system. For this system, we find the joint PDF that describes the temporal variability of the soil water, rainfall, and runoff processes. Since this approach does not account for the spatial variability of runoff, we introduce a probabilistic storage (ProStor) framework for constructing a lumped (unit area) rainfall-runoff response from the spatial distribution of watershed storage. This framework provides a basis for unifying and extending common event-based hydrology models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability Distributed (PDM) model, and TOPMODEL). In each case, we obtain simple equations for the fractions of the different source areas of runoff, the spatial variability of runoff and soil moisture, and the average runoff value (i.e., the so-called runoff curve). Finally, we link the temporal and spatial descriptions with a mean field approach for watershed hydrology. By applying this mean field approach, we upscale the point description with the spatial distribution of soil moisture and parameterize the numerous local interactions related to lateral fluxes of soil water in terms of its average. With this approach, we then derive PDFs that represent the space and time distribution of soil water and associated watershed fluxes such as evapotranspiration and runoff.
Item Open Access A Tapered Pareto-Poisson Model for Extreme Pyroclastic Flows: Application to the Quantification of Volcano Hazards(2015) Dai, FanThis paper intends to discuss the problems of parameter estimation in a proposed tapered Pareto-Poisson model for the assessment of large pyroclastic flows, which are essential in quantifying the size and risk of volcanic hazards. In dealing with the tapered Pareto distribution, the paper applies both maximum likelihood estimation and a Bayesian framework with objective priors and Metropolis algorithm. The techniques are further illustrated by an example of modeling extreme flow volumes at Soufriere Hills Volcano, and their simulation results are addressed.
Item Open Access Bayesian Analysis of Spatial Point Patterns(2014) Leininger, Thomas JeffreyWe explore the posterior inference available for Bayesian spatial point process models. In the literature, discussion of such models is usually focused on model fitting and rejecting complete spatial randomness, with model diagnostics and posterior inference often left as an afterthought. Posterior predictive point patterns are shown to be useful in performing model diagnostics and model selection, as well as providing a wide array of posterior model summaries. We prescribe Bayesian residuals and methods for cross-validation and model selection for Poisson processes, log-Gaussian Cox processes, Gibbs processes, and cluster processes. These novel approaches are demonstrated using existing datasets and simulation studies.
Item Open Access Continuous-Time Models of Arrival Times and Optimization Methods for Variable Selection(2018) Lindon, Michael ScottThis thesis naturally divides itself into two sections. The first two chapters concern
the development of Bayesian semi-parametric models for arrival times. Chapter 2
considers Bayesian inference for a Gaussian process modulated temporal inhomogeneous Poisson point process, made challenging by an intractable likelihood. The intractable likelihood is circumvented by two novel data augmentation strategies which result in Gaussian measurements of the Gaussian process, connecting the model with a larger literature on modelling time-dependent functions from Bayesian non-parametric regression to time series. A scalable state-space representation of the Matern Gaussian process in 1 dimension is used to provide access to linear time filtering algorithms for performing inference. An MCMC algorithm based on Gibbs sampling with slice-sampling steps is provided and illustrated on simulated and real datasets. The MCMC algorithm exhibits excellent mixing and scalability.
Chapter 3 builds on the previous model to detect specific signals in temporal point patterns arising in neuroscience. The firing of a neuron over time in response to an external stimulus generates a temporal point pattern or ``spike train''. Of special interest is how neurons encode information from dual simultaneous external stimuli. Among many hypotheses is the presence multiplexing - interleaving periods of firing as it would for each individual stimulus in isolation. Statistical models are developed to quantify evidence for a variety of experimental hypotheses. Each experimental hypothesis translates to a particular form of intensity function for the dual stimuli trials. The dual stimuli intensity is modelled as a dynamic superposition of single stimulus intensities, defined by a time-dependent weight function that is modelled non-parametrically as a transformed Gaussian process. Experiments on simulated data demonstrate that the model is able to learn the weight function very well, but other model parameters which have meaningful physical interpretations less well.
Chapters 4 and 5 concern mathematical optimization and theoretical properties of Bayesian models for variable selection. Such optimizations are challenging due to non-convexity, non-smoothness and discontinuity of the objective. Chapter 4 presents advances in continuous optimization algorithms based on relating mathematical and statistical approaches defined in connection with several iterative algorithms for penalized linear
regression. I demonstrate the equivalence of parameter mappings using EM under
several data augmentation strategies - location-mixture representations, orthogonal data augmentation and LQ design matrix decompositions. I show that these
model-based approaches are equivalent to algorithmic derivation via proximal
gradient methods. This provides new perspectives on model-based and algorithmic
approaches, connects across several research themes in optimization and statistics,
and provides access, beyond EM, to relevant theory from the proximal gradient
and convex analysis literatures.
Chapter 5 presents a modern and technologically up-to-date approach to discrete optimization for variable selection models through their formulation as mixed integer programming models. Mixed integer quadratic and quadratically constrained programs are developed for the point-mass-Laplace and g-prior. Combined with warm-starts and optimality-based bounds tightening procedures provided by the heuristics of the previous chapter, the MIQP model developed for the point-mass-Laplace prior converges to global optimality in a matter of seconds for moderately sized real datasets. The obtained estimator is demonstrated to possess superior predictive performance over that obtained by cross-validated lasso in a number of real datasets. The MIQCP model for the g-prior struggles to match the performance of the former and highlights the fact that the performance of the mixed integer solver depends critically on the ability of the prior to rapidly concentrate posterior mass on good models.