Exact Markov Chain Monte Carlo for a Class of Diffusions

Qi Wang, Purdue University

Abstract

This dissertation focuses on the simulation efficiency of the Markov process for two scenarios: Stochastic differential equations(SDEs) and simulated weather data.For SDEs, we propose a novel Gibbs sampling algorithm that allows sampling from a particular class of SDEs without any discretization error and shows the proposed algorithm improves the sampling efficiency by orders of magnitude against the existing popular algorithms. More specifically, SDEs or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler applies to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear on applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods. The work in this part of the dissertation was published in Wang et al. (2020) [1].In the weather data simulation study, we investigate how representative the simulated data are for three popular stochastic weather generators, which are widely used in hydrological, environmental, and agricultural applications to simulate weather time series. As it is a common practice to obtain data from only one simulation run, how representative it is has become an interesting question. In this dissertation, we study the impact of different numbers of realizations on the suitability of generated weather data. We generate 50 years of daily precipitation, and maximum and minimum temperatures for the weather stations in the Western Lake Erie Basin (WLEB) by using the weather generators (CLIGEN, LARSWG and WeaGETS) and compare them against the observed data. Our results suggest the need for more than a single realization when generating weather data to obtain suitable representations of climate. The work in this part of the dissertation was published in Guo et al. (2018) [2].

Degree

Ph.D.

Advisors

Rao, Purdue University.

Subject Area

Electrical engineering|Marketing|Meteorology|Operations research

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