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Particle and Ensemble Methods for State Space Models Open Access

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In modern time series analysis with state space models, two categories of time series are difficult to handle and of special interest: one is low-dimensional time series with severe nonlinearity and non-Gaussianity; the other is high-dimensional time series. The above problems become even more challenging in the presence of unknown parameters. This dissertation is motivated by the two research questions raised above.The first part of the dissertation is focused on using sequential Monte Carlo methods to solve the low-dimensional estimation problem. In Chapter 2, we propose two new smoothing algorithms to generate samples from the joint posterior distribution of the hidden states and parameters. We apply our new smoothing algorithms in three simulated examples: an AR(1) plus noise model, a non-stationary growth model and a nonlinear model from ecology. We compare the smoothing results from our methods to existing smoothing methods, including particle learning and smoothing (PLS), Markov chain Monte Carlo (MCMC) and particle marginal Metropolis-Hastings (PMMH), and show that our methods outperform or are highly competitive with all of these approaches. Finally, we apply our smoothing algorithm to a stochastic volatility model for daily returns on the S&P; 500 index and estimate the underlying volatility and parameters.The second part of the dissertation is focused on using the ensemble Kalman filter (EnKF) to provide approximate solutions to state and parameter estimation in high-dimensional state space models. In Chapter 3, we propose a new nearest-neighbor localization method to regularize the ensemble covariance matrix, and propose three algorithms to simultaneously or serially update the forecast ensembles. We show that our new nearest-neighbor localization method outperforms ordinary covariance tapering method in various situations through a simulation study.Finally, in Chapter 4, using our new localization and updating algorithms, we extend the particle marginal Metropolis-Hastings (PMMH) algorithm to high-dimensional models, and propose an ensemble marginal Metropolis-Hastings (EMMH) algorithm for high-dimensional state and parameter estimation. We apply our new smoothing algorithm to simulated data and a spatial-temporal cloud motion data set and estimate the cloud locations and parameters in the model. Our model is shown to provide accurate inference in this high-dimensional setting.

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