A canonical space-time state space model: state and parameter estimation

The maximum likelihood estimation of a dynamic spatiotemporal model is introduced, centred around the inclusion of a prior arbitrary spatiotemporal neighborhood description. The neighborhood description defines a specific parameterization of the state transition matrix, chosen on the basis of prior knowledge about the system. The model used is inspired by the spatiotemporal ARMA (STARMA) model, but the representation used is based on the standard state-space model. The inclusion of the neighborhood into an expectation-maximization based joint state and parameter estimation algorithm allows for accurate characterization of the spatiotemporal model. The process of including the neighborhood, and the effect it has on the maximum likelihood parameter estimate is described and demonstrated in this paper

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics

NONUNIFORMLY AND RANDOMLY SAMPLED SYSTEMS

Problems with missing data, sampling irregularities and randomly sampled systems are the topics covered by this dissertation. The spectral analysis of a series of periodically repeated sampling patterns is developed. Parameter estimation of autoregressive moving average models using partial observations and an algorithm to fill in the missing data are proved and demonstrated by simulation programs. Interpolation of missing data using bandlimiting assumptions and discrete Fourier transform techniques is developed. Representation and analysis of randomly sampled linear systems with independent and identically distributed sampling intervals are studied. The mean, and the mean-square behavior of a multiple-input multiple-output randomly sampled system are found. A definition of and results concerning the power spectral density gain are also given. A complete FORTRAN simulation package is developed and implemented in a microcomputer environment demonstrating the new algorithms

Fitting Nonlinear Ordinary Differential Equation Models with Random Effects and Unknown Initial Conditions Using the Stochastic Approximation Expectation–Maximization (SAEM) Algorithm

The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data. A stochastic approximation expectation–maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-h ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed

The Exact Discrete Time Representation of Continuous Time Models with Unequally Spaced Data

This thesis presents the exact discrete time representations of first order continuous time models with unequally spaced stocks, flows and mixed data. With unequally spaced data, given that the underlying continuous time models have constant coefficients and homeskedastic disturbances, the exact discrete time representations exhibit more complicated properties such as time-varying coefficients and heteroskedastic moving average disturbances, which arise due to the irregularity in sampling intervals. When data are purely stock variables, the exact discrete time representation follows a VAR(1) process with time-varying coefficients and serially uncorrelated heteroskedastic disturbances. When data are purely flow variables or a mixture of stocks and flows, the exact discrete time representation follows a VARMA(1, 1) process with time-varying coefficients and moving average heteroskedastic disturbances. Based on unequally spaced real life data, the empirical results show that the parameter estimates are different when accounting for the unequal sampling intervals compared to the approach that assumes data are equally spaced. In addition, the Monte Carlo evidences indicate that there are gains to be made in the estimation, such as smaller estimation bias, when the irregular sampling intervals are correctly accounted for