Combined Filtering and Parameter Estimation for Discrete-Time Systems Driven by Approximately White Gaussian Noise Disturbances

In the problem of combined filtering and parameter estimation one considers a stochastic dynamical system whose state x_t is only partially observed through an observation process y_t. The stochastic model for the process pair (x_t, y_t) depends furthermore on an unknown parameter theta. Given an observation history of the process y_t, the problem then consists in estimating recursively both the current state x_t of the system (filtering) as well as the value theta of the parameter (Bayesian parameter estimation). The problem is a rather difficult one: Even if, conditionally on a given value of theta, the process pair (x_t, y_t) satisfies a linear-Gaussian model so that the filtering problem for x_t can be solved via the familiar Kalman-Bucy filter; when theta is unknown, the problem becomes a difficult nonlinear filtering problem. The present paper, partly based on previous joint work of one of the authors, makes a contribution towards the solution of this problem in the case of discrete time and of a (conditionally on theta) linear model for x_t, y_t. The solution that is obtained is shown to be robust with respect to small variations in the a priori distributions in the model, in particular those of the disturbances

Linear and nonlinear filtering in mathematical finance: a review

Copyright @ The Authors 2010This paper presents a review of time series filtering and its applications in mathematical finance. A summary of results of recent empirical studies with market data are presented for yield curve modelling and stochastic volatility modelling. The paper also outlines different approaches to filtering of nonlinear time series

Parameter estimation for macroscopic pedestrian dynamics models from microscopic data

In this paper we develop a framework for parameter estimation in macroscopic pedestrian models using individual trajectories -- microscopic data. We consider a unidirectional flow of pedestrians in a corridor and assume that the velocity decreases with the average density according to the fundamental diagram. Our model is formed from a coupling between a density dependent stochastic differential equation and a nonlinear partial differential equation for the density, and is hence of McKean--Vlasov type. We discuss identifiability of the parameters appearing in the fundamental diagram from trajectories of individuals, and we introduce optimization and Bayesian methods to perform the identification. We analyze the performance of the developed methodologies in various situations, such as for different in- and outflow conditions, for varying numbers of individual trajectories and for differing channel geometries

Statistical Inference for Partially Observed Markov Processes via the R Package pomp

Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this paper is provided at the pomp package website: http://kingaa.github.io/pom