State Space Models: A New Look at Smoothing, Parameter Inference, and Model Choice

Harvard University, Cambridge MA

Investigators

Abstract

Time series are repeated observations of objects over time; for example, hourly prices of a financial asset, daily population sizes of plankton in the ocean, or monthly case counts during a disease outbreak. All of these examples are routinely studied using state space models, under which observations are treated as noisy measurements of an unobserved stochastic process. The underlying process can be modeled using any equation considered relevant for the object at hand, but flexibility comes at a computational cost. For instance, if the process is modeled using nonlinear differential equations, the associated statistical computations might become prohibitive. This is particularly acute for long and high-dimensional time series. This project introduces new methods for three important questions regarding state space models: how to estimate the unobserved process, how to estimate the model parameters, and how to compare multiple models. These historical questions will be revisited in the contexts of parallel computing hardware, of large datasets with vast amounts of missing values, and of limited prior information on the parameters. The toolbox under development will participate in a global effort to quantify uncertainty in scientific models for time series. It will be implemented in efficient and user-friendly software packages, made publicly available during the project. State space models form a very flexible class of time series models, under which observations are assumed conditionally independent given a latent stochastic process. The project concerns three fundamental questions in these models: latent process estimation, parameter estimation and model choice. For latent process estimation (1), an unbiased estimator of smoothing expectations is developed and investigated. The approach relies on a novel coupling of particle filters, using maximal couplings and ideas from optimal transport. The unbiasedness property allows a complete parallelization of the estimation procedure, and the construction of reliable confidence intervals. For parameter estimation (2), similar couplings of particle filters can be leveraged to drastically reduce the variance in score estimators and in Metropolis acceptance ratios, especially for long time series. For model comparison (3), a novel criterion is investigated following a predictive sequential approach. It is designed for situations where prior information on parameters is limited, contrarily to Bayes factors. The project aims at helping current users of state-space models, for instance in econometrics, genetics, neuroscience, ecology, and marine biogeochemistry, and to assist scientists for whom currently available methods are insufficient. On a more fundamental level, leveraging ideas from coupling and optimal transport for computational methods is of independent interest and paves the way to a wide range of new sampling and optimization methods.