Probabilistic scheme for joint parameter estimation and state prediction in complex dynamical systems

Many problems in physics demand the ability to calibrate the parameters and predict the time evolution of complex dynamical models using sequentially collected data. Here we introduce a general methodology for the joint estimation of the static parameters and the forecasting of the state variables of nonlinear stochastic dynamical models. The proposed scheme is essentially probabilistic. It aims at recursively computing the sequence of joint posterior probability distributions of the unknown model parameters and its (time-varying) state variables conditional on the available observations. This framework combines two layers of inference: In the first layer, a grid-based scheme is used to approximate the posterior probability distribution of the fixed parameters; in the second layer, filtering (or data assimilation) techniques are employed to track and predict different conditional probability distributions of the state variables. Various types of procedures (deterministic grids, Monte Carlo, Gaussian filters, etc.) can be plugged into both layers, leading to a wealth of algorithms. For this reason, we refer to the proposed methodology as nested hybrid filtering. In this paper we specifically explore the combination of Monte Carlo and quasi–Monte Carlo (deterministic) approximations in the first layer with Gaussian filtering methods in the second layer, but other approaches fit naturally within the framework. We prove a general convergence result for a class of procedures that use sequential Monte Carlo in the first layer. Then we turn to an illustrative numerical example. In particular, we apply and compare different implementations of the methodology to the tracking of the state, and the estimation of the fixed parameters, of a stochastic two-scale Lorenz 96 system. This model is commonly used to assess data assimilation procedures in meteorology. We show estimation and forecasting results, obtained with a desktop computer, for up to 5000 dynamic state variables.

Figure 1

MSE of the different NHFs depending on the number of particles used in the first-layer of the filter.

Figure 2

Comparison of the SQMC EKF (red lines) and SQMC EnKF (blue lines) in terms of (a) their running time and (b) their MSE as the state dimension $d_x$ increases, with a fixed gap between observations of $T=20$ discrete time steps.

Figure 3

Comparison of the SQMC EKF (red lines) and SQMC EnKF (blue lines) in terms of (a) their running time and (b) their MSE as the gap between observations $m$ increases, with fixed state dimension $d_x=100$.

Figure 4

Sequences of state values (black line) and estimates (dashed red line) in (a) $x_1$ and (b) $x_2$ over time. Variable $x_1$ is observed (in Gaussian noise), while $x_2$ is unobserved.

Figure 5

Posterior density of the parameters $\mathsf{a}={[{\mathsf{a}}_1,{\mathsf{a}}_2]}^{\top }$ and $F$ at $t=5$ in a 5000-dimensional Lorenz 96 model (red dashed lines). (a) The true value of $F$ is indicated by a black vertical line, while (b) the location of reference values of $\mathsf{a}$ is marked by a black cross. Note that there is no ground truth for the parameters in $\mathsf{a}$.