Basin structure of optimization based state and parameter estimation

TL;DR

This paper investigates how the basin structure affects the success of optimization-based state and parameter estimation in a Lorenz-96 model, highlighting the importance of initial guesses and measurement variables.

Contribution

It characterizes the basin size of the global minimum for optimization-based estimation and compares strategies for initial guesses in a high-dimensional chaotic system.

Findings

Initialization close to the true solution yields accurate estimates.

Local minima often trap the optimization when initial guesses are far from the true solution.

Measuring more variables improves the likelihood of successful estimation.

Abstract

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment).