An optimization based state and parameter estimation method is presented where the required Jacobian matrix of the cost function is computed via automatic differentiation. Automatic differentiation evaluates the programming code of the cost function
and provides exact values of the derivatives. In contrast to numerical differentiation it is not suffering from approximation errors and compared to symbolic differentiation it is more convenient to use, because no closed analytic expressions are required. Furthermore, we demonstrate how to generalize the parameter estimation scheme to delay differential equations, where estimating the delay time requires attention.
Features of the Jacobian matrix of the delay coordinates map are exploited for quantifying the robustness and reliability of state and parameter estimations for a given dynamical model using an observed time series. Relevant concepts of this approach
are introduced and illustrated for discrete and continuous time systems employing a filtered Henon map and a Rossler system.
Observability of state variables and parameters of a dynamical system from an observed time series is analyzed and quantified by means of the Jacobian matrix of the delay coordinates map. For each state variable and each parameter to be estimated a m
easure of uncertainty is introduced depending on the current state and parameter values, which allows us to identify regions in state and parameter space where the specific unknown quantity can (not) be estimated from a given time series. The method is demonstrated using the Ikeda map and the Hindmarsh-Rose model.
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 solution
s (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).
Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these dire
ctions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.
