Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally “best” trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.
Since the design of an experimental trajectory has a wide range of potential uses, there have been a number of contributions to the area from different fields. A large amount of literature on optimal experimental design exists in the fields of biology @cite_30 @cite_11 @cite_15 , chemistry @cite_8 , and systems @cite_31 @cite_28 @cite_25 @cite_35 . Many of these results focus on particular applications to experiments specific to their respective fields; however, the underlying principles of information theory remain the same.
@cite_30 @cite_35 @cite_31 @cite_8 @cite_28 @cite_15 @cite_25 @cite_11
2167485946 2343661315 1579213377 2110690917
To obtain a systems-level understanding of a biological system, the authors conducted quantitative dynamic experiments from which the system structure and the parameters have to be deduced. Since biological systems have to cope with different environmental conditions, certain properties are often robust with respect to variations in some of the parameters. Hence, it is important to use optimal experimental design considerations in advance of the experiments to improve the information content of the measurements. Using the MAP-Kinase pathway as an example, the authors present a simulation study investigating the application of different optimality criteria. It is demonstrated that experimental design significantly improves the parameter estimation accuracy and also reveals difficulties in parameter estimation due to robustness. The design of multiple experiments is commonly undertaken via suboptimal strategies, such as batch (open-loop) design that omits feedback or greedy (myopic) design that does not account for future effects. This paper introduces new strategies for the optimal design of sequential experiments. First, we rigorously formulate the general sequential optimal experimental design (sOED) problem as a dynamic program. Batch and greedy designs are shown to result from special cases of this formulation. We then focus on sOED for parameter inference, adopting a Bayesian formulation with an information theoretic design objective. To make the problem tractable, we develop new numerical approaches for nonlinear design with continuous parameter, design, and observation spaces. We approximate the optimal policy by using backward induction with regression to construct and refine value function approximations in the dynamic program. The proposed algorithm iteratively generates trajectories via exploration and exploitation to improve approximation accuracy in frequently visited regions of the state space. Numerical results are verified against analytical solutions in a linear-Gaussian setting. Advantages over batch and greedy design are then demonstrated on a nonlinear source inversion problem where we seek an optimal policy for sequential sensing. The optimal selection of experimental conditions is essential in maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. A general Bayesian framework for optimal experimental design with nonlinear simulation-based models is proposed. The formulation accounts for uncertainty in model parameters, observables, and experimental conditions. Straightforward Monte Carlo evaluation of the objective function - which reflects expected information gain (Kullback-Leibler divergence) from prior to posterior - is intractable when the likelihood is computationally intensive. Instead, polynomial chaos expansions are introduced to capture the dependence of observables on model parameters and on design conditions. Under suitable regularity conditions, these expansions converge exponentially fast. Since both the parameter space and the design space can be high-dimensional, dimension-adaptive sparse quadrature is used to construct the polynomial expansions. Stochastic optimization methods will be used in the future to maximize the expected utility. While this approach is broadly applicable, it is demonstrated on a chemical kinetic system with strong nonlinearities. In particular, the Arrhenius rate parameters in a combustion reaction mechanism are estimated from observations of autoignition. Results show multiple order-of-magnitude speedups in both experimental design and parameter inference. Motivation: Systems biology employs mathematical modelling to further our understanding of biochemical pathways. Since the amount of experimental data on which the models are parameterized is often limited, these models exhibit large uncertainty in both parameters and predictions. Statistical methods can be used to select experiments that will reduce such uncertainty in an optimal manner. However, existing methods for optimal experiment design (OED) rely on assumptions that are inappropriate when data are scarce considering model complexity. Results: We have developed a novel method to perform OED for models that cope with large parameter uncertainty. We employ a Bayesian approach involving importance sampling of the posterior predictive distribution to predict the efficacy of a new measurement at reducing the uncertainty of a selected prediction. We demonstrate the method by applying it to a case where we show that specific combinations of experiments result in more precise predictions. Availability and implementation: Source code is available at: http: bmi.bmt.tue.nl sysbio software pua.html Contact:j.vanlier@tue.nl; N.A.W.v.Riel@tue.nl Supplementary information:Supplementary data are available at Bioinformatics online.