Inference for Dynamic System Models

点击翻译按钮获取中文摘要

Uncertainty Quantification via Probabilistic Differential Equation Solver: The numerical solution of differential equations has been well developed in the field of numerical analysis, where Runga-Kutta solvers maintain a bound on the numerical solution approximation error for Lipschitz continuous initial value problems. However, extending beyond simple models the numerical error becomes important and complex. The proposed project considers a probabilistic solution to differential equation models as a way to quantify the uncertainty of the numerical analysis. This work proposes to use a probabilistic differential equation solver based on a Gaussian Process (GP) regression model with a non-stationary covariance structure incorporating the model dynamics to estimate the model solution while quantifying uncertainty probabilistically. The GP regression works simultaneously in the state and derivative function spaces while recursively estimating the system solution. First, a point in the state space is predicted ahead at the next time point. Then, the prediction and its uncertainty are smoothed in the state and derivative spaces, updating the prediction. Finally, the uncertainty in the point and derivative are carried forward into the prediction at the next time point. Applications for this work are models where discretization of numerical solvers induces noise. Chaotic systems are characterized by the divergence of two solutions which begin some small epsilon distance apart, consequently a probabilistic description of the solution uncertainty induced by the solver will lead to honest uncertainty quantification that is currently ignored. ABC and Design of Experiments: Approximate Bayesian Computation (ABC) allows inference on complex models where the likelihood is intractable or computationally infeasible. Assuming that the model can be evaluated for any parameter set, a pseudo likelihood is used to compare summary statistics for the model and the data. Parameter inference is then generally performed using Markov Chain Monte Carlo. The success of the inference depends on finding suitable summary statistics, which ideally would be sufficient statistics, although that is rarely possible or computationally feasible. This project concerns the situation where some data is available but we wish to augment that with new observations through a sequential experimental design. Standard Design of Experiments methods suggest taking new observations at design points that minimize a criterion such as the determinant of the posterior parameter covariance matrix or the largest eigenvalue of the posterior parameter covariance matrix, etc... However in ABC methods, the unique opportunity exists to not only select the design points to improve uncertainty, but to simultaneously choose to possibly include new summary statistics in the inference. Including new summary statistics allows for summaries that were not useful with the original data but may offer insights about new model attributes unaffected by the original set of summary statistics and/or data, and allows identifiability analysis to be incorporated into summary selection. Additionally, the sequential design proposal has related extensions into parameter estimation through sequential monte carlo methods.