Estimating Parameters in Complex Systems with Functional Outputs: A Wavelet-Based Approximate Bayesian Computation Approach

Abstract

We consider a family of parameter estimation problems involving functional data. In these problems, the relationship between functional data and the underlying parameters cannot be explicitly specified using a likelihood function. These situations often occur when functional data arises from a complex system and only numerical simulations (through a simulator) can be used to describe the underlying data-generating mechanism. To estimate the unknown parameters under these scenarios, we introduce a wavelet-based approximate Bayesian computation (wABC) approach that is likelihood-free and computationally scalable to functional data measured on a dense, high-dimensional grid. The proposed approach relies on near-lossless wavelet decomposition and compression to reduce the high-correlation between measurement points and the high-dimensionality. We adopt a Markov chain Monte Carlo algorithm with a Metropolis-Hastings sampler to obtain posterior samples of the parameters for Bayesian inference. To avoid expensive simulations from the simulator in the approximate Bayesian computation, a Gaussian process surrogate for the simulator is introduced, and the uncertainty of the resulting sampler is controlled by calculating the expected error rate of the acceptance probability. We motivate our approach and demonstrate its performance using the foliage-echo data generated by a sonar simulation system. Our Bayesian posterior inference provides the joint posterior distribution of all underlying parameters, which is otherwise intractable using existing analytical methods.