Gradient boosting machine assisted approximate Bayesian computation for uncertainty analysis of rainfall-runoff model parameters

Abstract

Bayesian inference is a well-regarded approach for diagnostic model evaluation that is often applied to hydrological models to constrain parameters and estimate uncertainty within a statistical framework. Typically, Bayesian uncertainty analysis is carried out using the Generalised Likelihood Uncertainty Estimation (GLUE) or Markov Chain Monte Carlo (MCMC) sampling. Approximate Bayesian Computation (ABC) is alternative set of likelihood-free Bayesian methods that have been gathering interest in many fields including astrophysics, population genetics and biology. The main appeal of ABC is that is the requirement for a formal likelihood function is replaced by one or more summary statistics that compare the simulated model to the observed data. ABC works in situations where an analytical likelihood function is either unavailable or intractable. Instead of evaluating the likelihood function, ABC only has to be able to sample from the likelihood function in an empirical fashion. This broadens the class of problems to which statistical inference can be applied. In practice, the appeal of the ABC method is limited somewhat due to its requirement for a large number of model evaluations. Because ABC is essentially a rejection sampling method, when the overlap between the prior and the posterior is poor, the sampling efficiency can be very low and it may be necessary for hundreds of thousands or even millions of model evaluations to be run to collect an appropriate number of accepted samples to construct a statistically informative posterior. If the model runtime is significant, ABC rejection sampling can easily be rendered impractical. In this paper a hybrid method is developed that serves to retain the flexibility of ABC while drastically reducing the computational effort required. The first component of the hybrid approach is to employ Sequential Monte Carlo sampling (SMC) to improve the sampling efficiency and reduce the total number of samples required. Secondly, the primitive model is replaced by a surrogate model that can accurately reproduce the results of the original model at a fraction of the computational cost. In this case, XGBoost, a gradient boosted regression tree machine learning algorithm, is used to construct the surrogate models. Employed together, SMC-ABC and XGBoost trained surrogate models offer an accurate and efficient framework for model parameter inference and uncertainty analysis. As a demonstration, the proposed method is applied to a four parameter GR4J distributed rainfall runoff model to estimate marginal model parameter probability density functions for parameter identification and uncertainty analysis.