Unsupervised local cluster-weighted bootstrap aggregating the output from multiple stochastic simulators

Abstract

In the present work, we consider the problem of combining the output from multiple stochastic computer simulators to make inference on a quantity of interest, as a means of reducing the inherent model-form uncertainty in the absence of any measurements. In most real-world situations, judging an individual stochastic simulator to be the “best” for any given point in the input space is highly doubtful. Thus, making inference by relying on the so-deemed best simulator may not be adequate, especially when the sampled data is limited. To this end, we propose an ensemble learning method based on local Clustering and bootstrap aggregation (Bagging), which rather than treating the stochastic predictions of the simulators as competing individual information sources, treats those as part of an ensemble, thus diversifying the hypothesis space. We call the proposed method: unsupervised local cluster-weighted bootstrap aggregation. Variational Bayesian Gaussian mixture clustering is the first step in this ensemble learning approach for discriminating the outputs, and deriving the probability map (weights) of the clustered simulators output. Clustering is performed on the stochastic output corresponding to the binned input space. Performing the clustering independently and deriving the probability map for each local region of the binned input space is a novelty that guarantees an adaptive solution, whereby certain simulators are potentially more fitting than others in corresponding regions of the input space. The second step consists in a local cluster-weighted Bootstrap Aggregation, which serves the purpose of weighted combination of the clustered ensemble of outputs from the individual simulators. Based on simulations, we demonstrate how the input bin size, sample size, output dispersion and level of agreement amongst the simulators affect the performance of the proposed method. We compare the unsupervised local cluster-weighted bootstrap aggregation method to classical Bagging, Bayesian Model Averaging and Stacking of predictive distributions. Finally, we demonstrate the method by evaluating the fatigue damage equivalent load on a wind turbine blade, using 10 finite element based simulators. The results point to the need for practitioners to consider this as a useful method, when model-form uncertainty is of concern and when output from multiple stochastic simulators are available.