A method of linear combination of multiple models for epistemic uncertainty minimization

In this study, we are evaluating methods to minimize the epistemic uncertainty that is usually caused due to the model selection and logic tree schemes. One of the methods is to simply select the best prediction model resulting in the least variation of errors if the variation is known, and the other one is to seek a multi-model weighting scheme through logic trees. For the latter case, selection of weights varies depending on the availability of distribution and correlation information among the models. Equal weights are often used when only mean predictions are available. Inverse-variance weighting methods are generally used when distributions for each model are available. Optimized weights can be determined by minimizing the prediction variance when distributions and correlations among the multiple models are available. In this study, we describe such methods with mathematical derivation and numerical solutions and apply to example cases with varying combination of variances and correlation levels to compare results. We find that the variance of combined model error is reduced when the models are less correlated, and the optimized weight scheme is more effective when the variance of each model is changing. Although such findings may not be claimed as new, we believe that this study can be considered as a benchmark. We also apply the linear combination method to proxy-based VS30 estimations using two regional data sets (California and Japan) considering three proxies for VS30: Slope, terrain, and geology. As these three proxy-based models are highly correlated, which results in two options for the best practice supposing that exact correlation is unknown: 1) use of the proxy with the least variance if variance is lower than others; 2) use of equal weight if variances are comparable.