Quantifying dependencies for sensitivity analysis with multivariate input sample data

We present a novel method for quantifying dependencies in multivariate datasets, based on estimating the R\'{e}nyi entropy by minimum spanning trees (MSTs). The length of the MSTs can be used to order pairs of variables from strongly to weakly dependent, making it a useful tool for sensitivity analysis with dependent input variables. It is well-suited for cases where the input distribution is unknown and only a sample of the inputs is available. We introduce an estimator to quantify dependency based on the MST length, and investigate its properties with several numerical examples. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. We apply our proposed method to an artificial testcase based on the Ishigami function, as well as to a real-world testcase involving sediment transport in the North Sea. The results are consistent with prior knowledge and heuristic understanding, as well as with variance-based analysis using Sobol indices in the case where these indices can be computed

Modelling of combustion processes and NO formation with reduced reaction mechanisms

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Quantifying dependencies for sensitivity analysis with multivariate input sample data

We present a novel method for quantifying dependencies in multivariate datasets, based on estimating the Rényi entropy by minimum spanning trees (MSTs). The length of the MSTs can be used to order pairs of variables from strongly to weakly dependent, making it a useful tool for sensitivity analysis with dependent input variables. It is well-suited for cases where the input distribution is unknown and only a sample of the inputs is available. We introduce an estimator to quantify dependency based on the MST length, and investigate its properties with several numerical examples. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. We apply our proposed method to an artificial testcase based on the Ishigami function, as well as to a real-world testcase involving sediment transport in the North Sea. The results are consistent with prior knowledge and heuristic understanding, as well as with variance-based analysis using Sobol indices in the case where these indices can be computed

Clustering-based collocation for uncertainty propagation with multivariate correlated inputs

In this article, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation (SC). The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, correlated input. The approach is particularly useful in situations where the probability distribution of the input is unknown, and only a sample from the input distribution is available. We examine several clustering methods and assess their suitability for stochastic collocation numerically using the Genz test functions as benchmark. The proposed methods work well, most notably for the challenging case of nonlinearly correlated inputs in higher dimensions. Tests with input dimension up to 16 are included. Furthermore, the clustering-based collocation methods are compared to regular SC with tensor grids of Gaussian quadrature nodes. For 2-dimensional uncorrelated inputs, regular SC performs better, as should be expected, however the clustering-based methods also give only small relative errors. For correlated 2-dimensional inputs, clustering-based collocation outperforms a simple adapted version of regular SC, where the weights are adjusted to account for input correlatio