[Paper Review] Data-Driven Sensitivity Indices for Models With Dependent Inputs Using the Polynomial Chaos Expansion

Zhanlin Liu, Youngjun Choe|arXiv (Cornell University)|Mar 29, 2018

Probabilistic and Robust Engineering Design4 citations

TL;DR

This paper proposes a data-driven method for computing variance-based sensitivity indices in models with dependent inputs using polynomial chaos expansion (PCE) without requiring prior knowledge of input dependence structures. By applying a modified Gram-Schmidt algorithm to empirical data, the method constructs orthogonal polynomials and derives interpretable sensitivity indices that quantify how dependent inputs contribute to output variance.

ABSTRACT

Uncertainties exist in both physics-based and data-driven models. Variance-based sensitivity analysis characterizes how the variance of a model output is propagated from the model inputs. The Sobol index is one of the most widely used sensitivity indices for models with independent inputs. For models with dependent inputs, different approaches have been explored to obtain sensitivity indices in the literature. Typical approaches are based on procedures of transforming the dependent inputs into independent inputs. However, such transformation requires additional information about the inputs, such as the dependency structure or the conditional probability density functions. In this paper, data-driven sensitivity indices are proposed for models with dependent inputs. We first construct ordered partitions of linearly independent polynomials of the inputs. The modified Gram-Schmidt algorithm is then applied to the ordered partitions to generate orthogonal polynomials with respect to the empirical measure based on observed data of model inputs and outputs. Using the polynomial chaos expansion with the orthogonal polynomials, we obtain the proposed data-driven sensitivity indices. The sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge on the dependence structure of the inputs. Three numerical examples are used to validate the proposed approach.

Motivation & Objective

• Address the challenge of variance-based sensitivity analysis in models with dependent inputs, where traditional Sobol indices are not directly applicable.
• Overcome limitations of existing methods that require assumptions about input dependence structures or conditional probability density functions.
• Develop a data-driven framework that leverages observed input-output data to estimate sensitivity indices without prior knowledge of input dependencies.
• Enable intuitive interpretation of input contributions to output variance through orthogonal polynomial expansions derived from empirical measures.
• Introduce conditional order-based sensitivity indices to hierarchically explain output variability from dependent inputs.

Proposed method

• Construct ordered partitions of linearly independent polynomials from input data to form a basis for polynomial chaos expansion.
• Apply the modified Gram-Schmidt algorithm to orthogonalize the polynomial basis with respect to the empirical measure derived from observed input-output data.
• Build a polynomial chaos expansion (PCE) model using the resulting orthogonal polynomials to approximate the model response.
• Derive data-driven sensitivity indices from the PCE coefficients, enabling variance-based decomposition of output uncertainty.
• Introduce conditional order-based sensitivity indices to provide hierarchical interpretation of input contributions to output variance.
• Use numerical examples to validate the method’s accuracy and robustness in estimating sensitivity indices without assuming input dependence structure.

Experimental results

Research questions

• RQ1 How can variance-based sensitivity indices be reliably computed for models with dependent inputs when no prior knowledge of their dependence structure is available?
• RQ2 Can a data-driven PCE-based method produce interpretable and accurate sensitivity indices without requiring transformation of dependent inputs into independent ones?
• RQ3 How does the modified Gram-Schmidt algorithm improve numerical stability in constructing orthogonal polynomials from empirical data compared to standard PCE approaches?
• RQ4 To what extent can the proposed method estimate sensitivity indices comparable to those from transformation-based methods without invoking their restrictive assumptions?
• RQ5 How do the proposed conditional order-based sensitivity indices enhance the interpretability of input contributions in models with dependent inputs?

Key findings

• The proposed method successfully computes interpretable sensitivity indices for models with dependent inputs using only observed input-output data, without requiring knowledge of the input dependence structure.
• The modified Gram-Schmidt algorithm enhances numerical stability in constructing orthogonal polynomials from empirical data, outperforming standard PCE approaches in terms of robustness.
• The method estimates sensitivity indices comparable to those from transformation-based approaches (e.g., [16], [17]) without requiring assumptions about conditional densities or input distributions.
• In Example 3, the analytical and data-driven sensitivity indices for the product terms $X_1X_2$, $X_3X_4$, and $X_5X_6$ show close agreement, validating the method’s accuracy.
• In Example 4, the sensitivity indices derived from the data-driven PCE model align well with reference values obtained from a response surface model, demonstrating robustness under complex dependence structures.
• The conditional order-based sensitivity indices provide a hierarchical decomposition of output variance, enabling clearer interpretation of how groups of dependent inputs contribute to uncertainty.