[Paper Review] A new kernel-based index for the global sensitivity analysis of models with correlated inputs

Troy Arron Larsen, Alen Alexanderian|arXiv (Cornell University)|Mar 1, 2026

Probabilistic and Robust Engineering Design0 citations

TL;DR

The paper introduces the total HSIC sensitivity index, a bounded, moment-independent, monotone under marginalization kernel-based measure for global sensitivity analysis with correlated and possibly function-valued inputs and outputs.

ABSTRACT

We present an HSIC-based approach for global sensitivity analysis of broad classes of models with correlated and possibly function-valued inputs and outputs. To this end, we define the total HSIC sensitivity index: a bounded, interpretable, and moment-independent analogue to the total-effect Sobol' index. These desirable qualities hinge upon the key property of monotonicity under marginalization for the HSIC. We rigorously establish this monotonicity property by using a suitable class of augmented kernels. Furthermore, we provide an efficient algorithm for computing an empirical estimator of the HSIC that significantly reduces computational complexity and storage requirements. The effectiveness and interpretability of the total HSIC sensitivity indices are demonstrated through computational experiments on models that feature nonlinear relationships, correlated inputs, and functional outputs.

Motivation & Objective

• Motivate global sensitivity analysis (GSA) for models with correlated inputs and possibly function-valued inputs/outputs.
• Introduce a kernel-based framework using HSIC that handles dependence among inputs.
• Ensure monotonicity under marginalization by augmenting kernels so the total HSIC can be interpreted like Sobol’ total effects.
• Define a bounded, interpretable, moment-independent total HSIC sensitivity index and establish its theoretical properties.
• Develop an efficient algorithm to estimate HSIC empirically and demonstrate its effectiveness on nonlinear, correlated, and functional-output models.

Proposed method

• Adopt a reproducing kernel Hilbert space (RKHS) framework and use characteristic kernels to measure dependence via HSIC.
• Construct augmented kernels that ensure the RKHS contains the constant function, enabling monotonicity under marginalization (Theorem 3.9).
• Define the total HSIC sensitivity index T_A(f) = 1 - HSIC(X_{ ilde A}, Y) / HSIC(X, Y) to quantify the share of dependence attributed to X_A.
• Provide an empirical estimator for HSIC using centered Gram matrices and Gaussian kernels with a median heuristic for bandwidths.
• Prove that T_A(f) is bounded between 0 and 1 under the augmented kernel setup (Lemma 4.1).
• Outline an efficient computation strategy for HSIC and demonstrate the approach with computational experiments on nonlinear, correlated, and functional-output models.

Experimental results

Research questions

• RQ1 How can HSIC-based measures be extended to provide a total sensitivity index for models with correlated inputs?
• RQ2 Can augmented kernels ensure monotonicity under marginalization to make HSIC-based indices interpretable like Sobol’ total effects?
• RQ3 Is the total HSIC sensitivity index bounded and interpretable across scalar, vector, and function-valued inputs/outputs?
• RQ4 How can HSIC be efficiently estimated from data to handle high-dimensional inputs and correlated structures?

Key findings

• Introduction of the total HSIC sensitivity index, a bounded, interpretable, and moment-independent analogue of the total-effect Sobol’ index.
• Proof of monotonicity under marginalization for HSIC via augmented kernels, enabling meaningful subset-based sensitivity rankings.
• An efficient estimator for HSIC using centered Gram matrices and Gaussian kernels, with reduced computational complexity and storage.
• Demonstration through computational experiments that the total HSIC captures nonlinear relationships, input correlations, and functional outputs.
• Theoretical guarantees that the augmented kernel construction preserves the characteristic property and enables monotonicity (Theorem 3.9, Corollary 3.7).
• The framework unifies kernel-based GSA with Sobol’-style interpretability under correlated-input scenarios.