Given-Data Sobol' Indices

• Given-Data Sobol’ Index Method is a variance-based sensitivity analysis approach that estimates sensitivity indices from observed data using flexible, distribution-aware partitions.
• Its streaming algorithm efficiently updates per-bin statistics for high-dimensional models, reducing memory usage while accommodating tens of thousands of parameters.
• A noise symmetrization heuristic filters out spurious sensitivities by reflecting negative indices and applying a conservative threshold to report only statistically meaningful contributions.

The Given-Data Sobol' Index Method refers to a suite of statistical techniques for estimating variance-based sensitivity indices directly from observational or simulation data, as opposed to ensemble-based methods that require repeated model runs with controlled input variation. The method has become central for global sensitivity analysis in cases where the model is computationally expensive or the dimensionality of the input space is extremely high. Its developments have focused on estimator design, computational scalability, bias and noise characterization, robustness to distributional assumptions, and practical applicability to domains such as neural networks with $>10^4$ parameters.

1. Generalized Estimator Formulation for Arbitrary Partitions

Traditional given-data methods often employ equiprobable partitions of the input domain for each variable, defining bins $A_1, \ldots, A_M$ so that each bin has equal empirical probability under the available data. The crucial advancement described in (Portone et al., 11 Sep 2025) is a fully general definition that accommodates arbitrary partitions—allowing for equidistant bins, quantile bins, or bins adapted to repeated values in nonstandard distributions:

$$\Omega_{X_i} = \bigcup_{k=1}^M A_k,\quad A_k \cap A_j = \emptyset ~~\text{for}~ j \neq k$$

The empirical probability for bin $A_k$ is $P_k = n_k / N$ for $n_k$ samples in $A_k$, where $N$ is the total sample count. The between-bin component (numerator) of the first-order Sobol' index for $X_i$ is then

$$\mathbb{E}^P [V] = \sum_{k=1}^M s^2(X_i \in A_k) \cdot P_k$$

where $s^2(X_i \in A_k)$ is the within-bin variance of $Y$. The corresponding estimator is

$$\hat{S}_i = 1 - \frac{\sum_k s^2(X_i \in A_k)\, (n_k/N)}{\tilde{V}}$$

where $\tilde{V}$ is the overall sample variance. This generalization overcomes restrictions of equiprobable bins, providing flexibility critical for models with non-uniform, heavy-tailed, or discrete input distributions.

2. Streaming Algorithm for Scalable Computation

Processing all input-output samples simultaneously is intractable for large $d$ (number of inputs) due to the $O(Nd)$ memory footprint. To address this, (Portone et al., 11 Sep 2025) introduces a streaming algorithm based on incremental update formulas (Chan et al., Pebay):

• An initial sample defines bin edges by quantiles, kernel density estimates, or equidistant division.
• As new batches arrive, per-bin and global means and unscaled variances are updated by

$$n = n_1 + n_2; \quad \delta = \mu_2 - \mu_1; \quad \mu = \mu_1 + \frac{n_2}{n}\delta; \quad U = U_1 + U_2 + \frac{n_1 n_2}{n} \delta^2$$

• Final indices are computed using the combined statistics after all samples are processed.

This approach achieves sublinear memory usage in $N$ and $d$ and enables analysis of models with $10^4$–$10^5$ parameters. Empirical tests show batch-processed indices closely match all-at-once estimators while maintaining computational efficiency.

3. Noise Symmetrization Heuristic for Filtering Spurious Sensitivities

In high-dimensional settings, the noise in the estimator can result in many indices near zero, with some even negative due to sampling error. To distinguish meaningful sensitivity indices from indistinguishable-from-zero ones, a noise-filtering heuristic is proposed:

• All negative indices (statistically assumed to arise from noise) are reflected about zero to construct an empirical noise distribution.
• The noise standard deviation $\sigma$ is estimated from this distribution.
• A conservative rule is applied: indices smaller than $4\sigma$ are considered statistically indistinguishable from zero.

This process ensures that only indices above the detection threshold—taking into account sample-induced variability—are reported as significant. The approach is particularly applicable when $d \gg 1$ and the effective number of influential variables is a small fraction of $d$.

4. Partition-Induced Bias and the Limitations of Equiprobable Binning

Standard equiprobable partitions generally fail when the input distribution is non-uniform or highly multimodal (e.g., "spike-slab" from analog circuit inputs). Bias emerges when rapid changes in $\mathbb{E}[Y|X]$ align with bin boundaries or when sample sparsity in the partitions yields poor conditional variance approximations.

Numerical analysis in (Portone et al., 11 Sep 2025) demonstrates:

• Equiprobable bins can produce pronounced bias, especially near discontinuities or heavy-tailed regions.
• Equidistant partitions, which fix bin locations in the input domain, often reduce such bias for normal and multi-point support scenarios.
• Approximating the conditional variance as $V(Y|X)$ within a bin $A_k$ using sample statistics can be inaccurate when $|A_k|$ is small or $f$ is non-smooth.

Empirical evaluation confirms failure of the equiprobable approach in several synthetic distributions, motivating the shift to generalized, distribution-aware partition strategies.

5. Numerical and Application Studies: High-Dimensional and Deep Models

Extensive benchmarking supports these methodological extensions:

• Analytical test functions (polynomials, Sobol' G, Ishigami) provide ground truth for controlled assessment. Comparison of streaming and batch estimators, across partition strategies, confirms that the equidistant and adaptive bin methods both reduce bias compared to equiprobable partitioning.
• The standard deviation of the noise distribution converges as $\mathcal{O}(1/N)$, validating the $4\sigma$ threshold's effectiveness as sample size increases.
• In a neural network for analog satellite object detection ($d \approx 10^4$) and a CIFAR-10 classifier ($d = 175,000$), the streaming and filtering methods scale to modern model sizes. Index estimates reveal that only $<1\%$ of weights are statistically significant in variance contribution, consistent with the underlying network structure (many nearly-zero weights due to initialization or sparsity induction).

6. Implications for Sensitivity Analysis in Modern Models

These scalable extensions fundamentally broaden the applicability of given-data Sobol' index estimation to domains where memory and computational constraints previously made variance-based sensitivity analysis infeasible. By enabling flexible, partition-aware, and streaming implementations with robust noise quantification, the method supports:

• Identification and interactive filtering of influential parameters in ultra-high-dimensional models.
• Accurate sensitivity analysis for models with discrete, multimodal, or otherwise atypical input support.
• Deployment in resource-constrained inference environments, including hardware-embedded neural networks and real-time experimental workflows.

7. Summary

The advancements in (Portone et al., 11 Sep 2025) introduce a general, distribution-agnostic, and memory-efficient methodology for Sobol' index estimation from given data. The formal estimator supports arbitrary binning, achieves streaming updates through incremental statistics, and incorporates a robust statistical filtering technique to separate genuinely influential from noise-induced indices. Combined with explicit demonstrations of equiprobable partition bias and application to large-scale neural models, these developments render global sensitivity analysis feasible in domains characterized by massive parameter spaces and non-trivial input distributions.