Variance-Based Sensitivity and Sobol Indices

Summary

Sobol’s method decomposes the output variance $V(Y)$ into contributions from individual factors and their interactions (the ANOVA / HDMR / Sobol-Hoeffding decomposition), assuming inputs are independent and uncorrelated. The first-order index $S_i=V_i/V(Y)$ measures the variance attributable to $X_i$ alone; the total-effect index $S_{Ti}$ sums all terms involving $X_i$, capturing its main effect plus every interaction. The gap $S_{Ti}-S_i$ quantifies how much of a factor’s influence operates through interactions, and ${\sum }_i{S}_i=1$ exactly for a purely additive model.

Overview

Variance-based methods rest on the assumption (Saltelli et al.) “that variance is sufficient to describe the output uncertainty.” The rationale: examine the variance of the conditional expected output given a factor. Sobol’s method “relies on decomposition of the model output variance under the assumption that inputs are independent and uncorrelated.” See Global Sensitivity Analysis - Overview.

Main Content

ANOVA / Sobol variance decomposition ^anova-decomposition

For $Y=f(X_1,\dots ,X_p)$ with independent inputs, the total variance decomposes into orthogonal terms of increasing order:

$$V(Y)=\sum _{i=1}^p{V}_i+\sum _{1\le i<j\le p}{V}_{ij}+\dots +V_{1,2,\dots ,p}$$

where the partial (first-order) variance of factor $X_i$ is

$$V_i=V(\mathbb{E}(Y\mid X_i))$$

and the second-order interaction variance of $X_i,X_j$ is

$$V_{ij}=V(\mathbb{E}(Y\mid X_i,X_j))-V_i-V_j.$$

$V_i$ is the expected reduction in output variance if $X_i$ were fixed; $V_{ij}$ is the additional joint effect beyond the two main effects.

First-order (main effect) Sobol index ^first-order-index

$$S_i=\frac{V_i}{V(Y)}=\frac{V(\mathbb{E}(Y\mid X_i))}{V(Y)}$$

$S_i\in [0,1]$ is the fraction of output variance caused by factor $X_i$ acting alone (its main effect, averaged over all other factors). Used for factor prioritization: a large $S_i$ means determining $X_i$ more precisely most reduces output uncertainty. The analogous second-order index is $S_{ij}=V_{ij}/V(Y)$, the share due to the $X_i$–$X_j$ interaction.

Total-effect (total-order) Sobol index ^total-effect-index

$$S_{Ti}=\sum _{k\in \#i}{S}_k$$

the sum over all index combinations $k$ that contain $i$ (its main effect plus every interaction it participates in). Equivalently $S_{Ti}=1-S_{\sim i}$, the share of variance left when all factors except $X_i$ are fixed. Used for factor fixing: $S_{Ti}\approx 0\Rightarrow X_i$ can be frozen anywhere in its range with negligible effect on $Y$.

Interaction detection and budget identities ^interaction-identities

Two diagnostic relations follow directly:

• ${\sum }_{i=1}^p{S}_i\le 1$, with equality iff the model is purely additive (no interactions). A deficit $1-{\sum }_i{S}_i$ measures total interaction strength.
• $S_{Ti}\ge S_i$ always; the gap $S_{Ti}-S_i$ is the portion of $X_i$‘s effect mediated by interactions with other factors. $S_{Ti}=S_i$ means $X_i$ acts additively.
• ${\sum }_i{S}_{Ti}\ge 1$, with equality iff additive (interactions are counted once per participating factor, so they are double/multiply counted in the total-effect sum).

The paper notes Sobol “assesses the impact of each input parameter, both in isolation and in conjunction with other parameters,” yielding first-, second-, total-, and higher-order indices. In the MNIST case study the Sobol $S_T$ index was among the most reliable importance measures.

Examples

Suppose a 3-parameter ABM gives $S_1=0.40,S_2=0.20,S_3=0.05$ and $S_{T1}=0.45,S_{T2}=0.50,S_{T3}=0.35$.

• $\sum S_i=0.65<1$ → 35% of output variance comes from interactions (non-additive model).
• $X_1$: $S_{T1}-S_1=0.05$ → mostly acts alone; it is the dominant main driver.
• $X_3$: $S_3=0.05$ but $S_{T3}=0.35$ → almost all of $X_3$‘s influence is through interactions. A local OAT scan holding others fixed would wrongly conclude $X_3$ is unimportant (see Local vs Global Sensitivity Analysis).
• $X_2$ has the largest total effect ($S_{T2}=0.50$) despite a modest main effect — a key interacting factor that must not be fixed.

Connections

• Global Sensitivity Analysis - Overview — places variance-based methods among the four GSA families.
• Sampling and Estimation for Sobol Indices — how $V_i$, $S_{Ti}$ are estimated (Saltelli Monte-Carlo, FAST spectral).
• Morris Elementary Effects Screening — cheap screening; Morris ${\mu }^{\ast }$ correlates with $S_{Ti}$ for ranking.
• Local vs Global Sensitivity Analysis — total-effect indices are exactly what OAT cannot capture.
• Uncertainty Quantification for ABM Calibration — variance decomposition complements output UQ.

See Also

• History Matching for ABMs; Approximate Bayesian Computation for ABMs — $S_{Ti}$ guides which parameters to keep active.
• Sobol (2001), “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.”
• Saltelli et al. (2010), “Variance based sensitivity analysis of model output: design and estimator for the total sensitivity index.”