Variance components and generalized Sobol' indices

1 Introduction

• Sobol’ indices are sums of variance components used to assess variable-subset importance in global sensitivity analysis.Linear combinations of indices also measure effective dimension for quasi-Monte Carlo integration.Source: §1 · p. 1
• Fixing methods estimate Sobol’ indices from pairs of function evaluations sharing selected input values while independently sampling the remaining inputs.Sample variances and covariances of the pairs are used for estimation.Source: §1 · pp. 1–2
• For the five-variable example, the second index includes interactions between the first three and last two variables, unlike the first index.Its sampling design matches the first index’s design.Source: §1 · p. 2
• The paper defines generalized Sobol’ indices as linear combinations of cross-moments from function evaluations with selected inputs fixed, then searches systematically for estimators of interpretable quantities.The search targets desirable computational or statistical properties.Source: §1 · p. 2
• The systematic search yields estimators with reduced computational cost or sampling variance and includes Saltelli’s (2002) strategy for estimating all two-factor interaction mean squares.The Saltelli strategy appears as a special case.Source: §1 · p. 2
• The paper organizes its estimator study around contrasts, squares, sums of squares, and bilinear GSIs, including applications to variance components and high-dimensional mean dimensions.It also presents bias correction for GSIs that are not contrasts and compares alternative methods.Source: §1 · p. 3

2 Background and notation

• The ANOVA decomposition assigns variance to effects for subsets of factors; its extension applies to square-integrable functions on tensor-product domains.The paper uses subset notation and defines effects recursively by subtracting sub-effects and averaging over variables outside the subset.Source: §2 · pp. 3–4

2.1 Sobol’ indices

• The lower and upper Sobol’ indices differ in interaction coverage: the lower index sums components for a set and its subsets, while the upper includes every component touching that set.The upper index therefore includes interactions between the selected inputs and their complement.Source: §2.1 · p. 4
• Sobol’ indices can be estimated from paired evaluations at hybrid points, where selected inputs are shared and the remaining inputs are independently sampled.The covariance between the function values at these points gives the variance component associated with the selected subset.Source: §2.1 · p. 5
• An estimator using an estimated mean can require 3 function evaluations per pair rather than 2; an unbiased variant uses 2 evaluations per pair.For finite fourth moment, the cited bias is asymptotically negligible, but it may matter when the index is small.Source: §2.1 · pp. 5–6

2.2 Related indices

• The paper also describes superset importance and effective dimension in the superposition and truncation senses as related measures of variable importance.Sums of ANOVA components are relevant to quasi-Monte Carlo performance, which depends on how variance is distributed across interaction orders.Source: §2.2 · p. 6
• Mean dimension is a linear combination of variance components, making it easier to estimate than effective dimension and more finely resolved in the example given.Two functions with effective dimension 2 can have mean dimensions 1.03 and 1.05.Source: §2.2 · p. 6

3 Generalized Sobol’ indices

• A generalized Sobol’ index is a linear combination of second-order moments formed from function evaluations with two subsets of inputs fixed.The moments are indexed by pairs of subsets and represented by the Sobol’ matrix Θ.Source: §3 · p. 7
• The family of GSIs has 2^(2d) dimensions, although only 2^d − 1 ANOVA components need estimation, so a target variance combination can have multiple GSI representations.This motivates selecting special forms with desirable properties.Source: §3.1 · p. 8

3.1 Special GSIs

• Contrasts are unaffected by the function mean and yield unbiased estimates of variance-component combinations; squares and sums of squares are nonnegative.Sparse square forms can be inexpensive to compute, while bilinear forms combine function pairs using at most the nonzero counts of their two coefficient vectors in function calls.Source: §3.1 · p. 8

3.2 Sample GSIs

• Sample GSIs are formed from pairs of independently sampled input vectors and a sample matrix of hybrid-point function products.The estimator takes the matrix inner product with the GSI coefficient matrix.Source: §3.2 · p. 9

3.3 Cost per (x, z) pair

• The per-pair evaluation cost counts the needed hybrid evaluations across nonzero rows and columns, subtracting evaluations counted in both.The paper treats the number of function evaluations as the dominant cost.Source: §3.3 · p. 9

4 Squares and sums of squares

• An unbiased nonnegative estimate of a GSI that is zero must return exactly zero.Squares and sums of squares yield nonnegative estimates, making this property especially valuable.Source: §4 · p. 10
• The paper notes that a sample version of Liu and Owen’s square expression for superset importance was judged best among four estimators by Fruth et al. (2012).The cited comparison concerns the superset importance measure Υ²_u.Source: §4 · p. 10
• No unbiased square or sum-of-squares GSI can estimate a truncation index τ²_u when |u| < d.The result follows because an unbiased sum of squares must include the highest-order variance component σ²_D.Source: §4 · p. 10

5 Specific variance components

• For w = {1, 2, 3}, equation (16) requires 6 function evaluations per (x, z) pair, versus 9 for equation (13).The paper also notes analogous variants of equation (15) that single out variables 2 and 3.Source: §5 · p. 11
• The bilinear estimate for a variance component uses 2^k + 2^|w|−k evaluations per pair for 1 ≤ k < |w|, compared with 2^|w| for a simple GSI.The cost expression is given for the specific variance component indexed by w.Source: §5 · p. 11
• A bilinear estimator of σ²_w is obtained from Theorem 3, while a similar argument gives a bilinear estimator of superset importance Υ²_w.Theorem 3 establishes the variance-component identity for a nonempty subset w.Source: §5 · pp. 12–13
• The bilinear estimator of superset importance costs 2|w₁| + 2|w₂| − 1 evaluations because one evaluation can be shared between the two terms.Here w₁ and w₂ partition w, and the shared evaluation is f(x_{wᶜ}:z_w).Source: §5 · pp. 12–13

6 GSIs with O(d) function evaluations per pair

• Some problems, including mean dimension, can use O(d) integrals rather than the O(2^d) needed to estimate all ANOVA components.The section focuses on GSIs based on O(d) carefully chosen evaluations per (x_i, z_i) pair.Source: §6 · p. 13

6.1 Cardinality restricted GSIs

• Restricting u and v to subsets of cardinality 0, 1, d − 1, or d makes several indices accessible, including all d(d−1)/2 second-order interactions at cost C = 2d + 2.The section also describes computing all d first-order indices at total cost C = d + 2.Source: §6.1 · pp. 13–14

6.2 Consecutive index GSIs

• For a discrete-time process, consecutive subsets represent inputs before and after a time j, allowing index values to describe forgetting of early inputs or lasting initial-condition effects.A small first-segment index indicates that the first j inputs are nearly forgotten; a large τ²_(0,j] indicates lasting initial-condition effects.Source: §6.2 · p. 15
• Consecutive-index subsets provide GSIs for quantities related to mean dimension in the truncation sense.The section also introduces a contrast measuring the extent to which indices at distant time lags contribute important interactions.Source: §6.2 · pp. 15–16

• When estimating a linear combination of variance components, the corresponding GSI is a contrast, though estimating a contrast can sometimes require an extra function evaluation per pair.Source: §7 · p. 16
• The estimator in (22) requires three function evaluations per pair, compared with two for Janon et al.’s (2012) biased estimator.Source: §7 · p. 16
• Proposition 3 gives a bias-corrected version of Janon et al.’s (2011) estimator of τ²_u using only two function evaluations per pair.Source: §7 · p. 16
• For non-contrast GSIs, the term µ²1ᵀΩ1 can be nonzero, and a bias correction is available for tr(ΩᵀΘ).Source: §7 · p. 17
• The unbiased estimator in Proposition 4 requires no additional function evaluations, but its confidence intervals are more complicated than those for contrasts.Jackknife or bootstrap methods can be used; contrast estimators are simple averages.Source: §7 · p. 17

8 Comparisons

• The space of GSIs has dimension 2^(2d), while linear combinations of variance components occupy a 2^d − 1 dimensional space, allowing multiple estimators for a target combination.The theoretical comparisons do not lead to unambiguous choices; GSI variance depends on generally unknown fourth moments, and no closed-form estimator appears known even for the two-way crossed model with interaction.Source: §8 · pp. 17–18, §8.1 · p. 18

8.1 Minimum variance estimation

• The GSI setting extends MINQE’s proxy-norm idea, but estimating minimum-variance choices is complicated by unknown fourth moments of ANOVA contributions.The GSI context includes variance components for crossed random-effects models with interactions of all orders.Source: §8.1 · p. 18
• A cost–variance proxy favors the original Sobol’ estimator for τ²_u and alternative estimator (16) for σ²_{ {1,2,3} }.The proxy compares C(Ω) × V(Ω); the reported values are 2 versus 6 for the τ²_u estimators and costs 6 versus 9 for the two σ²_{ {1,2,3} } estimators, which both have V(Ω) = 8.Source: §8.1 · p. 18

8.2 Test cases

• The comparisons use product-form test functions and a minimum function, with Monte Carlo estimation aimed at functions inexpensive to evaluate.The authors note that smoothness of the component functions does not affect these Monte Carlo comparisons.Source: §8.2 · p. 19

8.3 Estimation of σ2

• For product-form functions, all four estimators give the same answer for any specific set of sampled (x, z) pairs, while three bilinear estimators require 6 rather than 9 evaluations per pair.Source: §8.3 · p. 19
• On the minimum function, bilinear estimators had about half the simple estimator’s standard error, corresponding to about 5.1 times the statistical efficiency.The comparison is based on 1,000,000 function evaluations.Source: Table 1 · p. 20, §8.3 · p. 20

8.4 Estimation of τ

• 0.5–2.5: contrast efficiency relative to the simple estimator spans this range, with the contrast better for the small τ²_u effect.Neither estimator is always more efficient, so the proxy cannot reliably predict which is better for a specific problem.Source: §8.4 · p. 20, Table 2 · p. 21

8.5 Estimation of Υ2

• The square estimator’s advantage over the bilinear estimator more than compensates for its higher evaluation cost, especially for the smaller effect.The bilinear estimate uses 7 function evaluations versus 16 for the square; the bilinear cost advantage grows with |w|, while the square’s variance advantage can be extreme for small effects.Source: §8.5 · p. 21, Table 3 · p. 22

9 Conclusions

• Generalized Sobol’ indices estimate arbitrary linear combinations of variance components, and a target can have multiple estimators with different efficiencies.Square GSIs are effective when available; bilinear or low-rank GSIs can reduce function evaluations when no square or sum of squares is available.Source: §9 · p. 22
• Contrasts are simpler to work with than other GSIs because they avoid bias corrections.Source: §9 · p. 22
• Square GSIs are effective when available; otherwise, bilinear or low-rank GSIs can save function evaluations.Source: §9 · p. 22