Deriving Sobol Indices from Functional ANOVA

Why Decompose Output Variance?

A QSP model can respond to several uncertain inputs at once. A global sensitivity analysis asks how variation in a specified model output is apportioned across those inputs, including variation that arises through their interactions. Sobol indices answer this question by decomposing output variance into contributions associated with individual inputs and input subsets.

The decomposition is not a property of the equations alone. It depends on the probability distributions assigned to the inputs. In particular, the standard functional ANOVA decomposition identifies distinct variance contributions when the inputs are mutually independent. This tutorial derives that result and the indices that follow from it.

Square-integrability ensures that the component variances are finite. Independence means the joint input distribution is the product of the marginal distributions. Inputs may be continuous, discrete, or a mixture. The model output and input distributions must be fixed for the indices to have a defined meaning. If \(V=0\), variance-based indices with denominator \(V\) are undefined.

Constructing the Functional ANOVA Components

For a subset \(u\) of input indices, write \(X_u\) for the inputs whose indices belong to \(u\). The empty-set component is the overall mean. The component for a nonempty set \(u\) is defined recursively: take the conditional mean of the model output given \(X_u\), then subtract the components already assigned to all proper subsets of \(u\).

For a single input, this gives \(f_i(X_i)=E[Y\mid X_i]-E[Y]\), the centered main effect. For a pair, it gives \(f_{ij}(X_i,X_j)=E[Y\mid X_i,X_j]-f_{\varnothing}-f_i(X_i)-f_j(X_j)\), the part of the conditional response not accounted for by the mean and the two main effects. Components involving three or more inputs are defined in the same way.

The recursion gives the pointwise decomposition, up to sets of probability zero:

The key property is that each component has conditional mean zero when any one of its own inputs is averaged out. For \(i\in u\), this is \(E[f_u(X_u)\mid X_{u\setminus\{i\}}]=0\). The property follows from the recursive definition and independence: integrating over \(X_i\) in \(E[Y\mid X_u]\) produces \(E[Y\mid X_{u\setminus\{i\}}]\), and the lower-order components subtract exactly the terms in that conditional mean.

Why the Variances Add

The conditional-mean-zero property makes components for different subsets orthogonal. To see this, choose distinct subsets \(u\) and \(v\). At least one input index belongs to one subset but not the other; without loss of generality, let \(i\in u\) and \(i\notin v\). Then \(f_v\) does not depend on \(X_i\). Conditional on all inputs other than \(X_i\), \(f_v\) is fixed, while the conditional mean of \(f_u\) over \(X_i\) is zero. Therefore \(E[f_u f_v]=0\). This also applies when one component is the constant mean and the other is centered.

Squaring the decomposition and taking expectations now leaves no cross-products between distinct components. Since each nonconstant component has mean zero, its second moment is its variance. Thus the output variance is the sum of the component variances:

This is the substantive role of independence: it makes the conditional-mean construction yield orthogonal components, so each \(V_u\) is a nonnegative, separately identifiable contribution to total variance. With dependent inputs, the same formulas do not generally yield orthogonal components or a unique allocation of variance to subsets. A correlated-input analysis therefore needs a different allocation framework rather than treating these standard Sobol components as if independence still held.

First-, Second-, and Total-Order Indices

A first-order index is the fraction of output variance assigned to one input's main-effect component. A second-order index is the fraction assigned to the component for a specified pair, after the mean and both main effects have been removed. More generally, \(S_u\) is the fraction assigned to subset \(u\).

The total-order index for input \(i\) sums the variance fractions of every component that includes \(i\), including its main effect and all its interactions with other inputs. It can also be expressed using a conditional variance:

Here \(X_{-i}\) denotes all inputs except \(X_i\). To justify the conditional-variance form, condition on \(X_{-i}\). Components not involving \(i\) are fixed under that conditioning; the components involving \(i\) have conditional mean zero. Their conditional variances add because they remain orthogonal when averaged over \(X_i\). Averaging over \(X_{-i}\) gives \(\sum_{u:i\in u}V_u\). The final equality follows from the law of total variance: \(V=E[\operatorname{Var}(Y\mid X_{-i})]+\operatorname{Var}(E[Y\mid X_{-i}])\).

Assumptions, Diagnostics, and Alternatives

Before interpreting a decomposition, check that it matches the scientific question and the uncertainty representation. Independence is a substantive assumption about the joint distribution used for the analysis, not a consequence of the model equations. If calibrated parameter draws are dependent, the standard independent-input decomposition is not automatically an identifiable attribution among those parameters. Also check that the output is a scalar quantity with finite, nonzero variance under the chosen input distributions.

Theoretical component variances cannot be negative, and their sum must equal \(V\). First-order indices lie between zero and one; total-order indices do as well, and \(S_i\leq S_{T_i}\). For a complete decomposition, the sum of all subset indices is one, while the first-order indices alone generally sum to less than or equal to one. If numerical estimates violate these properties, possible explanations include finite-sample error, coding or scaling errors, or a mismatch between the assumed input distribution and the one actually sampled. A computed zero component can reflect genuine absence of that modeled contribution, but it can also arise from a particular output and input distribution that suppresses variation.

The introductory material on Morris elementary effects motivates screening with a relatively economical measure, but its elementary effects are not the same variance partition as Sobol indices. Local derivatives describe response near particular input values and can be useful when local behavior is the target; they do not by themselves allocate global output variance, especially when interactions or nonlinearities matter. Functional ANOVA instead provides a global variance accounting under independent inputs, at the cost of requiring an appropriate input distribution and a well-defined output variance.