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Complete orthonormal tensor expansions group into Sobol components

Statement

Assume Countable Choice. Let $X = (X_{1}, \dots, X_{d})$ have a finite product law. For each coordinate $j$ , let $(ϕ_{j, k})_{k \in A_{j}}$ be a finite or countable orthonormal family of real-valued functions in the real space $L^{2} (μ_{j})$ with $0 \in A_{j}$ , $ϕ_{j, 0} = 1$ , and $E ϕ_{j, k} (X_{j}) = 0$ for $k \neq 0$ . For $α \in A_{1} \times \dots \times A_{d}$ , put $ψ_{α} (x) = \prod_{j} ϕ_{j, α_{j}} (x_{j})$ and $supp (α) = {j : α_{j} \neq 0}$ . Assume the tensor family $(ψ_{α})_{α}$ is a complete orthonormal family in $L^{2} (μ)$ , and write $c_{α} = ⟨ f, ψ_{α} ⟩$ for real $f \in L^{2} (μ)$ . Then $Var (f) = \sum_{α \neq 0} c_{α}^{2}, V_{u} = \sum_{supp (α) = u} c_{α}^{2} (u \neq \emptyset) .$ Consequently, if $V = Var (f) > 0$ , $S_{i} = \frac{\sum_{supp (α) = {i}} c_{α}^{2}}{V}, S_{T_{i}} = \frac{\sum_{i \in supp (α)} c_{α}^{2}}{V} .$ The completeness assumption is separate from finite polynomial-chaos orthogonality; no law is asserted here to have a complete polynomial family.

Facts & Assumptions

Given: Countable Choice, the real product-law tensor family and completeness hypotheses in the Statement, and real $f \in L^{2} (μ)$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed only for the Parseval supplier below.

[F1]

Independent product inputs have an orthogonal Hoeffding decomposition: $f_{u} = \sum_{v \subseteq u} (- 1)^{∣ u ∣ - ∣ v ∣} P_{v} f$ and the variance components are $V_{u} = ∥ f_{u} ∥_{2}^{2}$ .

[F2]

Product-coordinate projections commute and contract in L2: every $P_{v}$ is an $L^{2}$ contraction.

[F3]

Parseval equivalences for an orthonormal family: under Countable Choice, finite-subset projections of a complete orthonormal family converge in norm to $f$ and their squared coefficients sum to $∥ f ∥_{2}^{2}$ .

[F4]

Orthonormal families, complete orthonormal systems and Hilbert bases: completeness means that the closed linear span is the whole Hilbert space.

[F5]

Fubini's theorem for L^1 functions on a sigma-finite product: integrable product sections may be integrated one coordinate at a time.

[F6]

$L^{2}$ with the integral pairing is a Hilbert space: on real $L^{2} (μ)$ , $⟨ f, g ⟩ = \int f g d μ$ , so the constant coefficient is $E f$ .

[F7]

First-order and total Sobol indices: positive output variance is required for the normalized indices.

Proof

Given: The hypotheses and facts [A1], [F1]–[F7].

1.1F5given

For a tensor $ψ_{α}$ and $J \subseteq [d]$ , the product integral defining $P_{J} ψ_{α}$ factors by [F5]. If every nonconstant coordinate of $α$ lies in $J$ , every integrated factor is $ϕ_{j, 0} = 1$ , so $P_{J} ψ_{α} = ψ_{α}$ . If some $j \notin J$ has $α_{j} \neq 0$ , its factor has mean zero, so $P_{J} ψ_{α} = 0$ .

2.1F1step 1.1algebra

Let $F$ be a finite set of multi-indices and $f_{F} = \sum_{α \in F} c_{α} ψ_{α}$ . By [F1] and step 1.1, the coefficient of $c_{α} ψ_{α}$ in $(f_{F})_{u}$ is zero when $supp (α) \subseteq̸ u$ ; otherwise it is $\sum_{supp (α) \subseteq v \subseteq u} (- 1)^{∣ u ∣ - ∣ v ∣} = (1 - 1)^{∣ u ∣ - ∣ supp (α) ∣}$ . This equals one exactly when $supp (α) = u$ and zero otherwise. Hence $(f_{F})_{u} = \sum_{α \in F : supp (α) = u} c_{α} ψ_{α}$ .

3.1A1F2F3F4step 2.1

By [F4], completeness means the tensor family's closed span is all of $L^{2} (μ)$ . With [A1], Parseval [F3] gives $f_{F} \to f$ in $L^{2}$ as the finite-subset net increases through the tensor family. Each $Q_{u} : f \mapsto f_{u}$ is a finite signed sum of contractions $P_{v}$ by [F2], so $Q_{u} f_{F} \to Q_{u} f = f_{u}$ in $L^{2}$ . Orthonormality and step 2.1 give $∥ Q_{u} f_{F} ∥_{2}^{2} = \sum_{α \in F : supp (α) = u} c_{α}^{2}$ . These nondecreasing finite subsums converge to the nonnegative sum over all indices with support $u$ ; continuity of the norm then yields $V_{u} = \sum_{supp (α) = u} c_{α}^{2}$ .

4.1F3F6F7step 3.1algebra∎

The constant tensor is $ψ_{0} = 1$ , so the real $L^{2}$ pairing [F6] gives $c_{0} = ⟨ f, 1 ⟩ = E f$ . Parseval [F3] applied to $f - c_{0}$ gives $Var (f) = \sum_{α \neq 0} c_{α}^{2}$ . Grouping these nonnegative terms by support and substituting the component identities from step 3.1 into [F7] proves both normalized formulas when $V > 0$ .

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Sources

Art B. Owen, Variance Components and Generalized Sobol' Indices, §2 (standard reference, not scraped)
Thierry Klein, Introduction to Sensitivity Analysis: The Statistical Study of Sobol Indices, complete lecture notes, §2.2 (standard reference, not scraped)