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arxiv: 2604.19165 · v1 · submitted 2026-04-21 · 📊 stat.ML · cs.LG· cs.NA· math.NA

Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions

Shijie Zhong , Jiangfeng Fu This is my paper

Pith reviewed 2026-05-10 01:53 UTC · model grok-4.3

classification 📊 stat.ML cs.LGcs.NAmath.NA
keywords conditional Sobol indicespolynomial chaos expansionsensitivity analysisuncertainty quantificationbasis decompositionglobal surrogate modelconditional variancealgebraic post-processing
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The pith

A single pre-trained global polynomial chaos expansion contains closed-form expressions for conditional Sobol indices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that conditional Sobol indices, which quantify input influence under fixed conditions, are already encoded in the basis of any sufficiently accurate global PCE surrogate. By decomposing the tensor-product basis, the global expansion separates into coefficient fields that depend explicitly on the conditioning variables. Orthogonality of the basis is preserved when the probability measure is restricted to those conditions, so conditional variances and indices follow directly from algebraic operations on the original coefficients. This replaces the standard practice of fitting separate models or running new samples at each condition point. A sympathetic reader would care because the change turns an expensive, repeated modeling task into a cheap post-processing step that also maintains consistency across the full parameter space.

Core claim

For a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step.

What carries the argument

Tensor-product decomposition of the PCE basis functions that isolates conditioning variables and yields analytical coefficient fields whose variances are obtained from preserved orthogonality under the conditional measure.

If this is right

  • Conditional sensitivity analysis becomes a post-processing step on an existing global model rather than a series of separate constructions.
  • Sensitivity measures remain consistent and physically coherent across the entire parameter space instead of being recomputed independently at each condition.
  • No new sampling or model training is required when the conditioning variables change, lowering the total computational cost for parameterized responses such as spatial fields.
  • Numerical robustness improves because the same global coefficients are reused rather than re-estimated from limited local data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algebraic nature of the extraction could allow conditional indices to be updated on the fly inside iterative design or optimization loops whenever operating conditions shift.
  • The same decomposition principle may apply to other orthogonal polynomial bases or tensor-structured surrogates beyond standard PCE, provided orthogonality survives conditioning.
  • For high-dimensional or spatially distributed outputs the method removes the scaling barrier that point-wise refitting imposes, making routine conditional analysis feasible where it was previously prohibitive.

Load-bearing premise

The orthogonality of the PCE basis functions is preserved when the probability measure is conditioned on the chosen variables, and the global model remains accurate everywhere in the domain.

What would settle it

On a low-dimensional benchmark function, compute the conditional Sobol indices at several fixed condition values both with the closed-form algebraic expressions from one global PCE and with independent point-wise PCE models trained at those same condition values; the two sets of numbers must agree within sampling tolerance.

Figures

Figures reproduced from arXiv: 2604.19165 by Jiangfeng Fu, Shijie Zhong.

Figure 1
Figure 1. Figure 1: Analytical spatially correlated variance and Sob [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: As observed, both the MC and point-wise PCE approaches yield sensitivity maps characterized by pronounced spatially uncorrelated fluctuations. This noise arises because each spatial location is treated as an isolated entity; the methods lack a mechanism to enforce the underlying physical continuity of the field. Consequently, the resulting S12 fields are not strictly continuous, and the error distributions… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of spatial distributions of the intera [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that conditional Sobol' indices can be extracted in closed form from a pre-trained global Polynomial Chaos Expansion (PCE) via tensor-product basis decomposition into analytical coefficient fields that depend on the conditioning variables. Leveraging asserted preservation of orthogonality under conditional probability measures, the method yields algebraic expressions for conditional variances and indices as a post-processing step, avoiding repeated modeling or sampling. Numerical benchmarks are presented to show physical coherence, robustness, and efficiency gains over point-wise approaches.

Significance. If the central derivation is valid under the stated conditions, the work offers a computationally efficient and consistent framework for conditional sensitivity analysis in uncertainty quantification, particularly useful for parameterized responses such as spatial fields. It builds directly on standard PCE tensor-product and orthogonality properties without introducing new fitted parameters or entities, and the algebraic post-processing aspect is a clear practical strength.

major comments (2)
  1. [Abstract / derivation of conditional variances] Abstract and the section deriving the closed-form expressions: the claim that orthogonality of the PCE basis is preserved under the conditional probability measure is asserted without an explicit statement or proof of the required independence assumption between conditioning and unconditioned variables. When inputs are correlated or when the global PCE contains interaction terms mixing conditioned and unconditioned variables, the univariate orthogonal polynomials no longer remain orthogonal under the conditional measure, so the algebraic extraction of exact conditional variances no longer holds. This assumption is load-bearing for the central claim of closed-form expressions.
  2. [Derivation of closed-form expressions] The derivation section (around the reformulation into coefficient fields and conditional indices): no explicit step-by-step derivation, error bounds, or handling of edge cases (e.g., when conditioning variables coincide with interaction terms or when the global PCE truncation error interacts with conditioning) is provided. Without these, it is not possible to verify that the post-processing step yields exact conditional Sobol' indices rather than approximations whose accuracy depends on the original PCE quality.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief, explicit list of the assumptions (input independence, sufficient global PCE accuracy) required for the orthogonality preservation to hold.
  2. [Method / numerical benchmarks] Notation for the conditional coefficient fields and the resulting conditional variance expressions could be clarified with a small example in the main text rather than only in the appendix.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive major comments. We address each point below with clarifications on the underlying assumptions and planned expansions to the derivation. The revisions will strengthen the manuscript without altering its core contribution.

read point-by-point responses

  1. Referee: [Abstract / derivation of conditional variances] Abstract and the section deriving the closed-form expressions: the claim that orthogonality of the PCE basis is preserved under the conditional probability measure is asserted without an explicit statement or proof of the required independence assumption between conditioning and unconditioned variables. When inputs are correlated or when the global PCE contains interaction terms mixing conditioned and unconditioned variables, the univariate orthogonal polynomials no longer remain orthogonal under the conditional measure, so the algebraic extraction of exact conditional variances no longer holds. This assumption is load-bearing for the central claim of closed-form expressions.

    Authors: We thank the referee for highlighting the need to make the independence assumption explicit. The method is developed under the standard PCE assumption that all input random variables are mutually independent; this is required for the tensor-product construction using univariate orthogonal polynomials. Under independence, conditioning on a subset of variables leaves the joint distribution (and thus the orthogonality measure) of the remaining variables unchanged, so the univariate bases remain orthogonal under the conditional measure. We will revise the abstract and derivation section to state this assumption explicitly and include a concise proof of preserved orthogonality. On interaction terms: mixed terms are handled by the tensor-product structure; the components depending only on unconditioned variables retain their orthogonality, allowing exact algebraic extraction of conditional variances. The claim does not extend to correlated inputs (which would require non-tensor bases), and we will clarify the scope. This preserves the validity of the closed-form expressions under the stated conditions. revision: yes

  2. Referee: [Derivation of closed-form expressions] The derivation section (around the reformulation into coefficient fields and conditional indices): no explicit step-by-step derivation, error bounds, or handling of edge cases (e.g., when conditioning variables coincide with interaction terms or when the global PCE truncation error interacts with conditioning) is provided. Without these, it is not possible to verify that the post-processing step yields exact conditional Sobol' indices rather than approximations whose accuracy depends on the original PCE quality.

    Authors: We agree that the derivation benefits from greater explicitness. In the revised manuscript we will expand the section with a complete step-by-step derivation: starting from the global PCE, reformulating into analytical coefficient fields via the tensor-product property, and arriving at the closed-form conditional variance and Sobol' index expressions. We will add error bounds showing that the extracted indices are exact for the PCE surrogate, with the discrepancy to the true model controlled by the same L2 truncation error as in the unconditional case. Edge cases will be addressed: when conditioning variables appear in interaction terms, the basis is partitioned by dependence on conditioned versus unconditioned variables; truncation error propagates identically to the unconditional setting. We will also add a short discussion noting that the indices are precisely those of the PCE model (standard for surrogate-based sensitivity analysis). These changes will be incorporated as an expanded theoretical section with examples. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard PCE tensor-product and orthogonality properties

full rationale

The paper's central derivation reformulates a global PCE via its tensor-product basis into coefficient fields over conditioning variables, then applies the preservation of orthogonality under conditional measures to obtain closed-form conditional variances and Sobol' indices. These steps rest on the algebraic structure of orthogonal polynomial bases and conditional expectation, which are external mathematical facts rather than self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No equations reduce to their own inputs by construction, and the abstract explicitly frames the result as algebraic post-processing of a pre-trained model. The independence assumption noted by the skeptic affects correctness but is not a circularity issue per the guidelines.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard properties of PCE constructions and one domain assumption about conditional measures; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • standard math Tensor-product structure of multivariate PCE basis functions
    Invoked to separate the expansion into conditioning-variable-dependent coefficient fields and the remaining orthogonal part.
  • domain assumption Preservation of orthogonality of PCE bases under conditional probability measures
    Required to obtain closed-form expressions for conditional variances and Sobol' indices without recomputation.

pith-pipeline@v0.9.0 · 5480 in / 1292 out tokens · 39193 ms · 2026-05-10T01:53:55.028566+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Global sensitivity analysis using polyno mial chaos expansions

    Bruno Sudret. Global sensitivity analysis using polyno mial chaos expansions. Reliability engineering & system safety, 93(7):964–979, 2008

    work page 2008

  2. [2]

    The wiener–askey polynomial chaos for stochastic differential equa- tions

    Dongbin Xiu and George Em Karniadakis. The wiener–askey polynomial chaos for stochastic differential equa- tions. SIAM Journal on Scientific Computing , 24(2):619–644, 2002

    work page 2002

  3. [3]

    Bayesian compress ive sensing

    Shihao Ji, Y a Xue, and Lawrence Carin. Bayesian compress ive sensing. IEEE Transactions on signal processing, 56(6):2346–2356, 2008

    work page 2008

  4. [4]

    Bi-fidelity adaptive sparse reconstruction of polynomial chaos using bayesian compressive sensing

    Mohamad Sadeq Karimi, Ramin Mohammadi, and Mehrdad Rais ee. Bi-fidelity adaptive sparse reconstruction of polynomial chaos using bayesian compressive sensing. Engineering with Computers, 41(4):2461–2481, 2025

    work page 2025

  5. [5]

    Signal recovery from pa rtial information via orthogonal matching pursuit

    Joel Tropp, Anna C Gilbert, et al. Signal recovery from pa rtial information via orthogonal matching pursuit. IEEE Trans. Inform. Theory, 53(12):4655–4666, 2007

    work page 2007

  6. [6]

    Krishnaprasad

    Ramin Rezaiifar and P . Krishnaprasad. Orthogonal match ing pursuit: Recursive function approximation with applications to wavelet decomposition. 06 1995

    work page 1995

  7. [7]

    Adaptive sparse polyno mial chaos expansion based on least angle regression

    Géraud Blatman and Bruno Sudret. Adaptive sparse polyno mial chaos expansion based on least angle regression. J. Comput. Phys., 230:2345–2367, 2011

    work page 2011

  8. [8]

    Stochastic m odeling and control of circulatory system with a left ventricular assist device

    Jeongeun Son, Dongping Du, and Y uncheng Du. Stochastic m odeling and control of circulatory system with a left ventricular assist device. In 2019 American Control Conference (ACC), pages 5408–5413, 2019

    work page 2019

  9. [9]

    Shape optimization and uncertainty assessment of a centrifugal pump

    Alessia Fracassi, Remo De Donno, Antonio Ghidoni, and Pi etro Marco Congedo. Shape optimization and uncertainty assessment of a centrifugal pump. Engineering Optimization, 54(2):200–217, 2022

    work page 2022

  10. [10]

    Cervantes, and Ahmad Nourbakhsh

    Saeed Salehi, Mehrdad Raisee, Michel J. Cervantes, and Ahmad Nourbakhsh. Efficient uncertainty quantifica- tion of stochastic cfd problems using sparse polynomial cha os and compressed sensing. Computers & Fluids , 154:296–321, 2017. ICCFD8

    work page 2017

  11. [11]

    Cervantes, and Ahmad Nourbakhsh

    Saeed Salehi, Mehrdad Raisee, Michel J. Cervantes, and Ahmad Nourbakhsh. An efficient multifidelity ℓ1- minimization method for sparse polynomial chaos. Computer Methods in Applied Mechanics and Engineering , 334:183–207, 2018

    work page 2018

  12. [12]

    On the flow field and performance of a centrifugal pump under operational and geometrical uncertainties

    Saeed Salehi, Mehrdad Raisee, Michel J Cervantes, and A hmad Nourbakhsh. On the flow field and performance of a centrifugal pump under operational and geometrical uncertainties. Applied Mathematical Modelling, 61:540– 560, 2018

    work page 2018

  13. [13]

    Stochas- tic simulation of the fda centrifugal blood pump benchmark

    Mohamad Sadeq Karimi, Pooya Razzaghi, Mehrdad Raisee, Patrick Hendrick, and Ahmad Nourbakhsh. Stochas- tic simulation of the fda centrifugal blood pump benchmark. Biomechanics and Modeling in Mechanobiology , 20(5):1871–1887, 2021

    work page 2021

  14. [14]

    Approximation of quantities of interest in stochastic pdes by the random discrete lˆ2 proje ction on polynomial spaces

    Giovanni Migliorati, Fabio Nobile, Erik von Schwerin, and Raúl Tempone. Approximation of quantities of interest in stochastic pdes by the random discrete lˆ2 proje ction on polynomial spaces. SIAM Journal on Scientific Computing, 35(3):A1440–A1460, 2013

    work page 2013

  15. [15]

    Uncertainty propagation and sensitivit y analysis in mechanical models–contributions to struc- tural reliability and stochastic spectral methods

    Bruno Sudret. Uncertainty propagation and sensitivit y analysis in mechanical models–contributions to struc- tural reliability and stochastic spectral methods. Habilitationa diriger des recherches, Université Blaise P ascal, Clermont-Ferrand, France, 147:53, 2007. 11

    work page 2007