Pith. sign in

REVIEW 2 major objections 5 minor 59 references

Effect Decomposition of Functional-Output Computer Experiments via Orthogonal Additive Gaussian Processes

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A Gaussian process model splits functional outputs into orthogonal effects without predefined basis functions.

desk verdict A solid incremental extension of OAGP to functional outputs, with a real proof gap in Theorem 1 that should be fixed, but the core idea is sound. read the letter →

arxiv 2506.12701 v1 pith:343BLS5D submitted 2025-06-15 stat.ME stat.ML

Yu Tan , Yongxiang Li , Xiaowu Dai , Kwok-Leung Tsui This is my paper · ORCID
classification stat.MEstat.ML MSC 62R1062G08
keywords functional-outputcomputerexperimentsfunctionalANOVAeffectdecompositionorthogonaladditiveGaussianprocessSobol'indicesexpectedconditionalvariancesensitivityanalysisKriging
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a Gaussian process model, called FOAGP, that decomposes a functional output y(x,t) into additive effects fu|t(xu,t) that are exactly orthogonal at every output position t, without any predefined basis functions or distributional assumptions. The orthogonality is achieved by enforcing a conditional zero-mean constraint on a separable prior whose kernel is a product of an output-position kernel and input kernels, yielding a data-driven functional-output orthogonal additive kernel. On top of this decomposition the model gives closed-form local Sobol' indices and expected conditional variance sensitivity indices, so global and local variable importance come out analytically from the training data. The authors demonstrate on two simulations and a fuselage shape control experiment that the decomposed effects and sensitivity indices match the theoretical values, and that the model's computational cost stays comparable to a standard GP. If correct, the paper would make functional-output FANOVA practical and fully data-driven for black-box computer experiments.

What carries the argument

The load-bearing object is the functional-output orthogonal additive kernel, defined as the output kernel kt(t,t') times the product over i in u of the orthogonal kernels k-tilde-i, where each k-tilde-i is obtained by conditioning a GP with kernel ki kt on the zero-integral constraint. This kernel is embedded in an additive GP with a Hadamard-product covariance that can be replaced by a Kronecker product on grid designs, reducing inverse and determinant computation from O($N^{3}$) to O($m^{3}$+$n^{3}$). It carries the argument because the same kernel delivers the effect predictions, the local variance formula at each t, and the global variance formula after integrating over t.

What would settle it

Fit FOAGP to data generated from a function with explicit input-output interaction, for example f(x1,t) = x1 times t or f(x1,x2,t) = (x1+x2)(1+sin(2πt)), where the interaction is part of the truth rather than noise, and check whether the estimated effects remain orthogonal and whether the ECV indices match numerical Sobol' integrals of the true function; any systematic mismatch at a fixed output position t would show that the exact-decomposition claim fails outside the separability assumption.

Watch

Extended reading notes

Core claim

The central claim is that exact orthogonal effect decomposition of functional outputs can be built into the model structure itself: if each effect follows a separable GP prior ku|t = kt(t,t') times the product over i in u of ki(xi,xi'), and is conditioned on the constraint that the integral of fu|t(xu,t) with respect to Fi(xi) is zero, the conditional process remains a GP with reconstructed orthogonal kernels. Because the effects have zero conditional mean and are pairwise orthogonal at every output position t, the total variance splits into per-effect variances locally and globally, and the paper derives analytical formulas for the local Sobol' indices and the expected conditional variance sensitivity indices. This makes the decomposition exact rather than approximate, data-driven rather than basis-dependent, and it absorbs the effect decomposition into the modeling step, eliminating a separate post-processing step.

Load-bearing premise

The entire decomposition rests on the prior assumption that there is no interaction between an input variable and the output position t: each effect kernel is exactly a product of an output kernel and input kernels, so if the true function couples x and t, the conditional orthogonality constraint is misspecified and the estimated effects and indices become biased.

Editorial extensions

If this is right

  • Functional-output FANOVA can be performed without choosing basis functions, so the effect decomposition adapts to the empirical distribution of the training inputs.
  • Local Sobol' indices and ECV sensitivity indices follow analytically from the fitted model, so sensitivity analysis adds no extra simulation or Monte Carlo step.
  • On grid designs the Kronecker structure makes the method usable for large functional outputs, as in the 10,000-instance fuselage study with 100 output positions.
  • Because the model includes all high-order effects, the decomposition accounts for interactions among inputs while keeping them orthogonal to main effects.
  • A unified global-local sensitivity analysis emerges, distinguishing effects that share the same global variance but behave differently at particular output positions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The conditioning argument suggests an immediate extension to multivariate functional outputs by replacing the single output kernel with a tensor product of output-coordinate kernels, preserving the same orthogonality reasoning coordinate-wise.
  • When input-output interactions exist, the separable prior is misspecified; a natural test is to fit FOAGP on an interaction-dominated function and compare the estimated local variances with numerical integrals, which would show bias outside the separability assumption.
  • Because the sensitivity estimators use the training data as the empirical integration measure, the indices should be read as sensitivity with respect to the observed input distribution; a different sampling design would yield different indices for the same fitted model.
  • Since bijective input transformations leave Sobol' indices invariant, users can map inputs to a hypercube before fitting to improve numerical conditioning without changing the sensitivity conclusions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a functional-output orthogonal additive Gaussian process (FOAGP) for computer experiments whose responses are functions of an output position. The model is an additive GP whose effect kernels are products of an output kernel and input kernels that are orthogonalized by subtracting a rank-one term based on expectations over the input distribution. The authors claim that this construction yields an exact conditional orthogonal effect decomposition (Theorem 2), local and global variance decompositions (Theorem 3), and analytical formulas for local Sobol' and expected conditional variance (ECV) sensitivity indices (Theorem 4). The methodology is validated on two simulation examples and a fuselage shape control case study.

Significance. If the theoretical foundation is made rigorous, the paper fills a practical gap: it provides a fully nonparametric, basis-free approach to functional-output FANOVA for black-box simulations, with sensitivity indices computed directly from the training data. The Hadamard and Kronecker product structures are an attractive computational feature, and the real case study demonstrates scalability. The authors are transparent about the no-input-output-interaction assumption. The main novelty is the adaptation of existing orthogonal kernel ideas (Durrande et al., 2013; Plumlee and Joseph, 2018) to functional outputs and the derivation of the corresponding local and global sensitivity indices; the orthogonal kernel itself is not new, as the paper acknowledges.

major comments (2)
  1. [Supplemental Material, Proof of Theorem 1 (Eq. (14))] The proof of Theorem 1 is not valid for general input distributions. The 'without loss of generality' reduction to a uniform distribution on [0,1] is incorrect because the grid points x_{i,u}^{(n)} = u/n and t_v^{(n)} = v/n are equally spaced in value, so the Riemann sums converge to integrals with respect to Lebesgue measure, not with respect to an arbitrary distribution F_i. Simply replacing dxi by dF_i(x_i) in the text does not repair the argument, since an equally spaced grid does not approximate the integral with respect to F_i. Because Eq. (14) is the foundation for Theorems 2-4, this is a load-bearing gap. The authors should either give a rigorous proof for general F_i (e.g., by using quantile-transformed grids or a direct conditional GP computation with general F_i) or explicitly restrict the theorem's scope. The result is known for scalar outputs from Durrande et al. (2013) and Plumlee and Joseph (2018), so the statement is likely correct, but the proof as written is incomplete.
  2. [Section 3.2, Eqs. (12)-(14), and Theorems 2-3] The orthogonality achieved by the reconstructed kernel \tilde k_i is relative to the distribution used to compute the expectations in Eq. (14). In practice, the paper estimates these expectations as empirical averages over the training data, as clearly stated in Section 3.2. The theorems and abstract should therefore say 'orthogonality with respect to the empirical distribution of the training data' rather than implying exact orthogonality under the true data distribution. The claim 'data-driven orthogonality without requiring prior distributional assumptions' overstates what is proven: exact orthogonality under the true distribution is not established, and it generally fails when the kernel is constructed from empirical expectations. The simulations showing convergence to theoretical values do not remove this distinction; they only show asymptotic behavior as the empirical distribution approaches the true one.
minor comments (5)
  1. [Section 2.1] The phrase 'cumulative cumulative distribution function' should be corrected to 'cumulative distribution function'.
  2. [Section 2 (Rougier description)] The sentence 'which which captures complex correlations between outputs and integrates multi-fidelity' contains a repeated 'which'; elsewhere, 'the the' appears in Section 4.1 and 'have a higher have a higher' in Example 1.
  3. [Section 3.5] The illustrative example text 'sint+1' should read 'sin(t)+1' and 'sin(10t)+1'.
  4. [Eqs. (30) and (31)] The notation K_i^2 and R_i^2 should be explicitly defined as elementwise squaring, because the standard matrix multiplication interpretation would be incorrect and is not what is intended.
  5. [Section 3.5, Theorem 4] The 'analytical formulas' in Theorem 4 involve expectations E_{x_i}[k_i k_i^T] and E_t[k_t k_t^T] that generally have no closed form; the paper later supplies empirical estimators in Eq. (30). The abstract's phrasing 'analytical formulations' is therefore somewhat overstated and should be clarified to mean formulas that are analytical in the kernel and the distribution, with empirical estimators used in practice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the orthogonal-kernel construction is proved in the paper from first principles and validated against external, independently computed benchmarks.

full rationale

The paper's central claim is that the functional-output orthogonal additive kernel in Eq. (14) yields exact orthogonal effect decomposition and analytical sensitivity indices. This is not circular. Theorem 1 derives Eq. (14) by conditioning a separable Gaussian process on the zero-integral constraint; the derivation is carried out in the Supplemental Material and does not assume the conclusion. The paper explicitly notes that the resulting orthogonal kernels coincide with those of Durrande et al. (2013) and Plumlee and Joseph (2018), which are external prior results rather than self-citations. Theorems 2-4 then follow algebraically from Eq. (14) and the mutual independence assumption; they are not restatements of the model definition. The sensitivity indices in Eqs. (26)-(29) are computed from the fitted model, and the simulation studies compare them against analytical values (Example 1) or values computed with Wolfram Mathematica (Example 2); these targets are not fitted parameters, so there is no fitted-input-called-prediction loop. The real case study is illustrative and not used as evidence that a fitted parameter was predicted. The paper does contain self-references, such as Liu et al. (2024) and Jiang et al. (2021), but these are contextual and not load-bearing for the theoretical derivation. Two limitations are acknowledged in the conclusion: the method handles only one-dimensional functional outputs and assumes no input-output interactions. A reviewer-level concern is that the proof of Theorem 1 in the supplement assumes a uniform grid and claims that arbitrary distributions follow by replacing dx_i with dF_i(x_i); that step is not a fully rigorous Riemann-sum argument for general F_i, but this is a mathematical proof gap rather than circularity. The derivation does not reduce to its inputs or to a self-citation chain, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard GP regression machinery, a stated independence assumption among inputs and output position, the separable prior that rules out input-output interactions, and the empirical distribution standing in for the unknown true distribution. All model hyperparameters are fitted by maximum marginal likelihood, so the sensitivity indices describe the fitted surrogate, not an independent external prediction.

free parameters (3)
  • kernel lengthscale θ_i for each input and θ_t for output position = estimated by MLE
    Every dimension has a lengthscale (and period T for t in the real case). These are optimized by maximizing the marginal likelihood (Eq. 19 or 23).
  • variance allocation parameters δ_0, δ_1,...,δ_d, δ_t = estimated by MLE
    These adjust the relative variance across the noise, each input, and the output position in the additive covariance (Eq. 16). They are part of the hyperparameters ϕ.
  • noise variance σ^2 = closed-form estimator, Eq. 18
    Given ϕ, σ^2 is estimated as γ^T y/N (Eq. 18).
assumptions (5)
  • domain assumption The input variables x_i and output position t are mutually independent.
    Stated in Theorems 1-4 as 'the elements in the vector [x^T, t]^T are mutually independent.' This is required for the factorization of expectations and the variance formulas.
  • domain assumption No input-output interactions: the separable prior process with kernel k_t * product k_i (Eq. 11).
    The kernel structure assumes that the input and output kernels share no common variables, so the effect of an input on the functional output is independent of the output position. This is acknowledged as a limitation in the conclusion.
  • domain assumption The output is one-dimensional functional (scalar output position).
    Stated in Sec. 3.1: 'we only consider one-dimensional functional output case for simplicity.'
  • domain assumption The empirical distribution of training inputs approximates the true input distribution for computing orthogonality.
    The paper estimates E_x[k] in Eq. (14) using training data 'as an empirical distribution (Lu et al., 2022).' The 'data-driven orthogonality' therefore holds with respect to the empirical distribution, not necessarily the true distribution.
  • standard math Standard Gaussian process regression and representer theorem (Kimeldorf and Wahba, 1971), used to show the posterior mean is in the span of the kernel functions.
    Invoked in the proof of Theorem 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effect Decomposition of Functional-Output Computer Experiments via Orthogonal Additive Gaussian Processes." pith.science (2026). https://pith.science/paper/343BLS5D

@misc{pith2026250612701,
  author       = {Pith},
  title        = {Pith review of: Effect Decomposition of Functional-Output Computer Experiments via Orthogonal Additive Gaussian Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/343BLS5D}},
  note         = {Machine review of arXiv:2506.12701}
}
read the original abstract

Functional ANOVA (FANOVA) is a widely used variance-based sensitivity analysis tool. However, studies on functional-output FANOVA remain relatively scarce, especially for black-box computer experiments, which often involve complex and nonlinear functional-output relationships with unknown data distribution. Conventional approaches often rely on predefined basis functions or parametric structures that lack the flexibility to capture complex nonlinear relationships. Additionally, strong assumptions about the underlying data distributions further limit their ability to achieve a data-driven orthogonal effect decomposition. To address these challenges, this study proposes a functional-output orthogonal additive Gaussian process (FOAGP) to efficiently perform the data-driven orthogonal effect decomposition. By enforcing a conditional orthogonality constraint on the separable prior process, the proposed functional-output orthogonal additive kernel enables data-driven orthogonality without requiring prior distributional assumptions. The FOAGP framework also provides analytical formulations for local Sobol' indices and expected conditional variance sensitivity indices, enabling comprehensive sensitivity analysis by capturing both global and local effect significance. Validation through two simulation studies and a real case study on fuselage shape control confirms the model's effectiveness in orthogonal effect decomposition and variance decomposition, demonstrating its practical value in engineering applications.

Figures

Figures reproduced from arXiv: 2506.12701 by the authors.

Figure 1
Figure 1. Two different local variance with the same global variance. [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Illustration of FOED of Example 1 using FOAGP and HDMR. The theoretical [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Illustration of FOED of Example 2 using FOAGP and HDMR. The theoretical [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Illustration of the local variance decomposition of Example 1 using FOAGP. [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Illustration of the local variance decomposition of the Example 2 using FOAGP. [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Illustration of the estimated global sensitivity indices and RMSE using FOAGP. [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: Composite fuselage simulation in Ansys 2024. [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: Box plot of the ECV sensitivity indices of the main effects. [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]
Figure 9
Figure 9. Figure 9: The deformation cased by the main effects of the forces. To illustrate the defor [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 57 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

  3. [3]

    Apley, D. W. and Zhu, J. (2020), Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82, 1059--1086

    work page 2020

  4. [4]

    Bolin, D., Verendel, V., Berghauser Pont, M., Stavroulaki, I., Ivarsson, O., and H kansson, E. (2021), Functional ANOVA Modelling of Pedestrian Counts on Streets in Three European Cities , Journal of the Royal Statistical Society Series A: Statistics in Society, 184, 1176--1198

    work page 2021

  5. [5]

    E., and Wilk, M

    Burt, D., Rasmussen, C. E., and Wilk, M. V. D. (2019), Rates of Convergence for Sparse Variational Gaussian Process Regression , in Proceedings of the 36th International Conference on Machine Learning , PMLR, pp. 862--871

    work page 2019

  6. [6]

    M., Grasso, M

    Centofanti, F., Colosimo, B. M., Grasso, M. L., Menafoglio, A., Palumbo, B., and Vantini, S. (2023), Robust Functional ANOVA with Application to Additive Manufacturing, Journal of the Royal Statistical Society Series C: Applied Statistics, 72, 1210--1234

    work page 2023

  7. [7]

    C., Lundberg, S

    Chen, H., Covert, I. C., Lundberg, S. M., and Lee, S.-I. (2023), Algorithms to Estimate Shapley Value Feature Attributions, Nature Machine Intelligence, 5, 590--601

    work page 2023

  8. [8]

    and Lu, Z

    Cheng, K. and Lu, Z. (2019), Time-Variant Reliability Analysis Based on High Dimensional Model Representation, Reliability Engineering & System Safety, 188, 310--319

    work page 2019

Show all 59 references
  1. [9]

    (2024), Sensitivity Analysis in the Wavelet Domain: A Comparison Study, Stochastic Environmental Research and Risk Assessment, 38, 1669--1684

    Chiogna, G., Marcolini, G., Engel, M., and Wohlmuth, B. (2024), Sensitivity Analysis in the Wavelet Domain: A Comparison Study, Stochastic Environmental Research and Risk Assessment, 38, 1669--1684

    2024

  2. [10]

    (2010), Functional ANOVA in Computer Models With Time Series Output , Technometrics, 52, 430--437

    Drignei, D. (2010), Functional ANOVA in Computer Models With Time Series Output , Technometrics, 52, 430--437

    2010

  3. [11]

    (2012), Additive Covariance Kernels for High-Dimensional Gaussian Process Modeling, Annales de la Facult \'e des sciences de Toulouse : Math \'e matiques , 21, 481--499

    Durrande, N., Ginsbourger, D., and Roustant, O. (2012), Additive Covariance Kernels for High-Dimensional Gaussian Process Modeling, Annales de la Facult \'e des sciences de Toulouse : Math \'e matiques , 21, 481--499

    2012

  4. [12]

    (2013), ANOVA Kernels and RKHS of Zero Mean Functions for Model-Based Sensitivity Analysis, Journal of Multivariate Analysis, 115, 57--67

    Durrande, N., Ginsbourger, D., Roustant, O., and Carraro, L. (2013), ANOVA Kernels and RKHS of Zero Mean Functions for Model-Based Sensitivity Analysis, Journal of Multivariate Analysis, 115, 57--67

    2013

  5. [13]

    K., Nickisch, H., and Rasmussen, C

    Duvenaud, D. K., Nickisch, H., and Rasmussen, C. (2011), Additive Gaussian Processes , in Advances in Neural Information Processing Systems , Curran Associates, Inc., vol. 24

    2011

  6. [14]

    and Stein, C

    Efron, B. and Stein, C. (1981), The Jackknife Estimate of Variance , The Annals of Statistics, 9, 586--596

    1981

  7. [15]

    (2005), Design and modeling for computer experiments, Chapman and Hall/CRC

    Fang, K.-T., Li, R., and Sudjianto, A. (2005), Design and modeling for computer experiments, Chapman and Hall/CRC

    2005

  8. [16]

    Faraway, J. J. (1997), Regression Analysis for a Functional Response , Technometrics, 39, 254--261

    1997

  9. [17]

    and Bouneffa, M

    Garouani, M. and Bouneffa, M. (2024), Automated Machine Learning Hyperparameters Tuning through Meta-Guided Bayesian Optimization, Progress in Artificial Intelligence

    2024

  10. [18]

    (2016), On ANOVA Decompositions of Kernels and Gaussian Random Field Paths , in Monte Carlo and Quasi-Monte Carlo Methods , eds

    Ginsbourger, D., Roustant, O., Schuhmacher, D., Durrande, N., and Lenz, N. (2016), On ANOVA Decompositions of Kernels and Gaussian Random Field Paths , in Monte Carlo and Quasi-Monte Carlo Methods , eds. Cools, R. and Nuyens, D., Cham: Springer International Publishing, pp. 315--330

    2016

  11. [19]

    (2021), Spatio- Temporal Variational Gaussian Processes , in Advances in Neural Information Processing Systems , Curran Associates, Inc., vol

    Hamelijnck, O., Wilkinson, W., Loppi, N., Solin, A., and Damoulas, T. (2021), Spatio- Temporal Variational Gaussian Processes , in Advances in Neural Information Processing Systems , Curran Associates, Inc., vol. 34, pp. 23621--23633

    2021

  12. [20]

    Hooker, G. (2004), Discovering Additive Structure in Black Box Functions, in Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , Seattle WA USA: ACM, pp. 575--580

    2004

  13. [21]

    --- (2007), Generalized Functional ANOVA Diagnostics for High-Dimensional Functions of Dependent Variables , Journal of Computational and Graphical Statistics, 16, 709--732

    2007

  14. [22]

    (2014), Asymptotic Normality and Efficiency of Two Sobol Index Estimators, ESAIM: Probability and Statistics, 18, 342--364

    Janon, A., Klein, T., Lagnoux, A., Nodet, M., and Prieur, C. (2014), Asymptotic Normality and Efficiency of Two Sobol Index Estimators, ESAIM: Probability and Statistics, 18, 342--364

    2014

  15. [23]

    Jiang, F., Tan, M. H. Y., and Tsui, K.-L. (2021), Multiple-Target Robust Design with Multiple Functional Outputs, IISE Transactions, 53, 1052--1066

    2021

  16. [24]

    (2020), Vecchia Approximations of Gaussian-Process Predictions , Journal of Agricultural, Biological and Environmental Statistics, 25, 383--414

    Katzfuss, M., Guinness, J., Gong, W., and Zilber, D. (2020), Vecchia Approximations of Gaussian-Process Predictions , Journal of Agricultural, Biological and Environmental Statistics, 25, 383--414

    2020

  17. [25]

    and Wahba, G

    Kimeldorf, G. and Wahba, G. (1971), Some Results on Tchebycheffian Spline Functions, Journal of Mathematical Analysis and Applications, 33, 82--95

    1971

  18. [26]

    and Reimherr, M

    Kokoszka, P. and Reimherr, M. (2017), Introduction to Functional Data Analysis, Texts in Statistical Science Series, Boca Raton: CRC Press, Taylor & Francis Group

    2017

  19. [27]

    (2022), Separable Spatio-Temporal Kriging for Fast Virtual Sensing, Applied Stochastic Models in Business and Industry, 38, 806--829

    Lambardi di San Miniato , M., Bellio, R., Grassetti, L., and Vidoni, P. (2022), Separable Spatio-Temporal Kriging for Fast Virtual Sensing, Applied Stochastic Models in Business and Industry, 38, 806--829

    2022

  20. [28]

    Liu, Q., Tong, N., Wu, X., Han, X., and Chen, C. (2021), A Generalized Sensitivity Analysis Method Based on Variance and Covariance Decomposition of Summatory Functions for Multi-Input Multi-Output Systems, Computer Methods in Applied Mechanics and Engineering, 385, 114009

    2021

  21. [29]

    (2024), Latent Functional Gaussian Process Incorporating Output Spatial Correlations , IISE Transactions, 0, 1--24

    Liu, Z., Li, Y., Yue, X., and Pan, E. (2024), Latent Functional Gaussian Process Incorporating Output Spatial Correlations , IISE Transactions, 0, 1--24

    2024

  22. [30]

    N., Nielsen, H

    Lophaven, S. N., Nielsen, H. B., and S ndergaard, J. (2002), Aspects of the matlab toolbox DACE, Citeseer

    2002

  23. [31]

    (2022), Additive Gaussian Processes Revisited , in Proceedings of the 39th International Conference on Machine Learning , PMLR, pp

    Lu, X., Boukouvalas, A., and Hensman, J. (2022), Additive Gaussian Processes Revisited , in Proceedings of the 39th International Conference on Machine Learning , PMLR, pp. 14358--14383

    2022

  24. [32]

    A., Hobbs, J., Song, J

    Ma, P., Mondal, A., Konomi, B. A., Hobbs, J., Song, J. J., and Kang, E. L. (2022), Computer Model Emulation with High-Dimensional Functional Output in Large-Scale Observing System Uncertainty Experiments , Technometrics, 64, 65--79

    2022

  25. [33]

    MacKay, D. J. (1998), Introduction to Gaussian processes, NATO ASI Series F Computer and Systems Sciences, 168, 133--166

    1998

  26. [34]

    and Leiber, M

    Marnissi, Y. and Leiber, M. (2024), A Unified View of FANOVA : A Comprehensive Bayesian Framework for Component Selection and Estimation , in Forty-First International Conference on Machine Learning

    2024

  27. [35]

    (2012), Data-Driven Kriging Models Based on FANOVA-decomposition , Statistics and Computing, 22, 723--738

    Muehlenstaedt, T., Roustant, O., Carraro, L., and Kuhnt, S. (2012), Data-Driven Kriging Models Based on FANOVA-decomposition , Statistics and Computing, 22, 723--738

    2012

  28. [36]

    Owen, A. B. (2014), Sobol' Indices and Shapley Value , SIAM/ASA Journal on Uncertainty Quantification, 2, 245--251

    2014

  29. [37]

    Owen, A. B. and Prieur, C. (2017), On Shapley Value for Measuring Importance of Dependent Inputs , SIAM/ASA Journal on Uncertainty Quantification, 5, 986--1002

    2017

  30. [38]

    J., Mohamed, S., and Lakshminarayanan, B

    Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., and Lakshminarayanan, B. (2021), Normalizing Flows for Probabilistic Modeling and Inference, Journal of Machine Learning Research, 22, 57:2617--57:2680

    2021

  31. [39]

    and Joseph, V

    Plumlee, M. and Joseph, V. R. (2018), Orthogonal Gaussian Process Models , Statistica Sinica, 28, 601--619

    2018

  32. [40]

    Rao, A. R. and Reimherr, M. (2023), Modern Non-Linear Function-on-Function Regression, Statistics and Computing, 33, 130

    2023

  33. [41]

    (2008), Efficient Emulators for Multivariate Deterministic Functions , Journal of Computational and Graphical Statistics, 17, 827--843

    Rougier, J. (2008), Efficient Emulators for Multivariate Deterministic Functions , Journal of Computational and Graphical Statistics, 17, 827--843

    2008

  34. [42]

    J., Williams, B

    Santner, T. J., Williams, B. J., and Notz, W. I. (2018), The Design and Analysis of Computer Experiments , Springer Series in Statistics , New York, NY: Springer New York

    2018

  35. [43]

    Q., Wang, B., Murray-Smith , R., and Titterington, D

    Shi, J. Q., Wang, B., Murray-Smith , R., and Titterington, D. M. (2007), Gaussian Process Functional Regression Modeling for Batch Data , Biometrics, 63, 714--723

    2007

  36. [44]

    (2018), Cross-Covariance Based Global Dynamic Sensitivity Analysis, Mechanical Systems and Signal Processing, 100, 846--862

    Shi, Y., Lu, Z., Li, Z., and Wu, M. (2018), Cross-Covariance Based Global Dynamic Sensitivity Analysis, Mechanical Systems and Signal Processing, 100, 846--862

    2018

  37. [45]

    (1993), Sensitivity estimates for nonlinear mathematical models, Math

    Sobol', I. (1993), Sensitivity estimates for nonlinear mathematical models, Math. Model. Comput. Exp., 1, 407

    1993

  38. [46]

    Sobol', I. M. (1990), On sensitivity estimation for nonlinear mathematical models, Matematicheskoe modelirovanie, 2, 112--118

    1990

  39. [47]

    --- (2001), Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates, Mathematics and Computers in Simulation, 55, 271--280

    2001

  40. [48]

    M., Lawrence, N., and Borgwardt, K

    Stegle, O., Lippert, C., Mooij, J. M., Lawrence, N., and Borgwardt, K. (2011), Efficient Inference in Matrix-Variate Gaussian Models with textbackslash Iid Observation Noise, in Advances in Neural Information Processing Systems , Curran Associates, Inc., vol. 24

    2011

  41. [49]

    (2020), Multiresolution Functional ANOVA for Large-Scale , Many-Input Computer Experiments , Journal of the American Statistical Association, 115, 908--919

    Sung, C.-L., Wang, W., Plumlee, M., and Haaland, B. (2020), Multiresolution Functional ANOVA for Large-Scale , Many-Input Computer Experiments , Journal of the American Statistical Association, 115, 908--919

    2020

  42. [50]

    (2020), Efficient Spatio-Temporal Gaussian Regression via Kalman Filtering, Automatica, 118, 109032

    Todescato, M., Carron, A., Carli, R., Pillonetto, G., and Schenato, L. (2020), Efficient Spatio-Temporal Gaussian Regression via Kalman Filtering, Automatica, 118, 109032

    2020

  43. [51]

    and Pianosi, F

    Wagener, T. and Pianosi, F. (2019), What Has Global Sensitivity Analysis Ever Done for Us? A Systematic Review to Support Scientific Advancement and to Inform Policy-Making in Earth System Modelling, Earth-Science Reviews, 194, 1--18

    2019

  44. [52]

    Wang, Z., Xing, W., Kirby, R., and Zhe, S. (2021), Multi- Fidelity High-Order Gaussian Processes for Physical Simulation , in Proceedings of The 24th International Conference on Artificial Intelligence and Statistics , PMLR, pp. 847--855

    2021

  45. [53]

    H., and Shi, J

    Wen, Y., Yue, X., Hunt, J. H., and Shi, J. (2018), Feasibility Analysis of Composite Fuselage Shape Control via Finite Element Analysis, Journal of Manufacturing Systems, 46, 272--281

    2018

  46. [54]

    (2018), Multivariate Global Sensitivity Analysis for Dynamic Models Based on Wavelet Analysis, Reliability Engineering & System Safety, 170, 20--30

    Xiao, S., Lu, Z., and Wang, P. (2018), Multivariate Global Sensitivity Analysis for Dynamic Models Based on Wavelet Analysis, Reliability Engineering & System Safety, 170, 20--30

    2018

  47. [55]

    R., Bolin, D., Rue, H., and Wang, X.-F

    Yue, Y. R., Bolin, D., Rue, H., and Wang, X.-F. (2019), Bayesian Generalized Two-way ANOVA Modeling for Functional Data Using INLA , Statistica Sinica, 29, 741--767

    2019

  48. [56]

    G., and Castruccio, S

    Zhang, J., Crippa, P., Genton, M. G., and Castruccio, S. (2024), Sensitivity Analysis of Wind Energy Resources with Bayesian Non- Gaussian and Nonstationary Functional ANOVA , The Annals of Applied Statistics, 18, 23--41

    2024

  49. [57]

    (2023), An Efficient Implementation for Spatial--Temporal Gaussian Process Regression and Its Applications, Automatica, 147, 110679

    Zhang, J., Ju, Y., Mu, B., Zhong, R., and Chen, T. (2023), An Efficient Implementation for Spatial--Temporal Gaussian Process Regression and Its Applications, Automatica, 147, 110679

    2023

  50. [58]

    (2019), A New Test for Functional One-Way ANOVA with Applications to Ischemic Heart Screening, Computational Statistics & Data Analysis, 132, 3--17

    Zhang, J.-T., Cheng, M.-Y., Wu, H.-T., and Zhou, B. (2019), A New Test for Functional One-Way ANOVA with Applications to Ischemic Heart Screening, Computational Statistics & Data Analysis, 132, 3--17

    2019

  51. [59]

    (2017), Matrix Analysis and Applications , Cambridge University Press, 1st ed

    Zhang, X.-D. (2017), Matrix Analysis and Applications , Cambridge University Press, 1st ed

    2017

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.