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Sensitivity Analysis on Policy-Augmented Graphical Hybrid Models with Shapley Value Estimation

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For linear Gaussian policy-augmented knowledge-graph models of bioprocesses, the Shapley value of each random factor on expected cumulative reward has an exact closed form, and a recursive computation plus variance-reduced permutation…

desk verdict Useful sampler and computational tricks, but the exact Shapley formula for correlated Gaussian residuals is wrong as stated. read the letter →

arxiv 2411.13396 v2 pith:KOWWGW63 submitted 2024-11-20 stat.ML stat.CO

Junkai Zhao , Wei Xie , Jun Luo This is my paper · ORCID
classification stat.MLstat.CO
keywords BiomanufacturingsensitivityanalysisShapleyvalueBayesiandynamicnetworksquasi-MonteCarlosimulationlinearGaussianapproximationvariancereduction
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

The paper argues that Shapley value (SV) sensitivity analysis, which fairly attributes output differences and risks to input variables, can be made practical for policy-augmented knowledge-graph (pKG) hybrid models of biomanufacturing processes. For the common case where a heavily instrumented process is monitored faster than its dynamics, a linear Gaussian pKG model applies, and the paper proves an exact SV formula for random shocks (Theorem 1) and gives recursive algorithms that cut the computational cost of estimating SVs of policy and model parameters by factors of $O(H)$ and $O(H^3)$. It also introduces TFWW-VRT, a permutation-sampling scheme that combines the TFWW hypersphere transformation with randomized quasi-Monte Carlo and antithetic sampling, yielding lower estimation error than existing BMT/SCT sampling at the same budget. If these claims hold, researchers and practitioners can rank the variation sources, control settings, and uncertain parameters that most affect product quality and profit, and respond faster in process development and online control.

What carries the argument

The load-bearing objects are: the pKG model, a Bayesian-network-style hybrid model whose nodes are process states, actions from a parametric policy, rewards, and random shocks; the linear Gaussian approximation of the state transition, reward, and policy; the pathway coefficient $R_{h,t} = \prod_{j=t}^{h} ((\beta^s_j)^\top + (\beta^a_j)^\top \theta^\top_j)$ that propagates a shock at period $h$ to period $t+1$; the value functions $g(U)$ defined either as conditional expectation or expected remaining variance of the output given a subset of inputs; and the permutation estimator of Shapley values (Castro's ApproShapley). The machinery that carries the efficiency argument is the recursive reuse of these pathway products and of block covariance terms, which turns sums over long horizons into a short recurrence; the sampling machinery is the TFWW-VRT permutation sampler, which maps low-discrepancy points from a unit cube to a hypersphere, orders coordinates to get permutations, and appends reversed permutations as antithetic pairs.

What would settle it

Measure the wall-clock time of Algorithm EC.2 for variance-based SV on the linear Gaussian pKG as $H$ goes from 4 to 32 at fixed $Q$ and $D$: the claimed $O(H^3)$ speedup predicts the runtime should grow roughly 8-fold when $H$ doubles; if it grows 64-fold (the brute-force rate), the reuse argument fails.

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Extended reading notes

Core claim

The central claim is that for linear Gaussian pKG models, the SV of a single random factor $e^k_h$ on the expected cumulative reward is exactly $E_w[\sum_{t=0}^{H-1} (b^\top_{t+1}\theta^\top_{t+1}+c^\top_{t+1}) R_{h,t} e^k_h 1_k]$, i.e., the propagated pathway coefficient product times the shock; the paper proves this as Theorem 1 and its variance-analog as Theorem 2. For the SVs of policy parameters and model parameters, where no closed form exists, the paper proves that recursive reuse of intermediate pathway and covariance computations reduces the cost of the SV estimates from $O(QDH^3(n^4m+n^3m^2))$ to $O(QDH^2(n^4m+n^3m^2))$ for predictive analysis and from $O(QDH^6(n^4m+n^3m^2))$ to $O(QDH^3(n^4m+n^3m^2))$ for variance-based analysis. It further claims its TFWW-VRT sampler generates more uniform permutations than BMT and SCT with linear-in-dimension cost, and that with randomized QMC and antithetic pairs it achieves smaller MSE in SV estimation than the baselines.

Load-bearing premise

The whole edifice of exact formulas and complexity savings rests on the assumption that a linear Gaussian pKG model (linear transitions, linear reward, linear policy, Gaussian noises) faithfully approximates the real bioprocess on the fast monitoring time scale; if that approximation is poor, the exact formulas and gains do not transfer to the true process.

Editorial extensions

If this is right

  • Sensitivity rankings of random shocks, policy gains, and model parameters for a linearized bioprocess can be computed exactly or with rigorous error bounds rather than by brute-force simulation, so sensitivity checks can run in time for real-time process control.
  • The $O(H^3)$ speedup for variance-based SV means that processes with hundreds of monitored time steps, previously intractable, become routine; the same structure applies to any linear-Gaussian policy-augmented dynamic network, not only fermentation.
  • The TFWW-VRT sampler with its variance reduction gives smaller SV estimation error at fixed permutation budget, so the number of model simulations needed to reach a target accuracy can be cut substantially.
  • Model-parameter SVs expose which uncertain kinetic constants most inflate output variance, guiding where to spend scarce lab experiments.

Reading between the lines

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

  • A direct diagnostic falls out of the exact formula: because the SV of $e^k_h$ is linear in the product of pathway matrices and the shock value, the dominant sensitivity path can be read off term-by-term, which could be used to identify which intermediate transition step amplifies a given variation most.
  • TFWW-VRT is not tied to bioprocesses; it could be dropped into any Shapley-value estimation pipeline (e.g., feature attribution in ML models) where permutation sampling is the bottleneck, provided the same uniformity gains transfer.
  • The paper's complexity savings assume the linear-Gaussian approximation; a natural extension would be to adapt the recursive reuse to piecewise-linear or locally linearized pKG models, preserving some speedup beyond the exact linear regime.
  • A falsifiable prediction of the approach is that, on a real fed-batch data set with enough trajectories, the linear-Gaussian Shapley rankings should match the simulation-based rankings from the nonlinear model on coarse qualitative patterns (e.g., which shocks matter); if they diverge sharply, the fast-monitoring approximation is not adequate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper develops a Shapley-value-based sensitivity analysis framework for policy-augmented knowledge graphical (pKG) hybrid models used in biomanufacturing. The framework defines value functions for three input types (random factors, policy parameters, and model parameters) and uses permutation sampling with pathway reuse to estimate Shapley values. For linear Gaussian pKG models, the authors derive closed-form expressions for the Shapley values of random factors (Theorems 1 and 2) and propose recursive algorithms that reduce the computational cost by O(H) for predictive analysis and O(H^3) for variance-based analysis. They also introduce TFWW-VRT, a permutation sampling algorithm combining the TFWW hypersphere transformation with randomized quasi-Monte Carlo and antithetic sampling, and they empirically compare it with BMT and SCT. The paper includes proofs of estimator properties, complexity analyses, and a publicly available code repository.

Significance. If correct, the linear-Gaussian results provide a substantial computational improvement for sensitivity analysis in heavily instrumented bioprocesses: the O(H) and O(H^3) reductions are practically meaningful, and the TFWW-VRT method appears to improve estimation accuracy at no extra cost. The paper also contributes a clean formulation of Shapley-value sensitivity analysis for a class of hybrid stochastic dynamic models, with attention to model uncertainty through the posterior distribution of parameters. The authors are to be credited for supplying full proofs of the estimator properties (Propositions 1 and 2), complexity statements (Propositions 3-5), and for making the source code and data publicly available. However, the central exact-formula claim (Theorems 1 and 2) is currently only valid under an unstated independence condition, and the empirical validation does not directly test these formulas. The practical relevance of the linear Gaussian approximation for the real fermentation process is also not assessed.

major comments (3)
  1. [Section 4.1 / EC.1.3] Theorem 1 is not correct for the general multivariate Gaussian residuals permitted by the model statement in Section 2.1.2. The proof in EC.1.3 computes the marginal contribution as E[s_{t+1}|U∪{o}] - E[s_{t+1}|U] = R_{h,t} e^k_h 1_k, but this step holds only if e^k_h is independent of every other residual component. Since the covariance matrix of eee is part of the model parameters and is not assumed diagonal, conditioning on a subset U changes the conditional means of the unfixed residuals when they are correlated with those in U. A concrete counterexample is H=2, n=1, β^s_1=β^a_1=0, reward r_2=s_2, and (e_1,e_2) jointly Gaussian with mean zero and covariance ρσ_1σ_2; the Shapley value of e_1 on r_2 is (1/2)(σ_2/σ_1)ρ e_1, while Theorem 1 gives 0. The same independence issue affects Theorem 2: the expression V_{t+1} ⊙ (1/2(1^l+1^{l⊤})) equals the conditional covariance only when the excluded coordinates are independent of the included ones; for correlated residuals, a Schur complement of the covariance matrix on the unfixed coordinates is required. The paper either needs to state an explicit independence assumption for the residuals (which would make the cross-covariance terms in Theorem 2 vanish) or derive the correct conditional-expectation/conditional-variance formulas for the general Gaussian case.
  2. [Section 6.4] The empirical study does not validate Theorems 1 and 2. Figure 4 reports MSE for policy parameters only, and Figures 5-9 show Shapley value point estimates for random factors without any error bars or comparison to a ground truth. Given that these theorems are the central analytical contribution, the paper should verify them on a small linear Gaussian pKG model by comparing the formula against exhaustive enumeration of the Shapley value definition in Equation (7), ideally for both diagonal and non-diagonal residual covariance matrices. The sample sizes Q, D, and L used in the experiments are also not reported for the results in Figures 5-9, which makes the empirical results difficult to reproduce or interpret.
  3. [Section 2.1.2 / Section 6.4] The linear Gaussian pKG model is introduced as an approximation of the nonlinear bioprocess on a fast monitoring time scale, but the approximation error is never quantified. The linear Gaussian case study uses synthetic data generated according to the linear model itself, so it cannot establish that the linear structure faithfully represents the real fermentation dynamics. Without an error analysis or a comparison to the nonlinear model on real experimental data, the practical significance of the exact Shapley value formulas for the biomanufacturing process remains unclear. The paper should at least discuss the sensitivity of the SV estimates to the linearization error and ideally provide an empirical assessment of this error.
minor comments (4)
  1. [Figure 8] The caption of Figure 8 states 'SV of selected policy parameters and outputs in linear pKG model', but the text and the heatmap describe random factors; the caption should say 'random factors'.
  2. [Table 1] The header 'Discrepency' in Table 1 is a typo and should be 'Discrepancy'.
  3. [Section 6.3] In Section 6.3 the random factors for the predictive analysis are fixed at e^k_h = E[e^k_h] + σ(e^k_h); for nonlinear models the marginal contribution depends on the chosen conditioning value, and the paper should state whether the reported results are sensitive to this choice.
  4. [Algorithm 3] The notation in Algorithm 3, especially the use of c^{(k)} in Step 1 of the TFWW transformation in Appendix EC.2.3, is dense; a short worked example for a small s would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's exact Shapley formulas are derived from the model equations rather than fitted or assumed, and the efficiency claims are complexity counts and benchmark comparisons.

full rationale

The core mathematical claims are self-contained. Theorem 1 is obtained in Appendix EC.1.3 by substituting the linear state/reward equations (4)-(6) into the Shapley marginal-contribution definition (7), and Theorem 2 is obtained in EC.1.4 by the same covariance algebra; neither theorem fits a parameter to data and then renames it a prediction. The value functions in Section 3.1 are definitions, not circular restatements of the SVs they feed into. The zero-baseline convention in Eq. (12) for policy parameters is explicitly borrowed from explainability literature (Covert et al. 2021, Ancona et al. 2019, Lundberg and Lee 2017), and the paper does not present this convention as a derived fact. The linear Gaussian pKG model is introduced as an approximation inherited from Zheng et al. (2023); this is a stated modeling assumption, not an unverified uniqueness theorem invoked to rule out alternatives, and the paper's exact formulas are conditional on that model. The computational savings claims (Propositions 3-5, Tables 3-4) are operation-count and runtime comparisons, not predictions that reduce to fitted values. One non-circularity caveat is that the proof of Theorem 1 in EC.1.3 appears to rely on an independence of residuals that Section 2.1.2 does not state; if residuals are merely jointly Gaussian with general covariance, the conditional expectations are not constant across subsets. That is a correctness risk, not a circularity, because the derivation still attempts to compute the Shapley value from the model rather than assuming the formula.

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

No new entities are introduced. The analysis rests on the pKG model structure and the linear Gaussian approximation, plus the zero-baseline convention for policy parameters. The empirical study has fitted model parameters and author-chosen baselines that affect the illustrative results but not the method itself.

free parameters (3)
  • model parameters ms, rL, beta_LCmax, mu_max = posterior distribution estimated from 8 real batches
    In the nonlinear pKG case study (Section 6.1), these four kinetic parameters are learned from limited real-world data; the sensitivity results depend on their posterior, though the method itself does not require specific values.
  • baseline means mu^a_t, mu^s_t = set by the authors
    Section 6.1 states 'the parameters mu^a_t and mu^s_t are set by ourselves'; these choices affect the policy parameter SV values.
  • sample sizes Q, D, L = not fully reported (D=1000 used in timing comparisons)
    The actual Q, D, and L used for the SV estimates in Figures 5-9 are not reported, only that they follow Equation (17).
assumptions (3)
  • domain assumption The pKG model with transition s_{t+1}=f_t(s_t,a_t;w_t)+e_{t+1}, parametric policy a_t=h_t(s_t;theta_t), and additive reward captures the biomanufacturing decision process
    Section 2.1.1; the entire sensitivity analysis is defined relative to this model; if the mechanistic structure is wrong, the SVs describe the model, not the process.
  • domain assumption Linear Gaussian approximation of the transition, reward, and policy is accurate on fast monitoring time scales
    Section 2.1.2: 'the general nonlinear pKG model can be approximated by linear Gaussian pKG model'; this is load-bearing for Theorems 1-2 and Algorithms EC.1/2.
  • ad hoc to paper Setting excluded policy parameters to zero is a valid baseline for Shapley value explanation
    Equation (12)/(13) define g(U|w)=E[Y|theta_{O/U}=0;w]; the choice of zero baseline is standard in ML explainability but not uniquely justified.

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Pith. "Pith review of Sensitivity Analysis on Policy-Augmented Graphical Hybrid Models with Shapley Value Estimation." pith.science (2026). https://pith.science/paper/KOWWGW63

@misc{pith2026241113396,
  author       = {Pith},
  title        = {Pith review of: Sensitivity Analysis on Policy-Augmented Graphical Hybrid Models with Shapley Value Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOWWGW63}},
  note         = {Machine review of arXiv:2411.13396}
}
read the original abstract

Driven by the critical challenges in biomanufacturing, including high complexity and high uncertainty, we propose a comprehensive and computationally efficient sensitivity analysis framework for general nonlinear policy-augmented knowledge graphical (pKG) hybrid models that characterize the risk- and science-based understandings of underlying stochastic decision process mechanisms. The criticality of each input (i.e., random factors, policy parameters, and model parameters) is measured by applying Shapley value (SV) sensitivity analysis to pKG (called SV-pKG), accounting for process causal interdependences. To quickly assess the SV for heavily instrumented bioprocesses, we approximate their dynamics with linear Gaussian pKG models and improve the SV estimation efficiency by utilizing the linear Gaussian properties. In addition, we propose an effective permutation sampling method with TFWW transformation and variance reduction techniques, namely the quasi-Monte Carlo and antithetic sampling methods, to further improve the sampling efficiency and estimation accuracy of SV for both general nonlinear and linear Gaussian pKG models. Our proposed framework can benefit efficient interpretation and support stable optimal process control in biomanufacturing.

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Works this paper leans on

50 extracted references · 48 canonical work pages

  1. [1]

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

    ENTRY address author booktitle chapter doi edition editor eid howpublished institution isbn issn journal key month note number organization pages publisher school series title type url 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...

  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 "" FUNCTION format.date year ...

  3. [3]

    International Conference on Machine Learning, 272--281

    Ancona M, Oztireli C, Gross M (2019) Explaining deep neural networks with a polynomial time algorithm for shapley value approximation. International Conference on Machine Learning, 272--281

    work page 2019

  4. [4]

    Tutorials in Operations Research: Advancing the Frontiers of OR/MS: From Methodologies to Applications 52--81

    Borgonovo E (2023) Sensitivity analysis. Tutorials in Operations Research: Advancing the Frontiers of OR/MS: From Methodologies to Applications 52--81

    work page 2023

  5. [5]

    European Journal of Operational Research 248(3):869--887

    Borgonovo E, Plischke E (2016) Sensitivity analysis: A review of recent advances. European Journal of Operational Research 248(3):869--887

    work page 2016

  6. [6]

    Computers & Operations Research 36(5):1726--1730

    Castro J, G \'o mez D, Tejada J (2009) Polynomial calculation of the shapley value based on sampling. Computers & Operations Research 36(5):1726--1730

    work page 2009

  7. [7]

    Science 355(6320):aag0804

    Clomburg JM, Crumbley AM, Gonzalez R (2017) Industrial biomanufacturing: the future of chemical production. Science 355(6320):aag0804

    work page 2017

  8. [8]

    Journal of Machine Learning Research 22(209):1--90

    Covert I, Lundberg S, Lee SI (2021) Explaining by removing: A unified framework for model explanation. Journal of Machine Learning Research 22(209):1--90

    work page 2021

Show all 50 references
  1. [9]

    SIAM/ASA Journal on Uncertainty Quantification 4(1):708--738

    De Lozzo M, Marrel A (2016) Estimation of the derivative-based global sensitivity measures using a gaussian process metamodel. SIAM/ASA Journal on Uncertainty Quantification 4(1):708--738

    2016

  2. [10]

    Operations Research 72(2):660--683

    Ding L, Zhang X (2024) Sample and computationally efficient stochastic kriging in high dimensions. Operations Research 72(2):660--683

    2024

  3. [11]

    Doran PM (2013) Bioprocess engineering principles (Academic Press, London)

    2013

  4. [12]

    Biotechnology Progress 39(6):e3378

    Drobnjakovic M, Hart R, Kulvatunyou B, Ivezic N, Srinivasan V (2023) Current challenges and recent advances on the path towards continuous biomanufacturing. Biotechnology Progress 39(6):e3378

    2023

  5. [13]

    Fang KT, Wang Y (1993) Number-theoretic Methods in Statistics, volume 51 (CRC Press)

    1993

  6. [14]

    Technical report, U.S

    FDA (2009) Q8 pharmaceutical development. Technical report, U.S. Food & Drug Administration, Silver Spring, MD

    2009

  7. [15]

    Journal of the American Statistical Association 95(452):1300--1304

    Gelfand AE (2000) Gibbs sampling. Journal of the American Statistical Association 95(452):1300--1304

    2000

  8. [16]

    Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian data analysis (Chapman and Hall/CRC)

    1995

  9. [17]

    Biometrics 61(3):781--788

    Golightly A, Wilkinson DJ (2005) Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61(3):781--788

    2005

  10. [18]

    Computers & Chemical Engineering 110:106--114

    Hong MS, Severson KA, Jiang M, Lu AE, Love JC, Braatz RD (2018) Challenges and opportunities in biopharmaceutical manufacturing control. Computers & Chemical Engineering 110:106--114

    2018

  11. [19]

    Biomanufacturing for Sustainable Production of Biomolecules, 175--188 (Springer)

    Kamaludin NH, Feisal NAS (2023) Biomanufacturing for sustainable production of biomolecules: Pseudomonas putida cell factory. Biomanufacturing for Sustainable Production of Biomolecules, 175--188 (Springer)

    2023

  12. [20]

    Manufacturing & Service Operations Management 25(1):108--125

    Koca Y, Martagan T, Adan I, Maillart L, Van Ravenstein B (2023) Increasing biomanufacturing yield with bleed--feed: Optimal policies and insights. Manufacturing & Service Operations Management 25(1):108--125

    2023

  13. [21]

    Mathematics of Operations Research 41(4):1248--1275

    Lam H (2016) Robust sensitivity analysis for stochastic systems. Mathematics of Operations Research 41(4):1248--1275

    2016

  14. [22]

    Springer Series in Statistics, 1 edition

    Lemieux C (2009) Monte Carlo and Quasi-Monte Carlo Sampling. Springer Series in Statistics, 1 edition

    2009

  15. [23]

    o vey H, R \

    Le \"o vey H, R \"o misch W (2015) Quasi-monte carlo methods for linear two-stage stochastic programming problems. Mathematical Programming 151:315--345

    2015

  16. [24]

    Advances in Neural Information Processing Systems 30:4765--4774

    Lundberg SM, Lee SI (2017) A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems 30:4765--4774

    2017

  17. [25]

    Current Opinion in Chemical Engineering 33:100705

    Luo Y, Kurian V, Ogunnaike BA (2021) Bioprocess systems analysis, modeling, estimation, and control. Current Opinion in Chemical Engineering 33:100705

    2021

  18. [26]

    L’Ecuyer P (2018) Randomized Quasi-Monte Carlo: An Introduction for Practitioners (Springer)

    2018

  19. [27]

    INFORMS Journal on Applied Analytics 53(1):85--95

    Martagan T, Adan I, Baaijens M, Dirckx C, Repping O, van Ravenstein B, Yegneswaran P (2023) Merck animal health uses operations research methods to transform biomanufacturing productivity for lifesaving medicines. INFORMS Journal on Applied Analytics 53(1):85--95

    2023

  20. [28]

    IIE Transactions 48(5):440--461

    Martagan T, Krishnamurthy A, Maravelias CT (2016) Optimal condition-based harvesting policies for biomanufacturing operations with failure risks. IIE Transactions 48(5):440--461

    2016

  21. [29]

    Journal of Machine Learning Research 23(43):1--46

    Mitchell R, Cooper J, Frank E, Holmes G (2022) Sampling permutations for shapley value estimation. Journal of Machine Learning Research 23(43):1--46

    2022

  22. [30]

    SIAM/ASA Journal on Uncertainty Quantification 6(4):1584--1599

    Morris MD (2018) Decomposing functional model inputs for variance-based sensitivity analysis. SIAM/ASA Journal on Uncertainty Quantification 6(4):1584--1599

    2018

  23. [31]

    Communications of the ACM 2(4):19--20

    Muller ME (1959) A note on a method for generating points uniformly on n-dimensional spheres. Communications of the ACM 2(4):19--20

    1959

  24. [32]

    Journal of Process Control 98:41--51

    Natarajan P, Moghadam R, Jagannathan S (2021) Online deep neural network-based feedback control of a lutein bioprocess. Journal of Process Control 98:41--51

    2021

  25. [33]

    Metabolic Engineering 66:31--40

    O'Brien CM, Zhang Q, Daoutidis P, Hu WS (2021) A hybrid mechanistic-empirical model for in silico mammalian cell bioprocess simulation. Metabolic Engineering 66:31--40

    2021

  26. [34]

    SIAM/ASA Journal on Uncertainty Quantification 1(1):19--41

    Owen AB (2013) Variance components and generalized sobol'indices. SIAM/ASA Journal on Uncertainty Quantification 1(1):19--41

    2013

  27. [35]

    SIAM/ASA Journal on Uncertainty Quantification 2(1):245--251

    Owen AB (2014) Sobol'indices and shapley value. SIAM/ASA Journal on Uncertainty Quantification 2(1):245--251

    2014

  28. [36]

    Computers & Chemical Engineering 125:232--248

    Papathanasiou MM, Burnak B, Katz J, Shah N, Pistikopoulos EN (2019) Assisting continuous biomanufacturing through advanced control in downstream purification. Computers & Chemical Engineering 125:232--248

    2019

  29. [37]

    Computers & Chemical Engineering 133:106649

    Petsagkourakis P, Sandoval IO, Bradford E, Zhang D, del Rio-Chanona EA (2020) Reinforcement learning for batch bioprocess optimization. Computers & Chemical Engineering 133:106649

    2020

  30. [38]

    Current Opinion in Biotechnology 71:225--230

    Reardon KF (2021) Practical monitoring technologies for cells and substrates in biomanufacturing. Current Opinion in Biotechnology 71:225--230

    2021

  31. [39]

    Naval Research Logistics 70(7):753--769

    Reich D, Sanchez SM (2023) Sensitivity analysis of hybrid microgrids with application to deployed military units. Naval Research Logistics 70(7):753--769

    2023

  32. [40]

    Management Science 68(10):7457--7479

    Singal R, Besbes O, Desir A, Goyal V, Iyengar G (2022) Shapley meets uniform: An axiomatic framework for attribution in online advertising. Management Science 68(10):7457--7479

    2022

  33. [41]

    SIAM/ASA Journal on Uncertainty Quantification 4(1):1060--1083

    Song E, Nelson BL, Staum J (2016) Shapley effects for global sensitivity analysis: Theory and computation. SIAM/ASA Journal on Uncertainty Quantification 4(1):1060--1083

    2016

  34. [42]

    Trends in Biotechnology 40(7):831--842

    Tellechea-Luzardo J, Otero-Muras I, Goñi-Moreno A, Carbonell P (2022) Fast biofoundries: coping with the challenges of biomanufacturing. Trends in Biotechnology 40(7):831--842

    2022

  35. [43]

    Nature Biotechnology 40(12):1722--1760

    Walsh G, Walsh E (2022) Biopharmaceutical benchmarks 2022. Nature Biotechnology 40(12):1722--1760

    2022

  36. [44]

    Bernoulli 28(4):2816--2832

    Wiesel JC (2022) Measuring association with wasserstein distances. Bernoulli 28(4):2816--2832

    2022

  37. [45]

    European Journal of Operational Research 310(1):238--248

    Xie W, Barton RRB, Nelson BL, Wang K (2023) Stochastic simulation uncertainty analysis to accelerate flexible biomanufacturing process development. European Journal of Operational Research 310(1):238--248

    2023

  38. [46]

    Naval Research Logistics 69(3):461--483

    Xie W, Wang B, Li C, Xie D, Auclair J (2022) Interpretable biomanufacturing process risk and sensitivity analyses for quality-by-design and stability control. Naval Research Logistics 69(3):461--483

    2022

  39. [47]

    Journal of Process Control 24(10):1556--1569

    Ye J, Xu H, Feng E, Xiu Z (2014) Optimization of a fed-batch bioreactor for 1,3-propanediol production using hybrid nonlinear optimal control. Journal of Process Control 24(10):1556--1569

    2014

  40. [48]

    IISE Transactions 50(5):431--447

    Zeng L, Deng X, Yang J (2018) Constrained gaussian process with application in tissue-engineering scaffold biodegradation. IISE Transactions 50(5):431--447

    2018

  41. [49]

    2023 Winter Simulation Conference (WSC), 3553--3564

    Zhao J, Luo J, Xie W (2023) Policy-augmented bayesian network optimization with global convergence. 2023 Winter Simulation Conference (WSC), 3553--3564

    2023

  42. [50]

    INFORMS Journal on Computing 35(1):66--82

    Zheng H, Xie W, Ryzhov IO, Xie D (2023) Policy optimization in dynamic bayesian network hybrid models of biomanufacturing processes. INFORMS Journal on Computing 35(1):66--82

    2023

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