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REVIEW 3 major objections 5 minor 61 references

gsaot: an R package for Optimal Transport-based sensitivity analysis

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Optimal transport ranks model inputs straight from data

desk verdict A genuinely useful R package for OT-based sensitivity analysis, blighted by a few fixable reproducibility mistakes and an unacknowledged binning bias. read the letter →

arxiv 2507.18588 v1 pith:MAM6QGPX submitted 2025-07-24 stat.CO

Leonardo Chiani , Emanuele Borgonovo , Elmar Plischke , Massimo Tavoni This is my paper · ORCID
classification stat.CO MSC 62H2062-0449Q22
keywords globalsensitivityanalysisoptimaltransportWassersteindistancemultivariateoutputcorrelatedinputsgiven-dataestimationentropicregularizationRpackage
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 introduces gsaot, an R package that computes Optimal Transport-based global sensitivity indices directly from an input-output dataset, without requiring a special sampling design or access to the simulation model. Its central claim is that gsaot is the first package able to handle both multivariate outputs and correlated inputs in this given-data setting. The indices it computes are normalized measures of statistical association with the properties of zero-independence and max-functionality, and they decompose an input's effect into components acting on the mean, the covariance, and higher-order moments of the output distribution. If the estimators work as claimed, any stored collection of model runs—from a climate simulator, an ODE model, or field data—can be post-processed into input rankings and local influence curves.

What carries the argument

The load-bearing object is the OT-based sensitivity index $\iota_K(Y,X_i) = E[K(P_Y, P_{Y|X_i})] / M_K[Y]$, where $K$ is the optimal transport cost under a user-chosen ground metric and $M_K[Y] = E[c(Y, Y')]$ is the expected cost between independent output replicas. The estimator replaces the conditioning event $X_i = x_i$ with membership in one of $H$ bins covering the input support, then averages the empirical OT costs over bins. This single plug-in device connects three computational routes: a closed-form quantile solution in one dimension, the Wasserstein-Bures closed form for elliptical distributions, and general empirical OT solved by network-flow or Sinkhorn algorithms.

What would settle it

Take a model with known true OT indices (for example, a Gaussian model as in the paper) and add a small-amplitude, high-frequency oscillation to the conditional output as a function of $X_i$, so that within every chosen bin the conditional distribution varies appreciably. Then compute the package's estimate at increasing $N$ with $H$ fixed; if the estimate stays biased away from the truth as $N$ grows while an exact-conditioning Monte Carlo estimate converges, the binning approximation—and with it the package's reliability—is refuted.

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Extended reading notes

Core claim

The paper's central claim is that the Wasserstein distance between the marginal output distribution and the output distribution conditional on an input defines a global sensitivity index that can be estimated from data alone, and that this index preserves the strong axiomatic properties of genuine measures of statistical association. Specifically, fixing input $X_i$ at $x_i$, the local separation $\zeta_K(x_i) = K(P_Y, P_{Y|X_i=x_i})$ is averaged over $X_i$ and divided by the maximum attainable cost $M_K[Y]$, yielding $\iota_K(Y,X_i)$ in $[0,1]$. The index is zero exactly when $Y$ and $X_i$ are independent and one exactly when $Y$ is a deterministic function of $X_i$. The paper implements this for one-dimensional outputs through a quantile formula, for elliptical or Gaussian outputs through the closed-form Wasserstein-Bures metric, and for general multivariate outputs through numerical optimal transport solvers, including entropic Sinkhorn regularization as a fast approximation.

Load-bearing premise

The whole method rests on replacing "fix $X_i$ at a precise value" with "$X_i$ falls in one of a few broad bins" and averaging the results over those bins; if this replacement is biased at finite sample sizes, the computed indices and rankings are unreliable.

Editorial extensions

If this is right

  • Analysts can rank input importance for any black-box model from a stored dataset, removing the need for purpose-built sampling designs or model reruns.
  • Multivariate and time-series outputs are treated as whole distributions, so an input that changes the shape or covariance of the output is detected even when marginal means do not move.
  • Correlated inputs do not invalidate the index; the ranking and its zero and one extremes remain well defined.
  • The index decomposes into advective (mean-shift), diffusive (covariance-shift), and higher-order components, allowing analysts to say not only how much but in what way an input affects the output.
  • Entropic regularization provides a fast, lower-bounded approximation that converges to the classical index as the regularization parameter goes to zero, making large-$N$ analyses feasible.

Reading between the lines

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

  • If the binning estimator is unbiased in practice, gsaot could be used as a screening tool on archival datasets in fields where experimenter-controlled sampling is impossible, such as epidemiology or observational climate analysis.
  • The mean/covariance decomposition suggests a diagnostic not developed in the paper: comparing advective with diffusive components across inputs can reveal which inputs drive heteroscedasticity, something variance-based indices miss.
  • Because the population index satisfies max-functionality, the package could double as an independence test for multivariate dependence, but the paper does not provide distributional results or calibration for such testing.
  • The dummy-variable irrelevance threshold is heuristic; a formal false-discovery or calibration analysis would strengthen its use for variable screening, but the paper does not propose one.
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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 / 5 minor

Summary. The manuscript presents gsaot, an R package for global sensitivity analysis based on Optimal Transport (OT) distances. It defines OT-based sensitivity indices from Borgonovo et al. (2024), including the Wasserstein-Bures decomposition into advective, diffusive, and higher-moment components, and implements given-data estimators that partition each input into H bins and average OT costs between the marginal output distribution and conditional output distributions. The package supports one-dimensional outputs, Gaussian/Bures closed-form estimation, classical OT via the transport package, and entropic OT via custom C++ Sinkhorn solvers, together with bootstrap confidence intervals, plotting, and an irrelevance-threshold tool based on dummy inputs. Three applications are shown: a linear Gaussian model with known analytical values, a spruce budworm ODE model with time-dependent output, and a DICE-based climate model with a custom ground cost.

Significance. If the estimates are reliable, the package is a useful contribution: it is model-agnostic, works directly from a dataset, supports correlated inputs and multivariate outputs, and implements indices with zero-independence and max-functionality. The paper also ships reproducible R code, makes the package available on CRAN, and checks one analytical benchmark. The main weakness is that the reliability of the central given-data estimator is not established beyond heuristics, and the benchmark contains a matrix inconsistency, so the validation currently does not close that gap.

major comments (3)
  1. [Section 3.1] The given-data estimator in Eq. (16) is the backbone of the package, yet the paper provides no bias, variance, or consistency analysis for it. For fixed H, by convexity of the Kantorovich cost in the second marginal, K(P_Y, P_{Y|X_i∈X^h_i}) ≤ (1/P(X^h_i)) ∫_{X^h_i} K(P_Y, P_{Y|X_i=x_i}) dP(x_i), so the binning estimator is downward biased and converges as N→∞ to a bin-average, not to ξ_K(Y,X_i). Because different inputs generally have different within-bin heterogeneity, rankings can change even in large samples. The heuristics in the text (at least 100 points per bin, the Strong-Oakley plateau, the dummy threshold) do not quantify this error. The Gaussian example in Section 3.1 cannot settle the question because the benchmark itself is compromised (see next comment). I ask the authors to add either a consistency result with H=H(N)→∞, or a systematic numerical study showing that the bias is negligible and rankings are stable for representative nonlinear and non-Gaussian models.
  2. [Section 3.1] The validation against Table 2 is not a clean check because the matrix used in the R code differs from the matrix stated in the text. The text defines A = [[4,2,3],[2,5,-1]], but the code 'A <- matrix(data = c(4,-2,1,2,5,-1), nrow = 2, byrow = TRUE)' produces A = [[4,-2,1],[2,5,-1]]. As a result, the column for X3 (and the correlated-input structure) differs between the claimed analytical model and the simulated data. Please reconcile the two definitions and report which matrix was actually used to generate the results; the comparison between ot_indices_wb, networkflow, and Sinkhorn remains informative only after this correction.
  3. [Equation (10)] Equation (10) is missing the normalization by M_K[Y] in the remainder term. Since the first two terms are defined in Eqs. (11)–(12) as normalized indices, the decomposition should read ι_K = ι_V + ι_Σ + M_K[Y]^{-1} E[Γ(P_Y,P_{Y|X_i})]; as written, the right-hand side mixes normalized and unnormalized quantities. This is a local fix, but the equation is part of the theoretical basis of the package's decomposition property and should be corrected.
minor comments (5)
  1. [Eq. (17)] The one-dimensional formula in Eq. (17) is written as a sum over j=1,...,N of the p-th differences of empirical quantile functions at j/N, but the conditional sample has N_h < N points; please state how the conditional quantile function is evaluated at the finer grid and confirm that the expression equals the L_p^p Wasserstein cost between the two empirical measures.
  2. [Section 2.4 / code examples] The number of partitions is denoted H in Eq. (16) and in the statement 'partitioned into H subsets,' but the code and the later discussion use M throughout; please unify the notation.
  3. [Section 3.3] The text says the example uses 'the fourth power of Minkowski distance of order 4 (L3^3)', while the code custom_metric uses dist(..., method='minkowski', p=3)^3; please correct the verbal description to match the implemented L_3^3 cost.
  4. [Section 3.2, Figure 4] The third panel in Figure 4 is labeled K_e, and the text says the plot shows the three most important inputs; the subsequent sentence about local separations for T_e should refer to a variable that is actually plotted, or the figure/ranking should be changed.
  5. [Throughout] There are several spelling and style slips, e.g., 'inizialize' should be 'initialize', and the sentence in Section 3.3 beginning 'We inizialize' should be reworded; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the package implements independently published OT-based indices and validates them against external analytical benchmarks; self-citations are not load-bearing.

full rationale

The derivation chain is self-contained rather than circular. The index iota_K is defined through the OT cost K(P_Y, P_Y|Xi=xi), with upper bound M_K[Y] = E[c(Y,Y')] and normalization iota_K = xi_K / M_K[Y]. The estimator in Eq. (16) replaces the conditioning event with a partition cell and averages empirical OT costs; this is an explicit approximation, not a restatement of the index. No parameter is fitted to force agreement: the number of partitions H is user-chosen with heuristic guidance, and the denominator is a U-statistic. The Gaussian validation in Section 3.1 is an external ground truth, albeit weakened by the code's matrix A (c(4,-2,1,2,5,-1)) differing from the text's A ([[4,2,3],[2,5,-1]]); that is a reproducibility/correctness issue, not circularity. The paper relies on the authors' prior work for the indices' zero-independence and max-functionality properties, but those properties are published results in peer-reviewed venues and are not themselves the target of this paper's derivation; they are also corroborated by independent work on Wasserstein-based dependence measures. No uniqueness theorem is imported to forbid alternatives, and no ansatz is smuggled in via self-citation. The binning estimator's possible downward bias under convex OT costs is a statistical accuracy concern, not evidence that the prediction reduces to its inputs by construction. Section 4 promises a discussion of limitations but does not actually list limitations; this is a completeness issue, not a circular step.

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

The package builds entirely on prior OT and sensitivity-analysis theory; no new physical or mathematical entities are introduced. The main user-controlled quantities (H, epsilon) are free parameters that affect the reported indices, and the validity of the partition estimator is assumed rather than demonstrated in this paper.

free parameters (3)
  • Number of partitions H (M) = M = 20 (Gaussian), M = 25 (budworm), M = 15 (DICE)
    Users must choose how many bins partition each input support; the estimate depends on this choice. The paper offers heuristic guidance (at least 100 points per bin, Strong-Oakley plateau) but no data-driven optimum.
  • Sinkhorn regularization epsilon = 0.001 (DICE), 0.05 and 0.001 (Gaussian comparison)
    The entropic index is defined for a user-chosen epsilon; small epsilon approximates the classical index, but the paper does not analyze sensitivity to epsilon beyond one comparison.
  • Sinkhorn solver tolerance and iteration limits = numIterations = 1e+06 or 1e+05, maxErr = 0.001
    These numerical tolerance parameters affect the computed entropic OT costs; they are implementation details but not fully exposed in the theoretical discussion.
assumptions (5)
  • standard math Kantorovich formulation of OT and Wasserstein metric properties
    Background for Eq. (4) and the definition of separation measures; referenced to Villani, Santambrogio, Peyre and Cuturi.
  • standard math Gelbrich's closed-form L2 Wasserstein formula for elliptical distributions with equal generators
    Used for the Wasserstein-Bures solver in Eq. (18) and for the equality in Eq. (10); referenced to Gelbrich (1990).
  • standard math Entropic OT converges to classical OT as epsilon tends to zero
    Basis for using Sinkhorn with small epsilon as a fast proxy for exact indices; referenced to Carlier et al. (2023).
  • domain assumption Partition-based plug-in approximation of conditional OT cost is a valid estimator
    Eq. (16) replaces Xi=xi by bins; no formal consistency or bias result is given, only heuristic guidance from Strong and Oakley (2013).
  • domain assumption Properties of the indices (zero-independence, normalization, max-functionality)
    Listed in Section 2.2 and attributed to Borgonovo et al. (2024, 2025); the paper does not re-derive them.

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Cite this review

Pith. "Pith review of gsaot: an R package for Optimal Transport-based sensitivity analysis." pith.science (2026). https://pith.science/paper/MAM6QGPX

@misc{pith2026250718588,
  author       = {Pith},
  title        = {Pith review of: gsaot: an R package for Optimal Transport-based sensitivity analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAM6QGPX}},
  note         = {Machine review of arXiv:2507.18588}
}
read the original abstract

gsaot is an R package for Optimal Transport-based global sensitivity analysis. It provides a simple interface for indices estimation using a variety of state-of-the-art Optimal Transport solvers such as the network simplex and Sinkhorn-Knopp. The package is model-agnostic, allowing analysts to perform the sensitivity analysis as a post-processing step. Moreover, gsaot provides functions for indices and statistics visualization. In this work, we provide an overview of the theoretical grounds, of the implemented algorithms, and show how to use the package in different examples.

Figures

Figures reproduced from arXiv: 2507.18588 by the authors.

Figure 1
Figure 1. OT-based indices of the Gaussian model. The bars in [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Comparison of different OT-based indices of the Gaussian model. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. OT-based indices of the spruce buworm and forest model. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Local separations of the three most relevant inputs in the spruce budworm and [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Temperature anomaly trajectories of the analyzed climate model [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: OT-based indices of the climate model [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Local separations of the last three inputs per importance in the climate model. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.