{"id":"54a14546-01fc-4fec-9631-51eaf402291f","arxiv_id":"2411.18575","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A continuous-game Shapley value defines a pointwise relevance function for functional predictors in scalar-on-function regression, available in the ShapleyFDA package.","lead":"This paper introduces a way to identify which parts of a functional input, such as a spectrum or sensor curve, most influence a pre-trained prediction model's outputs. The method adapts the Shapley value from cooperative game theory to an infinite set of input points and is released as the open source Python package ShapleyFDA.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The continuous-game interpretation is unsupported: Section 3.3 shows the game in Eq. (3) is not a vector measure game, and Section 2.3's existence conditions for the asymptotic Shapley value are never verified for this game.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: the paper relies on the existence and uniqueness of an asymptotic value for the specific game ν defined by Eq. (3), but never verifies the relevant conditions. My reading of Sections 2.3 and 3.3 confirms this is not a cosmetic omission. Section 3.3 is the decisive passage: it derives that even in the simplest linear functional model, ν(S) depends on S through countably many S-dependent measures and an S-dependent function ρ^S, so ν is not a vector measure game and the diagonal formula is unavailable. Since the paper offers no proof that ν lies in any other class with a guaranteed asymptotic value, the finite partition and random permutation estimator in Eq. (4) is not justified as an estimator of a continuous Shapley value. This is a genuine correctness risk for the central claim, not merely a request for more citations.\n\nThat said, I do not think the concern warrants rejection. The paper is honest about the lack of vector-measure structure, the empirical results are coherent, and the method may work as a practical heuristic even if the continuous-game interpretation is provisional. The conditional verdict already asks for the missing convergence justification, and that is the right level of scrutiny. The concrete check I propose would settle the question directly: compute relevance curves for refining partitions and observe whether they stabilize. This test is simple, uses the authors' released code, and would distinguish a stable functional quantity from a partition artifact. It is also consistent with the reader's suggested direction, so I mark agreement as 'agree' and recommend keeping the verdict unchanged at CONDITIONAL.","tokens_in":23531,"tokens_out":4616,"duration_ms":48625,"concrete_test":"Use one simulation scenario (e.g., Fourier X, linear unimodal Υ_lu, m=200, η=0.05, trained FLM) and compute the Shapley relevance curves for nested partitions with n = 4, 8, 16, 32, 64 using the released ShapleyFDA code, with exact Shapley enumeration for n ≤ 8 and a large fixed permutation budget for larger n. Measure the normalized L1 distance between successive curves, d_n = ∫ |R_f^{P_n}(t) − R_f^{P_{n/2}}(t)| dt / R^2(I), and track the argmax location. If d_n does not decrease toward 0 as n doubles, or if the argmax shifts by more than one interval width, the finite approximation has not converged to a unique continuous value and the 'continuous Shapley' interpretation is unsupported. If d_n → 0, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that R_f(t) is a continuous Shapley value relevance function. For this to hold, the payoff game ν(S) = \\tilde R^2(S) defined in Eq. (3) must belong to a class of games for which the Neyman/Aumann-Shapley asymptotic value exists, is unique, and equals the limit of finite Shapley values over refining partitions. Section 2.3 only states that 'under certain conditions' such a limit exists and cites Neyman (1994); it does not state or verify those conditions for this ν. The finite partition computation in Eq. (4) with a fixed n=20 and 1000 random permutations is therefore not shown to approximate a well-defined continuous quantity.\n\nThe problem is not merely a missing proof. Section 3.3 explicitly analyzes ν(S) under a simple linear functional model and shows that ν(S) depends on S through a countably infinite collection of S-dependent signed measures τ_k^{S+}, τ_k^{S-} and an S-dependent function ρ^S. The paper concludes that ν is not a vector measure game, so the diagonal formula cannot be invoked, and no alternative membership in a known value class (such as pNA) is established. Thus the title's 'continuous Shapley value' may depend on the chosen partition and permutation sample; the pointwise relevance curves could be an artifact of n=20 rather than a stable functional object. The practical heuristic can still be useful, but the central theoretical interpretation is not secured.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a global, model-agnostic interpretability method for scalar-on-function regression. For a trained predictor f and a test sample, it defines a set function ν(S) = ᷈R²(S), the coefficient of determination of f when the functional predictor is reconstructed on Sᶜ by its conditional expectation given the values on S (the ghost-variable idea). Using the asymptotic approach to Shapley values for games with a continuum of players, the authors define an interval Shapley value relevance function R_f(t) via finite interval partitions and random permutations, implement it in the Python package ShapleyFDA, and illustrate it on simulated scenarios and the Tecator data set. The central claim is that R_f(t) is a continuous Shapley value relevance function that identifies the points t most relevant for prediction, without retraining the model.","tokens_in":23897,"tokens_out":4892,"duration_ms":49867,"significance":"If the central claim is secured, the paper would be a useful contribution: it is, to my knowledge, one of the first systematic attempts at global, model-agnostic interpretability for functional predictors, and it offers pointwise relevance curves at no retraining cost. The ghost-variable imputation is a sensible way to avoid leave-one-point-out refitting, and the open-source package plus the reproducible simulation code are valuable assets. The simulations with known ground truth (linear, nonlinear, and discrete targets) give encouraging qualitative evidence that the method identifies relevant regions, and the Tecator example is a convincing illustration. However, the theoretical interpretation as a continuous Shapley value is not established in the manuscript, and several load-bearing assumptions of the continuous-game framework are left unverified.","major_comments":[{"comment":"The paper's central claim is that R_f(t) is a continuous Shapley value relevance function obtained by the asymptotic approach, but the required convergence is never established. Section 2.3 states only that 'under certain conditions' the limit of finite Shapley values over refining partitions exists and is unique, citing Neyman (1994); it does not state those conditions, and Section 3 does not verify them for the specific game ν(S)=᷈R²(S) in Eq. (3). The experiments in Section 5.1 fix n=20 intervals and |Π0|=1000 permutations, so the quantity computed in Eq. (4) is a finite-game Shapley value for one fixed partition. Without a proof that the finite Shapley values converge as n→∞, or at least a stability analysis in n, the pointwise curve R_f(t) is not shown to be a well-defined continuous object independent of the chosen partition and permutation sample.","section":"Section 2.3 and Section 3, Eq. (4)"},{"comment":"Section 3.3 shows that the game ν is not a vector measure game, so the diagonal formula cannot be used. This is a useful negative result, but it does not supply the positive membership needed for the asymptotic approach. The displayed expression for ν(S) depends on an S-dependent family of signed measures τ_k^{S+}, τ_k^{S-} and an S-dependent function ρ^S; no argument is given that the resulting game belongs to any class, such as pNA or another space on which the Neyman asymptotic value is known to exist and be unique. Consequently, the structural analysis in Section 3.3 leaves the convergence question open rather than resolving it.","section":"Section 3.3"},{"comment":"The continuous-game framework in Section 2.3 defines games as nonnegative and monotone set functions ν: B → R+, but the proposed payoff ν(S)=᷈R²(S) is a coefficient of determination, which can be negative for poorly fitting models on subsets S. Indeed, Table 1 reports mean R²(I) values close to 0 for FLM under the nonlinear target, and subset R² values can fall below 0. The paper does not verify that ν is monotone or nonnegative, nor does it argue that the cited value theory extends to such set functions. This is a load-bearing assumption for invoking Aumann–Shapley theory.","section":"Section 2.3 and Eq. (3)"},{"comment":"The experimental section provides no sensitivity analysis with respect to the two approximation choices on which the continuous interpretation depends: the number of intervals n and the number of random permutations |Π0|. All simulation scenarios use n=20 and |Π0|=1000 (5000 for Tecator), and no experiment varies the partition. Since the theoretical claim is about a limit over refining partitions, reporting results for, say, n ∈ {5,10,20,40} and a few permutation counts would be needed to support the claim that the observed curves are stable functional objects rather than artifacts of a particular finite partition.","section":"Section 5.1 and Section 5.2"}],"minor_comments":[{"comment":"The notation ρ^S({τ_k^{S+}(S), τ_k^{S-}(S): k≥1}) is not precise: ρ^S is said to be defined on ℝ^N, but the argument is a countably infinite sequence of scalar values; the domain should be stated as a sequence space, or the displayed expression should be rewritten accordingly.","section":"Section 3.3"},{"comment":"The step replacing the finite-sample average by the population variance is written with '≈' and the reconstruction error is assumed to have zero mean. This is plausible given Kneip and Liebl (2020), but the approximation should be stated as a formal assumption or a limiting statement as m→∞, since the subsequent conclusion about the structure of ν rests on it.","section":"Section 3.3"},{"comment":"The complexity analysis is useful, but the statement that E(T_O)=E(T_M)=T/2 should be made conditional on the uniform random permutation model; otherwise the expectation over the random subset of intervals used in the Shapley sum is implicit.","section":"Section 4"},{"comment":"The sentence 'founded by MICIU/AEI/...' should read 'funded by MICIU/AEI/....'","section":"Acknowledgments"},{"comment":"The main-text figures show only the m=200, η=0.05 scenarios and the remaining cases are deferred to Appendix B; it would help the reader if the figure captions stated explicitly that the displayed curves are means over 100 replications and that the corresponding pointwise standard deviations are provided in the appendix.","section":"Section 5.1"},{"comment":"The definition of an S-admissible sequence uses subalgebras (P_j) and requires S ∈ P_1; when the finite approximations are based on interval partitions, the paper does not connect this definition to the specific choice of partitions in Eq. (4), which would clarify the notational link between ν_P and ν_{I_i}.","section":"Section 2.3"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is legitimate and lands on the central claim. The paper's own Section 3.3 rules out the easy vector-measure route, but no alternative convergence argument is supplied, and the fixed choice n=20 appears throughout. I would not reject the paper: the finite-partition method and the software are usable, and the simulations give genuine evidence of practical value. However, the title and abstract promise a 'continuous Shapley value', and that promise is not currently supported. The authors should either prove the relevant convergence (or identify a value class containing their game), or explicitly reframe the contribution as a finite-interval Shapley method with a validation of partition stability. The latter would be a lesser but still publishable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the method is coherent, the code and experiments are real, but the central mathematical claim in the title is not actually established. The authors show in Section 3.3 that their game is not a vector-measure game, and Section 2.3 only gestures at \"certain conditions\" for the asymptotic Shapley value without checking them for this specific ν. So the R_f(t) computed with n=20 and 1000 random permutations is, as far as the paper shows, a finite-partition heuristic, not a proven continuous Shapley value. That is a real gap, not a cosmetic one.\n\nWhat is genuinely new: ghost-variable imputation to define coalition payoffs for functional predictors, combined with finite Shapley values over interval partitions, gives a global, model-agnostic way to rank points of a functional predictor. The algorithm is clearly specified, the complexity analysis is useful, and the simulations—where the relevant regions are known—show the heuristic recovers the expected structure. The Tecator example is also sensible, and shipping a Python package plus code for the experiments is exactly the kind of reproducibility that should be credited.\n\nThe soft spots, in rough order of importance. First, the convergence gap: the paper never shows that the finite Shapley values converge to a well-defined continuous quantity as the partition refines, and Section 3.3 demonstrates why standard tools (the diagonal formula) do not apply. Either prove convergence for a restricted but useful class of games, or re-label the object as a finite-partition Shapley relevance measure and leave the continuous interpretation as motivation. Second, as written, Eq. (3) does not give ν(∅)=0: at the empty coalition, the reconstructed curve is the unconditional mean, and the resulting R^2 is generally not zero. Since the Shapley formula uses empty coalitions, the computed values sum to R^2(I) − ν(∅), not R^2(I); the constant cancels in marginal contributions, so the shape of the relevance curve is probably unaffected, but the efficiency interpretation is off and this needs a convention or a fix. Third, there is no uncertainty quantification on the relevance curves, and the sensitivity to n and to the number of permutations is not reported; a small study varying n would help. Fourth, the \"first global agnostic FDA interpretability\" claim should be tempered with a nod to the global sensitivity-analysis literature on Shapley effects for functional inputs.\n\nOverall, this is a useful paper with a load-bearing weakness in its theoretical framing. The practical method stands as a well-tested heuristic, and the package makes it usable. I would send it to peer review, with a referee who knows non-atomic game theory, and require either a convergence proof for a meaningful game class or a more honest characterization of what is being computed.","headline":"A well-built and useful functional-data interpretability heuristic whose central label—'continuous Shapley value'—is not backed by a verified convergence argument; the paper deserves serious refereeing, but the theory needs either proof or a more modest framing.","tokens_in":24339,"tokens_out":5105,"would_cite":true,"duration_ms":53007,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12","62R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A continuous-game Shapley value turns every point of a functional input into a measurable player, giving any trained regression model a pointwise relevance curve.","keywords":["functional data analysis","model-agnostic interpretability","Shapley value","continuous games","scalar-on-function regression","ghost variables","functional relevance","ShapleyFDA"],"falsifier":"Run the method on a deliberately discontinuous prediction rule, such as $f(X)=\\max_t X(t)$, and refine the partition of $[0,1]$ while increasing the number of random permutations; if the resulting relevance function does not converge as the partition becomes finer, the assumed asymptotic value for $\\nu(S)=\\tilde R^2(S)$ does not exist for that model, and the finite approximation is not computing a well-defined continuous Shapley value.","tokens_in":23335,"feed_emoji":"📈","tokens_out":10220,"duration_ms":81804,"temperature":0.7,"pith_summary":"This paper proposes a way to answer, for a trained scalar-on-function prediction model, which points of the functional input actually matter for its predictions. The authors define a game whose players are the points $t$ in the interval where the predictor is observed, and whose payoff for a coalition $S$ is the coefficient of determination $R^2$ achieved when only the values of $X$ on $S$ are kept and the rest are replaced by their conditional expectation given $S$. The Shapley value of this continuous game yields a relevance function $R_f(t)$ that assigns an importance score to every point $t$, without retraining the model. If the construction is valid, any fitted functional predictor becomes auditable at pointwise resolution, and the method is presented as the first global, model-agnostic interpretability tool for functional data regression. The paper illustrates the method with simulated data and with the Tecator spectra.","feed_headline":"Shapley values now explain functional data point by point","feed_subtitle":"A new relevance function tells which regions of a curve drive any trained prediction model, with no retraining.","key_machinery":"The load-bearing object is the Shapley value relevance function $R_f(t)$, built from the continuous game $\\nu(S)=\\tilde R^2(S)$, where $\\tilde R^2(S)$ is the test coefficient of determination of the trained model $f$ evaluated on reconstructed curves $\\tilde X_j^S(t)=X_j(t)\\mathbf{1}_S(t)+\\dot X_j(t)\\mathbf{1}_{S^c}(t)$, with $\\dot X_j(t)$ the conditional expectation of $X(t)$ given the observed values on $S$, estimated under a Gaussian-process assumption. The finite version of the game on a partition $\\{I_1,\\dots,I_n\\}$ has Shapley value $\\varphi_{\\nu,i}=(1/n!)\\sum_{\\pi\\in\\Pi}[\\tilde R^2(p_i^\\pi\\cup I_i)-\\tilde R^2(p_i^\\pi)]$, averaged over random permutations in practice, and the relevance function $R_f(t)=\\sum_i \\varphi_{\\nu,i}(b_i-a_i)^{-1}\\mathbf{1}_{I_i}(t)$ is the histogram density of these interval contributions. This machinery converts an infinite-player relevance problem into a sequence of finite Shapley-value computations, with the asymptotic theory for continuous games supplying the intended limit as the partition is refined.","core_discovery":"On the paper's own terms, the central discovery is that Shapley values, originally defined for finitely many players, can be lifted to the infinite setting of functional regression by treating the continuum of points in the input interval as players and defining the payoff of each measurable subset $S$ as the test-set coefficient of determination of the already trained model when all points outside $S$ are replaced by conditional expectations. The resulting relevance function is computed through the asymptotic approach: for a partition of the interval into $n$ pieces, the finite Shapley values of the restricted game are averaged over random permutations, and the histogram of those interval contributions, renormalized by interval length, is the relevance curve $R_f(t)$. The paper also shows that this game is generally not a vector-measure game (a game whose payoff depends only on finitely many measures of the coalition), so no closed-form diagonal formula is available and the permutation-based finite approximation is not merely an algorithmic convenience but the operative definition.","pith_inferences":["Beyond the paper, the same game construction should extend to functional classification by replacing $R^2$ with test accuracy or log-loss, and to multiple functional regressors by letting players carry a type label; the authors list these only as future directions.","Beyond the paper, the Gaussian-process assumption on the reconstruction step is the likely weak point: on strongly non-Gaussian functional data the conditional expectations become best linear predictions, so the method may rank points by their ability to predict missing segments rather than by their role in the response; a retraining-based LOCO comparison on simulated non-Gaussian data would test ","Beyond the paper, the relevance curve offers a model-sanity check for functional linear models: discrepancies between $R_f(t)$ and the magnitude of the estimated coefficient $\\beta(t)$ would flag regions where the linear approximation is distorted, a diagnostic that currently requires manual inspection of $\\hat\\beta$.","Beyond the paper, the difference between relevance curves computed at two partition resolutions could serve as a stability measure, revealing whether the model's signal is localized near specific points or spread over broad regions."],"forward_implications":["Any trained functional predictor — linear, nearest-neighbour, or neural — can be audited with a single test set and no retraining, because the method needs only evaluations of the fitted model on reconstructed curves.","The relevance function inherits the Shapley efficiency property: integrated over the whole interval it recovers the model's total test $R^2$, so the curve is a genuine allocation of explained variability among regions of the input.","When the fitted model has no predictive skill on a target (a linear model facing a strongly non-linear map, for instance), the method returns a flat relevance curve, which serves as a diagnostic of model failure rather than a misleading ranking.","On the Tecator data, the relevance functions of all three algorithms agree that the interval $[1040,1050]$ dominates fat-content prediction and $[970,980]$ is negligible, matching the observed variability of the second-derivative curves.","The expected computational cost scales as $O(T^3|\\Pi'|n)+O(mT^2|\\Pi'|n)+O_f(m,T)|\\Pi'|n$, driven by the number of random permutations and grid points; the accompanying ShapleyFDA package caches coalition values shared across permutations to reduce the work."],"supporting_citations":[{"why":"Supplies the original finite Shapley value formula and axioms that the permutation average in Eq. (4) is built on.","marker":"Shapley (1953)"},{"why":"Defines the continuous-game value and its efficiency, symmetry, and linearity axioms, which the relevance function is intended to satisfy.","marker":"Aumann and Shapley (1974)"},{"why":"Supplies the asymptotic approach and the conditions under which the limit of finite-game Shapley values exists and is unique, the theoretical route the finite approximation leans on.","marker":"Neyman (1994)"},{"why":"Introduces values of games with a continuum of players as limits of finite games, the construction the paper follows.","marker":"Kannai (1966)"},{"why":"Introduces the ghost-variable idea of replacing an unobserved covariate by its conditional expectation given the rest, which motivates Eq. (2) and avoids retraining.","marker":"Delicado and Peña (2023)"},{"why":"Provides the optimal linear reconstruction operator for partially observed functional data, which the paper shows its conditional-expectation formula implements.","marker":"Kneip and Liebl (2020)"},{"why":"Decomposes $R^2$ among correlated regressors by Shapley values, the finite precedent for using $R^2$ as the payoff.","marker":"Lipovetsky and Conklin (2001)"}],"fun_headline_variants":["Shapley value goes continuous for functional data explainability","Continuous Shapley values pinpoint which curve parts drive predictions","Shapley relevance function assigns credit to each point in functional data","Explain any functional regression model with a continuous Shapley curve"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on the way payoff is assigned to coalitions of points being one of the continuous games for which the Shapley value exists and is unique; the paper states that 'under certain conditions' the limit exists but does not verify those conditions for its specific payoff before using a finite approximation.","fun_headline_variants_meta":{"raw":{"variants":["Shapley value goes continuous for functional data explainability","Continuous Shapley values pinpoint which curve parts drive predictions","Shapley relevance function assigns credit to each point in functional data","Explain any functional regression model with a continuous Shapley curve"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000628,"raw_usage":{"total_tokens":2864,"prompt_tokens":863,"completion_tokens":2001,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":1940}},"tokens_in":479,"tokens_out":2001,"duration_ms":12994,"temperature":1.0,"reasoning_tokens":1940,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:04:15.423680+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the method on a deliberately discontinuous prediction rule, such as $f(X)=\\max_t X(t)$, and refine the partition of $[0,1]$ while increasing the number of random permutations; if the resulting relevance function does not converge as the partition becomes finer, the assumed asymptotic value for $\\nu(S)=\\tilde R^2(S)$ does not exist for that model, and the finite approximation is not computing a well-defined continuous Shapley value.","supporting_citations":[{"cited_title":"Values of games with a continuum of players","cited_arxiv_id":null,"evidence_quote":"Introduces values of games with a continuum of players as limits of finite games, the construction the paper follows."}],"review_version":1}