{"id":"59429eb1-5437-4df1-8299-0605f933e376","arxiv_id":"1908.05272","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces functional FDR and a continuous Benjamini-Hochberg procedure with proven FDR control, along with an adjusted p-value function.","lead":"This paper defines false discovery rate for functional data and proposes a continuous Benjamini-Hochberg procedure that controls it. A generalist might read it to understand a new way to find statistically significant regions on curves and maps while limiting false positives.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The fFDR guarantee hinges on an unverified infinite-dimensional PRDS assumption; the climate p-values may violate it.","rationale":"The core theorem is a conditional statement and its proof is structurally sound: finite-grid BH controls FDR under PRDS, convergence Q_k→Q is shown under equations (3.5)-(3.6) and assumptions (a1)-(a3), and dominated convergence transfers the bound. The simulations in Section 4 provide empirical support, especially the 2D Matern-field study where positive dependence is plausible. The weakest link is therefore not the proof machinery but the transfer of the theorem to the headline application. In the climate analysis the input p-values may be miscalibrated because temporal autocorrelation is ignored, and PRDS is not checked; both are needed for the claimed FDR guarantee on the reported significance areas. This does not invalidate the methodological contribution, but it warrants a conditional acceptance in which the authors verify or relax the dependence/validity requirements for the application. The scaling typo in equations (3.5)-(3.6) noted by the reader is real but secondary; it is a statement-level normalization issue and does not change this assessment.","tokens_in":16739,"tokens_out":21477,"duration_ms":235859,"concrete_test":"Simulate a null version of the climate study: estimate the spatial covariance and AR(1) temporal autocorrelation from the 64800-point temperature residuals, generate 25-year null fields with no trend, recompute pointwise one-sided t-test p-values, apply fBH at α=0.05 on the same 180×360 grid, and repeat at least 100 times. If the empirical FDR exceeds roughly 0.05 by more than two simulation standard errors under this complete-null setting, the PRDS/validity assumptions fail and the reported significance areas are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.9 controls fFDR only if the pointwise p-value process is PRDS on the true-null set U (Definition 2.5) and the pointwise p-values are uniform under H0. The climate application (Section 5) supplies neither. The t-statistics are computed from 25 yearly observations per grid point with an ordinary linear model, ignoring temporal autocorrelation; if the errors are autocorrelated, the p-values are not U(0,1) under H0, so no FDR statement applies. Even granting the p-values, PRDS is a joint distributional property over all finite subsets of the 64,800 grid points; the paper offers no diagnostic, and Section 6 only asserts that 'a certain degree of positive association' is required. The proof's finite-grid step (Proposition 3.7) applies the Benjamini-Yekutieli bound to each grid, E[Q_k]≤α#(S_k∩U)/#S_k; if PRDS fails at any resolution, the limiting dominated-convergence step has no valid bound, so Table 2's significance areas are not covered by the theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the Benjamini-Hochberg (BH) false discovery rate framework to functional data with a continuum of pointwise hypotheses. It defines a functional FDR via a measure-weighted proportion of false rejections, proposes a functional BH (fBH) procedure and an adjusted p-value function, and proves that, under positive regression dependence (PRDS) of the pointwise p-value process on the true-null set plus regularity conditions, fBH controls the functional FDR at level αν(U)/ν(T). The proof approximates the continuum by nested finite grids, applies the classical Benjamini-Yekutieli bound on each grid, and passes to the limit by dominated convergence. Two simulation studies (one 1D, one 2D) and an application to satellite temperature data are presented.","tokens_in":16923,"tokens_out":7315,"duration_ms":76054,"significance":"If the main theorem is correct, this is a useful and elegant contribution: it gives a principled, practically implementable way to control FDR for local inference on functional data, with the finite-grid approximation reducing to ordinary BH. The proof is self-contained and rests on the classical BY result rather than new model-specific machinery. The explicit assumptions are a strength, and the paper is careful to state what is needed for the guarantee. The main limitations are that the central definition is formally misstated, the load-bearing PRDS assumption is not checked in the simulations or the application, and the climate application may violate the uniformity-of-p-values assumption. These are fixable in revision and do not undermine the theoretical core.","major_comments":[{"comment":"The definition of α* as 'arg max_r [ν({s : p(s) ≤ r})/ν(T) ≥ α^{-1} r]' is not a well-posed maximization, because the bracketed expression is a logical condition rather than a numerical objective function. The intended definition appears to be α* = sup{r ∈ [0,1] : a(r) ≥ r/α}, with the supremum attained under assumption (a2). The same correction is needed for b_k in Propositions 3.5 and 3.7. As written, the central object of the procedure is formally undefined, and Theorem 3.9 cannot be evaluated as stated.","section":"Section 3.2, Definition 3.3 (and Eqs. (3.4), (3.7))"},{"comment":"The assumptions of Theorem 3.9 are not verified for the climate application. The pointwise p-values are obtained from one-sided t-tests in a linear model with 25 yearly observations per grid cell; if the errors are temporally autocorrelated, the p-values are not U(0,1) under H0, so the FDR bound does not apply. The paper also provides no diagnostic for the infinite-dimensional PRDS condition over the 64,800 grid points; Section 6 only asserts that 'a certain degree of positive association' is required. Consequently, the adjusted significance areas in Table 2 are not covered by the theorem. Please either re-frame the application as illustrative and outside the theorem's assumptions, or add checks (e.g., block/bootstrap or simulations with realistic dependence) that support the assumptions.","section":"Sections 5.1, 5.2, and 6"},{"comment":"The assumption that 'p(t) is PRDS wrt the set of true null hypotheses with probability one' is stated as if PRDS were a sample-path property. PRDS is a property of the joint distribution of the p-value process (Definition 2.5), so 'with probability one' is inappropriate. What the proof needs is that, for each finite grid S_k, the joint law of {p(t) : t ∈ S_k} is PRDS on S_k ∩ U. This should be restated, because the finite-grid step applies Theorem 2.6 to those finite-dimensional laws.","section":"Section 3.5, Theorem 3.9"}],"minor_comments":[{"comment":"The acronym is inconsistently written as 'PDRS' in several places (e.g., Definition 2.5, Theorem 3.9, and Appendix A.1); it should be 'PRDS' uniformly.","section":"Throughout"},{"comment":"The text says that 'all hypotheses are rejected eventually' when b_k = 0 and min p(t) > 0, but the displayed conclusion #A_k = 0 means no hypotheses are rejected; the correct wording is 'all hypotheses are accepted eventually'.","section":"Appendix, Lemma A.5"},{"comment":"The phrase 'Qk behaves asymptotically as the functional false discovery proportion' is imprecise; it would be clearer to write 'Q_k → Q almost surely'.","section":"Section 3.4.1"},{"comment":"The dotted lines with slopes 1 and 0.717 are mentioned in the caption but not labeled in the figure; adding a legend or explicit annotation would make the comparison with the nominal bounds much easier to read.","section":"Figure 4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the reduction to the classical BH/BY result is a sound strategy. My concerns are formal (the misstated definition of α*) and empirical (unverified PRDS and p-value uniformity in the application); both are addressable in a revision. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things. First, this paper genuinely extends BH to an infinite-dimensional functional setting: it defines fFDR, a continuous fBH procedure, and an adjusted p-value function, and proves that the continuous procedure is the limit of finite-grid BH under dominated convergence. That is a real contribution, and the proof strategy—reducing to Benjamini-Yekutieli on grids—is sound. Second, the theory rests on an infinite-dimensional PRDS assumption that the paper never verifies in simulations or in the climate application, so the headline guarantee is conditional.\n\nWhat's new: the definition of fFDR as the expected proportion of ν-measure of false discoveries, the fBH threshold via the cumulated p-value function, and the adjusted p-value function that lets you read off significance for any α. The limit argument is careful about assumptions (a1)-(a3) and grid approximation; the appendix fills in the details. I believe Theorem 3.9 is correct as stated. The simulations show the expected behavior, including FDR control under the scenarios tested.\n\nWhere it's soft: the PRDS assumption is the load-bearing condition, and it is simply asserted for the applications. The climate data uses yearly averages over 25 years per grid point with a linear model, ignoring temporal autocorrelation; if the errors are autocorrelated, pointwise p-values are not uniform under the null and the theorem does not apply. That is a real gap between theory and the showcase application. Also, equations (3.5) and (3.6) in the weighted case have a scaling typo—sums over grid points divided by #S_k do not converge to integrals of f; this needs fixing. Minor: in Lemma A.5 the text says \"all hypotheses are rejected\" when it means \"none are rejected\" (#A_k = 0), a wording slip.\n\nWhether you believe the central result: yes, the proof holds up under its assumptions. But the paper overclaims slightly by applying the method to data where the assumptions are not checked.\n\nWho this is for: anyone doing local inference on functional or spatial data who wants an FDR analog. It deserves a serious referee, and the authors should be asked to fix the typos, add a diagnostic or at least a discussion of PRDS in the climate example, and address the temporal dependence issue.","headline":"The fBH procedure is a genuine extension of BH to functional data with a credible control proof, but the theory only applies under an infinite-dimensional PRDS assumption that the applications do not check.","tokens_in":17473,"tokens_out":2214,"would_cite":true,"duration_ms":22115,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a continuous Benjamini-Hochberg procedure controls the functional false discovery rate when the p-value function is positively regression dependent and the finite-grid approximation is regular.","keywords":["false discovery rate","functional data analysis","local inference","Benjamini-Hochberg procedure","multiple testing","positive regression dependence","adjusted p-value function","climate temperature data"],"falsifier":"Generate functional data on a one-dimensional domain with a known true-null region $U$, using pointwise test statistics from a Gaussian process with nonnegative covariance so the PRDS assumption is satisfied, and construct the p-value function accordingly. Apply the fBH procedure at level $\\alpha$ on a sequence of increasingly fine uniform grids; across many replications the empirical functional false discovery proportion should not exceed $\\alpha\\nu(U)/\\nu(T)$ if Theorem 3.9 is correct.","tokens_in":16542,"feed_emoji":"📊","tokens_out":10431,"duration_ms":97203,"temperature":0.7,"pith_summary":"The paper addresses local inference for functional data, where a null hypothesis is tested at every point of a continuous domain, so there are infinitely many tests. It defines the functional false discovery rate as the expected proportion of the rejected region that is falsely rejected, measured by a weight on the domain, and introduces a continuous version of the Benjamini-Hochberg procedure: reject the points whose p-value falls below an adjusted threshold built from the measure of the p-value sublevel set. The central result is that, under positive regression dependence on the true-null region and mild regularity of the p-value function, this functional BH procedure controls the functional FDR at level $\\alpha\\nu(U)/\\nu(T)$, where $U$ is the true-null region. Because the procedure can be approximated by ordinary BH on a fine grid, it gives practitioners a practical way to correct for multiple testing over curves and surfaces. The simulations and an application to satellite temperature data illustrate the method and show that large significant regions remain after adjustment.","feed_headline":"False-discovery control now reaches infinite hypothesis sets","feed_subtitle":"A continuous Benjamini-Hochberg threshold bounds wrong rejections by the true-null share of the domain.","key_machinery":"The load-bearing object is the cumulated p-value function $a(r)=\\nu(\\{s:p(s)\\le r\\})$, which compresses infinitely many pointwise tests into one sublevel-set curve. The fBH threshold is the largest crossing of $a(r)$ with the line $r/\\alpha$, and the procedure rejects every point whose p-value lies below that crossing. The convergence proof works by applying ordinary weighted BH to $p$ evaluated on a dense uniform grid $S_k$, obtaining a discrete false-discovery proportion $Q_k$ that converges almost surely to the functional $Q$, so the finite-sample PRDS bound can be passed through the limit by dominated convergence.","core_discovery":"On a domain $T$ with measure $\\nu$, the paper defines $V$ as the set of points where a true null is rejected, $S$ as the set where a false null is rejected, and the functional false discovery rate as $\\mathrm{fFDR}=\\mathbb{E}[\\nu(V)/(\\nu(V\\cup S))]$. The functional BH procedure rejects $H_0^t$ at all points $t$ with $p(t)\\le \\alpha^*$, where $\\alpha^*$ is the largest $r$ such that the cumulated p-value function $a(r)=\\nu(\\{s:p(s)\\le r\\})$ satisfies $a(r)/\\nu(T)\\ge r/\\alpha$. Theorem 3.9 states that if the p-value function is PRDS on the true-null set $U$ and the stated level-set regularity conditions hold with probability one, then $\\mathrm{fFDR}\\le \\alpha\\,\\nu(U)/\\nu(T)\\le \\alpha$. Equivalently, the adjusted p-value function $\\tilde p(t)=\\min_{s\\ge p(t)} \\nu(T)s/\\nu(\\{p\\le s\\})$ yields the same rejection region for every $\\alpha$.","pith_inferences":["A useful next step would be a practical diagnostic for the PRDS assumption, since the paper's proof assumes it but does not check it in the simulations or the climate application.","The same sublevel-set threshold construction could define functional analogues of weighted FDR, positive FDR, or q-value curves, giving a continuous-data counterpart to standard FDR-related quantities.","The convergence result suggests a grid-selection rule: keep refining the grid until the fBH threshold and the measure of the rejection region stop changing, since the proof requires uniform approximation of p-value level sets."],"forward_implications":["A dense-grid application of ordinary BH converges to the infinite-domain fBH threshold, so local inference on curves and surfaces can be FDR-adjusted with standard software.","The adjusted p-value function lets a researcher compute significance after correction for every $\\alpha$ at once, rather than re-running the procedure threshold by threshold.","Because fBH controls FDR rather than FWER, it is more sensitive than methods such as Fmax in the simulations, at the cost of only weak family-wise control.","With a weighted measure such as $\\cos(\\text{latitude})$ on the sphere, the procedure adapts to non-uniform domains; in the climate application 15.0% of Earth's area remains significant at the 5% level after adjustment, compared with 32.4% before."],"supporting_citations":[{"why":"Introduces the false discovery rate and the original finite Benjamini-Hochberg threshold that the continuous version generalises.","marker":"[2]"},{"why":"Supplies the PRDS condition and the finite-sample bound $E[Q]\\le m_0\\alpha/m$ that Theorem 3.9 invokes on each finite grid.","marker":"[4]"},{"why":"Defines the unadjusted p-value function for functional local inference, the input object of the fBH procedure.","marker":"[12]"},{"why":"Earlier spatial FDR formulation whose rejection-region measure definition the paper relates its functional FDR to.","marker":"[16]"},{"why":"Fmax procedure used as the FWER-controlling comparison baseline in the one-dimensional simulation study.","marker":"[10, 17]"},{"why":"Freedman-Lane permutation method used to compute the pointwise p-values in the 1D simulation.","marker":"[8]"},{"why":"Provides the peak-detection FDR setting that inspired the two-dimensional simulation design.","marker":"[6]"}],"fun_headline_variants":["Continuous BH controls false discovery rate on infinite domains","FDR for functional data: a new continuous Benjamini-Hochberg","Infinite-test FDR control via functional BH procedure","Functional FDR: taming infinite multiple comparisons"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guarantee depends on the pointwise p-value function being positively regression dependent on the true-null region, so that raising a true-null p-value can never make rejection of the set easier.","fun_headline_variants_meta":{"raw":{"variants":["Continuous BH controls false discovery rate on infinite domains","FDR for functional data: a new continuous Benjamini-Hochberg","Infinite-test FDR control via functional BH procedure","Functional FDR: taming infinite multiple comparisons"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000397,"raw_usage":{"total_tokens":2102,"prompt_tokens":994,"completion_tokens":1108,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":1042}},"tokens_in":610,"tokens_out":1108,"duration_ms":9403,"temperature":1.0,"reasoning_tokens":1042,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:42:19.872357+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate functional data on a one-dimensional domain with a known true-null region $U$, using pointwise test statistics from a Gaussian process with nonnegative covariance so the PRDS assumption is satisfied, and construct the p-value function accordingly. Apply the fBH procedure at level $\\alpha$ on a sequence of increasingly fine uniform grids; across many replications the empirical functional false discovery proportion should not exceed $\\alpha\\nu(U)/\\nu(T)$ if Theorem 3.9 is correct.","supporting_citations":[{"cited_title":"and Hochberg, Y","cited_arxiv_id":null,"evidence_quote":"Introduces the false discovery rate and the original finite Benjamini-Hochberg threshold that the continuous version generalises."},{"cited_title":"and Yekutieli, D","cited_arxiv_id":null,"evidence_quote":"Supplies the PRDS condition and the finite-sample bound $E[Q]\\le m_0\\alpha/m$ that Theorem 3.9 invokes on each finite grid."},{"cited_title":"and Vantini, S","cited_arxiv_id":null,"evidence_quote":"Defines the unadjusted p-value function for functional local inference, the input object of the fBH procedure."},{"cited_title":", Reich, B","cited_arxiv_id":null,"evidence_quote":"Earlier spatial FDR formulation whose rejection-region measure definition the paper relates its functional FDR to."},{"cited_title":"and Lane, D","cited_arxiv_id":null,"evidence_quote":"Freedman-Lane permutation method used to compute the pointwise p-values in the 1D simulation."},{"cited_title":", Schwartzman, A., and others","cited_arxiv_id":null,"evidence_quote":"Provides the peak-detection FDR setting that inspired the two-dimensional simulation design."}],"review_version":1}