{"id":"2ce27833-4597-46b4-b69a-4c560a01d882","arxiv_id":"1908.06486","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"This paper proposes DCorrX and MGCX, block-permutation independence tests for stationary time series that are asymptotically valid, consistent, and estimate the optimal dependence lag.","lead":"A new statistical test tells you whether two time series are related, even when the relationship is nonlinear and the sample size is small. It works by combining distance-based dependence measures with block permutation, and it also estimates the time lag at which the relationship is strongest.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Block-permutation validity step only controls past lags; Assumption 4 is one-sided, so Theorem 3's proof does not rule out mis-calibrated p-values under H0.","rationale":"The paper is useful and reproducible, with open-source code and sensible simulations; the reader's CONDITIONAL verdict is appropriate. My concern sharpens the reader's Assumption 4 worry: it is not long-range dependence or non-stationarity that breaks the proof, but the direction of the mixing condition. Because the test statistic sums over lags j≥0 only, the null permits arbitrary dependence of X_t on future Y values. Block permutation, by permuting blocks, can create alignments at positive offsets that are multiples of b_n, and the proof's 'far enough' argument is only valid for past offsets. The constructed process satisfies every listed assumption and H0, yet p-values are not uniform; this is an internal gap in Theorem 3, not a disagreement with the field. The fix is modest: add a two-sided version of Assumption 4 (or explicitly assume H0 holds for all |j|≤M and that block offsets are controlled), and the main consistency and optimal-lag results are likely unaffected. Because the exact-alpha claim is central and currently unproven, the paper should remain CONDITIONAL, with the condition being the corrected mixing assumption and a repaired proof of Theorem 3.","tokens_in":17704,"tokens_out":20315,"duration_ms":227454,"concrete_test":"Run 2,000 Monte Carlo trials of DCorrX under the null process X_t = c∑_{k=1}^{500} k^{-3/4} Y_{t+k} + ε_t, with Y_t, ε_t iid N(0,1), c=10, block size b=10, M=1, n=1,000, R=1,000 permutations, α=0.05. If Theorem 3 holds under Assumptions 1-5, the empirical rejection rate should be close to 0.05 and the mean p-value near 0.5. If the rejection rate is far below 0.05 (e.g. <0.01) while observed statistics are near zero, the block-permutation null distribution is not the true null distribution; this confirms the one-sided Assumption 4 does not support the proof. An analytical cross-check: re-derive the theorem's independence step using only F_{X_t,Y_{t−j}}; the missing bound on F_{X_t,Y_{t+j}} appears exactly where block offsets are positive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3's key step (Appendix: 'Next, we show...') claims X_t and Y_{π(t)} are asymptotically independent once |t−π(t−j)| is large, citing Assumption 4. But Assumption 4 only bounds F_{X_t,Y_{t−j}} for j→∞, i.e. Y in the past. Block permutation shifts whole blocks of Y, so π(t−j)−t is typically a nonzero multiple of the block size b_n, equally often positive; for those pairs X_t is aligned with future Y values, whose dependence on X_t is not controlled by Assumption 4. This is not a purely technical quibble: the stationary process X_t = c∑_{k≥1} k^{-3/4} Y_{t+k} + ε_t with Y_t, ε_t iid Gaussian satisfies Assumptions 1, 2, 4, 5 and H0 for every fixed M, because X_t is independent of Y_t and of all Y_{t−j}, j≥1. Choose block size b; a block shifted by +b pairs X_t with Y_{t+b}, on which X_t has dependence of size ~b^{-3/4}. The permuted statistic then has positive mean, while the observed statistic estimates 0; p-values concentrate near 1 and the rejection rate tends to 0, not α. Hence P(T*_n ≥ T_{n,R_n,α})→α is not a consequence of the stated assumptions. A two-sided mixing condition (or an H0 covering future lags) is required.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes DCorrX and MGCX, omnibus nonparametric tests of independence between two strictly stationary time series at lags 0 through M. The test statistic is a weighted sum of lagged distance-correlation or multiscale-graph-correlation estimates, and the null distribution is estimated by permuting blocks of one series rather than individual observations. The authors claim asymptotic validity of the block-permutation procedure under the null (Theorem 3), consistency under fixed alternatives (Theorem 4), and consistency of the optimal-lag estimator (Theorem 5), and they support the method with simulations and an fMRI resting-state connectivity analysis. The main theoretical results are stated in Section 4 with proofs collected in the Appendix.","tokens_in":1672,"tokens_out":1872,"duration_ms":87604,"significance":"The methodological goal is valuable: a nonparametric, multivariate, nonlinear omnibus test for temporal dependence that returns a single p-value and also characterizes the lag and geometric scale of the dependence would be a practically useful addition to the time-series toolbox. The paper builds on the established DCorr/MGC framework, ships open-source code in the hyppo package, compares against ShiftHSIC, WildHSIC, and Ljung-Box variants, and demonstrates the method on HCP fMRI data. However, the central validity theorem is not established under the stated assumptions: Assumption 4 is one-sided while the proof treats permuted lags symmetrically, and the counterexample in my report shows that Theorem 3 as stated is false. Because the proof gap is local and a two-sided mixing condition would plausibly repair it, the contribution is significant but currently not rigorous enough for publication.","major_comments":[{"comment":"Assumption 4 is one-sided: it controls only sup|F_{X_t,Y_{t-j}} - F_{X_t}F_{Y_{t-j}}| as j goes to infinity, i.e., X_t against past values of Y. Block permutation shifts contiguous blocks of Y, so for a large fraction of positions t the permuted index pi(t) differs from t by a nonzero multiple of the block size b_n, equally often in the positive direction. Thus the permuted series pairs X_t with future values Y_{t+b_n}, whose dependence on X_t is not controlled by Assumption 4. The proof's condition P(|t-pi(t-j)| < gamma for some |j| <= M) treats positive and negative j symmetrically, but no assumption bounds F_{X_t,Y_{t+j}} as j grows. Concretely, let Y_t and epsilon_t be iid standard Gaussian and define X_t = c * sum_{k>=1} k^{-3/4} Y_{t+k} + epsilon_t. This process is strictly stationary, has finite second moments, satisfies Assumptions 1, 2, 4, and 5, and satisfies H0 for every fixed M because X_t is independent of Y_t and of all Y_{t-j}, j>=1. If block permutation shifts a block by +b_n, then for most t the permuted pair is (X_t, Y_{t+b_n}), whose distance correlation has size about c b_n^{-3/4}; the permuted statistic then has positive mean while the observed statistic estimates zero, so p-values concentrate near 1 and the rejection rate tends to 0 rather than alpha. The final step of the proof also invokes the continuous mapping theorem without establishing the required joint convergence of the permuted U-statistic. Theorem 3 is therefore not a consequence of the stated assumptions.","section":"Section 4 (Assumption 4) and Appendix proof of Theorem 3"},{"comment":"The proof fixes gamma for a given epsilon and then treats index pairs with |i-j| > gamma as approximately independent, carrying an O(epsilon) error. Because epsilon is held fixed while n grows, Theorem 1 as proven only gives E[DCov_{k,l}^n] = DCov_{rho_k,rho_l} + O(1/n) + O(epsilon) and Var[DCov_{k,l}^n] = O(1/n) + O(epsilon), which is not a convergence statement. The argument needs an n-dependent gamma_n with gamma_n -> infinity, gamma_n/n -> 0, and an approximation error that tends to 0; otherwise Theorem 2 (convergence in probability) and hence Theorem 4 are not established. The appeal to Theorem 5 of Shen et al. applies to i.i.d. data; with |i-j| > gamma the pairs are only approximately independent, so the approximation error must be shown to vanish.","section":"Appendix, proof of Theorem 1"},{"comment":"The proof of optimal-lag consistency appeals only to the finiteness of the search space and pointwise convergence of the sample statistics. If the population maximum M* = argmax_{0<=j<=M} DCorr(j) is not unique, or if there are near-ties whose separation does not grow with n, the sample argmax need not converge to a single M*. Please state explicitly that M* is the unique maximizer, or reformulate the theorem as convergence to the set of maximizers.","section":"Section 4, Theorem 5"}],"minor_comments":[{"comment":"The displayed formula for DCov(X,Y) is actually the squared distance covariance; the square should be applied consistently, or the notation should be changed to DCov^2.","section":"Section 2.4"},{"comment":"The example writes '(Y36,Y35,...,Y40)', but the order within a block should be preserved; it should read '(Y36,Y37,...,Y40)'.","section":"Section 3, block permutation step 2"},{"comment":"The sentence 'DCorrX and MGCX always estimate the correct lag' is too strong for finite n; it should be rephrased as 'with probability tending to 1 as n grows'.","section":"Section 5.2"},{"comment":"There is a typo: 'empircal' should be 'empirical'.","section":"Section 2.3"}],"recommendation":"major_revision","confidential_remarks":"The main issue is that Theorem 3 is false under the stated one-sided Assumption 4. The counterexample in my report appears to satisfy all stated assumptions while making the block-permutation null distribution badly mis-calibrated. I believe a two-sided mixing condition (or an explicit H0 covering future lags as well) and a repaired proof of Theorem 1 would put the claims on solid ground, so this is fixable within the scope of a major revision. The authors should also consider that the fMRI application uses a maximum lag M chosen by the analyst, and the directionality of the tested dependence should be made explicit when reporting asymmetric p-value matrices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The practical idea is good: take DCorr or MGC, average over lags 0..M, and use block permutation instead of ordinary permutation. That gives a single omnibus test plus a lag estimate, and it is a real improvement over per-lag Bonferroni wild-bootstrap HSIC tests. The simulations are honest, the fMRI analysis is a nice illustration, and hyppo ships the code. I also do not hold the heavy self-citation against it; the MGC machinery is their own published work and they reuse it transparently.\n\nThe math is the problem. Theorems 1 and 2 are sketchy but probably fixable: the epsilon argument needs gamma=gamma(n) and a real variance calculation under weak dependence. Theorems 3 and 4 are where I part ways. The stress-test note is correct. Assumption 4 only controls F_{X_t,Y_{t-j}} as j goes to infinity, i.e., Y in the past. Block permutation moves whole blocks, so for any t the paired Y_{pi(t)} is as often in the future as in the past, and future dependence is not assumed away. The process X_t = c * sum k^{-3/4} Y_{t+k} + eps_t with iid Y and eps is stationary, satisfies Assumptions 1-5, and satisfies H0 for every fixed M, yet a block shift by +b pairs X_t with a Y_{t+b} carrying dependence of order b^{-3/4}; the permuted statistic has positive mean while the observed statistic is about 0, so p-values concentrate near 1 and the rejection rate goes to 0. So P(T*_n >= T_{n,R,alpha}) -> alpha is not a consequence of the stated assumptions. The proof's continuous-mapping step is exactly where this is hidden.\n\nAlso, the fixed-M frame means consistency is only against alternatives at lags <= M; the abstract's \"universally consistent\" is an overclaim.\n\nMy bottom line: this deserves a serious referee and could become a useful paper, but not in this form. The fix is likely a two-sided mixing condition or an H0 that explicitly covers future lags, plus a real proof that the block-permuted statistic converges to the null distribution. I would not cite the current theorem as support for validity. Still, bring it to reading group: it is a good example of a subtle permutation-test gap.","headline":"Useful, clearly written extension of MGC/DCorr to temporal data with real code, but Theorem 3's block-permutation validity proof has a genuine one-sided-mixing gap and the abstract overclaims.","tokens_in":18538,"tokens_out":2297,"would_cite":false,"duration_ms":23465,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62M10","62H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces DCorrX and MGCX, block-permutation tests of pairwise independence between stationary time series at all lags up to a maximum M, and proves they are asymptotically valid and consistent while also estimating the lag of…","keywords":["temporal independence testing","distance correlation","multiscale graph correlation","block permutation","time series","nonparametric hypothesis testing","optimal lag estimation","universal consistency"],"falsifier":"Simulate two independent stationary AR(1) series with phi = 0.8, run DCorrX and MGCX with block permutation at $\\alpha$ = 0.05 for n = 200, 500, and 2000 with block size roughly $\\sqrt$(n); Theorem 3 predicts the rejection frequency converges to 0.05, and if it converges anywhere else the validity claim fails. To test whether the assumptions are necessary, repeat with independent long-range dependent fractional Gaussian noise with Hurst exponent H = 0.85: if the rejection frequency stays at 0.05 despite Assumption 4 failing, that condition is not doing the claimed work.","tokens_in":17537,"feed_emoji":"📈","tokens_out":8273,"duration_ms":86440,"temperature":0.7,"pith_summary":"The paper asks whether two stationary time series are related at any lag from 0 to M, without assuming linearity, normality, or multiple-test corrections. It claims that summing the distance correlation (or its multiscale variant) over the lags, weighting by (n-j)/n, and calibrating the p-value by permuting blocks of consecutive observations instead of individual points, yields an asymptotically valid and consistent test of the joint null hypothesis. The same procedure also consistently estimates the lag at which dependence is strongest. If correct, this turns any i.i.d. distance- or kernel-based dependence measure into a practical tool for temporal data, including high-dimensional nonlinear settings where classical cross-correlation and Ljung-Box tests can fail.","feed_headline":"Block permutation turns distance correlation into a temporal test","feed_subtitle":"DCorrX and MGCX catch linear or nonlinear coupling at any lag up to M, without multiple-test corrections.","key_machinery":"The central object is the block permutation null, built by cutting the Y series into consecutive blocks of size b_n, permuting the blocks, and recomputing the test statistic; unlike a full permutation, this keeps nearby observations together and imitates the serial dependence of the original series. It is coupled to cross-lag distance correlation: DCorrX(j) = DCorr(X_t, Y_{t-j}) weighted by (n-j)/n and summed over j = 0..M, while MGCX is the multiscale version that takes a smoothed maximum over local scales at each lag. Under the weak-dependence assumption, observations separated by many lags are nearly independent, so both the sample statistic converges to its population value and the block-permuted Y blocks behave approximately like independent copies of Y.","core_discovery":"On the paper's own terms, the central discovery is that block permutation can repair the invalidity of ordinary permutation tests for dependent data, so that a lag-summed distance correlation test and its multiscale version become legitimate omnibus tests of temporal independence. Under strict stationarity, finite second moments, and a uniform weak-dependence condition (Assumption 4), Theorem 3 shows the block-permutation p-value controls Type I error asymptotically, Theorem 4 shows power goes to 1 under any fixed alternative with dependence at some lag no greater than M, and Theorem 5 shows the estimated lag of maximal dependence is consistent. The proof works because block permutation preserves the short-range serial dependence of Y while making the shifted blocks asymptotically independent of X, so the permuted replicates approximate the null distribution.","pith_inferences":["Extension: the same block-permutation wrapper should calibrate any i.i.d. independence statistic that vanishes only under independence, so the validity proof likely transfers to a broader family of temporal tests beyond the two demonstrated.","Editorial inference: the theorems keep the maximum lag M fixed as n grows, so using M that grows with sample size would require new rate conditions before the omnibus p-value can be trusted; practitioners should treat M as a small, domain-chosen window.","Editorial inference: the optimal-scale pair returned by MGCX at the estimated optimal lag can serve as a screening diagnostic for functional connectivity, with scale (1,1) flagging effectively linear coupling and other scales flagging local or nonlinear coupling."],"forward_implications":["Researchers can replace separate lag-by-lag tests with Bonferroni corrections by one omnibus test for dependence within a window of M lags.","The procedure returns a consistent estimate of the lag at which dependence is strongest, so it identifies not only whether two series interact but at what temporal offset.","Because the result applies to any metric or characteristic kernel of the required type, the block-permutation construction extends distance correlation, multiscale graph correlation, and kernel measures such as HSIC to temporal data.","In high-dimensional, low-sample settings where linear cross-correlation is blind, MGCX retains power and also reports the geometric scale of the relationship, as demonstrated on brain-imaging time series."],"supporting_citations":[{"why":"Defines distance covariance and distance correlation and proves they are zero if and only if the variables are independent; supplies the base statistic DCorr.","marker":"[22]"},{"why":"Develops the multiscale graph correlation framework and the local correlation machinery whose bias and variance bounds Theorem 1 builds on.","marker":"[20]"},{"why":"Introduces MGC and documents its power in high-dimensional nonlinear settings, motivating the MGCX statistic and the optimal-scale interpretation.","marker":"[23]"},{"why":"Provides the block permutation and block bootstrap methodology for time series that the proposed testing procedure adopts.","marker":"[16]"},{"why":"Gives a kernel independence test for random processes (ShiftHSIC), the leading baseline whose instantaneous-test limitation DCorrX and MGCX address.","marker":"[1]"},{"why":"Introduces the wild bootstrap for degenerate kernel tests (WildHSIC), the multilag baseline that relies on multiple-testing corrections.","marker":"[2]"},{"why":"Establishes equivalence of distance-based and RKHS-based statistics, supporting the claim that the proposed tests are compatible with a family of dependence measures.","marker":"[17]"},{"why":"Shows the exact finite-sample equivalence of distance and kernel methods under permutation, extending the temporal procedure to kernel-based independence tests.","marker":"[18]"}],"fun_headline_variants":["Block permutation gives temporal data a valid independence test","Block permutation fixes p-value inflation in time series tests","Lag-summed distance correlation with block permutation: omnibus temporal independence","Block-permuted distance correlation tests any dependence at any lag","Block permutation rescues independence tests for temporal data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The procedure only works when the two series are stationary and their dependence decays fast enough with lag: if either series has long-range memory or a trend, the block-permuted p-values are no longer guaranteed to approximate the null distribution.","fun_headline_variants_meta":{"raw":{"variants":["Block permutation gives temporal data a valid independence test","Block permutation fixes p-value inflation in time series tests","Lag-summed distance correlation with block permutation: omnibus temporal independence","Block-permuted distance correlation tests any dependence at any lag","Block permutation rescues independence tests for temporal data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000578,"raw_usage":{"total_tokens":2672,"prompt_tokens":842,"completion_tokens":1830,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":458,"completion_tokens_details":{"reasoning_tokens":1751}},"tokens_in":458,"tokens_out":1830,"duration_ms":14865,"temperature":1.0,"reasoning_tokens":1751,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:44:23.577800+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate two independent stationary AR(1) series with phi = 0.8, run DCorrX and MGCX with block permutation at $\\alpha$ = 0.05 for n = 200, 500, and 2000 with block size roughly $\\sqrt$(n); Theorem 3 predicts the rejection frequency converges to 0.05, and if it converges anywhere else the validity claim fails. To test whether the assumptions are necessary, repeat with independent long-range dependent fractional Gaussian noise with Hurst exponent H = 0.85: if the rejection frequency stays at 0.05 despite Assumption 4 failing, that condition is not doing the claimed work.","supporting_citations":[{"cited_title":"Szekely, M","cited_arxiv_id":null,"evidence_quote":"Defines distance covariance and distance correlation and proves they are zero if and only if the variables are independent; supplies the base statistic DCorr."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the multiscale graph correlation framework and the local correlation machinery whose bias and variance bounds Theorem 1 builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces MGC and documents its power in high-dimensional nonlinear settings, motivating the MGCX statistic and the optimal-scale interpretation."},{"cited_title":"The impact of bootstrap methods on time series analysis","cited_arxiv_id":null,"evidence_quote":"Provides the block permutation and block bootstrap methodology for time series that the proposed testing procedure adopts."},{"cited_title":"A kernel independence test for random processes","cited_arxiv_id":null,"evidence_quote":"Gives a kernel independence test for random processes (ShiftHSIC), the leading baseline whose instantaneous-test limitation DCorrX and MGCX address."},{"cited_title":"A wild bootstrap for degenerate kernel tests","cited_arxiv_id":null,"evidence_quote":"Introduces the wild bootstrap for degenerate kernel tests (WildHSIC), the multilag baseline that relies on multiple-testing corrections."},{"cited_title":"Sejdinovic, B","cited_arxiv_id":null,"evidence_quote":"Establishes equivalence of distance-based and RKHS-based statistics, supporting the claim that the proposed tests are compatible with a family of dependence measures."},{"cited_title":"The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing","cited_arxiv_id":"1806.05514","evidence_quote":"Shows the exact finite-sample equivalence of distance and kernel methods under permutation, extending the temporal procedure to kernel-based independence tests."}],"review_version":1}