{"id":"d05d0748-361f-4bab-84cd-802d08471ffc","arxiv_id":"2607.03186","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Sub-sampling a CRW at rate 2^k transforms its turning-angle PDF by a known convolution (exact for rate 2 with fixed steps; approximate otherwise via a fitted tortuosity parameter p).","lead":"The paper derives exact and approximate formulas linking a correlated random walk's turning-angle and step-length distributions to those obtained after regular sub-sampling. The formulas let analysts test whether observed tracks are pure CRWs, biased walks, or multi-state processes without large-scale simulation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the paper's own openly quantified limitations.","rationale":"The paper’s strongest claim is carefully scoped: exact analytic form for fixed-length steps and r=2 (Theorem 1), high-accuracy multi-fold convolution once a single fitted p is supplied for r=2^k (Theorem 2). The isosceles-triangle and constant-p assumptions are the only non-rigorous steps, yet they are stated as such, restricted to the approximate regime, and checked a posteriori. Because the exact special case already stands alone and the residual error of the approximation is quantified, those assumptions do not undermine the central claim. The reader’s CONDITIONAL verdict already reflects precisely this residual modelling gap; no further adjustment is required.","tokens_in":30516,"tokens_out":460,"duration_ms":4940,"concrete_test":"Recompute the multi-fold convolution of Eq. (16) for k=3 (r=8) using the exact (non-isosceles) geometry obtained by sampling a CRW whose step lengths are drawn from a high-variance log-normal (e.g. µ=1, σ=1) and compare the resulting density to the p-fitted prediction; if the L2 discrepancy remains of the same order as the already-reported residual in Fig. 7, the approximation is confirmed to be non-load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags Remark 2 and the constant-p recursion of Theorem 2 / Claim 1. Those are real modelling choices: successive steps are treated as equal-length so that the composite turning angle remains a linear combination of original angles, and a single tortuosity parameter p is used at every recursion level. Both are required for the closed-form coefficients. However, the paper itself treats them as approximations, supplies the exact special case (Theorem 1, fixed step length, r=2) that needs neither, and quantifies the residual error by extensive simulation (Figs. 7–9, Appendix C). The central claim therefore already distinguishes the exact statement from the approximate one; the approximations do not silently underwrite the strongest claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies how temporal sub-sampling changes the turning-angle and step-length distributions of a discrete-time correlated random walk (CRW). For fixed step length and sampling every second point (r=2), Theorem 1 gives an exact circular density via convolution of the original turning-angle law with a rescaled double convolution; Corollary 1 gives the exact induced step-length density via the cosine rule. For sampling rates r=2^k (k≥2), Theorem 2 supplies a multi-fold convolution formula that is exact once successive steps are treated as equal length and a single tortuosity parameter p is held constant across recursion levels; p is then fitted by L2 matching to simulated histograms. The authors verify both results by large Monte-Carlo experiments, explore the sensitivity of p, and illustrate two qualitative uses in movement ecology: distinguishing biased from correlated walks and flagging multi-state trajectories.","tokens_in":30733,"tokens_out":1143,"duration_ms":21279,"significance":"The exact r=2 result (Theorem 1 and Corollary 1) is a clean, parameter-free contribution that closes a long-standing gap left by earlier simulation studies (Codling & Hill 2005; Rosser et al. 2013; Nams 2013). The geometric derivation also supplies an explicit mechanism for the artificial turn autocorrelation noted by Nams. The approximate formula for powers of two, while dependent on a fitted tortuosity parameter, is still useful for practitioners who routinely thin telemetry tracks. The two application sketches are modest but immediately actionable and do not require large-scale simulation. Overall the work strengthens the mathematical toolkit available for scale-aware analysis of animal movement and related CRW models.","major_comments":[{"comment":"§3.3 and Theorem 2: the single free parameter p is obtained by minimising L2 distance between the convolution formula and the same simulated turning-angle histograms later used for visual validation (Figs. 7–9). Once p is fixed the formula is an independent prediction, but the fitting step is not cross-validated (e.g., fit on one ensemble, score on a held-out ensemble) nor linked a priori to a measurable sinuosity index. A short cross-validation or an explicit map from Benhamou-type sinuosity to p would remove residual circularity and make the approximation claim fully predictive.","section":null},{"comment":"Remark 2 / Claim 1: the closed-form coefficients rest on treating every composite triangle as isosceles and on replacing the level-dependent sign probabilities p_j by a single constant p. Appendix C shows that step-length variance has only a small visual effect, yet no quantitative error bound (e.g., total-variation or Kolmogorov distance as a function of step-length CV and ρ) is supplied. For a math.ST audience a brief analytic or numerical bound on the approximation error would substantially strengthen Theorem 2.","section":null}],"minor_comments":[{"comment":"Abstract and Introduction: the phrase “descriptive distributions” is slightly ambiguous; “turning-angle and step-length distributions” would be clearer on first use.","section":null},{"comment":"Eq. (8) and surrounding text: the four possible sign combinations are asserted to leave the distribution unchanged by symmetry; a one-line appeal to the evenness of f_∘ would make the step fully rigorous.","section":null},{"comment":"Figure 4 caption: “y-axis has been log transformed” — state whether natural or common log and whether the density itself or only the plot scale is transformed.","section":null},{"comment":"Section 4.1.2: the Marsh–Jones classifications are cited from supplementary material of Bailey et al. (2021a); a one-sentence reminder of the numerical thresholds used would help readers who do not have that file open.","section":null},{"comment":"Appendix A, triangle distribution: the re-scaling formula for a>π is correct but the notation F(|x|>a) is non-standard; writing 2(1-F(a)) would avoid confusion.","section":null},{"comment":"Typographical: “Onenaturalproblem” (p. 2), “whichaCRWissampledateveryotherpoint” (p. 2), and several missing spaces after commas in the Introduction should be corrected in copy-editing.","section":null},{"comment":"Conjectures 1–2 are interesting but sit outside the proved results; moving them to a short “Open questions” paragraph would keep the main claims cleanly separated from speculation.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript sits at the interface of mathematical statistics and movement ecology. For a pure math.ST venue the approximate character of Theorem 2 and the fitted parameter p may feel light; for an applied probability or ecological-modelling journal the exact r=2 theory plus the practical illustrations are already a solid contribution. I see no integrity or citation-pattern concerns."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is Theorem 1: for fixed step length and any symmetric unimodal turning-angle density, the density after every-other-point sampling is exactly the wrapped convolution 4 f ◦ f^{*2}(2·). That is elementary geometry plus convolution, parameter-free, and matches 10^8-step Monte Carlo to machine precision. Theorem 2 then gives a recursive multi-fold convolution for rates 2^k that works once a single tortuosity scalar p is fixed; residual error is small and openly shown (Figs 7–9).\n\nWhat the paper does well is turn a routine data-processing step into a cheap model check. The beetle and elephant examples recover independent classifications without large simulation campaigns, and the appendices show that step-length distribution has only second-order effects on the turning-angle result. Citations to Codling & Hill, Rosser et al., and Nams are accurate; the analytic objects really are new relative to those simulation-only or restricted-assumption results.\n\nSoft spots are real but proportionate. For r>2 the isosceles-triangle assumption and the constant-p recursion are approximations; p is fitted by L2 to the same histograms later used for visual validation, so there is mild circularity. The paper already separates the exact case from the approximate one and quantifies the mismatch, so the central claim does not rest on silent error. The two conjectures at the end are left open and do not affect the theorems.\n\nThis is for movement ecologists and anyone who regularly thins telemetry tracks. The math is transparent, the simulations extensive, and the diagnostic is immediately usable. I would send it to peer review; a referee can push on general r and on a less ad-hoc treatment of p, but the core contribution is solid enough to deserve that time. Worth engaging if you work with CRWs or sampling artefacts.","headline":"Clean exact convolution for r=2 plus a usable recursive approximation for 2^k; practical diagnostic value, with the free parameter p as the main soft spot.","tokens_in":31298,"tokens_out":482,"would_cite":true,"duration_ms":5441,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G50","62H11","92D50"],"pacs":[],"model":"grok-4.5","headline":"Sub-sampling a correlated random walk maps its turning-angle density to an explicit convolution of the original density.","keywords":["correlated random walk","sub-sampling","turning angle distribution","circular convolution","step-length distribution","movement ecology","tortuosity"],"falsifier":"Generate long CRWs with highly variable step lengths (e.g. exponential or log-normal with large variance), sub-sample at rates 4 and 8, and test whether the observed turning-angle histograms still match the multi-fold convolution of Theorem 2 for the optimally fitted p; systematic mismatch would falsify the equal-step approximation.","tokens_in":31389,"feed_emoji":"🔄","tokens_out":820,"duration_ms":19233,"temperature":0.7,"pith_summary":"Correlated random walks are the default model for persistent movement, yet almost every data set is recorded or analysed at a coarser temporal scale than the true reorientation events. This paper derives the exact probability density of the new turning angles (and the accompanying step lengths) that appear after every second location is kept, for any symmetric circular turning-angle law and constant step length. For sampling rates that are higher powers of two it supplies a close multi-fold convolution once a single geometric tortuosity parameter is fixed. The resulting closed-form maps let an analyst compare an observed sub-sampled histogram with the predicted density and thereby decide, even on short tracks, whether the path is better described as pure correlation, directional bias, or multi-state behaviour. The same formulae also quantify how artificial auto-correlation is injected by the act of sampling itself.","feed_headline":"Subsampling maps CRW angles to exact convolutions","feed_subtitle":"Closed-form densities let short tracks diagnose pure correlation, bias or multi-state movement","key_machinery":"Recursive isosceles triangulation: each composite turning angle is written as a signed linear combination of the original angles by repeatedly bisecting successive triangles; the distribution then follows by ordinary convolution of the scaled densities, with signs averaged into the single geometric weight p.","core_discovery":"For a constant-step CRW whose turning angles follow any zero-centred symmetric circular density f_◦, the turning-angle density after every-other-point sampling is exactly the circular wrapping of 4 f_◦(θ) convolved with the double convolution of the scaled density f_◦(2θ). For sampling rate 2^k the same geometric decomposition yields an explicit multi-fold convolution whose coefficients are determined by a single tortuosity parameter p; the match to simulation is already high once p is fitted.","pith_inferences":["A simple regression or look-up table for the tortuosity weight p against mean resultant length and sampling rate would turn the approximation into a routine diagnostic tool.","The overlapping triangles supply the precise linear dependence that produces the artificial turn auto-correlation previously observed only in simulation.","The same path-counting graph can be re-weighted for non-dyadic sampling rates, closing the gap between theory and arbitrary telemetry frequencies."],"forward_implications":["An observed sub-sampled turning-angle peak that is sharper than the original density diagnoses bias rather than pure correlation.","A sub-sampled peak that sits between the original density and the pure-CRW prediction flags multi-state movement without needing hidden-Markov fitting.","The exact rate-2 step-length formula supplies a parameter-free check on speed statistics under down-sampling.","No non-uniform circular family is closed under the rate-2 map, so re-fitting the same parametric family after sub-sampling is formally inconsistent."],"fun_headline_variants":["Subsampling maps CRW turns to exact multi-fold convolutions","Every-other sample of CRWs yields closed-form turning densities","Sampled CRW angles equal geometric wraps of original f_◦","2^k sampling of CRWs gives p-driven multi-convolutions of angles","Subsampling links CRW step-turn densities via exact convolutions"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"Even when original step lengths vary, successive steps are treated as equal so that every composite triangle remains isosceles and the same tortuosity weight p can be used at every recursion level.","fun_headline_variants_meta":{"raw":{"variants":["Subsampling maps CRW turns to exact multi-fold convolutions","Every-other sample of CRWs yields closed-form turning densities","Sampled CRW angles equal geometric wraps of original f_◦","2^k sampling of CRWs gives p-driven multi-convolutions of angles","Subsampling links CRW step-turn densities via exact convolutions"]},"model":"grok-4.5","effort":"low","cost_usd":0.005688,"raw_usage":{"total_tokens":1488,"prompt_tokens":714,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":56880000,"prompt_tokens_details":{"text_tokens":714,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":676,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":714,"tokens_out":98,"duration_ms":6435,"temperature":1.0,"reasoning_tokens":676,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T04:15:52.969347+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Generate long CRWs with highly variable step lengths (e.g. exponential or log-normal with large variance), sub-sample at rates 4 and 8, and test whether the observed turning-angle histograms still match the multi-fold convolution of Theorem 2 for the optimally fitted p; systematic mismatch would falsify the equal-step approximation.","supporting_citations":[],"review_version":1}