{"id":"c1f19124-9846-4761-ad62-8388d0349147","arxiv_id":"2411.15822","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors introduce a Möbius-link linear-circular regression with an area-based loss to estimate the timing of daily highs and lows in cryptocurrency data.","lead":"This paper proposes a linear-circular regression model in which a Möbius transformation maps a real-valued predictor to a circle, and parameters are estimated by minimizing an area-based loss derived from the geometry of a torus. The method is applied to the timing of daily price extremes in Bitcoin and Ethereum data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The estimator is not well-defined: the loss in Eq. (3.1) uses A_C^0(ψ) based on the torus area element (1.3) with radii R and r never specified; since the loss changes with r/R, the reported estimates are unreproducible.","rationale":"The reader's weakest assumption identifies the missing torus radii R and r, and this is indeed the single most load-bearing concern about the paper's central claim. The proposed method is defined by minimizing the mean square angle error based on A_C^0, yet that function is never given in closed form nor are the geometric constants R and r specified. I independently derived the natural expression from Eq. (1.3) and found that the loss depends on r/R, not merely on an overall scale. Therefore, unless the authors supply these constants (or a specific normalization), different readers using different implicit torus geometries would obtain different parameter estimates on the same data. This blocks reproduction of all simulation and data-analysis results. Other concerns noted by the reader, such as the lack of comparison with existing models and the selection of results from 1,000 optimizations, are secondary: they affect the strength of the empirical claims but do not call into question the basic definition of the estimator. The missing geometry makes the central construction itself underspecified. The REJECT verdict is appropriate; I would not change it, hence UNCHANGED.","tokens_in":13261,"tokens_out":8985,"duration_ms":78538,"concrete_test":"Take a fixed simulated dataset (e.g., n=100, b0=0, b1=1.5, b2=0.5, errors from vM(0,1)). Using the analytic form A_C^0(ψ) = Rrψ² + r²ψ sinψ derived from Eq. (1.3), minimize Eq. (3.1) twice with the same L-BFGS-B implementation and starting values: once with (R,r)=(1,0.1) and once with (R,r)=(1,10). If the estimated (b0,b1,b2) differ by more than a small tolerance (e.g., 0.05 in b1 or b2), then the estimator depends on the unspecified torus radii and the method is not well-defined as written. If the intended A_C^0 differs from the coordinate-square form, derive the analogous expression from Eq. (1.3) and repeat the same comparison.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central methodological contribution is the area-based loss L(β0,β1) = (1/n)Σ A_C^0(ψ_i) in Eq. (3.1), where A_C^0(ψ) is the minimum area between (0,0) and (ψ,ψ) on the curved torus (1.1). The area element (1.3) is dA = r(R + r cos θ)dϕ dθ, but R and r are never assigned anywhere in the manuscript. Even under the most direct interpretation—the coordinate square [0,ψ]×[0,ψ]—one obtains A_C^0(ψ) = Rrψ² + r²ψ sinψ. The two terms have different ψ-dependence, so the minimizer of Σ A_C^0(ψ_i) changes with the ratio r/R; a common scale factor is irrelevant, but the ratio is not. Because R and r are unspecified, the loss is not a fixed function of the data, the numerical optimization in §3.1 is not reproducible, and the simulation (Tables 1–4) and real-data results (Table 5) cannot be independently verified. The citation to Biswas et al. (2024b) for A_C^0 does not remedy this: the current paper must define its own loss or explicitly state which torus geometry it inherits. This is the load-bearing assumption because every reported estimate depends on an unspecified choice of geometry.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a semi-parametric linear-circular regression model in which a linear predictor x is mapped to the unit circle by a Möbius transformation, θ = arg(M(x;β0,β1)) + ε mod 2π, and parameters are estimated by minimizing an area-based loss function derived from the intrinsic geometry of a curved torus. The loss function is defined as the mean 'square of an angle' A_C^0(ψ) between observed and fitted angles, where A_C^0 is said to be the minimum area between (0,0) and (ψ,ψ) on the torus. The paper presents simulation studies under von Mises and wrapped Cauchy errors, a comparison with existing models, and applications to Bitcoin and Ethereum data.","tokens_in":13604,"tokens_out":5327,"duration_ms":45978,"significance":"The proposed use of a Möbius transformation as a flexible link function for linear-circular regression is a reasonable idea, and the attempt to construct an area-based loss that parallels least squares is conceptually interesting. If the method were well defined and validated, it could offer a meaningful alternative to parametric circular regression models, particularly because the error distribution is left unspecified. However, the current manuscript does not achieve this: the loss function is not actually defined because the torus radii R and r are never specified, so the estimator is not reproducible and the simulation and data-analysis results cannot be verified. The paper also contains internal inconsistencies and questionable data-analysis practices that would need to be addressed. As submitted, the central methodological contribution is not sufficiently concrete to support the claims.","major_comments":[{"comment":"The loss function L(β0,β1) = (1/n)Σ A_C^0(ψ_i) is not fully specified because A_C^0(ψ) depends on the torus area element dA = r(R + r cos θ)dϕ dθ in Eq. (1.3), yet the paper never assigns values to R and r. Even under the natural interpretation of A_C^0(ψ) as the area of the coordinate square [0,ψ]×[0,ψ], the area is Rrψ² + r²ψ sinψ, which depends on the ratio r/R beyond a common scale factor; different choices of R and r yield different optimizers. Since R and r are never given, the estimator is not uniquely defined, the numerical optimization in Section 3.1 is not reproducible, and the simulation results in Tables 1–4 and the data analysis in Table 5 cannot be independently verified. A citation to Biswas et al. (2024b) does not remedy this: the current paper must either define the loss explicitly in terms of a stated torus geometry or clearly state which geometry it inherits and justify that choice.","section":"Section 3.1, Eq. (3.1); Eq. (1.3)"},{"comment":"The paper states that β1 lies in the upper half-plane, but the simulation section includes a setting with b2 = −1.8 (after Table 4), which violates this condition since β1 = b1 + ib2 is then in the lower half-plane. This inconsistency makes the parameter space ambiguous and affects the interpretation of the regression model and the simulation results. Additionally, the decomposition of the Möbius transformation into 'translation, inversion, scaling' is presented incompletely and does not correctly reflect the structure of Eq. (2.2) when β1 is complex; this should be corrected or removed.","section":"Section 2.1, Eqs. (2.1)–(2.3); Section 4 (additional simulations)"},{"comment":"The data analysis selects the best of 1,000 optimization runs based on 'minimum standard error and a reasonably good QQ-plot'. This selection procedure is data-dependent and invalidates the reported standard errors for the final estimates, because the distribution of the selected estimator under repeated optimization is not the same as the reported standard error of a single run. The paper does not adjust for this multiplicity and does not provide a principled criterion for choosing among the 1,000 results. Unless the optimization is deterministic or the selection is accounted for, the estimates in Table 5 should not be treated as valid statistical results.","section":"Section 6.3, Table 5"}],"minor_comments":[{"comment":"The caption of Table 3 lists the true parameters as (b0,b1,b2) = (0,1.7,0.5), but the column header and the text in Section 4 indicate b1 = 1.5; one of these is a typo and should be corrected.","section":"Section 4, Tables 3 caption"},{"comment":"The formula for the angular response is written as θ = (arg max ...)/(24 × 60) ∈ [0,2π), but the denominator gives a value in [0,1], not radians; the intended mapping to [0,2π) is unclear and the formula should be rewritten.","section":"Section 6.2, formula for θ"},{"comment":"The paper relies on Biswas et al. (2024b) for the definition of A_C^0(θ) but does not reproduce or restate the definition; given that the paper's main contribution is the use of this quantity, the definition should be made self-contained.","section":"Section 1, Eq. (1.1)"},{"comment":"The statement 'if |β1| → 0 then arg[M(x;β0,β1)] → 0 (radian)' is not quite correct: when β1 → 0, the numerator and denominator both tend to x, so the argument tends to arg(β0) rather than 0; the claim should be fixed or qualified.","section":"Section 2.1, after Eq. (2.2)"},{"comment":"There are numerous typographical errors, including 'M ¨obius' with inconsistent spacing, 'Resudual' in Figure 3, and incomplete reference entries (e.g., Brown and Churchill lack the edition/year details, Lund et al. has an unusual volume/page string). A careful proofread is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The core problem is that the loss function is not defined because the torus radii are not specified; this is a fundamental issue with the method, not a presentation issue. Even if the authors later specify R and r, the simulation and data-analysis results would need to be recomputed and the sensitivity to the geometry examined, which is a substantial revision rather than a minor one. The paper also has inconsistencies in the parameter space and questionable practices in the data analysis. I cannot recommend acceptance or even a minor revision in the current state. The instructions say to reserve reject for load-bearing errors that cannot be fixed within the manuscript's scope; here the missing definition of the loss is load-bearing and the paper's claims cannot be evaluated without it. I see no alternative to rejection, though the underlying model idea is not without merit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [Name],\n\nThis paper's core idea—using a Möbius link for linear-circular regression with an area-based loss—is worth a look, but the loss function is never actually specified. The torus radii R and r appear in the area element and then vanish from the paper, leaving the estimator undefined.\n\nWhat's new: the combination of Möbius transformations (already used in circular regression by McCullagh, Downs, etc.) with an area-based loss from the authors' previous work. That specific pairing for linear-circular responses is new. The simulations are internally consistent and show parameter recovery when the sample is large and the error is concentrated. The Bitcoin/Ethereum application is a reasonable illustration.\n\nWhat's wrong: the loss in Eq. (3.1) uses A_C^0(ψ), which depends on R and r, but the paper never assigns them. Under the most direct reading, A_C^0(ψ) = Rrψ² + r²ψ sinψ, and the minimizer changes with r/R. So no one can reproduce Tables 1–5 without knowing the chosen geometry. That's not a nitpick; it's load-bearing. The citation to Biswas et al. (2024b) doesn't help because that paper doesn't fix the current one's geometry.\n\nThere are smaller problems. The claim that arg(M(x;β0,β1))→0 as |β1|→0 is false unless β0=1. The data analysis picks the best of 1,000 optimizations by examining standard errors and QQ-plots, which is cherry-picking. And despite the section title, there's no numerical comparison with Fisher-Lee or Kim-SenGupta. The table captions also have inconsistencies.\n\nIf the authors specify R and r (or reparameterize the loss to be scale-invariant), add at least a consistency argument, and stop selecting optimizations by outcome, the paper could become a modest but useful contribution. As it stands, the method is undefined and the results are unreproducible.\n\nThis paper is for someone in circular statistics who wants to see the idea, but they'll be frustrated. I wouldn't cite it until it's fixed. A desk reject is defensible, but if the editor is lenient, a \"major revision\" asking for the missing geometry would also be reasonable. I lean toward sending it back for revision rather than outright rejection, because the concept has some merit.\n\nRecommendation: desk reject or major revision with a clear request for the torus geometry.","headline":"A Möbius-area-loss regression idea that can't be evaluated because the torus radii R and r are never specified, leaving the estimator undefined and the results unreproducible.","tokens_in":14111,"tokens_out":4815,"would_cite":false,"duration_ms":40091,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","62J02"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Möbius link function and a torus-area loss give distribution-free linear-circular regression.","keywords":["linear-circular regression","Möbius transformation","angular data","torus area element","semi-parametric regression","mean square angle error","cryptocurrency","bootstrap intervals"],"falsifier":"For a fixed simulated dataset, compute the loss $L$ under two different choices of torus radii, say $R=r=1$ and $R=2,r=1$, and check whether the minimizers $\\hat{\\beta}_0,\\hat{\\beta}_1$ differ; if they do, the method as stated is not reproducible. A second check is whether the paper's simulation results can be reproduced only after fixing $R$ and $r$ to some hidden values.","tokens_in":13037,"feed_emoji":"📈","tokens_out":5477,"duration_ms":42580,"temperature":0.7,"pith_summary":"The paper proposes a regression model for an angular response θ given a real predictor x, with the regression curve built from a Möbius transformation that maps the real line onto the unit circle: θ = arg(M(x;β0,β1)) + ε mod 2π. The parameters are estimated by minimizing a mean square angle error in which the loss is the area swept out between two angles on a curved torus, a geometric stand-in for the usual squared residual. The angular error ε is only assumed to have zero mean direction and fixed variance, so the model is semi-parametric and does not commit to a von Mises or wrapped Cauchy distribution. Through simulations the paper shows the estimates converge to the true β0 and β1 under both error families, and it illustrates the method on the timing of daily high prices for Bitcoin and Ethereum.","feed_headline":"Torus-area loss makes circular regression distribution-free","feed_subtitle":"A semi-parametric model with a geometric loss recovers true parameters and reads Bitcoin and Ethereum price timing.","key_machinery":"The load-bearing object is the Möbius transformation $M(x;\\beta_0,\\beta_1)=\\beta_0\\frac{x-\\beta_1}{x-\\bar{\\beta}_1}$, which maps the real predictor axis onto the unit circle and whose argument gives the regression curve $g(x;\\beta_0,\\beta_1)$. The loss is built from $A_C^{(0)}$, defined as the minimum area between $(0,0)$ and $(\\theta,\\theta)$ on the curved torus with area element $dA=r(R+r\\cos\\theta)\\,d\\phi\\,d\\theta$, giving a quadratic-like penalty on angular residuals. The minimization is carried out numerically with a bounded L-BFGS-B optimizer, and bootstrap resampling of the angular residuals supplies confidence and prediction intervals.","core_discovery":"The central claim is that a loss function built from the intrinsic area of a curved torus, $A_C^{(0)}$, provides a well-behaved, distribution-free estimator for linear-circular regression when the link is the Möbius map $M(x;\\beta_0,\\beta_1)=\\beta_0\\frac{x-\\beta_1}{x-\\bar{\\beta}_1}$. Writing the parameters as $\\beta_0=e^{ib_0}$ and $\\beta_1=b_1+ib_2$, the paper argues that minimizing $L=\\frac{1}{n}\\sum_{i=1}^n A_C^{(0)}((\\theta_i-g(x_i;\\beta_0,\\beta_1))\\bmod 2\\pi)$ recovers the true parameters with small standard error for sample sizes from $n=50$ to $n=500$, with residuals that behave like a von Mises distribution even though no such parametric error family was imposed. The demonstration on cryptocurrency high-price timestamps, using the ratio of low to high price divided by the close-open spread as a linear predictor, yields residuals that pass a Watson test and align in a QQ plot.","pith_inferences":["The paper leaves the torus radii $R$ and $r$ unspecified; a natural inference is that the loss inherits a scale factor from these constants, so the estimator might be invariant to their ratio, but this is not shown and a concrete choice of $R$ and $r$ is needed to make the method reproducible.","The method could be tested on other circular regressors such as wind direction or protein dihedral angles, where the Möbius link's boundary-mapping property is equally natural.","A fully parametric version that couples the area loss with a von Mises error would let a user compare likelihood-based and geometric estimators on the same data.","The predictor $x = (\\text{low}/\\text{high})/(\\text{close}-\\text{open})$ used in the data analysis is ad hoc; an implication is that the method's practical value depends on choosing an informative linear feature, which the paper does not address."],"forward_implications":["If the claim holds, angular responses can be regressed on continuous predictors without choosing a circular error family, eliminating a major source of misspecification.","The torus-area loss gives a concrete geometric meaning to 'squared angular error', and the resulting estimator should be consistent as $n$ grows under the zero-mean, fixed-variance error condition.","The same loss can be reused for other Möbius-link regressions, including circular-circular settings, by substituting the appropriate map.","Bootstrap confidence and prediction intervals computed from residual resampling will have correct coverage if the angular errors are exchangeable.","In applied finance, the model offers a way to study the timing of intraday extremes as a function of price-derived variables."],"supporting_citations":[{"why":"Defines the 'square of an angle' $A_C^{(0)}$ as the minimum torus area between two angles, which the paper adopts as the loss.","marker":"Biswas et al. (2024b)"},{"why":"Establishes the Möbius transformation as a tool in directional statistics, motivating the link function.","marker":"McCullagh (1996)"},{"why":"Provides a baseline tangent-link linear-circular regression that the proposed model compares against.","marker":"Fisher and Lee (1992)"},{"why":"Another baseline model using stereographic projection, compared in Section 5.","marker":"Kim and SenGupta (2015)"},{"why":"Applies Möbius transformations as a link in circular-circular regression, a direct predecessor.","marker":"Jha and Biswas (2018)"},{"why":"Earlier circular regression via Möbius-type maps, cited as background for the link.","marker":"Downs and Mardia (2002)"}],"fun_headline_variants":["Torus-area loss frees circular regression from distributions","Semi-parametric circular regression via Möbius map and area loss","No distribution assumptions: torus-area regression for angles","Geometric loss on torus gives distribution-free angular fit","Möbius transformation + torus area = distribution-free circular regression"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The loss function $A_C^{(0)}$ is defined through a torus with radii $R$ and $r$, but the paper never assigns these values, so the estimator is not uniquely specified until that geometry is fixed.","fun_headline_variants_meta":{"raw":{"variants":["Torus-area loss frees circular regression from distributions","Semi-parametric circular regression via Möbius map and area loss","No distribution assumptions: torus-area regression for angles","Geometric loss on torus gives distribution-free angular fit","Möbius transformation + torus area = distribution-free circular regression"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1813,"prompt_tokens":908,"completion_tokens":905,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":524,"completion_tokens_details":{"reasoning_tokens":819}},"tokens_in":524,"tokens_out":905,"duration_ms":7849,"temperature":1.0,"reasoning_tokens":819,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:52:02.198602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed simulated dataset, compute the loss $L$ under two different choices of torus radii, say $R=r=1$ and $R=2,r=1$, and check whether the minimizers $\\hat{\\beta}_0,\\hat{\\beta}_1$ differ; if they do, the method as stated is not reproducible. A second check is whether the paper's simulation results can be reproduced only after fixing $R$ and $r$ to some hidden values.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Möbius transformation as a tool in directional statistics, motivating the link function."},{"cited_title":"and SenGupta, A","cited_arxiv_id":null,"evidence_quote":"Another baseline model using stereographic projection, compared in Section 5."},{"cited_title":"and Biswas, A","cited_arxiv_id":null,"evidence_quote":"Applies Möbius transformations as a link in circular-circular regression, a direct predecessor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier circular regression via Möbius-type maps, cited as background for the link."}],"review_version":1}