{"id":"3adc6884-ad58-48b8-b23b-672f88763695","arxiv_id":"2506.04441","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A distribution obtained by square-rooting Dirichlet variables is corrected, characterized, and fitted to text-mining term-frequency vectors.","lead":"This paper defines the Spherical-Dirichlet Distribution, a probability model for unit vectors with positive coordinates, by taking the square root of a Dirichlet random vector, and derives its density, moments, mode, and parameter estimates. It also fits the model to simulated vectors and to term frequencies from an email text-mining sample.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The SDD density and moment algebra check out; the load-bearing weakness is that the text-mining application supplies no goodness-of-fit evidence, so the central 'useful model' claim is not actually supported.","rationale":"The mathematical core is sound. I re-derived the change of variables z_i=x_i^2 on S^{p-1}_+: for p=2, the surface element is dtheta=dz/(2*x1*x2), and in general domega=dz/(2^{p-1}*prod x_i), which gives exactly the density in Eq. (3). The moments, including E(x_i^2)=alpha_i/alpha_0 and E(x_i*x_j)=mu_i*mu_j/alpha_0 for i not equal j, follow from Dirichlet moment identities, and the covariance simplification checks out algebraically. The mode formula with the stated condition alpha_i>1/2 is the interior stationary point; boundary cases are not fully characterized but are outside the stated condition. The load-bearing weakness is therefore not in the derivation but in the application. The real-data section is an exercise in parameter estimation on one small, privately held sample, with no comparison or goodness-of-fit. Because the distribution is a one-to-one transform of Dirichlet, a direct test of the implied Dirichlet model on z=x^2 is feasible and would settle whether the practical claim holds. The reader's conditional verdict already requires such validation, so I do not move the verdict.","tokens_in":8905,"tokens_out":13639,"duration_ms":119545,"concrete_test":"Run a goodness-of-fit test on the same 160 text vectors: transform to z_i=x_i^2, fit Dirichlet(beta) by MLE with beta_i=alpha_hat_i+1/2, and compute a Pearson chi-square statistic on a fixed partition of the simplex, obtaining a parametric bootstrap p-value under the fitted Dirichlet. If the bootstrap p-value is below 0.05, the real-data example does not establish the SDD as useful for text data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 'Text Mining Example' is the only real-data evidence for the paper's central applied claim that the SDD is a useful model for text vectors. The section reports that MOM and MLE converge to similar estimates after the ad hoc transformation x=ln(1.10+w) with normalization, but it gives no baseline, goodness-of-fit test, or uncertainty quantification. Agreement between two estimators is not evidence of adequacy of the model; both can converge to a misspecified fit. Since the SDD is literally the Dirichlet distribution under z_i=x_i^2, the model can and should be checked by asking whether the transformed squared vectors follow Dirichlet with parameters alpha_hat+1/2. Without such a check, the real-data example supports only the internal consistency of the estimation procedures, not the distributional assumption on which the usefulness claim rests. In addition, several estimated alpha_i in Table 3 are below 1/2, so the interior mode formula in the section on the mode is not applicable to the fitted application; this is not a contradiction, but it underscores that the paper's practical claims do not exercise its theoretical mode result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript corrects and extends the author's earlier treatment of the Spherical-Dirichlet Distribution (SDD), defined by taking a Dirichlet random vector z on the simplex and setting x_i = sqrt(z_i), so that x lies in the positive orthant of the unit sphere. The paper derives the SDD density with respect to surface measure, computes first and second moments, variance, covariance, mode, and some limiting cases, and compares the SDD with the von Mises and Fisher-Bingham distributions. It then proposes method-of-moments (MOM) and maximum-likelihood (MLE) estimators, with an L-BFGS-B implementation, and reports a simulation study and a text-mining application using transformed term-frequency vectors. The abstract and introduction claim that the SDD is a useful new model for positive-orthant directional data in text mining and gene-expression analysis.","tokens_in":9109,"tokens_out":11904,"duration_ms":105016,"significance":"The theoretical core of the paper is largely sound and useful. The moment formulas in Eqs. (16), (20), and (21) are transparent, are derived by kernel recognition, and agree with the corresponding Dirichlet moments after the transformation z_i = x_i^2, which is a genuine check on the normalizing constant. The simulation study is a useful sanity check for the proposed MOM and MLE procedures, and the correction of an earlier published error is of value to users of this distribution. However, the paper's applied claim that the SDD is a useful model for text data is not supported by the real-data section, which contains no goodness-of-fit assessment or baseline comparison. The manuscript therefore needs revision before the applied claims can be accepted.","major_comments":[{"comment":"The intermediate display f_SDir(x;alpha) = f_Dir(x^2) 2^{p-1} / sqrt(x_1^2 ... x_p^2) has the surface-measure factor inverted. Combining Eq. (2) as d omega = dz / (2^{p-1} sqrt(z_1...z_p)) with z_i = x_i^2 gives dz = 2^{p-1} sqrt(z_1...z_p) d omega = 2^{p-1}(x_1...x_p) d omega, so the SDD density must be f_SDir = f_Dir(x^2) 2^{p-1}(x_1...x_p), not f_Dir(x^2) 2^{p-1}/(x_1...x_p). The printed denominator would produce exponents 2 alpha_i - 3 and would contradict the correctly normalized density in Eq. (3). Please correct this display and state explicitly that dz in Eq. (2) is the (p-1)-dimensional Lebesgue measure on the simplex.","section":"§Basic Properties, display preceding Eq. (3)"},{"comment":"The text-mining section reports that MOM and MLE produce similar alpha estimates, but this does not provide evidence that the SDD actually fits the transformed normalized term-frequency vectors. Because the SDD is equivalent to a Dirichlet distribution on z = x^2, a natural and inexpensive check would be to test whether the squared, normalized data follow Dirichlet(alpha + 1/2), using marginal beta QQ plots, second-moment comparisons, or a likelihood-ratio comparison against a more flexible model. Without such a diagnostic, the close MOM/MLE agreement only shows internal consistency of the two estimation algorithms, not the distributional adequacy on which the paper's applied claim rests. The ad hoc transformation x_transf = ln(1.10 + x) also needs justification and some sensitivity analysis, since the conclusion may depend on the choice 1.10.","section":"§Text Mining Example, Table 3"},{"comment":"The interior-mode formula (28) is derived under the condition alpha_i > 1/2 for all i, and when any alpha_i <= 1/2 the mode should be analyzed on the boundary of the positive orthant. The text-mining estimates in Table 3 include several alpha_i below 1/2, so the fitted model in the application lies outside the regime where Eq. (28) applies. The paper should either restrict the mode discussion to the interior case and note its inapplicability to the fitted model, or develop the boundary-mode analysis. This is not a contradiction, but it is a gap between the theoretical section and the application.","section":"§Mode, Eq. (28)"}],"minor_comments":[{"comment":"The measure dz in Eq. (2) is not defined; please state that it is the flat Lebesgue measure on the (p-1)-dimensional simplex and add a reference or a short proof for the surface-measure formula, since it is load-bearing for the density derivation.","section":"§Probability Density Function and Normalizing Constant, Eq. (2)"},{"comment":"The phrase \"percentage error based on vector norm ratios\" is not defined; please specify the formula used to compute the reported percentage errors.","section":"§Simulation Example, Table 1"},{"comment":"Equation (35) appears to contain typesetting or algebraic errors: the term (1 - mu_alpha^2 / alpha) is dimensionally inconsistent with the variance expression (17), and the following display does not clearly follow. Please rewrite the symmetric covariance matrix carefully.","section":"§Limiting Behavior, Eq. (35)"},{"comment":"The abstract calls the SDD a \"novel probability distribution,\" but the paper is a correction and extension of Guardiola (2020); please phrase the novelty claim as a corrected and extended treatment rather than a fully new distribution.","section":"§Abstract and Introduction"},{"comment":"The section title promises a relationship between mode and mean, but the only result is an asymptotic equality as alpha tends to infinity; consider renaming the section or adding a finite-sample comparison for the symmetric case.","section":"§Mode and Relationship with the Mean"}],"recommendation":"major_revision","confidential_remarks":"The theoretical contribution is sound and the correction of the earlier error is valuable, but the real-data evidence is too thin to support the paper's applied claims. I would not require a completely new application, but the authors should add at least a minimal goodness-of-fit diagnostic for the text-mining example and correct the inverted surface-measure display before Eq. (3). If the journal is willing to treat the paper primarily as a correction and theoretical note, the bar for the application section could be correspondingly lower."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper's math is mostly right, but its claim to introduce a new distribution is a stretch, and the real-data application doesn't test the model. It's a careful, self-contained derivation of the density and moments for a Dirichlet distribution under the change of variables x_i = sqrt(z_i). That's the whole story.\n\nWhat's genuinely useful: Guardiola corrects his earlier published error, gives clean formulas for the normalizing constant, moments up to order two, covariance, mode, and MLE/MOM equations. I checked the moments against the Dirichlet results; E(x_i^2) = alpha_i/alpha_0 and the cross-moment formula mu_i mu_j / alpha_0 are correct. The derivation via surface measure (cited to Gupta) is the one non-obvious step, but it lands correctly. For readers working with positive-orthant unit vectors, this is a tidy reference.\n\nThe soft spots are in the framing and the applications. The SDD is not a new distribution; it is exactly the Dirichlet distribution reparameterized on squared coordinates. The paper's own equations show this—every result reduces to the corresponding Dirichlet moment. Calling it novel overstates the contribution. The text-mining example is the bigger issue: it compares MOM and MLE estimates and calls that validation. Agreement between two estimators is not evidence that the SDD fits the data. There is no goodness-of-fit test, no baseline, no standard errors. Since the model is just Dirichlet on squares, the author could have checked whether the squared data follow Dirichlet with adjusted parameters, but didn't. Also, several fitted alphas in Table 3 are below 1/2, so the mode formula derived earlier doesn't apply to the actual fit. That's not a contradiction, but it makes the application even thinner. No code or data is shipped, and the data availability statement is just \"available upon request.\"\n\nWho is this for? Someone who needs the corrected formulas and estimators for a Dirichlet-on-the-sphere model, or a referee who wants to see whether the 2020 error is fixed. As a standalone contribution it's modest, but the math is solid enough that it deserves a real peer review. I'd ask the author to add a genuine model check and a baseline comparison before publication.\n\nRecommendation: send it to review, but expect major revisions on the applied sections.","headline":"Solid corrected math for a known object, but the applied claims outrun the evidence.","tokens_in":9628,"tokens_out":2096,"would_cite":false,"duration_ms":19538,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H11","60E05","62F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the Spherical-Dirichlet Distribution, built as the square-root of a Dirichlet vector, correctly models unit vectors on the positive orthant of the hypersphere, with closed-form moments and mode and workable MLE and…","keywords":["spherical Dirichlet distribution","Dirichlet distribution","text mining","hypersphere","gene expressions","positive orthant","method of moments","maximum likelihood estimation"],"falsifier":"Integrate the claimed density over the positive orthant of $\\mathbb{S}^{p-1}$ for a concrete case, e.g. $p=3$, $\\alpha=(2,2,2)$, using the standard spherical surface element; if the result is not exactly 1 and the second moment does not reproduce $\\alpha_i/\\alpha_0$ for a simulated sample of vectors drawn uniformly from the orthant with $\\alpha_i=1/2$, the measure factor in the density is mistaken.","tokens_in":8693,"feed_emoji":"📐","tokens_out":6191,"duration_ms":50799,"temperature":0.7,"pith_summary":"This paper claims that the spherical Dirichlet distribution, obtained by taking square roots of a Dirichlet random vector, is a valid and useful model for unit vectors restricted to the positive orthant of the hypersphere. The paper corrects a technical error in earlier work, derives the density with its normalizing constant, gives closed-form expressions for the first two moments, the mode, and variances and covariances, and develops method-of-moments and maximum-likelihood estimators. It then shows the model fits simulated data and a real text-mining sample of email term-frequency vectors. If the claims hold, the distribution gives a simple alternative to mixtures and to whole-sphere models for text and gene-expression data, with no probability mass outside the feasible region.","feed_headline":"New distribution fits text vectors on the sphere's positive orthant","feed_subtitle":"Corrected spherical Dirichlet density, moments, and estimators match simulated and real email data.","key_machinery":"The central object is the square-root map $x_i=\\sqrt{z_i}$ from the $(p-1)$-dimensional Dirichlet simplex to the positive orthant of the unit sphere, together with the surface-measure element $d\\omega_{p-1}(x)=dz/(2^{p-1}\\sqrt{z_1\\cdots z_p})$ supplied by the paper's reference [3]. This factor, not the full Jacobian of the $p$-dimensional transformation, determines the density's $2^{p-1}\\prod_i x_i$ term and all subsequent moment and mode formulas. The inference machinery is the exponential-family form of the log-likelihood with sufficient statistics $\\sum_i\\log x_{ij}$, leading to moment and likelihood equations solved numerically by L-BFGS-B.","core_discovery":"The central claim is that the distribution with density $f_{\\mathrm{SDir}}(x;\\alpha) = \\frac{2^{p-1}\\Gamma(\\alpha_0)}{\\prod_{i=1}^p \\Gamma(\\alpha_i)}\\prod_{i=1}^p x_i^{2\\alpha_i-1}$ on the positive orthant of the unit sphere $\\mathbb{S}^{p-1}_+$, obtained from $x_i=\\sqrt{z_i}$ where $z$ is Dirichlet, is correctly normalized. The paper further claims that its first moment is $E(x_i)=\\Gamma(\\alpha_i+\\tfrac12)\\Gamma(\\alpha_0)/(\\Gamma(\\alpha_i)\\Gamma(\\alpha_0+\\tfrac12))$, its second moment is $E(x_i^2)=\\alpha_i/\\alpha_0$, and its mode is $\\sqrt{(2\\alpha_i-1)/(2\\alpha_0-p)}$, and that these formulas make the distribution a practical parametric family for vectors on the positive orthant. The corrected surface-measure factor $2^{p-1}\\prod x_i$ is what makes the density integrate to one.","pith_inferences":["A direct testable extension: for any sample of non-negative vectors, the SDD's fit could be checked by verifying that the sample mean of squared coordinates approximates $\\hat\\alpha_i/\\hat\\alpha_0$, a diagnostic the paper does not explicitly propose.","The same square-root transformation could be applied to other simplex-based models (e.g., logistic-normal or generalized Dirichlet) to generate new sphere-supported distributions with the same surface-measure correction.","The paper's log-shift transformation $\\ln(1.10+x)$ for zero counts is an ad hoc step; a dedicated zero-inflated SDD or a Bayesian treatment of zeros would make the model directly applicable to sparse term-frequency matrices.","Because the surface measure formula is imported without derivation, a careful check in the literature on the induced measure of the square-root map would settle whether the density's factor is $2^{p-1}$ or something else; the editorial instinct is to verify this before using the moments."],"forward_implications":["Text-mining and gene-expression data, once normalized to unit length, can be modeled with a single parametric distribution that assigns zero mass outside the positive orthant.","The MLE system solves to accurate estimates, with the reported simulations showing MLE converging in tens of iterations and lower error than MOM.","Setting all $\\alpha_i=1/2$ recovers the uniform distribution on the positive orthant, giving a baseline model with no fitted parameters.","As the common $\\alpha$ grows, the SDD concentrates to a point mass at the mean direction rather than approaching a von Mises or normal distribution on the sphere.","Because $E(x_i^2)=\\alpha_i/\\alpha_0$, the second empirical moments directly give a quick method-of-moments estimate of the parameter proportions."],"supporting_citations":[{"why":"Supplies the surface-measure element $d\\omega_{p-1}(x)=dz/(2^{p-1}\\sqrt{z_1\\cdots z_p})$ that yields the density's $2^{p-1}\\prod x_i$ factor.","marker":"[3]"},{"why":"Provides the gamma-function quotient limit used to show that mode and mean coincide asymptotically.","marker":"[4]"},{"why":"Gives the method-of-moments estimation approach adapted for the SDD.","marker":"[8]"},{"why":"Supplies the email text-mining dataset used in the real-data application.","marker":"[9]"},{"why":"Provides the standard Dirichlet density and its parameterization used in the transformation.","marker":"[2]"}],"fun_headline_variants":["Corrected spherical Dirichlet for positive-orthant vectors","Fixing the spherical Dirichlet: moments and estimators","Spherical Dirichlet, corrected: fits text and gene data","New density for sphere's positive orthant, now correct","Positive-orthant vectors: a corrected Dirichlet fit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the surface-measure formula $d\\omega_{p-1}(x)=dz/(2^{p-1}\\sqrt{z_1\\cdots z_p})$, stated without proof, is correct; if that factor is wrong, every derived density, moment, mode, and estimator shifts.","fun_headline_variants_meta":{"raw":{"variants":["Corrected spherical Dirichlet for positive-orthant vectors","Fixing the spherical Dirichlet: moments and estimators","Spherical Dirichlet, corrected: fits text and gene data","New density for sphere's positive orthant, now correct","Positive-orthant vectors: a corrected Dirichlet fit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000156,"raw_usage":{"total_tokens":1215,"prompt_tokens":942,"completion_tokens":273,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":192}},"tokens_in":558,"tokens_out":273,"duration_ms":3058,"temperature":1.0,"reasoning_tokens":192,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:43:50.535448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate the claimed density over the positive orthant of $\\mathbb{S}^{p-1}$ for a concrete case, e.g. $p=3$, $\\alpha=(2,2,2)$, using the standard spherical surface element; if the result is not exactly 1 and the second moment does not reproduce $\\alpha_i/\\alpha_0$ for a simulated sample of vectors drawn uniformly from the orthant with $\\alpha_i=1/2$, the measure factor in the density is mistaken.","supporting_citations":[{"cited_title":"Chapman and Hall/CRC, Boca Raton (2000) Guardiola Page 15 of 15","cited_arxiv_id":null,"evidence_quote":"Supplies the surface-measure element $d\\omega_{p-1}(x)=dz/(2^{p-1}\\sqrt{z_1\\cdots z_p})$ that yields the density's $2^{p-1}\\prod x_i$ factor."},{"cited_title":"The American Mathematical Monthly56(8), 529–535 (1949)","cited_arxiv_id":null,"evidence_quote":"Provides the gamma-function quotient limit used to show that mode and mean coincide asymptotically."},{"cited_title":"Computers and Mathematics with Applications24(10), 11–17 (1992)","cited_arxiv_id":null,"evidence_quote":"Gives the method-of-moments estimation approach adapted for the SDD."},{"cited_title":"https://www.cs.cmu.edu/afs/cs/project/theo-20/www/data/news20.html Accessed 2019-09-01","cited_arxiv_id":null,"evidence_quote":"Supplies the email text-mining dataset used in the real-data application."}],"review_version":1}