--- title: Sine-Skewed Models for Circular and Toroidal Data url: https://www.emergentmind.com/topics/sine-skewed-models type: topic --- # Sine-Skewed Models for Circular and Toroidal Data Searching arXiv for recent and foundational papers on sine-skewed circular and toroidal distributions. Sine-skewed models are circular and toroidal probability distributions obtained by perturbing a pointwise symmetric base density with a sine term that shifts mass directionally while preserving the original support. On the circle, a standard construction takes \[ f(\theta)=f_0(\theta)\bigl(1+2\lambda \sin\theta\bigr), \qquad -\pi<\theta<\pi,\quad -1<\lambda<1, \] whereas on the torus \(\mathbb{T}^d=[-\pi,\pi)^d\) one uses \[ g(\boldsymbol{x}-\boldsymbol{\mu};\boldsymbol{\vartheta},\boldsymbol{\lambda}) = f(\boldsymbol{x}-\boldsymbol{\mu};\boldsymbol{\vartheta}) \left( 1+\sum_{s=1}^{d}\lambda_s \sin(x_s-\mu_s) \right), \] with \(\boldsymbol{\lambda}\in[-1,1]^d\) and \(\sum_{s=1}^d |\lambda_s|\le 1\). These constructions belong to the literature on skew-symmetric distributions and directional statistics, and they are used to introduce asymmetry into baseline circular or toroidal laws such as the von Mises, cardioid, Sine, Cosine, and wrapped Cauchy families [1902.02579, 1910.13293]. ## 1. General construction and structural principles The basic sine-skewing mechanism starts from a symmetric density and multiplies it by a linear sine tilt. In the univariate circular setting, the construction in the sine-skewed von Mises study is explicitly described as “slightly different” from Abe and Pewsey (2011), but it retains the same general sine-skewing mechanism. The essential point is that the perturbation is odd, so it introduces asymmetry without changing the support \((-\pi,\pi)\), and when the skewness parameter vanishes the model reduces exactly to its symmetric parent [1902.02579]. In the toroidal setting, the construction is formulated for any pointwise symmetric density centered at \(\boldsymbol{\mu}\). The constraint \(\sum_{s=1}^d |\lambda_s|\le 1\) guarantees \[ 1+\sum_{s=1}^{d}\lambda_s \sin(x_s-\mu_s)\ge 0 \] for all \(\boldsymbol{x}\), so the skewed density is nonnegative. The same paper shows that the density integrates to 1 over \(\mathbb{T}^d\), and that the original normalizing constant of the base model is unchanged. This “no new normalizing constant” property is a central reason the construction is computationally attractive on the torus, where normalizing constants are often difficult to evaluate [1910.13293]. The mechanism is also structurally transparent in moment space. Because multiplication by a sine factor shifts Fourier or trigonometric components by one order, the skewing term alters directional moments in a controlled way rather than replacing the baseline model outright. This makes sine-skewed families parsimonious: in the examples treated in the literature here, concentration or dependence remains governed by the parent family, while \(\lambda\) or \(\boldsymbol{\lambda}\) governs asymmetry. ## 2. Canonical circular models Two representative one-dimensional models are the sine-skewed von Mises distribution and the sine-skewed cardioid distribution [1902.02579, 1811.10603]. | Model | Density | Distinctive point | |---|---|---| | Sine-skewed von Mises \(vM(\theta,\lambda,k)\) | \(\dfrac{e^{k\cos\theta}}{2\pi I_0(k)}\bigl(1+2\lambda\sin\theta\bigr)\) | Reduces to standard von Mises when \(\lambda=0\) | | Sine-skewed cardioid \(\text{ssc}(\lambda,\rho)\) | \(\dfrac{1}{2\pi}(1+\lambda\sin x)(1+\rho\cos x)\,1_{[-\pi,\pi]}(x)\) | Two-parameter circular law on \([-\pi,\pi]\) | For the sine-skewed von Mises model, the symmetric baseline is \[ f_0(\theta)=\frac{e^{k\cos\theta}}{2\pi I_0(k)}, \qquad -\pi\le \theta \le \pi,\quad k>0, \] so the skewed density becomes \[ f_{\text{svm}}(\theta;\lambda,k)=\frac{e^{k\cos\theta}}{2\pi I_0(k)}\bigl(1+2\lambda \sin\theta\bigr), \qquad -\pi<\theta<\pi,\quad -1<\lambda<1,\quad k\ge 0. \] The Bessel-series identity \[ e^{k\cos\theta}=I_0(k)+2\sum_{j=1}^{\infty} I_j(k)\cos(j\theta) \] makes the perturbation explicit: \[ f_{\text{svm}}(\theta;\lambda,k) =\frac{1}{2\pi I_0(k)} \left(I_0(k)+2\sum_{j=1}^{\infty} I_j(k)\cos(j\theta)\right) (1+2\lambda \sin\theta). \] When \(\lambda\neq 0\), the density is asymmetric; positive and negative \(\lambda\) induce opposite directional tilts, and if the mode for \(vM(\theta,\lambda,k)\) is at \(\theta_0\), then the mode for \(vM(\theta,-\lambda,k)\) is at \(-\theta_0\) [1902.02579]. The sine-skewed cardioid distribution is defined by \[ f(x)=\frac{1}{2\pi}(1+\lambda\sin x)(1+\rho\cos x)\,1_{[-\pi,\pi]}(x), \] or, after expansion, \[ f(x)=\frac{1}{2\pi}\left(1+\lambda\sin x+\rho\cos x+\frac{\lambda\rho}{2}\sin 2x\right), \qquad x\in[-\pi,\pi]. \] Its cumulative distribution function is explicit: \[ F(x)=\frac{1}{2}+\frac{1}{2\pi}\left(x-\lambda(\cos x+1)+\rho\sin x-\frac{\lambda\rho}{4}(\cos 2x-1)\right),\qquad x\in[-\pi,\pi]. \] Because the support is bounded above at \(\pi\), the paper derives endpoint expansions and concludes that the original cdf \(F\) lies in the Weibull domain \(D(H_{-1})\), while the transformed upper-tail law \(F^\star(x)=F(\pi-1/x)\) belongs to the Fréchet domain \(D(H_1)\) [1811.10603]. ## 3. Moments, shape, characterization, and estimation For sine-skewed circular models, moment calculations are unusually explicit. In the sine-skewed von Mises case, the \(p\)th circular moment \[ \Phi_p = E\!\left(e^{ip\theta}\right)=\int_{-\pi}^{\pi} e^{ip\theta} f(\theta)\,d\theta \] satisfies \[ \Phi_p(k,\lambda)=\Phi_p(k)+\lambda\bigl(\Phi_{p-1}(k)-\Phi_{p+1}(k)\bigr), \] where \(\Phi_p(k)\) is the \(p\)th circular moment of the standard von Mises distribution. The same paper derives a cumulative distribution function in series form and studies modal behavior by differentiating the density. The resulting modal equation is reported as \[ k^2\sin^2\theta-2\cos\theta+k\sin\theta=0, \] and the accompanying table shows that, for fixed \(k\), the modal location increases as \(\lambda\) increases, while for fixed \(\lambda\), the mode decreases as \(k\) increases [1902.02579]. A major theoretical contribution of the same work is characterization via truncated first moments. Under absolute continuity and differentiability assumptions, the sine-skewed von Mises density is characterized by identities of the form \[ E(\theta\mid \theta\le x)=g(x)\,f(x) \qquad\text{or}\qquad E(\theta\mid \theta\ge x)=h(x)\,\frac{f(x)}{1-F(x)}. \] Differentiation leads to the first-order differential equation \[ \frac{f'(x)}{f(x)} = -k\sin x+\frac{2\cos x}{1+2\sin x}, \] which integrates to \[ f(x)=c\,e^{k\cos x}\,(1+2\sin x), \] with \(c=1/(2\pi I_0(k))\). The paper presents these characterization theorems as its main theoretical result, aimed at model identification and goodness-of-fit reasoning in circular statistics [1902.02579]. The sine-skewed cardioid paper develops a more explicitly inferential treatment. From \[ m=\mathbb{E}X=\lambda\left(1-\frac{\rho}{8}\right), \qquad m_2=\mathbb{E}X^2=\frac{\pi^2}{3}-2\rho, \] the moment estimators are \[ \hat\rho_n=\frac{\pi^2/3-m_{2,n}}{2}, \qquad \hat\lambda_n=\frac{4\bar X_n}{4-\hat\rho_n} =\frac{8\bar X_n}{8-\pi^2/3+m_{2,n}}. \] The paper proves \[ \left(\sqrt{n}(\hat\rho_n-\rho),\sqrt{n}(\hat\lambda_n-\lambda)\right)\rightsquigarrow \mathcal{N}_2(0,\Sigma), \] using the functional empirical process and a delta-method style linearization. It also gives a closed-form characteristic function, \[ \psi(t)=\frac{1}{\pi}\left(\frac{1}{t}+\frac{\lambda}{i(t^2-1)}-\frac{\rho t}{t^2-1}-\frac{\lambda\rho}{i(t^2-4)}\right)\sin(\pi t), \] for \(t\notin\{-2,-1,0,1,2\}\), and recommends numerical inversion of the explicit cdf for simulation. By contrast, its conclusion on maximum likelihood is negative: ML estimators are “not well-defined in the usual way” for this model, so the paper emphasizes moment estimation instead [1811.10603]. ## 4. Toroidal sine-skewing and protein bioinformatics The multivariate toroidal construction extends the same idea from the circle to \(\mathbb{T}^d\). Starting from a symmetric toroidal density \(f(\boldsymbol{x}-\boldsymbol{\mu};\boldsymbol{\vartheta})\), the skewed version \[ g(\boldsymbol{x}-\boldsymbol{\mu};\boldsymbol{\vartheta},\boldsymbol{\lambda}) = f(\boldsymbol{x}-\boldsymbol{\mu};\boldsymbol{\vartheta}) \left( 1+\sum_{s=1}^{d}\lambda_s \sin(x_s-\mu_s) \right) \] is properly normalized without changing the base constant, and when \(\boldsymbol{\lambda}=\mathbf{0}\) it reduces exactly to the symmetric antecedent. The cdf is \[ G(\boldsymbol{x};\boldsymbol{\vartheta},\boldsymbol{\lambda}) = F(\boldsymbol{x};\boldsymbol{\vartheta}) + \sum_{s=1}^{d}\lambda_s \int_{-\pi}^{x_1}\cdots\int_{-\pi}^{x_d} \sin(t_s)\,f(\boldsymbol{t};\boldsymbol{\vartheta})\,d\boldsymbol{t}, \] with \[ G(\boldsymbol{\pi};\boldsymbol{\vartheta},\boldsymbol{\lambda}) - G(-\boldsymbol{\pi};\boldsymbol{\vartheta},\boldsymbol{\lambda}) = 1. \] The same framework yields explicit trigonometric moments: \[ \alpha_{\boldsymbol{p}}=\alpha^0_{\boldsymbol{p}}, \qquad \beta_{\boldsymbol{p}} = \frac12\sum_{s=1}^{d}\lambda_s \left( \alpha^0_{[p_1,\ldots,p_s-1,\ldots,p_d]} - \alpha^0_{[p_1,\ldots,p_s+1,\ldots,p_d]} \right), \] so the cosine moments are inherited from the base distribution, while the sine moments are modified by the skewing parameters [1910.13293]. The toroidal paper also establishes an important caveat about marginals. Marginal distributions are not always sine-skewed again. They remain sine-skewed if the base density is pointwise symmetric in the marginalized-out coordinates, or if \(\lambda_s=0\) for all excluded coordinates; otherwise, the marginal need not be sine-skewed, and skewness can appear even in coordinates with \(\lambda_s=0\) because of dependence. This directly addresses a common misconception that sine-skewing acts independently coordinate by coordinate [1910.13293]. Random generation is particularly simple. If \(\boldsymbol{Y}\sim f(\cdot-\boldsymbol{\mu};\boldsymbol{\vartheta})\) and \(U\sim \text{Uniform}(0,1)\), then \[ \boldsymbol{X}= \begin{cases} \boldsymbol{Y}, & U \le \frac{1+\sum_{s=1}^{d}\lambda_s \sin(Y_s-\mu_s)}{2},\\[6pt] -\boldsymbol{Y}+2\boldsymbol{\mu}, & \text{otherwise} \end{cases} \] has the sine-skewed density. For inference, the log-likelihood for a sample \(\mathcal{X}=(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_n)\) is \[ \ell(\boldsymbol{\mu},\boldsymbol{\vartheta},\boldsymbol{\lambda}) = \sum_{i=1}^{n} \log\left( 1+\sum_{s=1}^{d}\lambda_s\sin(X_{i;s}-\mu_s) \right) + \sum_{i=1}^{n}\log f(\boldsymbol{X}_i-\boldsymbol{\mu};\boldsymbol{\vartheta}), \] and under standard regularity conditions with true parameters in the interior, \[ \sqrt{n}\left( (\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\vartheta}},\hat{\boldsymbol{\lambda}}) - (\boldsymbol{\mu}_0,\boldsymbol{\vartheta}_0,\boldsymbol{\lambda}_0) \right) \overset{d}{\to} N(\mathbf{0},\mathfrak{I}^{-1}). \] A likelihood ratio test for \[ H_0:\boldsymbol{\lambda}=\mathbf{0} \qquad\text{vs.}\qquad H_1:\boldsymbol{\lambda}\neq\mathbf{0} \] uses \[ -2\left( \ell(\hat{\boldsymbol{\mu}}_0,\hat{\boldsymbol{\vartheta}}_0,\mathbf{0}) - \ell(\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\vartheta}},\hat{\boldsymbol{\lambda}}) \right) \overset{a}{\sim} \chi^2_d. \] The main empirical application is protein bioinformatics. Using the Top500 protein database, 500 high-quality protein structures were initially considered and reduced to 402 proteins after removing chain breaks and parsing errors. The observed variables are the backbone dihedral-angle pairs \((\phi,\psi)\), treated as points on \(\mathbb{T}^2\) and modeled with mixtures of two toroidal components built from the bivariate Sine, Cosine, and wrapped Cauchy families and their sine-skewed counterparts SS, SC, and SWC. Model comparison uses log-likelihood, AIC, BIC, and the symmetry test at level \(\alpha=0.01\). Across most amino acids, the sine-skewed versions improve the fit; for serine the best AIC fit is the sine-skewed wrapped Cauchy, and for glycine and proline the best AIC fit is the sine-skewed Sine [1910.13293]. ## 5. Extended sine-skewed circular distributions A later extension was motivated by a limitation explicitly identified for classical sine-skewed circular distributions: they are identifiable, have easily-computable trigonometric moments, and admit a simple random-number generation algorithm, but they are known to have relatively low levels of asymmetry. The proposed remedy is an order-\(m\) family built from \[ g_m(x)= \frac{\Gamma\{2(m+1)\}}{2^{2m+1}\Gamma(m+1)^2} (1-x^2)^m, \qquad -1\le x\le 1, \] with cdf \[ G_m(x)=\int_{-1}^x g_m(t)\,dt. \] For integer \(m\), \[ G_m(x) = C_m \sum_{\ell=0}^{m}\binom{m}{\ell}\frac{(-1)^\ell}{2\ell+1}x^{2\ell+1} +\frac12, \qquad C_m= \frac{\Gamma\{2(m+1)\}}{2^{2m+1}\Gamma(m+1)^2}, \] and the extended sine-skewed density is \[ f_{\mathrm{ESS}^{(m)}}(\theta;\mu,\bm{\rho},\lambda) = 2f_0(\theta-\mu;\bm{\rho})\, G_m\!\bigl(\lambda\sin(\theta-\mu)\bigr). \] When \(m=0\), \(G_0(x)=(x+1)/2\), so the model reduces to the ordinary sine-skewed family; for \(m\ge 1\), the skewing function becomes steeper around 0 and permits stronger asymmetry [2402.09788]. The identifiable subfamilies treated explicitly are the ESS von Mises and ESS wrapped Cauchy families, denoted \(\mathcal{F}_{\mathrm{vM}^{(m)}}\) and \(\mathcal{F}_{\mathrm{WC}^{(m)}}\), for every integer \(m\ge 0\). For \(\mu=0\), the cosine moments are unchanged from the symmetric base: \[ \alpha_p=\alpha_{0,p}. \] For ESS-vM, \[ \alpha_p=\frac{I_p(\kappa)}{I_0(\kappa)}, \] while for ESS-WC, \[ \alpha_p=\rho^{|p|}. \] The paper gives explicit formulas for the sine moments \(\beta_p\) for \(m=1\) and \(m=2\), thereby preserving one of the main attractions of classical sine-skewing: analytic tractability of moments [2402.09788]. The same work develops a sign-flip simulation algorithm. If \(\Phi\sim f_0(\cdot;\bm{\rho})\) and \(U\sim \mathrm{Unif}(0,1)\), then \[ \Theta= \begin{cases} \Phi, & U0.99\) occur more often when the true \(|\lambda|\) is close to 1; and increasing \(m\) reduces the frequency of boundary estimates. In real-data analyses of two termite mound orientation datasets, higher-order ESS models improve fit relative to the ordinary sine-skewed case, and ESS-WC often fits concentrated data better than ESS-vM [2402.09788]. ## 6. Inferential regularity, Fisher information singularity, and limitations Sine-skewed models are not uniformly regular near symmetry. In the \(d\)-dimensional toroidal setting, asymmetry is introduced through \[ f(\theta;\mu,\lambda) := f_0(\theta-\mu)\left(1+\sum_{j=1}^{d}\lambda_j \sin(\theta_j-\mu_j)\right), \] or through the related multiplicative form \[ f_0(\theta-\mu)\prod_{j=1}^{d}\left(1+\lambda_j \sin(\theta_j-\mu_j)\right). \] The score vector near symmetry is \[ S_{f_0} = \left( -\frac{\frac{\partial}{\partial \theta_1}f_0(-)}{f_0(-)}, \dots, -\frac{\frac{\partial}{\partial \theta_d}f_0(-)}{f_0(-)}, \sin(\theta_1-\mu_1),\dots,\sin(\theta_d-\mu_d) \right)^\top, \] and the Fisher information matrix is singular iff the location-score and skewness-score components are linearly dependent [2603.04369]. The general characterization theorem states that for \(f_0\in\mathcal{F}\) satisfying the differentiability assumption, singularity in the vicinity of symmetry occurs if and only if there exists \(\alpha=(\alpha_1,\ldots,\alpha_d)^\top\in\mathbb{R}^d\), with \(\alpha_i\neq 0\) for all \(i\), such that \[ h_0(\theta) := f_0(\theta)\exp\left(\sum_{i=1}^d \gamma_i \cos(\theta_i-\mu_i)\right) \] satisfies \[ h_0(\theta+t)=h_0(\theta) \] for all \(t\in\mathbb{R}\) and all \(\theta\in[-\pi,\pi)^d\). Equivalently, \[ f_0(\theta) = h_0(\theta)\exp\left(-\sum_{i=1}^d \gamma_i \cos(\theta_i-\mu_i)\right), \] with \(h_0\) invariant under the common shift \(\theta\mapsto \theta+t\mathbf{1}\). The proof proceeds through the PDE \[ \sum_{i=1}^d \alpha_i \frac{\partial}{\partial \theta_i}f_0(\theta) = f_0(\theta)\sum_{i=1}^d \beta_i \sin(\theta_i-\mu_i). \] The theorem yields a sharp distinction among standard toroidal bases. The sine-skewed product of independent von Mises distributions, the Cosine distribution, and the multivariate extension of the Cosine distribution do suffer from singular Fisher information near symmetry. By contrast, the Sine distribution, the multivariate extension of the Sine distribution, the bivariate wrapped Cauchy distribution, and the trivariate wrapped Cauchy distribution do not. Thus singularity is not a generic feature of all sine-skewed toroidal models; it depends on the structure of the symmetric base density [2603.04369]. The inferential consequences are substantial. Singularity indicates lack of local identifiability; the usual \(\sqrt{n}\)-rate asymptotic normality of the MLE may fail; convergence can be slower than \(n^{-1/2}\); the likelihood may become irregular, for example bimodal; and standard inference tools such as Wald tests, confidence intervals, and likelihood-based procedures can break down. This limitation connects with more specific model-dependent issues already documented elsewhere in the sine-skewed literature: the sine-skewed cardioid paper reports that ML estimators are not well-defined in the usual way for that model, and the ESS paper notes that the Fisher information for ESS-vM can become singular at \(\lambda=0\). Taken together, these results show that sine-skewing is mathematically simple at the density level but can generate nontrivial inferential pathologies near symmetry or near the boundary of the parameter space [1811.10603, 2402.09788, 2603.04369]. Source: https://www.emergentmind.com/topics/sine-skewed-models