Sine-Skewing Mechanism Overview

• Sine-skewing is a method that multiplies a symmetric periodic density by a sine term, introducing controlled asymmetry while retaining the normalization constant.
• In the von Mises special case, the sine-skewed density exhibits explicit trigonometric moments and maintains its normalization, making it ideal for circular statistics.
• Extensions to d-dimensional torus and nonlinear dynamical systems demonstrate the mechanism's versatility in breaking symmetry and inducing directional responses.

The sine-skewing mechanism is a symmetry-breaking construction on periodic domains in which a central object is modified by a sine term while retaining its basic periodic structure. In circular and toroidal statistics, the mechanism takes a symmetric density and multiplies it by a factor linear in sine coordinates, thereby introducing asymmetry without altering the normalizing constant when the base law is central. In nonlinear dynamics, a related “sine-skew” construction mixes sinusoidal modes to produce a zero-mean but non-antiperiodic forcing, so that temporal asymmetry rather than mean bias drives a nonzero cycle-averaged response in nonlinear odd-resistance systems (Ahsanullah et al., 2019, Schutte et al., 4 Mar 2026, Hashemi et al., 2022).

1. Circular sine-skewing as a density transformation

Let $f_0(\theta;\mu,\kappa)$ be a circular probability density on $[\mu-\pi,\mu+\pi]$ that is symmetric about $\mu$, in the sense that

$$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$

For a scalar skewness parameter $\lambda$ with $|\lambda|<1$, the sine-skewed density is defined by

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$

for $\theta\in[\mu-\pi,\mu+\pi]$.

The key structural feature is that no additional normalizing constant is required. By symmetry,

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$

so the multiplicative perturbation integrates to one. Positivity is controlled entirely by the sine factor:

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$

which is guaranteed when $[\mu-\pi,\mu+\pi]$0 because $[\mu-\pi,\mu+\pi]$1 (Ahsanullah et al., 2019).

This construction isolates asymmetry in a single periodic odd component. The base density remains the carrier of concentration and location structure, whereas the multiplier $[\mu-\pi,\mu+\pi]$2 provides the skewing.

2. Specialization to the von Mises law

For the symmetric von Mises base,

$[\mu-\pi,\mu+\pi]$3

with $[\mu-\pi,\mu+\pi]$4 and $[\mu-\pi,\mu+\pi]$5 the modified Bessel function of the first kind, order $[\mu-\pi,\mu+\pi]$6, the sine-skewed construction yields

$[\mu-\pi,\mu+\pi]$7

with $[\mu-\pi,\mu+\pi]$8 and $[\mu-\pi,\mu+\pi]$9 (Ahsanullah et al., 2019).

The normalizing constant remains $\mu$0, because

$\mu$1

Positivity is again ensured by $\mu$2.

Within the one-dimensional toroidal setting, this family occupies a distinguished position in later inferential theory: the general characterization of Fisher-information singularity on $\mu$3 implies that on the circle, only the von Mises base leads to a singular Fisher information matrix in the sine-skewed extension (Schutte et al., 4 Mar 2026).

3. Trigonometric moments and directional summaries

For the sine-skewed von Mises law, the $\mu$4th complex moment is

$\mu$5

Using the Fourier-Bessel expansion

$\mu$6

one obtains

$\mu$7

In particular, for $\mu$8,

$\mu$9

and

$$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$0

These formulas give the standard directional summaries. The mean resultant length is

$$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$1

and the mean direction relative to $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$2 is

$$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$3

The paper also notes that an elementary “skewness” measure on the circle is the imaginary part of the second moment divided by $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$4real part, or else one may simply report $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$5 itself as the built-in skewness parameter (Ahsanullah et al., 2019).

4. Characterization by truncated first moments

A distinctive theoretical contribution for the sine-skewed von Mises family is a pair of truncated-moment characterizations. Let $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$6 have cdf $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$7 and density $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$8 on $$f_0(\mu+\phi;\mu,\kappa)=f_0(\mu-\phi;\mu,\kappa), \qquad \phi\in[-\pi,\pi].$$9, with $\lambda$0 finite, and set $\lambda$1, $\lambda$2.

For lower truncation, if for $\lambda$3

$\lambda$4

with $\lambda$5 continuously differentiable, then

$\lambda$6

with $\lambda$7 fixed by $\lambda$8.

For upper truncation, if

$\lambda$9

with $|\lambda|<1$0 continuously differentiable, then

$|\lambda|<1$1

The two main theorems establish that, exactly in the sine-skewed von Mises case, one recovers for suitable choices of $|\lambda|<1$2 and $|\lambda|<1$3 expressed in closed form the defining density

$|\lambda|<1$4

More precisely, Theorem 3.1 and Theorem 3.2 exhibit explicit functions $|\lambda|<1$5 and $|\lambda|<1$6, built out of partial sums of Bessel-series and polynomials, so that

$|\lambda|<1$7

and

$|\lambda|<1$8

hold if and only if $|\lambda|<1$9 is sine-skewed von Mises. These are stated as the first truncated-moment characterizations for that family (Ahsanullah et al., 2019).

5. Extension to the $$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$0-dimensional torus

On the $$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$1-dimensional torus $$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$2, let

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$3

be a central density satisfying periodicity and symmetry:

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$4

and

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$5

Writing

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$6

the sine-skewed extension with skewness vector $$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$7 is

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$8

where

$$f(\theta;\mu,\kappa,\lambda)=f_0(\theta;\mu,\kappa)\,[1+\lambda\sin(\theta-\mu)],$$9

To guarantee nonnegativity, one typically requires

$\theta\in[\mu-\pi,\mu+\pi]$0

and a simple sufficient condition is

$\theta\in[\mu-\pi,\mu+\pi]$1

The normalizing constant is

$\theta\in[\mu-\pi,\mu+\pi]$2

By symmetry, the integral of each $\theta\in[\mu-\pi,\mu+\pi]$3 vanishes, so $\theta\in[\mu-\pi,\mu+\pi]$4, and one may write

$\theta\in[\mu-\pi,\mu+\pi]$5

If $\theta\in[\mu-\pi,\mu+\pi]$6 is continuously differentiable, indeed $\theta\in[\mu-\pi,\mu+\pi]$7, then so is the sine-skewed density. The location parameter $\theta\in[\mu-\pi,\mu+\pi]$8 is only identified modulo $\theta\in[\mu-\pi,\mu+\pi]$9 in each coordinate. Under mild conditions on $$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$0, $$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$1 and $$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$2 are identifiable away from $$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$3. For moments,

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$4

where the first term vanishes by symmetry and

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$5

The 2026 toroidal study states that asymmetry is introduced through the sine-skewing mechanism, which is the only skewing mechanism that has been proposed on the hyper-torus in the literature (Schutte et al., 4 Mar 2026).

6. Fisher-information singularity and the dynamical “sine-skew” analogue

A central inferential issue for toroidal sine-skewed models is the possible singularity of the Fisher information matrix in the vicinity of symmetry. At $$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$6, with log-density

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$7

the score components are

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$8

$$\int_{\mu-\pi}^{\mu+\pi} f_0(\theta;\mu,\kappa)\sin(\theta-\mu)\,d\theta=0,$$9

and

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$0

The Fisher information matrix is

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$1

and at $$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$2 it is singular if and only if there exists a nontrivial linear combination of score components that is zero almost surely. This is equivalent to the partial differential equation

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$3

whose general solution is

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$4

with $$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$5 and $$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$6 satisfying

$$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$7

Accordingly, the Fisher information of the sine-skewed version of $$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$8 is singular at $$1+\lambda\sin(\theta-\mu)\ge 0 \quad \forall\,\theta,$$9 if and only if $[\mu-\pi,\mu+\pi]$00 can be written, up to location, in that form. On the circle, this recovers the statement that only the von Mises base leads to a singular Fisher information matrix in the sine-skewed extension (Schutte et al., 4 Mar 2026).

A distinct but structurally related use of “sine-skew” occurs in nonlinear transport. Hashemi et al. consider a two-mode sinusoid

$[\mu-\pi,\mu+\pi]$01

and in the special case

$[\mu-\pi,\mu+\pi]$02

Antiperiodicity over a full period $[\mu-\pi,\mu+\pi]$03 requires

$[\mu-\pi,\mu+\pi]$04

which with $[\mu-\pi,\mu+\pi]$05 holds only if $[\mu-\pi,\mu+\pi]$06, a ratio of two odd integers. Whenever $[\mu-\pi,\mu+\pi]$07 contains an even integer, the waveform is non-antiperiodic. Temporal asymmetry can be quantified by the “time-skew”

$[\mu-\pi,\mu+\pi]$08

or by the shift-symmetry residue

$[\mu-\pi,\mu+\pi]$09

with $[\mu-\pi,\mu+\pi]$10 if and only if the drive is time-skewed.

In the generic nonlinear equation

$[\mu-\pi,\mu+\pi]$11

a non-antiperiodic $[\mu-\pi,\mu+\pi]$12 can yield $[\mu-\pi,\mu+\pi]$13. The paper reports that drift appears first when the second mode introduces an even-order harmonic; numerically the largest drift occurs when $[\mu-\pi,\mu+\pi]$14, and smaller or zero drift occurs for odd/odd ratios. In the dimensionless stick-slip model,

$[\mu-\pi,\mu+\pi]$15

Fig. 2(e) shows $[\mu-\pi,\mu+\pi]$16 for $[\mu-\pi,\mu+\pi]$17 and $[\mu-\pi,\mu+\pi]$18 for $[\mu-\pi,\mu+\pi]$19, with sign reversals upon $[\mu-\pi,\mu+\pi]$20. The same odd/odd versus even-containing distinction appears for $[\mu-\pi,\mu+\pi]$21 and in colloidal electrophoresis under a two-mode AC voltage, where $[\mu-\pi,\mu+\pi]$22 for odd/odd ratios and $[\mu-\pi,\mu+\pi]$23 when $[\mu-\pi,\mu+\pi]$24 contains an even integer (Hashemi et al., 2022).

A plausible implication is that sine-skewing functions as a general periodic asymmetry template. In statistical models it breaks central symmetry through a sine multiplier while preserving normalization by symmetry; in driven nonlinear systems it breaks half-period shift symmetry while preserving zero temporal mean.