Skew Normal¶

Convolution and Integration¶

The skew-normal profile is the standard skew-normal distribution (Azzalini 1985) expressed in the unite erf parametrisation. Unlike the skew-Voigt, the convolution with a Gaussian LSF is exact: the convolved profile is again a skew-normal with a rescaled shape parameter, requiring no numerical correction. The pixel integral follows analytically from Owen’s T function. This document derives both results and connects them to integrate_skewNormal().

Definition¶

The intrinsic skew-normal profile centred at \(\mu\) with Gaussian width \(\sigma\) and shape parameter \(\alpha\) is

\[ f(x) = G_\sigma(x)\,\bigl[1 + \text{erf}\!\left(\tfrac{\alpha(x-\mu)}{w_0}\right)\bigr], \qquad w_0 = \sigma\sqrt{2}, \]

where \(G_\sigma(x) = ({\sigma\sqrt{2\pi}})^{-1}\exp[-(x-\mu)^2/(2\sigma^2)]\) is the Gaussian envelope. The equivalence to the standard form \(f = 2G_\sigma \Phi(\alpha(x-\mu)/\sigma)\) follows immediately from \(\Phi(u) = \tfrac{1}{2}[1+\text{erf}(u/\sqrt{2})]\).

Normalisation. Since \(G_\sigma\) is even and \(\text{erf}(\alpha(x-\mu)/w_0)\) is odd about \(\mu\), their product integrates to zero and \(\int f = \int G_\sigma = 1\) for any \(\alpha\).

The shape parameter \(\alpha > 0\) shifts flux toward the red; \(\alpha < 0\) toward the blue. At \(\alpha = 0\) the profile reduces to a pure Gaussian.

Exact Convolution with a Gaussian LSF¶

Let \(G_\text{lsf}(x) = \mathcal{N}(0, \sigma_\text{lsf}^2)\). Convolving \(f\) with the LSF (working with \(\mu = 0\) for clarity) splits into two terms:

\[ (f * G_\text{lsf})(x) = \underbrace{(G_\sigma * G_\text{lsf})(x)}_{G_{\sigma_\text{tot}}(x)} + \int_{-\infty}^\infty G_\sigma(t)\,\text{erf}\!\left(\tfrac{\alpha t}{w_0}\right) G_\text{lsf}(x-t)\,dt, \]

where \(\sigma_\text{tot} = \sqrt{\sigma^2 + \sigma_\text{lsf}^2}\). The first term is the standard Gaussian convolution. The second term — the skew correction \(I(x)\) — can be evaluated exactly using the Gaussian erf identity.

Gaussian erf identity¶

For \(W \sim \mathcal{N}(\mu_W, \tilde\sigma^2)\):

\[ \mathbb{E}[\text{erf}(W/c)] = \int_{-\infty}^\infty \mathcal{N}(w;\,\mu_W,\tilde\sigma^2)\,\text{erf}(w/c)\,dw = \text{erf}\!\left(\frac{\mu_W}{\sqrt{c^2 + 2\tilde\sigma^2}}\right). \]

Proof. Writing \(\text{erf}(w/c) = (2/\sqrt\pi)\int_0^{w/c} e^{-u^2}\,du\) and exchanging integrals, the inner integral over \(w\) at fixed \(u\) is the probability \(P(W > cu)\) of a Gaussian, which evaluates to \(\text{erfc}(\cdot)\). Completing the square in the resulting integral over \(u\) and converting back to erf gives the stated result.

Applying the identity to \(I(x)\)¶

Use the Gaussian product identity to write the integrand as the product of \(G_{\sigma_\text{tot}}(x)\) and a conditional Gaussian:

\[ G_\sigma(t)\,G_\text{lsf}(x-t) = G_{\sigma_\text{tot}}(x)\,\mathcal{N}\!\left(t;\;\frac{\sigma^2}{\sigma_\text{tot}^2}x,\; \tilde\sigma^2\right), \qquad \tilde\sigma^2 = \frac{\sigma^2\sigma_\text{lsf}^2}{\sigma_\text{tot}^2}. \]

Integrating over \(t\) amounts to taking the expectation of \(\text{erf}(\alpha t / w_0) = \text{erf}(t/c)\) with \(c = w_0/\alpha = \sigma\sqrt{2}/\alpha\) under this conditional Gaussian \(\mathcal{N}(\mu_{t|x}, \tilde\sigma^2)\) with \(\mu_{t|x} = \sigma^2 x/\sigma_\text{tot}^2\). Applying the identity:

\[ I(x) = G_{\sigma_\text{tot}}(x) \cdot \text{erf}\!\left(\frac{\sigma^2 x/\sigma_\text{tot}^2} {\sqrt{2\sigma^2/\alpha^2 + 2\sigma^2\sigma_\text{lsf}^2/\sigma_\text{tot}^2}}\right). \]

Factoring \(2\sigma^2/\alpha^2\) from the square root in the denominator and simplifying:

\[ I(x) = G_{\sigma_\text{tot}}(x) \cdot \text{erf}\!\left(\frac{\alpha_\text{eff}\,x}{w_0'}\right), \]

where \(w_0' = \sigma_\text{tot}\sqrt{2}\) and

\[ \boxed{ \alpha_\text{eff} = \frac{\alpha\,\sigma}{\sqrt{\sigma_\text{tot}^2 + \alpha^2\sigma_\text{lsf}^2}} = \frac{\alpha\,\sigma}{\sqrt{\sigma^2 + (1+\alpha^2)\,\sigma_\text{lsf}^2}} } \]

Result¶

\[ (f * G_\text{lsf})(x) = G_{\sigma_\text{tot}}(x)\,\bigl[1 + \text{erf}\!\left(\tfrac{\alpha_\text{eff}\,x}{w_0'}\right)\bigr]. \]

The convolved profile is exactly a skew-normal with the same functional form, wider width \(\sigma_\text{tot}\), and reduced shape parameter \(|\alpha_\text{eff}| < |\alpha|\). No numerical correction is needed.

Properties:

\(\alpha_\text{eff} \to \alpha\) as \(\sigma_\text{lsf} \to 0\) (no LSF)
\(\alpha_\text{eff} \to 0\) as \(\sigma_\text{lsf} \to \infty\) (LSF washes out skew)
\(|\alpha_\text{eff}| < |\alpha|\) for all \(\sigma_\text{lsf} > 0\)

CDF and Pixel Integration¶

The CDF of the skew-normal (Azzalini 1985) involves Owen’s T function \(T(h, a) = (2\pi)^{-1}\int_0^a (1+t^2)^{-1} e^{-h^2(1+t^2)/2}\,dt\):

\[ F(x) = \int_{-\infty}^x f(x')\,dx' = \Phi(z) - 2\,T(z,\,\alpha_\text{eff}), \qquad z = \frac{x}{\sigma_\text{tot}}, \]

where \(\Phi(z) = \tfrac{1}{2}[1 + \text{erf}(z/\sqrt{2})]\) is the standard normal CDF. The pixel integral over \([\lambda_l, \lambda_h]\) centred at \(c\) is then

\[ \int_{\lambda_l}^{\lambda_h} f(x)\,dx = \bigl[\Phi(z)\bigr]_{\lambda_l-c}^{\lambda_h-c} - 2\,\bigl[T(z,\alpha_\text{eff})\bigr]_{\lambda_l-c}^{\lambda_h-c}. \]

The Gaussian term reduces to the standard erf difference; the skew correction requires evaluating Owen’s T at the two bin edges.

Connection to the Code¶

In integrate_skewNormal(), the halfvar coordinate \(t = z/\sqrt{2}\) is used throughout so that erf can be called directly. With \(\sigma_\text{tot} = \sqrt{\sigma_g^2 + \sigma_\text{lsf}^2}\) and

\[ \alpha_\text{eff} = \frac{\alpha\,\sigma_g}{\sqrt{\sigma_g^2 + (1+\alpha^2)\,\sigma_\text{lsf}^2}}, \]

the pixel integral over \([l, h]\) relative to center \(c\) is

\[ \frac{\text{erf}(t_h) - \text{erf}(t_l)}{2} - 2\,\bigl[T(z_h,\,\alpha_\text{eff}) - T(z_l,\,\alpha_\text{eff})\bigr], \]

where \(t = (x - c)/(\sigma_\text{tot}\sqrt{2})\) and \(z = t\sqrt{2}\). Owen’s T is called directly from jax.scipy.special.owens_t.