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While working on an application of Gaussian process regression to ultrasound imaging, I came across an interesting similarity between the K-distribution and the Matérn covariance function.

Background

The K-distribution is proposed as a generalized model for scattering in (Jakeman and Pusey, 1978; Jakeman and Tough, 1987). It is the distribution of the product of two independent random variables, each having a Gamma distribution. The K-distribution has the form

\begin{equation} f(x) = \dfrac{x^{\nu + (n/2) - 1} \; b^{\nu + (n/2)}}{2^{\nu+(n/2)-2} \; \Gamma{\left(\nu\right)}\Gamma{\left(\dfrac{n}{2}\right)}} K_{\nu - (n/2)} \left( b x \right) ,\tag{1}\label{generalK} \end{equation}

where $\nu > -1$ is a shape parameter, $b = \dfrac{\sqrt{2n\nu}}{\rho}$ is the scaling parameter, $n$ is the number of dimensions, $\rho$ is the characteristic length scale, $\Gamma(\cdot)$ is the Gamma function, and $K_\nu(\cdot)$ is an order-$\nu$ modified Bessel function of the second kind.

The Matérn covariance function is defined as \begin{equation} k(x_i, x_k) = \sigma^2 \dfrac{2^{1-\nu}}{\Gamma(\nu)} \left( \dfrac{\sqrt{2\nu}}{l}r\right)^\nu K_\nu \left( \dfrac{\sqrt{2\nu}}{l}r \right) \tag{2}\label{matern} \end{equation} where $l$ is the characteristic length scale, $r$ is the distance between $x_i$ and $x_j$, and $\nu$ is the smoothness parameter.

Observation

Let's consider the case when $n=2$ for the K-distribution. Then \begin{equation} f(x) = 4\dfrac{\sqrt{\nu}}{\rho\Gamma{\left(\nu\right)}} \left(\dfrac{\sqrt{\nu}}{\rho} x\right)^{\nu} K_{\nu - 1} \left(\dfrac{2\sqrt{\nu}}{\rho} x \right) .\tag{3}\label{kPDF} \end{equation}

We notice that the the Matérn covariance function in \eqref{matern} and the PDF of the K-distribution \eqref{kPDF} look similar. In fact, the CDF of the K-distribution is, \begin{equation} F(x) = \int_0^x f(u) d u = 1 - \dfrac{2^{1-\nu}}{\Gamma (\nu )} \left(\dfrac{ 2 \sqrt{\nu} }{\rho} x \right)^{\nu} K_{\nu} \left(2 \dfrac{\sqrt{\nu}}{\rho} x \right) .\tag{4} \end{equation} Setting $\rho = \sqrt{2}\rho^*$, \begin{equation} 1 - F(x) = \dfrac{2^{1-\nu}}{\Gamma{(\nu)}} \left( \dfrac{\sqrt{2\nu}}{\rho^*}x \right) ^ \nu K_\nu \left( \dfrac{\sqrt{2\nu}}{\rho^*}x \right) ,\tag{5} \end{equation} which gives us the Matérn covariance function in \eqref{matern} with unit variance ($\sigma^2 = 1$).

Question

Now, I realize that the K-distribution is a scalar distribution and has no spatial dependence, whereas autocovariance is inherently related to spatial random processes. However, is there a way to explain the similarity between these two functions?

Thank you!

References

  • Jakeman, E., Pusey, P.N., 1978. Signi cance of K distributions in scattering experiments. Phys. Rev. Lett. 40, 546-550
  • Jakeman, E., Tough, R.J.A., 1987. Generalized K distribution: a statistical model for weak scattering. J. Opt. Soc. Am. A 4, 1764-1772.
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