Skew-Elliptical Cluster Processes

Abstract

This paper introduces skew-elliptical cluster processes. In contrast to the simple Gaussian isotropic structure of the distribution of the “children” events of a Thomas process, we propose an anisotropic structure by allowing the choice of a flexible covariance matrix and incorporating skewness or ellipticity parameters into the structure. Since the theoretical pair correlation functions of these processes are complex and analytically incomplete, and therefore the estimation of the parameters is computationally intensive, we propose reasonable approximations of the theoretical pair correlation functions of these cluster processes, which allow for a simpler parameter estimation. We present the estimation of their parameters using the minimum contrast method. For a data application, we use a fraction of the full redwood dataset. Our analysis shows that an elliptical cluster process can describe this point pattern better than a common Thomas process, since it is able to statistically model the non-circular shapes of the clusters in the data. The skew-elliptical cluster processes can be very meaningful for analyzing complex datasets in the field of spatial point processes since they provide more flexibility to detect interesting characteristics of the data.

Appendix

1.1 Table of Acronyms

Acronym	Complete words
AGOF	Adjusted goodness-of-fit
cdf	Cumulative distribution function
CP	Cluster process
CSR	Complete spatial randomness
df	Degrees of freedom
EST	Extended skew-t
GOF	Goodness-of-fit
MCM	Minimum contrast method
pcf	Pair correlation function
pdf	Probability density function
SPP	Spatial point pattern
SUE	Unified skew-elliptical
SUN	Unified skew-normal
TP	Thomas process

Skew-Elliptical-Normal Cluster Processes

According to the transformation in (1), the joint distribution f _R,Θ(r, θ) of (R,  Θ) is

$$\displaystyle \begin{aligned} f_{R, \Theta}(r, \theta) &= \frac{r \exp\left(-\frac{\sigma^2_2 r^2 \cos^2\theta + \sigma^2_1 r^2 \sin^2\theta}{4\sigma^2_1\sigma^2_2}\right)}{\pi \sigma_1 \sigma_2}{}\\ &\quad \times \Phi_2\left\{\frac{\left ( \frac{\alpha_1 r \cos\theta}{\sigma_1} + \frac{\alpha_2 r \sin\theta}{\sigma_2} \right ) \begin{pmatrix} 1\\-1\end{pmatrix}}{2\sqrt{1+\alpha^2_1+\alpha^2_2}};\frac{\begin{pmatrix} 2 + \alpha^2_1 + \alpha^2_2 & \alpha^2_1 + \alpha^2_2 \\ \alpha^2_1 + \alpha^2_2 & 2 + \alpha^2_1 + \alpha^2_2 \end{pmatrix}}{2(1+\alpha^2_1+\alpha^2_2)}\right\}. \end{aligned} $$

(A.1)

Elliptical-Normal Cluster Process

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} f_{d}(r)& =&\displaystyle \frac{r}{2 \sigma_1 \sigma_2} \exp\left\{- \;\frac {(\sigma^2_1 + \sigma^2_2)r^2}{8\sigma^2_1 \sigma^2_2}\right\} \mbox{BesselI}_0\left\{\frac {(\sigma^2_1 - \sigma^2_2)r^2}{8\sigma^2_1 \sigma^2_2}\right\}. \end{array} \end{aligned} $$

(A.2)

For a different parametrization, σ ₁ ≡ σ and σ ₂ = c _σ σ with c _σ > 0, the pdf f _d(r) can be rewritten as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} f_{d}(r)& =&\displaystyle \frac{1}{2 c_{ \sigma} \sigma^2} \exp\left\{- \;\frac {(1 + c_{ \sigma}^2)r^2}{8c_{ \sigma}^2\sigma^2}\right\} \mbox{BesselI}_0\left\{\frac {(1 - c_{ \sigma}^2)r^2}{8c_{ \sigma}^2 \sigma^2}\right\}. \end{array} \end{aligned} $$

(A.3)

Circular-Normal Cluster Process

Here, σ ₁ = σ ₂. We provide the pdf of R, f _R(r) = f _d(r) in the following:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} f_{R}(r)& =&\displaystyle \frac{r}{2 \sigma^2} \exp\left(- \;\frac {r^2}{4 \sigma^2}\right). \end{array} \end{aligned} $$

(A.4)

Skew-Normal Cluster Process

Following the transformation defined in (1),

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{R, \Theta}(r, \theta)& =&\displaystyle \frac{r}{2 c_0 \sqrt{\pi^2c_1 c_2 - 4 \alpha^2_1 \alpha^2_2}} {}\\ & &\displaystyle \times \:\exp\left(- \: \frac{\pi r^2 [\pi\{\cos^2\theta(c_2 - c_1) + c_1\} + 4 \alpha_1 \alpha_2 \cos \theta \sin \theta ]}{2 c_0 (\pi^2 c_1 c_2 - 4 \alpha^2_1 \alpha^2_1)}\right), \end{array} \end{aligned} $$

(A.5)

where $c_0=2\sigma ^2/(1+\alpha ^2_1 + \alpha ^2_2)$, $c_1=1+\alpha ^2_1(1-2/\pi ) + \alpha ^2_2$, and $c_2=1 + \alpha ^2_1+\alpha ^2_2(1-2/\pi )$. The pdf, f _d(r), is analytically complete only in the following two cases. First, $\alpha ^2_1=\alpha ^2_2$, i.e., (i) α ^T = α(1, 1), (ii) α ^T = α(−1, −1), (iii) α ^T = α(1, −1), or (iv) α ^T = α(−1, 1), assuming that α > 0. Consequently, c ₁ = c ₂,

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{d}(r)& =&\displaystyle \frac{\pi \sqrt{1+2\alpha^2} r}{2\sigma^2 \sqrt{\pi\{\pi(1+2\alpha^2) - 4 \alpha^2\}}} \exp\left[-\;\frac{\{\pi + 2\alpha^2(\pi - 1)\}r^2}{4\sigma^2\{\pi(1+2\alpha^2) - 4\alpha^2\}}\right] \qquad {}\\ & &\displaystyle \times \: \mbox{BesselI}_0\left[\frac{ \alpha^2 r^2}{2\sigma^2\{\pi(1+2\alpha^2) - 4\alpha^2\}}\right], \qquad \end{array} \end{aligned} $$

(A.6)

where $\mbox{BesselI}_0(x)=\sum _{n=0}^{\infty } (x/2)^{2n}/(n!)^{2}$ is a modified Bessel function of the first kind.

Second, suppose that α = (0, α)^T or α = (α, 0)^T. Then,

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{d}(r) & =&\displaystyle \frac{r(1+\alpha^2)}{2\sigma^2\sqrt{(1+\alpha^2)\{1+\alpha^2(1-2/\pi)\}}}\exp\left[-\;\frac{r^2\{1 + \alpha^2(1-1/\pi)\}}{4 \sigma^2\{1+\alpha^2(1-2/\pi)\}}\right]{} \\ & &\displaystyle \times \; \mbox{BesselI}_0\left[\frac{ \alpha^2 r^2}{4 \pi \sigma^2 \{1+\alpha^2(1-2/\pi)\}}\right]. \end{array} \end{aligned} $$

(A.7)

Skew-Elliptical-t Cluster Processes

For x = (x ₁, x ₂)^T,

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} f_{\mathbf{X}}(x_1, x_2) & =&\displaystyle \frac{T_1\left[ \frac{\tau}{\sqrt{1 + 2 (\alpha^2_1 + \alpha^2_2)}} \left\{\frac{\nu +2}{\nu + \left(x^2_1/\sigma^2_1 + x^2_2/\sigma^2_2 \right)/2} \right\}^{1/2};\nu + 2 \right]}{4 \pi\sigma_1 \sigma_2 \left(1 + \frac{x^2_1/\sigma^2_1 + x^2_2/\sigma^2_2}{2\nu}\right)^{(\nu+2)/2} T_1\left\{\frac{\tau}{\sqrt{1+2(\alpha^2_1 + \alpha^2_2)}};\nu\right\}}, \qquad \end{array} \end{aligned} $$

(A.8)

where T ₁(⋅;ν) denotes the cdf of the univariate t-distribution with ν degrees of freedom. According to the transformation in (1), the joint distribution of (R,  Θ) is

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} & &\displaystyle f_{R ,\Theta}(r, \theta)\\ & &\displaystyle \quad = \frac{r \: T_1\left[ \frac{\tau}{\sqrt{1 + 2 (\alpha^2_1 + \alpha^2_2)}} \left\{\frac{\nu +2}{\nu + \left(r^2 \cos^2 \theta/\sigma^2_1 + r^2 \sin^2 \theta/\sigma^2_2 \right)/2} \right\}^{1/2};\nu + 2 \right]}{4 \pi\sigma_1 \sigma_2 \left(1 + \frac{r^2 \cos^2 \theta/\sigma^2_1 + r^2 \sin^2 \theta/\sigma^2_2}{2\nu}\right)^{(\nu+2)/2} T_1\left\{\frac{\tau}{\sqrt{1+2(\alpha^2_1 + \alpha^2_2)}};\nu \right\}}. \end{array} \end{aligned} $$

(A.9)