A class of random fields with two-piece marginal distributions for modeling point-referenced data with spatial outliers

Abstract

In this paper, we propose a new class of non-Gaussian random fields named two-piece random fields. The proposed class allows to generate random fields that have flexible marginal distributions, possibly skewed and/or heavy-tailed and, as a consequence, has a wide range of applications. We study the second-order properties of this class and provide analytical expressions for the bivariate distribution and the associated correlation functions. We exemplify our general construction by studying two examples: two-piece Gaussian and two-piece Tukey-h random fields. An interesting feature of the proposed class is that it offers a specific type of dependence that can be useful when modeling data displaying spatial outliers, a property that has been somewhat ignored from modeling viewpoint in the literature for spatial point referenced data. Since the likelihood function involves analytically intractable integrals, we adopt the weighted pairwise likelihood as a method of estimation. The effectiveness of our methodology is illustrated with simulation experiments as well as with the analysis of a georeferenced dataset of mean temperatures in Middle East.

Appendix

1.1 7.1 Proof of Lemma 1

Proof

We make use of some special functions in this proof. In particular the parabolic cylinder function $D_{n}(x)$, the confluent hypergeometric function ${}_1F_1(a;b;x)$ and the Gaussian hypergeometric function ${}_2F_1(a;b;c;x)$ (see Gradshteyn and Ryzhik (2007) for the definitions of these functions). By definition, we have:

$$\begin{aligned}&\mathrm{I\! E}(|T_{h}(\varvec{s}_{i})||T_{h}(\varvec{s}_{j})|)\nonumber \\&\quad =\mathrm{I\! E}(|G(\varvec{s}_{i})e^{ \frac{h(G(\varvec{s}_{i}))^{2}}{2} }||G(\varvec{s}_{j})e^{\frac{h(G(\varvec{s}_{j}))^{2}}{2}}|)\nonumber \\&\quad =\int \limits _{\mathbb {R}^2_+} |g_{i}e^{ \frac{hg_i^{2}}{2} }| |g_{j}e^{ \frac{hg_{j}^{2}}{2}}| f_{|\varvec{G}_{ij}|}(g_i,g_j){\mathrm{d}}g_{i}{\mathrm{d}}g_{j}\nonumber \\&\quad =\frac{1}{\pi (1-\rho ^2(\varvec{h}))^{1/2}}\int \limits _{\mathbb {R}^2_+}g_ig_j e^{-\frac{1}{2(1-\rho ^2(\varvec{h}))}\left[ g_i^2+g_j^2-2\rho (\varvec{h})g_ig_j\right] }e^{\frac{hg_i^2}{2}+\frac{hg_j^2}{2}}{\mathrm{d}}g_i{\mathrm{d}}g_j \nonumber \\&\qquad +\frac{1}{\pi (1-\rho ^2(\varvec{h}))^{1/2}}\int \limits _{\mathbb {R}^2_+}g_ig_j e^{-\frac{1}{2(1-\rho ^2(\varvec{h}))}\left[ g_i^2+g_j^2+2\rho (\varvec{h})g_ig_j\right] }e^{\frac{hg_i^2}{2}+\frac{hg_j^2}{2}}{\mathrm{d}}g_i{\mathrm{d}}g_j\nonumber \\&\quad =A_1+A_2. \end{aligned}$$

(32)

Taking the first integral of (32) and using (3.462.1) of Gradshteyn and Ryzhik (2007), we obtain

$$\begin{aligned} A_1&=\frac{1}{\pi (1-\rho ^2(\varvec{h}))^{1/2}}\int \limits _{\mathbb {R}_+}g_je^{-\frac{[1-(1-\rho ^2(\varvec{h}))h]g^2_j}{2(1-\rho ^2(\varvec{h}))}} \left[ \int \limits _{\mathbb {R}_+}g_ie^{\left[ -\frac{[1-(1-\rho ^2(\varvec{h}))h]g^2_i}{2(1-\rho ^2(\varvec{h}))}+\frac{\rho (\varvec{h})g_ig_j}{(1-\rho ^2(\varvec{h}))}\right] }{\mathrm{d}}g_i\right] {\mathrm{d}}g_j\nonumber \\&=\frac{1}{\pi (1-\rho ^2(\varvec{h}))^{1/2}}\left[ \frac{(1-\rho ^2(\varvec{h}))}{1-(1-\rho ^2(\varvec{h}))h}\right] \int \limits _{\mathbb {R}_+} g_je^{-\left[ \frac{[1-(1-\rho ^2(\varvec{h}))h]}{2(1-\rho ^2(\varvec{h}))}-\frac{\rho ^2(\varvec{h})}{4(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}\right] g^2_j}\nonumber \\&\quad \times D_{-2}\left( -\frac{\rho (\varvec{h})g_j}{\sqrt{(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}}\right) {\mathrm{d}}g_j, \end{aligned}$$

(33)

where $D_{n}(x)$ is the parabolic cylinder function. Now, considering (9.240) of Gradshteyn and Ryzhik (2007):

$$\begin{aligned}&D_{-2}\left( -\frac{\rho (\varvec{h})g_j}{\sqrt{(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}}\right) \nonumber \\&\quad =e^{-\frac{\rho ^2(\varvec{h})g^2_j}{4(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}}\nonumber \\&\qquad \times {}_1F_1\left( 1;\frac{1}{2};\frac{\rho ^2(\varvec{h})g^2_j}{2(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}\right) \nonumber \\&\qquad +\frac{\sqrt{2\pi }\rho (\varvec{h})g_j}{2\sqrt{(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}}e^{-\frac{\rho ^2(\varvec{h})g^2_j}{4(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}}\nonumber \\&\qquad \times {}_1F_1\left( \frac{3}{2};\frac{3}{2};\frac{\rho ^2(\varvec{h})g^2_j}{2(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}\right) . \end{aligned}$$

(34)

by combining Eqs. (34) and the integral of (33) and using (7.621.4) of Gradshteyn and Ryzhik (2007), we obtain

$$\begin{aligned} A_1&=\frac{(1-\rho ^2(\varvec{h}))^{1/2}}{\pi [1-(1-\rho ^2(\varvec{h}))h]}\int \limits _{\mathbb {R}_+}g_j e^{-\frac{[1-(1-\rho ^2(\varvec{h}))h]g_j^2}{2(1-\rho ^2(\varvec{h}))}}\nonumber \\&\quad {}_1F_1\left( 1;\frac{1}{2};\frac{\rho ^2(\varvec{h})g^2_j}{2(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}\right) {\mathrm{d}}g_j\nonumber \\&\qquad +\frac{\sqrt{2\pi }\rho (\varvec{h})}{2\pi [1-(1-\rho ^2(\varvec{h}))h]^{3/2}}\int \limits _{\mathbb {R}_+}g_j^2 e^{-\frac{[1-(1-\rho ^2(\varvec{h}))h]g_j^2}{2(1-\rho ^2(\varvec{h}))}}\nonumber \\&\quad {}_1F_1\left( \frac{3}{2};\frac{3}{2};\frac{\rho ^2(\varvec{h})g^2_j}{2(1-\rho ^2(\varvec{h}))[1-(1-\rho ^2(\varvec{h}))h]}\right) {\mathrm{d}}g_j\nonumber \\&\quad =\frac{(1-\rho ^2(\varvec{h}))^{3/2}}{\pi [1-(1-\rho ^2(\varvec{h}))h]^2} {}_2F_1\left( 1,1;\frac{1}{2};\frac{\rho ^2(\varvec{h})}{[1-(1-\rho ^2(\varvec{h}))h]^2}\right) \nonumber \\&\qquad +\frac{\rho (\varvec{h})(1-\rho ^2(\varvec{h}))^{3/2}}{2[1-(1-\rho ^2(\varvec{h}))h]^3}{}_2F_1\left( \frac{3}{2},\frac{3}{2};\frac{3}{2};\frac{\rho ^2(\varvec{h})}{[1-(1-\rho ^2(\varvec{h}))h]^2}\right) . \end{aligned}$$

(35)

Similarly, the second integral of (32) is given by

$$\begin{aligned} A_2&=\frac{(1-\rho ^2(\varvec{h}))^{3/2}}{\pi [1-(1-\rho ^2(\varvec{h}))h]^2} {}_2F_1\left( 1,1;\frac{1}{2};\frac{\rho ^2(\varvec{h})}{[1-(1-\rho ^2(\varvec{h}))h]^2}\right) \nonumber \\&\quad -\frac{\rho (\varvec{h})(1-\rho ^2(\varvec{h}))^{3/2}}{2[1-(1-\rho ^2(\varvec{h}))h]^3} {}_2F_1\left( \frac{3}{2},\frac{3}{2};\frac{3}{2};\frac{\rho ^2(\varvec{h})}{[1-(1-\rho ^2(\varvec{h}))h]^2}\right) . \end{aligned}$$

(36)

Combining Eqs. (35), (36) in (32), we obtain

$$\begin{aligned} \mathrm{I\! E}\left( |T_{h}(\varvec{s}_{i})||T_{h}(\varvec{s}_{j})|\right)&=\frac{2\left( 1-\rho ^2(\varvec{h})\right) ^{3/2}}{\pi [1-(1-\rho ^2(\varvec{h}))h]^2} {}_2F_1\left( 1,1;\frac{1}{2};\frac{\rho ^2(\varvec{h})}{[1-(1-\rho ^2(\varvec{h}))h]^2}\right) . \end{aligned}$$

Finally, we use the identity:

$$\begin{aligned} {}_2F_1\left( 1,1;\frac{1}{2};x\right) =\frac{\sqrt{x}\arcsin (\sqrt{x})+\sqrt{1-x}}{(1-x)^{3/2}} \end{aligned}$$

$\square$