Spatio-temporal determinantal point processes

The covariance function of a stationary process can be represented as a Fourier transform of a positive finite measure. According to the Wiener-Khintchine theorem (see, e.g. Rasmussen and Williams, 2006), if the spectral density function $\varphi(\cdot,\cdot)$ exists, the covariance function $C(\cdot,\cdot)$ and the spectral density $\varphi(\cdot,\cdot)$ are Fourier duals of each other given by

	$\displaystyle C(u,t)$	$\displaystyle=\int_{\mathbb{R}^{d+1}}e^{2\pi i(\omega^{T}u+\tau t)}\varphi(\omega,\tau)\mathrm{d}\omega\mathrm{d}\tau,$		(6)
	$\displaystyle\varphi(\omega,\tau)$	$\displaystyle=\int_{\mathbb{R}^{d+1}}e^{-2\pi i(\omega^{T}u+\tau t)}C(u,t)\mathrm{d}u\mathrm{d}t,$		(7)

where $T$ stands for transpose, $\omega$ is the $d$-dimensional spatial component and $\tau$ is the temporal component.

A straightforward generalization of Proposition 3.1 in Lavancier et al. (2015) to the spatio-temporal setting, under continuity and stationarity of $C(u,t)$, when $C(u,t)\in\mathbb{L}^{2}(\mathbb{R}^{d}\times\mathbb{R}^{+})$, implies that a stdpp with kernel $C$ exists if the corresponding spectral density satisfies

$\displaystyle\varphi(\omega,\tau)\leq 1.$

(8)

A class of separable spatio-temporal covariance functions is usually given by

$\displaystyle C_{0}(u,t)\propto C_{0}^{s}(u)C_{0}^{t}(t),$

(9)

which is valid (i.e. a positive definite function) if both the spatial covariance function, $C_{0}^{s}(u)$, and the temporal covariance function, $C_{0}^{t}(t)$, are valid covariance functions. For a separable class of covariance functions, the spectral density $\varphi(\omega,\tau)$ has also a separable form, namely

$\displaystyle\varphi(\omega,\tau)\propto\Big{[}\int_{\mathbb{R}^{d}}e^{-2\pi i\omega^{T}u}C_{0}^{s}(u)\mathrm{d}u\Big{]}\times\Big{[}\int_{\mathbb{R}}e^{-2\pi i\tau t}C_{0}^{t}(t)\mathrm{d}t\Big{]}=\varphi_{0}^{s}(\omega)\varphi_{0}^{t}(\tau),$

where $\varphi_{0}^{s}(\omega)$ and $\varphi_{0}^{t}(\tau)$ are the spatial and temporal spectral densities, respectively. According to (8), the condition $\varphi_{0}^{s}(\omega)\varphi_{0}^{t}(\tau)<1$ must be satisfied for a stdpp with kernel (9) to exist. Further, for this class, the pair correlation function takes the following simple form

$$g(u,t)=1-\lvert R_{0}^{s}(\|u\|/\alpha_{s})\rvert^{2}\lvert R_{0}^{t}(\lvert t\rvert/\alpha_{t})\rvert^{2},$$

where $R_{0}^{s}$ and $R_{0}^{t}$ are the correlation functions in space and time corresponding to $C_{0}^{s}$ and $C_{0}^{t}$, respectively. Therefore, by (4) the corresponding $K$-function is given by

$$K(u,t)=\pi u^{2}t-\int_{0}^{u}u^{\prime}\lvert R_{0}^{s}(u^{\prime}/\alpha_{s})\rvert^{2}\mathrm{d}u^{\prime}\int_{0}^{t}\lvert R_{0}^{t}(t^{\prime}/\alpha_{t})\rvert^{2}\mathrm{d}t^{\prime}.$$

There are a large number of classes of valid spatial and valid temporal covariance functions in the literature, for example the Matérn, power exponential and Gaussian classes, to name a few (see, e.g. Cressie and Wikle, 2011). As an example, we consider the Gaussian covariance function $C_{0}^{s}(u)=\sqrt{\rho}\sigma^{2}_{s}\exp(-\|u\|^{2}/\alpha_{s})$, $u\in\mathbb{R}^{2}$, with spectral density $\varphi_{0}^{s}(\omega)=\sqrt{\rho}\pi\sigma^{2}_{s}\alpha_{s}^{2}\exp(-\pi^{2}\alpha_{s}^{2}\|\omega\|^{2})$, and the exponential covariance function $C_{0}^{t}(t)=\sqrt{\rho}\sigma^{2}_{t}\exp(-\lvert t\rvert/\alpha_{t})$, $t\in\mathbb{R}^{+},$ with spectral density $\varphi_{0}^{t}(\tau)=(2\sqrt{\rho}\sigma^{2}_{t}\alpha_{t})/(1+4\pi^{2}\alpha_{t}^{2}\lvert\tau\rvert^{2})$. Here $\sigma^{2}_{s}$ and $\sigma^{2}_{t}$ are the variance parameters of the spatial and time components, respectively, and $\alpha_{s}>0$ and $\alpha_{t}>0$ are the corresponding range parameters. A stationary stdpp with intensity $\rho$ and the separable covariance function (9) with these components, i.e

$\displaystyle C_{0}(u,t)=\rho\sigma^{2}_{s}\sigma^{2}_{t}\exp\Big{(}-\frac{\|u\|^{2}}{\alpha_{s}}-\frac{\lvert t\rvert}{\alpha_{t}}\Big{)}$

(10)

will exist if

$\displaystyle\varphi_{sep}(\omega,\tau)=\frac{2\pi\rho\alpha_{s}^{2}\alpha_{t}\sigma^{2}_{s}\sigma^{2}_{t}}{(1+4\pi^{2}\alpha_{t}^{2}\tau^{2})}\exp{\left(-\pi^{2}\alpha_{s}^{2}\|\omega\|^{2}\right)}<1.$

Since the maximum of the spectral density occurs at $(0,0)$, so a stdpp($C_{0}$) exists if $\varphi_{sep}(0,0)<1$, which implies that $\rho<(2\pi\alpha_{s}^{2}\alpha_{t}\sigma^{2}_{s}\sigma^{2}_{t})^{-1}$, and hence the maximal intensity is $\rho_{\max}=(2\pi\alpha_{s}^{2}\alpha_{t}\sigma^{2}_{s}\sigma^{2}_{t})^{-1}.$ For this process the pair correlation function is simply given by

$\displaystyle g(u,t)=1-\exp\Big{(}-\frac{2\|u\|^{2}}{\alpha_{s}}-\frac{2\lvert t\rvert}{\alpha_{t}}\Big{)}.$

(11)

The corresponding $K$-function also has a closed-form expression, which is given in A.

Figure 1 (top row) shows that the values of the pair correlation function (11) decrease by the increase of the spatial range $\alpha_{s}$ and the temporal delay $\alpha_{t}$. Thus, these parameters determine the degree of repulsion for the above separable model.

Following Fuentes et al. (2007), we consider the spatio-temporal spectral density

$\displaystyle\varphi_{\epsilon}(\omega,\tau)=\gamma(\alpha_{s}^{2}\alpha_{t}^{2}+\alpha_{t}^{2}\lvert\omega\rvert^{2}+\alpha_{s}^{2}\tau^{2}+\epsilon\lvert\omega\rvert^{2}\tau^{2})^{-\nu},$

(12)

which is an extension of the commonly used Matérn spectral density (Cressie and Wikle, 2011). Here, the non-negative parameter $\alpha_{s}^{-1}$ (spatial range) explains the rate of decay of the spatial correlation, the non-negative parameter $\alpha_{t}^{-1}$ (temporal delay) explains the rate of decay for the temporal correlation. Further, $\gamma>0$ is a scale parameter. The parameter $\nu$ measures the degree of smoothness of the process and it should be larger than $(d+1)/2$ to have a well-defined spectral density. The parameter $\epsilon$ controls the interaction between the spatial and temporal components. For $0\leq\epsilon<1$ the spectral density is non-separable while it is separable when $\epsilon=1$. The maximum of the spectral density is $\gamma(\alpha_{s}^{2}\alpha_{t}^{2})^{-\nu}$ and accordingly a stdpp with spectral density (12) exists if $\gamma<(\alpha_{s}^{2}\alpha_{t}^{2})^{\nu}$.

In the separable case, i.e. when $\epsilon=1$, the spectral density is given by

$\displaystyle\varphi_{\epsilon=1}(\omega,\tau)=\gamma(\alpha_{s}^{2}+\|\omega\|^{2})^{-\nu}(\alpha_{t}^{2}+\tau^{2})^{-\nu}=\varphi_{0}^{s}(\omega)\varphi_{0}^{t}(\tau).$

(13)

In this case, $\varphi_{0}^{s}(\omega)$ and $\varphi_{0}^{t}(\tau)$ are Matérn-type spectral densities in space and time, respectively. Consequently, the corresponding separable spatio-temporal covariance function, combining (6), and (13) and using the equations 6.726.4 and 8.432.5 in Gradshteyn and Ryzhik (2007) and setting $d=2$, is given by

	$\displaystyle C_{0}(u,t)$	$\displaystyle=\frac{4\gamma\pi\sqrt{\pi}}{2^{2\nu-1/2}(\Gamma(\nu))^{2}}\Big{(}\frac{2\pi\lvert t\rvert}{\alpha_{t}}\Big{)}^{\nu-1/2}\Big{(}\frac{2\pi\|u\|}{\alpha_{s}}\Big{)}^{\nu-1}$
		$\displaystyle\times\mathcal{K}_{\nu-\frac{1}{2}}\Big{(}2\pi\alpha_{t}\lvert t\rvert\Big{)}\mathcal{K}_{\nu-1}\Big{(}2\pi\alpha_{s}\|u\|\Big{)},$

where $\mathcal{K}_{\nu}(\cdot)$ is the modified Bessel function of the second kind of order $\nu$. $C_{0}(u,t)$ is proportional to the product of Matérn covariance functions in space and time. Using the special cases, $\mathcal{K}_{\frac{1}{2}}(r)=e^{-r}(2r/\pi)^{-1/2}$ and $\mathcal{K}_{\frac{3}{2}}(r)=e^{-r}(1+r^{-1})(2r/\pi)^{-1/2}$ (Abramowitz and Stegun, 1992), the above covariance function for $\nu=2$ can be presented as

$\displaystyle C_{0}(u,t)=\frac{\gamma\pi^{2}}{4\alpha_{s}^{2}\alpha_{t}^{3}}(2\pi\alpha_{t}\lvert t\rvert+1)\exp(-2\pi\alpha_{t}\lvert t\rvert)(2\pi\alpha_{s}\|u\|)\mathcal{K}_{1}(2\pi\alpha_{s}\|u\|).$

(14)

For this case, considering the fact that $\lim_{x\rightarrow 0}x\mathcal{K}_{1}(x)=1$ (Yang and Chu, 2017), the intensity of the process is $\rho=C_{0}(0,0)=(\gamma\pi^{2})/(4\alpha_{s}^{2}\alpha_{t}^{3})$. Hence, taking into account that $\gamma<\alpha_{s}^{4}\alpha_{t}^{4}$, a stdpp with kernel (14) exists if $4\rho\leq\pi^{2}\alpha_{s}^{2}\alpha_{t}$. For this separable case with $\epsilon=1$, considering (3), the pair correlation function is

$\displaystyle g(u,t)=1-(2\pi\alpha_{t}\lvert t\rvert+1)^{2}\exp(-4\pi\alpha_{t}\lvert t\rvert)\Big{[}(2\pi\alpha_{s}\|u\|)\mathcal{K}_{1}(2\pi\alpha_{s}\|u\|)\Big{]}^{2}.$

(15)

For $\epsilon\in(0,1)$ the spatio-temporal covariance function corresponding to (12) should be computed numerically as there is no exact closed-form expression. For the case $\epsilon=0$, the stationary non-separable spatial-temporal spectral density is given by

$\displaystyle\varphi_{\epsilon=0}(\omega,\tau)=\gamma(\alpha_{s}^{2}\alpha_{t}^{2}+\alpha_{t}^{2}\lvert\omega\rvert^{2}+\alpha_{s}^{2}\tau^{2})^{-\nu}.$

(16)

Combining (6) and (16), and using the equations 6.726.4 and 8.432.5 in Gradshteyn and Ryzhik (2007), the covariance function when $\epsilon=0$ is

	$\displaystyle C_{0}(u,t)$	$\displaystyle=\frac{\gamma\pi^{\frac{d+1}{2}}}{2^{\nu-\frac{d+1}{2}}\Gamma(\nu)\alpha_{s}^{(2\nu-d)}\alpha_{t}^{(2\nu-1)}}\left\{2\pi\alpha_{s}\Big{(}(\frac{\alpha_{t}}{\alpha_{s}}t)^{2}+\|u\|^{2}\Big{)}^{1/2}\right\}^{\nu-\frac{d+1}{2}}$
		$\displaystyle\times\mathcal{K}_{\nu-\frac{d+1}{2}}\left(2\pi\alpha_{s}\Big{(}(\frac{\alpha_{t}}{\alpha_{s}}t)^{2}+\|u\|^{2}\Big{)}^{1/2}\right).$

According to (8), for $d=2$ and $\nu=2$, there exists a stdpp with kernel

$\displaystyle C_{0}(u,t)=\frac{\gamma\pi^{2}}{2\alpha_{s}^{2}\alpha_{t}^{3}}\exp\left(-2\pi\Big{(}\alpha_{t}^{2}\lvert t\rvert^{2}+\alpha_{s}^{2}\|u\|^{2}\Big{)}^{1/2}\right)$

(17)

if and only if $\gamma<\alpha_{s}^{4}\alpha_{t}^{4}$. Further, it holds that $\rho=C_{0}(0,0)=(\gamma\pi^{2})/(2\alpha_{s}^{2}\alpha_{t}^{3})$ for the covariance functions (17). Thus, under the condition $\gamma<\alpha_{s}^{4}\alpha_{t}^{4}$, it holds that $2\rho\leq\pi^{2}\alpha_{s}^{2}\alpha_{t}$. Therefore, for a stdpp with the above covariance function, the intensity should be at most $\rho_{max}=\pi^{2}\alpha_{s}^{2}\alpha_{t}/2$. Further, for this process the pair correlation function is simply given by

$\displaystyle g(u,t)=1-\exp\left(-4\pi\Big{(}\alpha_{t}^{2}\lvert t\rvert^{2}+\alpha_{s}^{2}\|u\|^{2}\Big{)}^{1/2}\right).$

(18)

The expression for the corresponding $K$-function can be found in A.

Figure 1 (middle and bottom rows) shows that the values of the theoretical pair correlations (15) and (18) decrease as $\alpha_{s}^{-1}$ and $\alpha_{t}^{-1}$ increase. Thus, for these models, the parameters $\alpha_{s}^{-1}$ and $\alpha_{t}^{-1}$ play the role of spatial range and time delay that determine the degree of repulsion. Moreover, for fixed range parameters, the separable covariance model with (15) leads to smaller values of the pair correlation function and thus more repulsive patterns than the non-separable model with (18). While the separable covariance function controls to repulsiveness of points in space and time separately, in the non-separable case the points repel each other in the 3D space. This leads to the different small scale interactions.