A nonhomogeneous Poisson process geostatistical model

Abstract

This paper introduces a new geostatistical model for counting data under a space-time approach using nonhomogeneous Poisson processes, where the random intensity process has an additive formulation with two components: a Gaussian spatial component and a component accounting for the temporal effect. Inferences of interest for the proposed model are obtained under the Bayesian paradigm. To illustrate the usefulness of the proposed model, we first develop a simulation study to test the efficacy of the Markov Chain Monte Carlo (MCMC) method to generate samples for the joint posterior distribution of the model’s parameters. This study shows that the convergence of the MCMC algorithm used to simulate samples for the joint posterior distribution of interest is easily obtained for different scenarios. As a second illustration, the proposed model is applied to a real data set related to ozone air pollution collected in 22 monitoring stations in Mexico City in the 2010 year. The proposed geostatistical model has good performance in the data analysis, in terms of fit to the data and in the identification of the regions with the highest pollution levels, that is, the southwest, the central and the northwest regions of Mexico City.

Appendix: Prior and posterior distributions and interpolation

1.1 Prior distributions for the parameters $\beta$, $\alpha$, $\mathbf \Psi ,\sigma ^2$ and $\phi$

The Bayesian approach for the model defined by Eq. (1) to (7) must be complemented with the specification for the prior distributions of the parameters. In the same way as considered by Achcar et al. (1998), we consider the introduction of a latent variable $N_j'=N_j-n_j$. The introduction of such latent variable was proposed by Kuo and Yang (1996) to facilitate sampling from the joint posterior distributions. Similarly to the original paper, as shown in Sect. 1, the posterior distribution of $( \mathbf {N'},\Phi \mid D)$ is given directly, and the conditional distribution $(\Phi \mid D)$ is given by the marginal distribution. We assume the following prior distributions for $N_j',\beta ,\alpha ,$ $\mathbf \Psi ,\sigma ^2$ and $\phi$:

• $N'_j\sim Pois\left[ e^{W_j -\beta T_j^{\alpha }}\right]$,

• $\beta \sim G(a_{\beta },b_{\beta }),$ in which $a_{\beta }$, $b_{\beta }$ are known,

• $\alpha \sim G(a_{\alpha },b_{\alpha }),$ in which $a_{\alpha }$, $b_{\alpha }$ are known,

• ${\varvec {\Psi}} \sim N({\mathbf {m}},{\mathbf {v}}),$ in which $\mathbf {m}$, $\mathbf {v}$ are known,

• $\sigma ^2 \sim G(a_{\sigma ^2},b_{\sigma ^2}),$ in which $a_{\sigma ^2}$, $b_{\sigma ^2}$ are known,

• $\phi \sim G(a_{\phi }*\eta ,\eta ),$ in which $a_{\phi }=-2\log (0.05)/max(\mid \mathbf {s}_i-\mathbf {s}_j\mid )$, a prior distribution proposed by Schmidt and Gelfand (2003),

where $Pois(\lambda )$ denotes a Poisson distribution with parameter $\lambda$, G(a, b) denotes a gamma distribution with mean a / b and variance $a/b^2$; and $N(\mathbf {\mu },\mathbf {A})$ denotes a normal multivariate distribution with mean vector $\mathbf {\mu }$ and covariance matrix $\mathbf {A}$. Thus, by the definition of a Gaussian process, $\mathbf {W}\sim N(\mathbf {X}\mathbf \Psi ,\mathbf \Sigma )$, where $\mathbf {X}$ is an observed matrix of covariates and $\mathbf \Sigma$ is a covariance matrix in which its elements are given by $\Sigma _{ij}=\sigma ^2\rho _{\phi }(\mathbf {s}_i,\mathbf {s}_j),\;i,j=1,\ldots ,n.$ Thus, considering independence among the prior distributions of the other parameters, the prior distribution for $\mathbf \Phi$ is given by

$$\begin{aligned} \pi (\mathbf \Phi )=\pi (\beta )\pi (\alpha )\pi (\mathbf \Psi )\pi (\mathbf {W}\mid \phi ,\sigma ^2)\pi (\phi )\pi (\sigma ^2). \end{aligned}$$

(8)

1.2 Posterior distribution

Given the prior distribution (8), and combining this information with the likelihood function given in (7), the joint posterior distribution for $\mathbf \Phi$ and $N_j,j=1,\ldots ,n$ is given by

$$\begin{aligned} \pi ({\mathbf {N'}},{\varvec {\Phi}} \mid D)\propto L({\varvec {\Phi}} \mid D)\pi ({\varvec {\Phi}} )\prod _{j=1}^n\pi (N_j'), \end{aligned}$$

(9)

where $\mathbf {N'} = (N_1, \ldots , N_n)$. To overcome the difficulty of generating samples of the joint posterior distribution (9), we use MCMC simulation methods to obtain the posterior quantities of interest of this model.

1.3 Conditional distributions required for the simulation algorithm

The algorithm used to generate the posterior distribution samples in (9) is a Gibbs sampling algorithm with Metropolis–Hastings steps (Metropolis et al. 1953; Hastings 1970). The conditional posterior densities for the parameters required in the Metropolis–Hastings steps are given by

$$\begin{aligned} \pi (\beta \mid {\mathbf {N}}, {\varvec{\Phi}} _{-\beta })\propto \beta ^{ \sum _{j=1}^n n_j+a_{\beta }-1}\exp \left\{ -\beta \left( \sum _{j=1}^n \sum _{i=1}^{n_j}t_{ij}^{\alpha }+b_{\beta }+\sum _{j=1}^n N'_j T_j^{\alpha }\right) \right\} , \end{aligned}$$

$$\begin{aligned} \pi (\alpha \mid {\mathbf {N}}, {\varvec{\Phi}} _{-\alpha })\propto \alpha ^{ \sum _{j=1}^n n_j} \left[ \prod _{j=1}^n \prod _{i=1}^{n_j} t_{ij}^{\alpha } \right] \exp \left\{ -\beta \left( \sum _{j=1}^n\sum _{i=1}^{n_j} t_{ij}^{\alpha }+ \sum _{j=1}^n N'_j T_j^{\alpha }\right) \right\} \pi (\alpha ), \end{aligned}$$

$$\begin{aligned} \pi (W\mid {\mathbf {N}}, {\varvec {\Phi}} _{-W})\propto \exp \left\{ \sum _{j=1}^n n_j W_j+\sum _{j=1}^n N'_j W_j-\sum _{j}^n e^{W_j}-\frac{1}{2}(W-X\Psi)' \Sigma ^{-1}(W-X \Psi )\right\} , \end{aligned}$$

$$\begin{aligned} \pi (\Psi\mid {\mathbf {N}}, {\varvec {\Phi}} _{-\Psi})\propto \exp \left\{ -\frac{1}{2} ({\varvec{\Psi}}-{\mathbf{m}})^\prime{\mathbf{v}}^{-1} ({\varvec{\Psi}}-{\mathbf{m}}) -\frac{1}{2} ({\mathbf {W}}-{\mathbf {X}}{\varvec{\Psi}})^{\prime} {\varvec{\Sigma}} ^{-1} ({\mathbf {W}}-{\mathbf {X}}{\varvec{\Psi}})\right\} , \end{aligned}$$

$$\begin{aligned} \pi (\phi \mid {\mathbf {N}}, {\varvec {\Phi}} _{-\phi }) \propto \mid {\varvec {\Sigma}} \mid ^{-\frac{1}{2}}\exp \left\{ -\frac{1}{2}({\mathbf {W}}-{\mathbf {X}}{\varvec{\Psi }})' {\varvec \Sigma ^{-1}}({\mathbf {W}}-{\mathbf {X}}{\varvec{\Psi }}) -b_{\phi }\phi \right\} \phi ^{a_{\phi }-1}, \end{aligned}$$

$$\begin{aligned} \pi (\sigma ^2 \mid {\mathbf {N}}, {\varvec {\Phi}} _{-\sigma ^2}) \propto \mid {\varvec {\Sigma}} \mid ^{-\frac{1}{2}}\exp \left\{ -\frac{1}{2}({\mathbf {W}}-{\mathbf {X}}{\varvec{\Psi}})' {\varvec{\Sigma ^{-1}}}(\mathbf {W}-{\mathbf {X}}{\varvec{\Psi}}) -b_{\sigma ^2}\sigma ^2 \right\} ({\sigma ^2})^{a_{\sigma ^2}-1}, \end{aligned}$$

where $\Theta _{-\theta }$ denotes the set of parameters $\Theta$ excluding the parameter $\theta$.

1.4 Interpolation

The main objective of geostatistics is to make predictions or interpolation of a process under study at any location from the data observed only in a fixed finite number of locations. In this paper, the main goal is to estimate the intensity function of the process at any point in the region of interest A at a time t. Before interpolating the intensity function $\lambda$, it is necessary to interpolate the W’s values. The interpolations of W in the space A and of $\lambda (\mathbf {s},t)$ are done as follows:

1.4.1 Interpolation in $\mathbf {W}$

Let $\mathbf {W}^{NM}=(W_{n+1},\ldots ,W_{n+m})$ be a vector of values of the function $W(\cdot )$, corresponding to the geographic points $\mathbf{s_{n+1},\ldots ,s_{n+m} }\in A$ where the process is not observed. By definition of a Gaussian process, we have

$$\begin{aligned} \left( \begin{array}{l} \mathbf { W} \\ \mathbf { W}^{NM} \\ \end{array} \right) \sim N \left[ \left( \begin{array}{l} \mathbf { A_1} \\ \mathbf { A_2} \\ \end{array} \right) , \left( \begin{array}{ll} \mathbf { \Sigma _{A_{1}}} &{} \mathbf { \Sigma _{A_{12}}'} \\ \mathbf { \Sigma _{A_{12}}} &{} \mathbf { \Sigma _{A_2}} \\ \end{array} \right) \right] , \end{aligned}$$

where $\mathbf { A_1}=\mathbf { X_{A_1}} \mathbf {\Psi },$ $\mathbf { A_2}=\mathbf { X_{A_2}}\mathbf { \Psi }$, , in which $\mathbf { X_{A_1}}$ and $\mathbf { X_{A_2}}$ are the covariates matrices associated with $\mathbf {W}$ and $\mathbf { W^{NM}}$ respectively; $\mathbf {\Sigma _{A_1}}$ is the matrix the covariance of $\mathbf { W}$, $\mathbf {\Sigma _{A_2}}$ is the matrix of the covariance of $\mathbf { W^{NM}}$ and $\mathbf {\Sigma _{A_{12}}}$ is the matrix of the covariance between $\mathbf { W}$ and $\mathbf { W^{NM}}$. By the properties of the normal multivariate distribution we have

$$\begin{aligned} \mathbf { W^{NM}}\mid \mathbf { W} \sim N(\mathbf { A_2^*},\mathbf {\Sigma _{A_{2}}^*} ), \end{aligned}$$

in which $\mathbf { A_2^*}=\mathbf { A_2} +\mathbf {\Sigma _{A_{12}}'} \mathbf {\Sigma _{A_{1}}^{-1}}(\mathbf { W}-\mathbf { A_1})$ and $\mathbf {\Sigma _{A_{2}}^*}=\mathbf {\Sigma _{A_{2}}}-\mathbf {\Sigma _{A_{12}}'} \mathbf {\Sigma _{A_{1}}^{-1}}\mathbf {\Sigma _{A_{12}}}$.   

Samples of the posterior distribution of $\mathbf { W}^{NM}$ are generated from

$$\begin{aligned} \mathbf { {W^{NM}}^{(r)}}\mid \mathbf { W^{(r)}}\sim N(\mathbf { {A_2^*}^{(r)}},\mathbf {\Sigma _{A_{2}}^*}^{(r)} ), \end{aligned}$$

where $\mathbf { {A_2^*}}^{(r)}=\mathbf {{A_2}}^{(r)} +\mathbf {\Sigma _{A_{12}}'}^{(r)} \mathbf {\Sigma _{A_{1}}^{-1}}^{(r)}(\mathbf { W}^{(r)}-\mathbf { {A_1}}^{(r)})$ and $\mathbf {\Sigma _{A_{2}}^*}^{(r)}=\mathbf {\Sigma _{A_{2}}}^{(r)}-\mathbf {\Sigma _{A_{12}}'}^{(r)} \mathbf {\Sigma _{A_{1}}^{-1}}^{(r)}\mathbf {\Sigma _{A_{12}}}^{(r)}$, with $\mathbf {A_1}^{(r)}=\mathbf { X}_{A_1}\mathbf {\Psi }^{(r)},$ $\mathbf { A_2}=\mathbf { X}_{A_2}\mathbf { \Psi }^{(r)}$. The elements of $\mathbf {\Sigma _{A_{1}}}^{(r)},$ $\mathbf {\Sigma _{A_{2}}}^{(r)}$ and $\mathbf {\Sigma _{A_{12}}'}^{(r)}$ are calculated from ${\Sigma ^{(r)}}_{jl}={\sigma ^2}^{(r)}\exp [-\phi ^{(r)}\mid s_j-s_l\mid ]$, $j,l=1,\ldots ,n+m$, where $\mathbf { W}^{(r)},\mathbf {\Psi }^{(r)},\;{\sigma ^2}^{(r)},\;\phi ^{(r)}$ ($r=1,\ldots ,M$), are the samples of the posterior distribution obtained using the MCMC algorithm.

1.4.2 Interpolation in $\lambda$

The posterior mean of $\lambda (t,\mathbf {s}_l)$ is estimated by

$$\begin{aligned} E(\lambda (t,\mathbf {s}_l))=\frac{1}{M}\sum _{r=1}^M \exp \{{W_l}^{(r)}\} (1-\exp \{\beta ^{(r)} t^{\alpha ^{(r)}}\}) , l=n+1,\ldots ,n+m. \end{aligned}$$