An autoregressive point source model for spatial processes

Abstract

We suggest a parametric modeling approach for nonstationary spatial processes driven by point sources. Baseline near-stationarity, which may be reasonable in the absence of a point source, is modeled using a conditional autoregressive (CAR) Markov random field. Variability due to the point source is captured by our proposed autoregressive point source (ARPS) model. Inference proceeds according to the Bayesian hierarchical paradigm, and is implemented using Markov chain Monte Carlo (MCMC) methods. The parametric approach allows a formal test of effectiveness of the point source. Application is made to a real dataset on electric potential measurements in a field containing a metal pole and the finding is that our approach captures the pole’s impact on small-scale variability of the electric potential process.

1. INTRODUCTION

Stochastic processes across spatial domains are often modeled by decomposing them into trend and stationary error processes. However, it has become increasingly clear that an assumption of stationarity of the error process is driven more by mathematical convenience than by reality, and more practitioners are choosing to replace this convenient assumption with the more realistic assumption of nonstationarity; see, for example, Sampson and Guttorp (1992), Haas (1995), Cressie and Majure (1997), Higdon et al. (1999), Fuentes (2002), and Wikle (2004). Nonstationary error models are difficult to develop and estimate because deviations from stationarity can occur in many ways. They require direct or indirect methods for describing variance and correlation as they change over the index set of the process, while guaranteeing that linear transformations of the process will have non-negative variances. Auxiliary information, if available, can ease the difficulty associated with modeling nonstationary error processes.

In this paper, we consider the case where a point source impacts the stochastic process of interest. Auxiliary information provided by a point source and the resulting impact on the covariance function of a process has received only minimal attention. In epidemiology, relatively recent activities have led to a large body of literature relating point sources to disease “hot spots.” The primary goal is to test whether or not clusters of disease cases in space and/or time are significant, after accounting for chance variations. In their nomenclature, a “focused” test is used to detect significant clustering around a point source exposure to a pollutant that is presumed to increase disease risk. These efforts typically model disease incidence using the Cox process based on inhomogeneous Poisson processes. A few key references are Diggle and Rowlingson (1994), Bithell (1995), Lawson (1995, 2001), Lawson and Waller (1996), and Diggle et al. (1997). Hughes-Oliver et al. (1998a, 1998b), and Hughes-Oliver and Gonzalez-Farias (1999) discuss the effects of point sources on a deposition process in semiconductor manufacturing and measuring electric potential in a field.

We propose a hierarchical Bayesian approach to an extension of the process decomposition model suggested by Hughes-Oliver and Gonzalez-Farias (1999). There are three major benefits afforded by the Bayesian paradigm. First, it is able to account for uncertainty in parameter estimates when evaluating prediction uncertainty; frequentist approaches are highly criticized for their inability to deal with this issue. Second, Bayesian inference does not require derivation of asymptotic properties (as sample size increases) of estimators because results are based entirely on simulated observations from relevant posterior distributions. Third, the Bayesian paradigm allows the incorporation of prior information that may not be as easily incorporated in a frequentist approach. There are also benefits of the process decomposition model. This model provides a parametric method for decomposing the observed process into a trend surface, a baseline error process, and an additional error process (associated with the point source) that may be viewed as a “shock” to the baseline. Nonstationarity is easily modeled by this additional error process since no restrictions are placed on the forms of their covariance functions.

We also introduce a newly created autoregressive point source (ARPS) process capturing the effect of a point source on the error component of a process. This process is attractive for at least three reasons. First, it allows site-specific variances to be a function of proximity to source, where no restriction is placed on definition of distance; that is, any relevant distance metric may be used. Second, site-to-site correlations may be a function of proximity to source. Third, and possibly most attractive, this model is parametric and lends itself to testing the statistical significance of point source impacts on the error component of a process. We use the data from Hughes-Oliver and Gonzalez-Farias (1999) to illustrate our proposed method for modeling the effect of a point source.

Section 2 contains a description of the process decomposition approach. Section 3 introduces our approach, the so-called ARPS process, for modeling the effect of a point source on the covariance structure. The full, overall covariance structure is explored in Section 4. Section 5 contains modeling details for electric potential in a field containing a metal pole. Concluding remarks are presented in Section 6.

2. PROCESS DECOMPOSITION

Suppose {Y(s) : s ∈ D ⊆ R²} is the stochastic process of interest and D is the index set, such that Y(s) is the response at site s. That is, s = (x, y), where x may be longitude and y may be latitude. Covariates for this process are given as {X(s) : s ∈ D ⊆ R², X is q-dimensional}. These q covariates may be fixed or random and will often include distance to the point source located at P. Given covariates, the Y(·) process is decomposed into trend and error as

$$Y(s)=\mu (s)+Z(s),$$ (1)

where μ(s) = f (X(s), β) (the trend) is the expectation of Y(s) conditioned on covariates X(s) at site s and Z(s) (the error or detrended response) has zero expectation. Effect of the point source on trend or large-scale variation is modeled by judicious choice of the function f(·,·). This is commonly done using a linear or non-linear function where the functional form of f(·,·) is completely known and only the vector β is unknown. Generalized additive models GAMs, (Hastie and Tibshirani, 1990) may also be used as a first-stage approximation of the trend assuming errors are independent and identically distributed; see Holland et al. (2000) for an application of GAMs to environmental data.

It is also possible, however, that the point source affects the covariance or small-scale variation; see Hughes-Oliver et al. (1998a, 1998b), and Hughes-Oliver and Gonzalez-Farias (1999) for discussions and datasets. This effect leads to covariances that are functions of the distances (and possibly angles) between the sites and source. The resulting nonstationarity of Y(s), where the covariance is site-dependent, is modeled using Z(s).

Suppose that Z(s) is the result of a baseline zero-mean process Z₀(s) that has been “shocked” by the zero-mean point source process Z₁(s) in such a way that

$$Z(s)=Z_0(s)+Z_1(s).$$ (2)

Suppose further that processes Z₀ and Z₁ are independent and that Cov_i(·,·) is the covariance function for Z_i, that is, Cov_i(s, t) = Cov(Z_i(s), Z_i(t)). The covariance function for Z is then Cov(s, t) = Cov₀(s, t) + Cov₁(s, t).

Any valid covariance may be assigned to the Cov_i and this will automatically guarantee that the covariance of Z is valid, thus linear transformations of the Z process will have non-negative variances. The choice of Cov_i should be motivated by the data in such a way that the baseline process is well represented and parameters of Cov_i measure the strength of importance of the source. Specific features may be built into the Cov_i in a variety of ways. Suppose the effect of source P on error process Z is such that the detrended response is the same at all sites equi-distant from P; note this does not imply that the effect of the point source on the response is necessarily taken to be constant at all sites equi-distant from P, only that the errors are constant. Then the required two-dimensional covariance Cov_i(s,t) may be built from a one-dimensional covariance r₁(d_s, d_t), where d_s ≡ ‖s − P‖ is the Euclidean distance between site s and the source. This is because {Z₁(s) : s ∈ D ⊆ R²} may be written as a process $\{Z_1^{*}(d_s):d_s\in R\}$ on the real line; see Hughes-Oliver and Gonzalez-Farias (1999) for details. Another approach to building Cov₁(s, t) is to explore the local behavior of the process around the source by means of a conditional autoregressive (CAR) model based on special neighbor relationships, as in Heo and Hughes-Oliver (2004). In the next section, we propose a spatial autoregressive random effects model for Z₁ based on a point source.

3. AUTOREGRESSIVE POINT SOURCE MODELING

Building on the process decomposition model in Equations (1) and (2), we propose using an autoregressive random effects model to capture the effect of the point source on error process Z₁. For this point source at P, we create a sequence of concentric regions R_k ⊆ D for k = 1,…, r ≤ n, such that P ∈ R_r and the outer boundary of R_k is the inner boundary of R_k−1 for k = 2,…, r, where R_k ⋂ R_k′ = ϕ, for all k ≠ k′; see Figure 1 for an illustration of such concentric regions. Next, we assume that the (random) spatial effect due to point source P for the region R_k is captured by α_k. In other words, {R₁,…, R_r} forms a partition of the index set D and the process Z₁ is constant across each of these regions. It is important to realize that this assumption of constancy applies to the error process and not the response. In fact, the trend term of Equation (1) allows the mean response to vary within any particular region R_k. Finally, the random effects α₁,…, α_r arise as an AR(1) process. This assumption is based on the fact that the spatial effect due to region R_k depends on the two neighboring regions R_k−1 and R_k+1. More specifically, the error process caused by the point source P may be written as

$$\begin{array}{c}{Z}_1(s)={\displaystyle \sum _{k=1}^r{\alpha }_{k}I(s\in R_k)}\hfill \\ {\alpha }_1\sim N(0,{\sigma }_1^2), {\sigma }_1^2>0\hfill \\ {\alpha }_k|{\alpha }_{k-1}\sim N(\psi {\alpha }_{k-1},{\sigma }_2^2), {\sigma }_2^2>0, k=2,\cdots ,r\hfill \end{array}$$ (3)

Figure 1.

Illustration of regions for the autoregressive point source process

In the ensuing discussion, we will refer to the model in Equation (3) as ARPS $(\psi ,{\sigma }_1^2,{\sigma }_2^2)$, where ARPS stands for autoregressive point source.

To understand the impact of Equation (3) on the covariance Cov_i(·,·), let Z₁ = [Z₁(s₁),…, Z₁(s_n)]′ be the vector of responses from the point source process over all n sampling sites. Then ∑₁ ≡ Var(Z₁) has (i, j) th element

$${\displaystyle \sum _{k=1}^r{\displaystyle \sum _{m=1}^{r}I(s_i\in R_k)}I(s_j\in R_m)\text{Cov}({\alpha }_k,{\alpha }_m)}$$

where, assuming m ≥ k,

$$\text{Cov}({\alpha }_k,{\alpha }_m){\psi }^{m-k}({\sigma }_1^2{\psi }^{2(k-1)}+{\sigma }_2^2{\displaystyle \sum _{q=0}^{k-2}{\psi }^{2q}})$$ (4)

Suppose ${\mathbf{Z}}_1^{†}$ contains the reordered entries of Z₁ such that ${\mathbf{Z}}_1^{†}={[{\mathbf{Z}}_{1_1},{\mathbf{Z}}_{1_2},\cdots ,{\mathbf{Z}}_{1_r}]}^{\prime }$, where Z_1k corresponds to the vector for all n_k sites in region R_k. Then, the variance–covariance matrix for ${\mathbf{Z}}_1^{†}$ is the n × n matrix

$$\begin{array}{cc}{\sum }_1^{†}=\mathit{\text{Var}}(Z_1^{†})\hfill & =\left[\begin{array}{ccccc}\hfill {\delta }_{1,1}{J}_{n_1\times n_1}\hfill & \hfill {\delta }_{1,2}{J}_{n_1\times n_2}\hfill & \hfill {\delta }_{1,3}{J}_{n_1\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\delta }_{1,r}{J}_{n_1\times n_r}\hfill \\ \hfill {\delta }_{1,2}{J}_{n_2\times n_1}\hfill & \hfill {\delta }_{2,2}{J}_{n_2\times n_2}\hfill & \hfill {\delta }_{2,3}{J}_{n_2\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\delta }_{2,r}{J}_{n_2\times n_r}\hfill \\ \hfill {\delta }_{1,3}{J}_{n_3\times n_1}\hfill & \hfill {\delta }_{2,3}{J}_{n_3\times n_2}\hfill & \hfill {\delta }_{3,3}{J}_{n_3\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\delta }_{3,r}{J}_{n_3\times n_r}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\delta }_{1,r}{J}_{n_r\times n_1}\hfill & \hfill {\delta }_{2,r}{J}_{n_r\times n_2}\hfill & \hfill {\delta }_{3,r}{J}_{n_r\times n_3}\hfill & \hfill \dots \hfill & \hfill {\delta }_{r,r}{J}_{n_r\times n_r}\hfill \end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{ccccc}\hfill {\delta }_{1,1}{J}_{n_1\times n_1}\hfill & \hfill \psi {\delta }_{1,1}{J}_{n_1\times n_2}\hfill & \hfill {\psi }^2{\delta }_{1,1}{J}_{n_1\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\psi }^{r-1}{\delta }_{1,1}{J}_{n_1\times n_r}\hfill \\ \hfill \psi {\delta }_{1,1}{J}_{n_2\times n_1}\hfill & \hfill {\delta }_{2,2}{J}_{n_2\times n_2}\hfill & \hfill \psi {\delta }_{2,2}{J}_{n_2\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\psi }^{r-2}{\delta }_{2,2}{J}_{n_2\times n_r}\hfill \\ \hfill {\psi }^2{\delta }_{1,1}{J}_{n_3\times n_1}\hfill & \hfill \psi {\delta }_{2,2}{J}_{n_3\times n_2}\hfill & \hfill {\delta }_{3,3}{J}_{n_3\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\psi }^{r-3}{\delta }_{3,3}{J}_{n_3\times n_r}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\psi }^{r-1}{\delta }_{1,1}{J}_{n_r\times n_1}\hfill & \hfill {\psi }^{r-2}{\delta }_{2,2}{J}_{n_r\times n_2}\hfill & \hfill {\psi }^{r-3}{\delta }_{3,3}{J}_{n_r\times n_3}\hfill & \hfill \dots \hfill & \hfill {\delta }_{r,r}{J}_{n_r\times n_r}\hfill \end{array}\right],\hfill \end{array}$$

where J is a matrix with all elements equal to 1, ${\delta }_{1,1}={\sigma }_1^2,{\delta }_{k,k}={\psi }^2{\delta }_{k-1,k-1}+{\sigma }_2^2$ for k = 2, 3,…, r, and δk,m = ψδ_k,m−1 for k = 1,…, r − 1 and m = k + 1,…, r. If ψ = ±1, the variance–covariance matrix $\mathit{\text{Var}}({\mathbf{Z}}_1^{†})$ takes a simpler form with ${\delta }_{k,m}={\sigma }_1^2+(k-1){\sigma }_2^2$ if ψ = 1 and ${\delta }_{k,m}={(-1)}^{m-k}[{\sigma }_1^2+(k-1){\sigma }_2^2]$ if ψ = −1, for k ≤ m. Note that $\mathit{\text{Var}}({\mathbf{Z}}_1^{†})$ is always singular when r<n. Also note that there is relatively little information available for estimating ${\alpha }_1^2$, so in a Bayesian framework one needs to either specify a very informative prior for ${\alpha }_1^2$ or impose a constraint such as a relationship between ${\alpha }_1^2$ and one or more of the other parameters; we develop this idea more fully in Section 4.

Alternative expressions for the ARPS model allows it to benefit from insights commonly gained for mixed effect models. Specifically, we can write Z₁ as

$${\mathbf{Z}}_1=\mathrm{\Delta }\alpha ,$$

where α = (α₁,…, α_r)′, Δ = ((δ_ik)), and δ_ik = I(s_i ∈ R_k) for i = 1,…, n, k = 1,…, r. Consequently,

$${\sum }_1=\mathrm{\Delta }C\mathrm{\Delta }\prime$$

where C = ((c_km)) and c_km = Cov(α_k, α_m) is as specified in (4), for k,m = 1,…, r. Because rank(∑₁) = rank(C) = r, it is clear that ∑₁ is singular when r<n.

The ARPS model in Equation (3) allows for two types of nonstationarity that we loosely call variance nonstationarity and correlation nonstationarity. Variance nonstationarity is nonconstant variance and correlation nonstationarity is correlation that is not a function only of site-to-site distance. If ${\sigma }_1^2>{\sigma }_1^2{\psi }^2+{\sigma }_2^2$, then variance increases as sites are further from the source. On the other hand, variance is constant when ${\sigma }_1^2={\sigma }_1^2{\psi }^2+{\sigma }_2^2$ and decreases when ${\sigma }_1^2<{\sigma }_1^2{\psi }^2+{\sigma }_2^2$. If ψ ≠ 0, then correlation is a function of both site-to-site and site-to-source distances from P, no matter what relationships exist between ${\alpha }_1^2,{\alpha }_2^2$, and ψ. Consequently, we test the impact of point source P on small-scale variability by testing the null hypothesis that

$$\psi =0\text{and}{\sigma }_1^2={\sigma }_2^2$$

against an appropriate alternative; under these conditions, the process exhibits neither variance nonstationarity nor correlation nonstationarity. Using slightly different notation, let

$$\eta =\frac{{\sigma }_1^2}{{\sigma }_1^2{\psi }^2+{\sigma }_2^2}.$$

Then,

• η > 1 and ψ = 0 implies there is no correlation and the point source causes a single increase in variance from R₂ to R₁ (variance is the same in R₂, …, R_r);

• η > 1 and ψ ≠ 0 implies variance increases as sites are further from source P and correlation is a function of site-to-source distances;

• η = 1 and ψ = 0 implies that variance is constant and there is no correlation, that is, the point source does not affect small-scale variability of the process Y;

• η = 1 and ψ ≠ 0 implies that variance is constant and correlation is a function of site-to-source distances;

• η<1 and ψ = 0 implies there is no correlation and the point source causes a single decrease in variance from R₂ to R₁ (variance is the same in R₂,…, R_r); and

• η<1 and ψ ≠ 0 implies variance decreases as sites are further from source P and correlation is a function of site-to-source distances.

We can thus test the impact of point source P on small-scale variability by testing the null hypothesis that ψ ≠ 0 and η = 1 against any of the above bulleted alternatives.

4. RESTRICTIONS ON THE PARAMETER SPACE

Process decomposition, in particular Equation (2), separates the overall error process into two component processes, one representing the baseline (when no source acts on the system) and the other representing the point source. Point source process Z₁ was discussed in Section 3 with a comment made that restrictions are necessary for one of its parameters. This need is discussed here.

The baseline process Z₀ is assumed to include a measurement error (ME) component and a residual component. In other words, using ∑₀ to represent Var(Z₀), where Z₀ = [Z₀(s₁),…, Z₀(s_n)]′, we get

$${\sum }_0={\sum }_{00}+{\sum }_{01}={\sigma }_e^2{I}_n+{\sum }_{01},$$

and consequently

$$\sum ={\sigma }_e^2{I}_n+{\sum }_{01}+{\sum }_1$$

is the overall covariance matrix of the Y process based on Equations (1) and (2).

Let ${\mathbf{Z}}_0^{†}={[{\mathbf{Z}}_{0_1},{\mathbf{Z}}_{0_2},\cdots ,{\mathbf{Z}}_{0_r}]}^{\prime }$ follow the order of ${\mathbf{Z}}_1^{†}$ as described in Section 3. If the Z₀ process is i.i.d (making ∑₀₁ the zero matrix) and Z₁ is ARPS $(1,{\sigma }_1^2,{\sigma }_2^2)$, then

$$\begin{array}{cc}{\sum }^{†}\hfill & ={\sigma }_e^2{I}_n+{\sum }_1^{†}\hfill \\ \hfill & ={\sigma }_e^2{I}_n+\left[\begin{array}{ccccc}\hfill {\sigma }_1^2{J}_{n_1\times n_1}\hfill & \hfill {\sigma }_1^2{J}_{n_1\times n_2}\hfill & \hfill {\sigma }_1^2{J}_{n_1\times n_3}\hfill & \hfill \cdots \hfill & \hfill {\sigma }_1^2{J}_{n_1\times n_r}\hfill \\ \hfill {\sigma }_1^2{J}_{n_2\times n_1}\hfill & \hfill ({\sigma }_1^2+{\sigma }_2^2)J_{n_2\times n_2}\hfill & \hfill ({\sigma }_1^2+{\sigma }_2^2)J_{n_2\times n_3}\hfill & \hfill \cdots \hfill & \hfill ({\sigma }_1^2+{\sigma }_2^2)J_{n_2\times n_r}\hfill \\ \hfill {\sigma }_1^2{J}_{n_3\times n_1}\hfill & \hfill ({\sigma }_1^2+{\sigma }_2^2)J_{n_3\times n_2}\hfill & \hfill ({\sigma }_1^2+2{\sigma }_2^2)J_{n_3\times n_3}\hfill & \hfill \cdots \hfill & \hfill ({\sigma }_1^2+2{\sigma }_2^2)J_{n_3\times n_r}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\sigma }_1^2{J}_{n_r\times n_1}\hfill & \hfill ({\sigma }_1^2+{\sigma }_2^2)J_{n_r\times n_2}\hfill & \hfill ({\sigma }_1^2+2{\sigma }_2^2)J_{n_r\times n_3}\hfill & \hfill \cdots \hfill & \hfill ({\sigma }_1^2+[r-1){\sigma }_2^2]J_{n_r\times n_r}\hfill \end{array}\right]\hfill \end{array}$$

In this case, the scale parameter ${\sigma }_e^2$ always appears with scale parameter ${\sigma }_1^2$ as ${\sigma }_e^2+{\sigma }_1^2$. To improve identifiability, we can place a restriction on either ${\sigma }_e^2,{\sigma }_1^2$, or both. A commonly used restriction in time series analysis is one that guarantees stationarity of the one-dimensional autoregressive process, namely ${\sigma }_1^2={\sigma }_2^2/(1-{\psi }^2)$. This restriction would force η = 1 and consequently not allow variances of the Z₁ process to differ based on distance from source. Finding this unacceptable, we settled on the alternative restriction ${\sigma }_1^2={\sigma }_e^2$. This is quite reasonable given that ${\sigma }_e^2$ represents measurement variability of the baseline process Z₀. Region R₁, whose variance is ${\sigma }_1^2$, is farthest from the source and is thus least affected by that source. Depending on the strength of impact of the source and the relative size of study region D, we argue that R₁ should exhibit features very similar to the baseline process Z₀, thus supporting the restriction ${\sigma }_1^2={\sigma }_e^2$. We use this restriction throughout.

5. MODELING ELECTRIC POTENTIAL IN A FIELD CONTAINING A METAL POLE

5.1. Dataset and candidate models

To illustrate some details of our approach, we now describe a real dataset from Hughes-Oliver and Gonzalez-Farias (1999) in which a point source acts as a catalyst for the response. Quoting these authors (pp. 63–64), “measurements [of electric potential] are taken at sites falling on a regular grid, as shown in [Figure 2], where the sites are one meter apart in both the vertical and horizontal directions. …[electric potential] is expected to be fairly constant across the field, but an existing metal pole affects the measuring device so that the constant pattern in the field is not observable. It is in this sense that we consider the metal pole to be a point source. … [electric potential] appears to be a function only of distance to the point source, and because the contours are approximately circular, there is no apparent need for rotating or rescaling the axes.”

Figure 2.

Contours of electric potential measurements, g, in a field containing a metal pole. The measurement sites (●) fall on a regular grid, with spacings of one meter in both directions. The metal pole is located at (12,33.4) in the original coordinate system, and at (0,0) in the translated coordinate system. The contours represent the following sample percentiles: .7, 1, 2.5, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90

Let Y(s) denote the following transformed value of electric potential at site s, g(s)

$$Y(s)=\frac{46300-g(s)}{4630}$$

This simple rescaling moderates extreme behavior in our Markov chain Monte Carlo (MCMC) techniques discussed below. Another view of the data is given in Figure 3, where both electric potential and transformed electric potential are plotted as a function of distance from the point source. This figure clearly suggests that the mean and variance of Y(s) are decreasing functions of distance from the point source. Using d(s) to represent Euclidean distance between s and source P, we use Equation (1) with

$$\mu (s)={\beta }_0+\frac{{\beta }_1}{d(s)}$$

to describe the trend surface and

$$Z(s)=Z_{00}(s)+Z_{01}(s)+Z_1(s)$$

to describe the error process as in Equation (2) for Z₀(_s) = Z₀₀(_s) + Z₀₁(s). In this error process, Z₀₀ is an ME process and Z₀₁ captures the effect of spatial proximity in the baseline process; Z₀₀ and Z₀₁ together represent the baseline process Z₀ discussed in Sections 2 and 4. The process Z₁ captures the effect of the point source, based on r = 10 concentric regions from the metal pole. ARPS regions R₁, …, R₁₀ are indicated in Figure 3, along with boxplots of the transformed electric potential y within these regions. The processes Z₀₀, Z₀₁, and Z₁ all have mean zero and are assumed independent, conditioned on the mean function μ(·).

Figure 3.

Electric potential (plotted as bullets) as a function of distance between measurement sites and the metal pole. Locations of ARPS regions R₁, …, R₁₀ and boxplots of transformed electric potential within these regions are also shown

We begin by assuming normality of the conditional Y process. Normality can easily be replaced by a fat-tailed distribution like the Student’s t, but this would require an adjustment in interpretation of the variance–covariance matrix by incorporating a scale factor; we do not pursue this here. The Z₀₀ process is assumed to follow a normal distribution with constant variance ${\sigma }_e^2$; the Z₀₁ process is assumed to follow a normal CAR model; and the Z₁ process follows an ARPS model. More specifically, suppose D is an index set containing sites s₁,…, s_n on the regular grid of Figure 2, where n = 160. Then, the Bayesian hierarchical model may be written as follows:

$$\begin{array}{c}Y(s_j)\sim \text{Normal}(Y^{*}(s_j),{\sigma }_e^2),j=1,\dots ,n\hfill \\ Y^{*}(s_j)={\beta }_0+\frac{{\beta }_1}{d(s_j)}+Z_{01}(s_j)+Z_1(s_j)\hfill \\ Z_{01}(·)~\mathit{\text{CAR}}(\rho ,{\sigma }_c^2)\hfill \\ Z_1(·)~\mathit{\text{ARPS}}(\psi ,{\sigma }_1^2={\sigma }_e^2,{\sigma }_2^2)\hfill \end{array}$$ (5)

The ME variance is ${\sigma }_e^2$. While the $\text{CAR}(1,{\sigma }_c^2)$ process is more popular, we prefer the more general $\text{CAR}(\rho ,{\sigma }_c^2)$ process (Sun et al., 2000) for several reasons. First, because −1 ≤ ρ ≤ 1, the former is actually a special case of the latter. Second, the value of ρ indicates the strength of spatial correlation. And third, when −1 < ρ < 1, $\text{CAR}(\rho ,{\sigma }_c^2)$ yields a positive definite covariance structure for Z₀₁, unlike the semidefinite covariance offered by $\text{CAR}(1,{\sigma }_c^2)$. The $\text{CAR}(\rho ,{\sigma }_c^2)$ process specifies conditional distributions as

$$Z_{01}(s_j)|\{Z_{01}(s_k),j\ne k\}~\text{Normal}\left(\rho {\displaystyle \sum _{k=1}^n{b}_{\mathit{\text{jk}}}{Z}_{01}}(s_k),{\epsilon }_j^2\right),$$

where b_jj = 0, b_jk = (1/N_j)I (locations s_j and s_k are neighbors), N_j is the number of neighbors of s_j, ${\epsilon }_j^2={\sigma }_c^2/N_j$, and ${\sigma }_c^2$ is a scale parameter measuring smoothness. The overall variance of the Z₀₁ process is obtained from ${\sum }_{01}^{-1}=M^{-1}(I-\rho B)$, where $B={((b_{\mathit{\text{jk}}}))}_{j,k=1}^n$ and M is a diagonal matrix with elements ${\epsilon }_1^2,\cdots ,{\epsilon }_n^2$. Carlin and Banerjee (2002) argue that negative smoothness parameters are not desirable, so they use 0 < ρ < 1.

We use proper priors for all hyper parameters, $\theta =({\beta }_0,{\beta }_1,{\sigma }_e^2,\rho ,{\sigma }_c^2,\psi ,{\sigma }_2^2)$, as outlined below:

$$\begin{array}{l}{\beta }_0~\text{Normal}(0,\mathrm{10\; 000})\\ {\beta }_1~\text{Normal}(0,\mathrm{10\; 000})\\ {\sigma }_e^2~\text{Inverse}\text{Gamma}(0.001,0.001)\\ \rho ~\text{Uniform}(0,1)\\ {\sigma }_c^2~\text{Inverse}\text{Gamma}(.01,.01)\\ \psi ~\text{Uniform}(-0.1,2)\\ {\sigma }_2^2~\text{Inverse}\text{Gamma}(0.001,0.001).\end{array}$$

The six modeling scenarios considered here are summarized in Table 1. Model 1 recognizes no spatial patterns in small-scale variability, neither in the baseline process nor due to the point source; Model 1 uses ordinary least squares estimation. Models 2, 4, and 6 all include point source covariance models. For Model 2, no spatial proximity modeling is used for the baseline process. Models 3 and 4 both fit the singular CAR model, while Models 5 and 6 fit the more general CAR model.

Table 1.

Modeling cases for electric potential in a field

Model*	CAR(1,τ)	CAR(ρ,τ)	ARPS(ψ,σ2, ${\sigma }_2^2$)
(1) trend + ME
(2) trend + ME + ARPS			x
(3) trend + ME + CAR1	x
(4) trend + ME + CAR1 + ARPS	x		x
(5) trend + ME + CARρ		x
(6) trend + ME + CARρ + ARPS		x	x

Posterior distributions are obtained using MCMC in softwareWinBUGS Version 1.3 (Spiegelhalter et al., 2000). This version of WinBUGS provides a function for $\text{CAR}(\rho ,{\sigma }_c^2)$, but only for deterministic ρ. We hard-coded the cases where ρ was allowed to be random, namely, Models 5 and 6. WinBUGS offers many useful convergence measures that can be conveniently summarized using the CODA module (Best et al., 1996) running under R or SPlus.

Using this Bayesian hierarchical approach, we implement a Gibbs sampler method to sample from the full conditional distributions of all these parameters. While all hyperpriors are proper, Models 2, 4, and 6 contain the singular ARPS and Models 3 and 4 contain the singular $\text{CAR}(1,{\sigma }_c^2)$. Presence of the ME component, however, ensures that the overall covariance of Y, given by ∑, is a positive definite matrix and hence all posterior distributions are proper. Concerns raised by Sun et al. (1999, 2000, 2001) are not applicable here. In Table 2, we report summaries of the posterior densities for each of the six models, and in Table 3 we report model diagnostics. Results are based on three chains of 70 000 iterations each after a burn-in period of 30 000 iterations.

Table 2.

Summaries of posterior densities from modeling electric potential

Parameter	Mean	MC error		2.5%	Median	97.5%
			Model 1
β0	−0.0400	1.397E-5		−0.0477	−0.0400	−0.0324
β1	0.8059	3.904E-5		0.7848	0.8059	0.8271
${\sigma }_e^2$	0.001326	3.143E-7		0.001062	0.001315	0.001653
			Model 2
β0	−0.0211	2.796E-5		−0.0299	−0.0211	−0.0123
β1	0.7281	1.036E-4		0.6985	0.7281	0.7579
${\sigma }_e^2$	0.001011	3.097E-7		0.000806	0.001003	0.001266
ψ	1.330	3.213E-3		0.5619	1.3430	1.9280
η	0.4737	1.955E-3		0.2265	0.4259	1.0580
${\sigma }_2^2$	0.000586	1.710E-6		0.000179	0.000472	0.001671
			Model 3
β0	−0.0224	4.356E-5		−0.0312	−0.0224	−0.0137
β1	0.7339	1.712E-4		0.7012	0.7340	0.7660
${\sigma }_e^2$	0.000599	1.211E-6		0.000330	0.000593	0.000906
${\sigma }_c^2$	0.001775	3.568E-6		0.001051	0.001725	0.002776
			Model 4
β0	−0.0191	4.190E-5		−0.0282	−0.0191	0.0100
β1	0.7201	1.637E-4		0.6866	0.7202	0.7534
${\sigma }_e^2$	0.000665	1.173E-6		0.000392	0.000661	0.000959
${\sigma }_c^2$	0.001454	3.927E-6		0.000815	0.001398	0.002420
ψ	1.1470	4.478E-3		0.2223	1.182	1.866
η	0.5262	3.107E-3		0.2037	0.4479	1.384
${\sigma }_2^2$	0.000588	1.498E-6		0.000180	0.000476	0.001671
			Model 5
β0	−0.0292	1.067E-4		−0.0433	−0.0295	−0.0127
β1	0.7633	2.263E-4		0.7267	0.7632	0.8001
${\sigma }_e^2$	0.000596	1.779E-6		0.000279	0.000588	0.000962
ρ	0.8527	8.813E-4		0.5397	0.8817	0.9934
${\sigma }_c^2$	0.002221	6.378E-6		0.001143	0.002147	0.003705
			Model 6
β0	−0.0204	5.191E-5		−0.0307	−0.0204	−0.0098
β1	0.7252	1.719E-4		0.6928	0.7253	0.7574
${\sigma }_e^2$	0.000554	1.527E-6		0.000265	0.000549	0.000878
ρ	0.4643	1.339E-3		0.0330	0.4630	0.9254
${\sigma }_c^2$	0.002090	5.583E-6		0.001116	0.002031	0.003378
ψ	1.3320	2.858E-3		0.5457	1.3470	1.9250
η	0.3801	1.299E-3		0.1706	0.3494	0.7814
${\sigma }_2^2$	0.000592	1.645E-6		0.000180	0.000477	0.001690

Table 3.

Diagnostic measures from modeling electric potential

Model	Criteria
	DIC (Rank)	pD	pθ	D(θ̄)	D̄	SSE (Rank)
(1) trend + ME	−606 (6)	3	3	−612	−609	0.208 (6)
(2) trend + ME + ARPS	−642 (5)	10	15	−662	−652	0.158 (5)
(3) trend + ME + CAR1	−660 (2)	82	164	−824	−742	0.093 (3)
(4) trend + ME +CAR1 + ARPS	−648 (4)	74	176	−796	−722	0.103 (4)
(5) trend + ME + CARρ	−656 (3)	89	165	−834	−745	0.092 (2)
(6) trend + ME + CARρ + ARPS	−664 (1)	90	177	−845	−755	0.086 (1)

Several checks were made on convergence of the MCMC, including running three separate chains using different starting values and observing Gelman and Rubin’s (1992) diagnostic measure R; all values were very close to unity, indicating good mixing of the chains. Trace and posterior density plots for all three chains obtained for Model 6 (the most complicated model considered) are given in Figure 4. Additional details on the effect of multiple starting values are provided in Figure 5 for ψ of Model 6, where ψ is the parameter of greatest interest in capturing effect of the point source on the error process. Recall from Section 3 that if ψ ≠ 0 then correlation is a function of both site-to-site and site-to-source distances. There is strong evidence of convergence.

Figure 4.

MCMC trace and posterior density plots for Model 6 using three chains

Figure 5.

MCMC results for ψ of Model 6. Panel (a) is the autocorrelation function of ψ for each chain. Panel (b) is the posterior histogram of ψ for each chain. Panel (c) shows side-by-side boxplots of posterior values for ψ from each chain

Recent developments in Bayesian modeling of random effects suggest that results can be very sensitive to the priors on the associated variances (Gelman, 2004; Spiegelhalter et al., 2004, chapter 5). To investigate the effect of priors on Model 6, we considered two additional priors for ${\sigma }_c^2\text{and}{\sigma }_2^2$, leading us to nine combinations of priors for these two parameters based on individual priors of uniform(0,.1) for σ_c or σ₂ (most noninformative), inverse gamma(0.001,0.001) for ${\sigma }_c^2\text{or}{\sigma }_2^2$, and inverse gamma(.01,.01) for ${\sigma }_c^2\text{or}{\sigma }_2^2$ (most informative). Figure 6 shows histograms of posterior simulations of σ₂ from the nine combinations of priors. Inference on σ₂ clearly differs according to prior specification for σ₂ (but not for σ_c), with the most informative prior leading to larger values for σ₂, but none of the inverse gamma priors considered here constrain posterior inference to the degree presented in Gelman et al. (2003, Appendix C). The parameter σ₂ is not of direct interest when determining the impact of ARPS, however, so we once again focus attention to ψ.

Figure 6.

Histogram of posterior simulations of σ₂ from models with nine different prior distributions: σ_c distributed as uniform(0,.1) (top row), ${\sigma }_c^2$ distributed as inverse gamma(.001,.001) (middle row), ${\sigma }_c^2$ distributed as inverse gamma(.01,.01) (bottom row), σ₂ distributed as uniform(0,.1) (left column), ${\sigma }_2^2$ distributed as inverse gamma(.001,.001) (middle column), ${\sigma }_2^2$ distributed as inverse gamma(.01,.01) (right column)

Figure 7 shows posterior densities of c for each of the nine combinations considered. Given a prior for ${\sigma }_2^2({\sigma }_2)$, selection of prior for ${\sigma }_c^2({\sigma }_2)$ has little effect on the posterior distribution of ψ. On the other hand, choice of prior for ${\sigma }_2^2({\sigma }_2)$ has a big effect on the model. Vague or non-informative priors allow the data to dominate, and this leads to a stronger effect, or large values, of the point source parameter ψ. As expected, very informative priors that place little prior probability on a strong ARPS process lead to small values of ψ. It is important to note, however, that 95% credible sets for ψ never include 0, which suggests that the ARPS process adds an important component to the model for this data.

Figure 7.

Posterior densities of ψ obtained from nine different combinations of priors for ${\sigma }_c^2$ (σ_c) and ${\sigma }_2^2$ (σ₂) in Model 6. Label (a) indicates all three densities corresponding to σ₂ having a uniform(0,.1) prior, the most non-informative prior considered. Likewise, labels (b) and (c) indicate all three densities corresponding to ${\sigma }_2^2$ having inverse gamma(.001,.001) or inverse gamma(.01,.01) priors, respectively. Vertical reference lines indicate 2.5th posterior percentiles for ψ

For all models, the 2.5th posterior percentile of β₁ exceeds 0.68, thus suggesting that electric potential increases at sites further from the pole. Posterior regions for β₀, β₁, and ${\sigma }_e^2$ overlap in significant ways across all six models. The posterior mean of ${\sigma }_e^2$ under Models 1 and 2 are almost double the values under other models, thus suggesting that the residuals of Models 1 and 2 contain spatial variation that is later captured by the CAR process. The fitted ARPS process is most impressive when the CAR process is not modeled, as expected. In the presence of the strong spatial structure created by a degenerate $\text{CAR}(1,{\sigma }_c^2)$, ARPS parameters are closer to their null values, with posterior means of 1.147 for ψ (null value is 0) and 0.5262 for η (null value is 1) in Model 4. When strength of spatial dependence is estimated using $\text{CAR}(\rho ,{\sigma }_c^2)$ with random ρ, as in Model 6, posterior means are 1.332 for ψ, 0.3801 for η, and 0.4643 for ρ.

5.2. Model comparison

Many techniques have been suggested for choosing among competing models. Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), and Bayes factors are among the most often used, but these methods fail in our models because of the random effects and some improper priors (Spiegelhalter et al., 2002). Bayes factors are not interpretable with a $\text{CAR}(1,{\sigma }_c^2)$ prior for spatial effects (Han and Carlin, 2001). This $\text{CAR}(1,{\sigma }_c^2)$ prior creates n additional parameters that cannot be counted as n free parameters, thus making AIC and BIC inapplicable. For our model comparisons, we use the Deviance Information Criterion (DIC) proposed by Spiegelhalter et al. (2002). Sum of squared errors (SSE) is used as a secondary model comparison criterion. SSE due to prediction is computed as

$$\text{SSE}=E\{{\displaystyle \sum _{j=1}^n}{[Y(s_j)-\widehat{Y}(s_j)]}^2|Y(\cdot )\}$$

where the expectation is taken with respect to the posterior predictive distribution of Ŷ(·). See Gelfand and Ghosh (1998) for a decision-theoretic justication of this quantity as a model-choice criterion.

DIC, like AIC and BIC balances model adequacy against model complexity. Model adequacy is measured by the posterior mean D̄ = E_θ|y(D(θ)) of the deviance D(θ), where D(θ) = −2 log f (y|θ)+ 2 log(h(y)) and h(y) is unaffected by the model. Model complexity is measured by the effective numbers of parameters, p_D = D̄ − D(θ̄), where D(θ̄) is the deviance evaluated at the posterior mean, θ̄, of the parameters. Consequently, DIC = D + p_D and small values are desirable. DIC is effective in high-dimensional or very complicated models because of the method used for determining the effective number of parameters.

In Table 3, we report DIC and its components for each of the six models. Multiple runs yield differences of at most 1.9 in DIC. We also report p_θ, the total number of parameters and the posterior mean of SSE for all six models. Models 1 and 2 are clearly inadequate. Model 4 is better than Models 1 and 2 but is not as good as Models 3, 5, and 6. Model 6 is the best model by all measures.

Model 6 has many interesting features. While the CAR process enforces unequal variances only to address edge effects caused by differing numbers of neighbors, the ARPS process explicitly fits unequal variances as suggested by the data in Figure 3. Figure 8(a) shows fitted variances (obtained as the diagonal elements of ∑ evaluated at the posterior mean, θ̄, of the parameters) for Models 3, 5, and 6. The curve for Model 6 is much more variable than the curves for Models 3 and 5. Figure 8(b) shows fitted variances only for Model 5, and it is clear that these fitted variances are not consistent with the patterns in the data, as displayed in Figure 3. Corner sites (site numbers 1, 20, 141, and 160) have largest variances, followed by sites adjacent to corners, followed by other edge sites, and finally followed by interior sites. Figure 8(c) shows fitted variances only for Model 3, and again, these fitted variances are clearly not consistent with the data. In both Models 3 and 5, fitted variances are a function of number of neighbors and they increase away from the pole. There are much stronger spatial patterns and smoothness in Model 3 (ρ=.99 and ${\sigma }_c^2$ = .0018 in CAR) than in Model 5 (ρ=.85 and ${\sigma }_c^2$ = .0022 in CAR).

Figure 8.

Fitted variances from Models 3 (modified), 5, and 6. A modification is necessary for Model 3 because this fitted model is singular. For this model, we replace ρ = 1 with ρ = 0.99 to obtain invertible matrices that yield variances. Panel (a) shows variances for all models, while panel (b) shows variances only for Model 5 and panel (c) shows variances only for modified Model 3. Observation sites numbers are used as plotting symbols in panels (b) and (c), where site 1 is in the bottom left corner of Figure 2 and site 160 is in the top right corner

A second interesting feature of Model 6 is the relationship between correlation and site-to-source distances. Figure 9 shows correlation clouds from ∑ evaluated at θ̄ as a function of site-to-site and site-to-source distances for Models 3, 5, and 6. Figure 9(a) shows that for Model 5, correlations between sites closest to the pole (ARPS region R₁₀) are in general smaller than correlations between sites farthest from the pole (ARPS region R₁). There is, however, no clear separation between these groups of correlations. For Model 3, Figure 9(b) shows a greater degree of separation of correlations according to distance from pole, but the separation is still small. On the other hand, Model 6, as shown in Figure 9(c), clearly has correlations that are a function of distance to pole. Moreover, correlations between sites in R₁₀ are largest.

Figure 9.

Fitted correlations from Models 3 (modified), 5, and 6. A modification is necessary for Model 3 because this fitted model is singular. For this model, we replace ρ = 1 with ρ = 0.99 to obtain an invertible matrix that yields correlations. Panels (a), (b), and (c) show correlations for Models 5, 3, and 6, respectively. All panels display correlations between all pairs of sites where both sites are either in ARPS Region R₁ or R₁₀. Panel (c) also displays correlations between all pairs of sites where both sites are in ARPS Region R₂

Third, focusing only on the ARPS process, we consider the one-step correlations from regions R₁ to R₁₀, defined as

$$\begin{array}{c}\text{Corr}({\alpha }_1,{\alpha }_2)=\psi \sqrt{\frac{{\sigma }_1^2}{{\sigma }_1^2{\psi }^2+{\sigma }_2^2}},\hfill \\ \text{Corr}({\alpha }_k,{\alpha }_{k+1})=\psi \sqrt{\frac{{\sigma }_1^2{\psi }^{2(k-1)}+{\sigma }_2^2{\displaystyle \sum _{q=0}^{k-2}{\psi }^{2q}}}{{\sigma }_1^2{\psi }^{2k}+{\sigma }_2^2{\displaystyle \sum _{q=0}^{k-1}{\psi }^{2q}}}}\text{\hspace{1em}for}k=2,3,\dots ,9\hfill \end{array}$$

Evaluating at θ̄ for Model 6, we obtain Corr (α_k, α_k+1), k = 1, 2,…, 9 as: 0.790, .908, .954, .975, .987, .993, .996, .998, .999. These numbers change quickly from k = 1 to k = 3 and are primarily responsible for the differences seen in Figure 9(c).

Finally, as evidenced by the posterior density intervals for ψ and η given in Table 2, as well as additional results not presented here, there is clear evidence of statistical significance of the ARPS model. More specifically, the posterior interval for ψ does not contain the value 0 and the posterior interval for η does not contain the value 1.

6. CONCLUDING REMARKS

Our newly proposed ARPS process can improve models of stochastic phenomena across spatial domains because it takes advantage of auxiliary information provided by point sources. The result is more simplified, interpretable, parametric, yet realistic, nonstationary modeling of error processes. By restricting the initial variance ${\sigma }_1^2$ of the ARPS process to equal the ME variance ${\sigma }_e^2$, we strengthen the interpretation that ARPS captures extra variation due to the point source, and sites far from this source are well represented by the baseline error process. Additionally, we comment on the interplay between the ARPS process, the commonly used $\text{CAR}(1,{\sigma }_c^2)$ process, and the more desirable $\text{CAR}(\rho ,{\sigma }_c^2)$ process. This is all done in a Bayesian hierarchical framework where key objectives include simple tests for, and clear interpretations of, the impact of a point source.

A complete analysis of electric potential in a field containing a metal pole clearly shows the advantages of our approach. With ARPS, we capture the observed trend of decreasing variances away from the pole; this finding is consistent with Hughes-Oliver and Gonzalez-Farias (1999). We also achieve greater flexibility in modeling correlations within and across regions of various distances from the pole. Posterior density regions for parameters ψ and η clearly do not contain null values, thus implying ARPS is important in capturing statistical significance of the pole’s impact on small-scale variability of the electric potential process. Model selection is based on both the DIC and SSE.

The ARPS error process may, at first glance, appear unrealistically simple because it assigns the same random effect α_k across an entire region R_k, but it is actually a very flexible model. Regions R₁, …, R_r may be chosen to represent expert knowledge of the process being monitored (perhaps regions should be irregularly shaped and defined by geographic features such as rivers, valleys, or plateaus in the monitored area) with only minimal requirements that one must have enough data to be able to fit a reliable model for the specified number of parameters. Because the model is parametric, we allow testing for the contribution of ARPS toward improving fit of the observed data, as we have demonstrated for the data on electric potential. Also by letting r approach infinity, we can approximate any stochastic process by a sequence of simple stochastic processes. This idea is similar to approximating a measurable function by a sequence of simple functions (Billingsley, 1986, pp. 63–65).

There are, however, several issues that are currently being investigated. One such issue deals with dynamic regionalization for the ARPS process, where we observe a spatio-temporal process that can possibly alter the effect of a source over time. This is particularly relevant in spatio-temporal processes affected by meteorological conditions. On the other hand, regionalization for purely spatial processes may be done in a straightforward manner, with model fits from different choices for R₁,…, R_r compared for adequacy. Ideally, we desire r as large as possible, but this increases the number of parameters and reduces reliability of the model. Cost-benefits analyses, possibly using cross validation may be necessary for selecting r.

Another issue under current investigation is simultaneously accounting for the impact of multiple sources. Ranking the impact of different sources is particularly difficult when sites are affected by multiple sources in very different ways. The field of source apportionment (see, for example, Henry, 1997; Henry et al., 1997; Park et al., 2001; Christensen and Sain, 2002; Park et al., 2002) ranks the impact of multiple sources by effectively partitioning the observed variance matrix into factors that represent different categories of sources. Bayesian hierarchical modeling of multiple ARPS processes is a fundamentally different approach from the standard techniques of source apportionment. This will be the subject of future manuscripts.