Using a non-homogeneous Poisson model with spatial anisotropy and change-points to study air pollution data

Abstract

A non-homogeneous Poisson process is used to study the rate at which a pollutant’s concentration exceeds a given threshold of interest. An anisotropic spatial model is imposed on the parameters of the Poisson intensity function. The main contribution here is to allow the presence of change-points in time since the data may behave differently for different time frames in a given observational period. Additionally, spatial anisotropy is also imposed on the vector of change-points in order to account for the possible correlation between different sites. Estimation of the parameters of the model is performed using Bayesian inference via Markov chain Monte Carlo algorithms, in particular, Gibbs sampling and Metropolis-Hastings. The different versions of the model are applied to ozone data from the monitoring network of Mexico City, Mexico. An analysis of the results obtained is also given.

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Acknowledgements

The authors thank three anonymous reviewers for their careful reading of this work and for all the comments and suggestions that helped to improve the presentation of the results. In particular, they thank one of them for calling their attention to the work by Koop and Potter (2009). ERR and MHT were partially funded by the Projects PAPIIT-IN102713 and IN102416 of the Dirección General de Apoyo al Personal Académico of the Universidad Nacional Autónoma de México, Mexico (DGAPA-UNAM). This work started and ended during ERR two recent sabbatical visits to the Department of Statistics of the University of Oxford, UK, which was also partially supported by DGAPA-UNAM. ERR is grateful to the Departments of Statistics of the University of Oxford, UK, and of the Universidade Estadual Paulista “Júlio de Mesquita Filho” – Campus Presidente Prudente, Brazil, for support and hospitality during the development of this work. MHT thanks the Instituto de Matemáticas of the Universidad Nacional Autónoma de México, Mexico, for support and hospitality.

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Correspondence to Eliane R. Rodrigues.

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Appendix

Appendix

In this appendix we present the values of the hyperparameters of the prior distributions of the parameters present in the Poisson and spatial components of the model as well as some of the computational details of their estimation. We also present conditional posterior distributions of the change-points. The expressions for the posterior distributions corresponding to the other parameters may be obtained as in Rodrigues et al. (2015a). Additionally, the tables with the values of DIC and MLF, the results of the estimated means of the multivariate normal distributions of the parameters \(\log (\varvec{\alpha })\) and \(\log (\varvec{\beta })\), and the MC errors are also given together with the estimated correlation matrices of the change-points.

A.1. Hyperparameters of the prior distributions of the parameters in the rate function and the spatial model

We would like to draw attention to the fact that the initial prior elicitation was very superficial and led to very flat uniform priors. This allowed for physically unrealistic parameters. It was observed that the MCMC mixed very poorly at these values. On reflection, we revised the hyperparameters in our priors to restrict the distributions to more realistic values. Hence, the data played a role in informing the support of the prior distributions. This is not strictly Bayesian, but made the analysis computationally more tractable. Therefore, the chosen prior distributions for the several versions of the model are given as follows.

Model without change-points

The hyperparameters of the prior distributions of the parameters are given as follows. The normal prior distributions of the components of the corresponding vector \(\varvec{\mu }^{\alpha }\) have mean 0.1 when data from stations FAC/EAC, TLA, MER, UIZ, PED, CUA and PLA are used. In the case of the stations SAG and MON/CHA the value of the mean is − 0.05, and for station TAH its value is 0.2. In the case of vector \(\varvec{\mu }^{\beta }\), the normal prior distributions of its coordinates are more heterogeneous. Their means are − 0.35, − 0.1, 0.1, − 0.15, 0.07 and − 0.2 in the cases of data from stations FAC/EAC, SAG, MON/CHA, MER, UIZ and TAH, respectively. When we consider stations TLA and PLA, the mean has value − 0.25, and it is − 0.3 for stations PED and CUA. In all cases, we have a standard deviation of 0.1.

The hyperparameters of the Pareto prior distribution for the parameter \(\psi _{r}\) are \(c=3\) and \(d=1\). The prior distribution of \(\psi _{a}\) is, as specified in Sect. 3.3, i.e., it is a uniform distribution on \((0, \pi /2)\).

The prior distributions for \(\phi _{\alpha }\) and \(\phi _{\beta }\) are inverse gamma distributions IG(2.5, \(b_{\alpha }\)) and IG(3, \(b_{\beta }\)), respectively, where the hyperparameters \(b_{\alpha }\) and \(b_{\beta }\) are the solutions of the optimisation problem (6) corresponding to each particular \(\phi \). The values of the coordinates of the vectors \(\varvec{\sigma }_{\alpha }\) and \(\varvec{\sigma }_{\beta }\) are equal to 0.1.

The prior distribution of \(\varvec{\alpha }\) is, as specified in Sect. 3.3, a log-normal distribution with parameters \(\varvec{\mu }^{\alpha }\) and \(\Sigma ^{\alpha }\). Similar procedure is applied to the vector \(\varvec{\beta }\), but now using \(\varvec{\mu }^{\beta }\) and \(\Sigma ^{\beta }\).

All parameters were estimated using a sample of size 6000 drawn from three chains after a burn-in period of 10,000 iterations with values collected using a sampling interval of 10.

Model with one change-point

The prior distributions corresponding to the parameters \(\psi _{a}\), \(\psi _{r}\), \(\phi _{\alpha }\) and \(\phi _{\beta }\) will have hyperparameters as in the case where no change-points are present. The new parameter \(\phi _{\tau }\) will have as prior distribution an inverse gamma IG(3, \(b_{\tau }\)) where \(b_{\tau }\) is also the solution of (6) corresponding to \(\phi _{\tau }\).

If we consider the components of the vector \(\varvec{\mu }^{\alpha }_{1}\), then the mean of the normal prior distribution is − 0.27 for all stations with the exception of station PLA. In this case, the value of the mean is 0.27. If we look at the coordinates of the vector \(\varvec{\mu }^{\alpha }_{2}\), then the means of the normal prior distributions are all equal to − 0.27 for data from all stations. In the case of the vectors \(\varvec{\mu }^{\beta }_{1}\) and \(\varvec{\mu }^{\beta }_{2}\) the values of the means of the normal prior distributions of their coordinates varied from station to station. Hence, for the coordinates of \(\varvec{\mu }^{\beta }_{1}\) the value of the mean is 0.7 for all stations with the exception of station MON/CHA where the value is 0.09. When we take into account the coordinates of \(\varvec{\mu }^{\beta }_{2}\) the means of their normal prior distributions are equal to 1.3 for all stations with the exception of stations MON/CHA and CUA where the values were 0.1 and 1.0, respectively. The values of the coordinates of the vectors \(\varvec{\sigma }_{\alpha }\) and \(\varvec{\sigma }_{\beta }\) are equal to 0.1.

Estimation of all parameters, including the change-points, was performed using a sample of size 6000 obtained from three chains after a burn-in period of 10,000 steps and using a sampling interval of 10 steps.

Model with two change-points

In the case of Model 2_1, the hyperparameters of the prior distributions of \(\psi _{a}\), \(\psi _{r}\), \(\phi _{\alpha }\), \(\phi _{\beta }\) and \(\phi _{\tau }\) are as in the version with one change-point. For the coordinates of the vectors \(\varvec{\mu }^{\alpha }_{1}\), the means of their normal prior distributions are equal to 0.05 for all stations with the exception of stations SAG, MON/CHA and TAH where we have mean − 0.05 in the case of SAG and MON/CHA, and 0.15 in the case of TAH. For the coordinates of the vector \(\varvec{\mu }^{\alpha }_{2}\), we have mean 0.15 for stations FAC/EAC and TLA; − 0.05 for stations SAG and MON/CHA; 0.1 for station MER, UIZ and CUA; 0.25 for stations TAH and PED; and 0.2 for station PLA. In the case of the means of \(\varvec{\mu }^{\alpha }_{3}\) we have the following. For stations FAC/EAC, TLA and PLA we have normal prior distributions with means equal to 0.15; for stations SAG, MON/CHA, PED and TAH the respective means are − 0.05, − 0.07, 0 and 0.25; and in the case of stations MER, UIZ, and CUA we have means equal to 0.1.

Consider the coordinates of the vector \(\varvec{\mu }^{\beta }_{1}\). In this case the normal prior distribution of the coordinates corresponding to stations FAC/EAC, TLA, SAG, MER and CUA will have means − 0.05, − 1.05, 0.2, 0.05 and 0.1, respectively. In the case of stations MON/CHA and TAH the means are equal to 0.09 and are − 0.1 when we use data from stations UIZ, PED and PLA. Consider now the coordinates of vector \(\varvec{\mu }^{\beta }_{2}\). Their normal prior distributions have means − 0.35 in the case of stations FAC/EAC and TLA; 0.1 for stations SAG and MON/CHA; − 0.3 in the case of stations TAH, PED and PLA. For stations MER, UIZ and CUA, the corresponding means are − 0.15, 0.07 and − 0.2. The coordinates of the vector \(\varvec{\mu }^{\beta }_{3}\), have normal prior distributions with means associated to stations FAC/EAC, TLA and PLA equal to − 0.5; they are equal to 0.1 for stations SAG and MON/CHA; stations TAH and CUA have means equal to − 0.3; and they are − 0.1, 0, and − 0.6 in the cases of MER, UIZ and PED, respectively.

The hyperparameters related to Models 2_2 and 2_3 are as in Model 2_1 with the exception of the coordinate of \(\varvec{\mu }^{\beta }_{3}\) corresponding to station MER where the mean of its normal distribution is − 0.25 instead of − 0.1. The values of the coordinates of the vectors \(\varvec{\sigma }_{\alpha }\) and \(\varvec{\sigma }_{\beta }\) are equal to 0.1.

In all versions of the two change-point models, all parameters, including the change-points, were estimated using a sample of size 6000 drawn from three chains after a burn-in period of 10,000 steps using a sampling interval of 10.

A.2. Full posterior distributions of the change-points

In this section we present the expressions for the full conditional marginal posterior distributions from which the change-points were sampled. We will denote by \(\varvec{\theta }_{(- x)}\) the vector of parameters \(\varvec{\theta }\) without the component x. Values will be sampled through a Metropolis-Hastings step within the Gibbs sampling algorithm.

The change-points posterior distributions have contributions from the prior (\(P({\varvec{\tau }})\)) and the likelihood. The density \(P({\varvec{\tau }})\) is straightforward to simulate but awkward to write down in close form. It is defined by the simulation procedure given in Sect. 3.3. Hence,

$$\begin{aligned}&P({\varvec{\tau }}_{1} \, | \, {\varvec{\Theta }}_{(- {\varvec{\tau }}_{1})}, \mathbf{D}) \propto \prod _{i = 1}^{N} \left\{ \left( \frac{\alpha _{1}^{(i)}}{\beta _{1}^{(i)}}\right) ^{N^{(i)}_{\tau ^{(i)}_{1}}} \, \left( \frac{\alpha _{2}^{(i)}}{\beta _{2}^{(i)}} \right) ^{N^{(i)}_{\tau ^{(i)}_{2}} - N_{\tau _{1}^{(i)}}} \, \left[ \prod _{k=1}^{ N^{(i)}_{\tau _{1}^{(i)}}} \left( \frac{d_{k,i}}{\beta _{1}^{(i)}} \right) ^{\alpha _{1}^{(i)} - 1} \right] \right. \\&\quad \left. \left[ \prod _{k=N^{(i)}_{\tau _{1}^{(i)}}+1}^{ N^{(i)}_{\tau _{2}^{(i)}}} \left( \frac{d_{k,i}}{\beta _{2}^{(i)}} \right) ^{\alpha _{2}^{(i)} - 1} \right] \exp \left( - \left[ \left( \frac{\tau _{1}^{(i)}}{\beta _{1}^{(i)}} \right) ^{\alpha _{1}^{(i)}} - \left( \frac{\tau _{1}^{(i)}}{\beta _{2}^{(i)}} \right) ^{\alpha _{2}^{(i)}} \right] \right) \right\} \, P({\varvec{\tau }}), \end{aligned}$$

and for \(j = 2, 3, \ldots , J\)

$$\begin{aligned}&P({\varvec{\tau }}_{j} \, | \, {\varvec{\Theta }}_{(- {\varvec{\tau }}_{j})}, \mathbf{D}) \propto \prod _{i = 1}^{N} \left\{ \left( \frac{\alpha _{j}^{(i)}}{\beta _{j}^{(i)}} \right) ^{N^{(i)}_{\tau _{j}^{(i)}} - N^{(i)}_{\tau _{j-1}^{(i)}}} \, \left( \frac{\alpha _{j+1}^{(i)}}{\beta _{j+1}^{(i)}} \right) ^{N^{(i)}_{\tau _{j+1}^{(i)}} - N^{(i)}_{\tau _{j}^{(i)}}} \right. \\&\quad \left[ \prod _{k = N^{(i)}_{\tau ^{(i)}_{j-1}} +1}^{N^{(i)}_{\tau ^{(i)}_{j}}} \left( \frac{d_{k,i}}{\beta _{j}^{(i)}} \right) ^{\alpha _{j}^{(i)} - 1} \right] \, \left[ \prod _{k = N^{(i)}_{\tau ^{(i)}_{j}} +1}^{N^{(i)}_{\tau ^{(i)}_{j+1}}} \left( \frac{d_{k,i}}{\beta _{j+1}^{(i)}} \right) ^{\alpha _{j+1}^{(i)} - 1} \right] \\&\quad \left. \exp \left( - \left[ \left( \frac{\tau _{j}^{(i)}}{\beta _{j}^{(i)}} \right) ^{\alpha _{j}^{(i)}} - \left( \frac{\tau _{j}^{(i)}}{\beta _{j+1}^{(i)}} \right) ^{\alpha _{j+1}^{(i)}} \right] \right) \right\} \, P({\varvec{\tau }}), \end{aligned}$$

where we take \(N^{(i)}_{\tau ^{(i)}_{J+1}} = K_{i}\).

A.3. Some of the figures

In this appendix we give some of the figures discussed in the main text (Figs. 2, 3 and 4). First, we have the plots of the daily maximum ozone concentrations during the observational period in all stations used in the present work. Then, we have the plots of the differences between the accumulated observed and estimated means using all versions of the model. This is followed by the estimated rate function for all stations when the selected model is used.

Fig. 2
figure2

Daily maximum ozone concentrations for all stations during the period ranging from 01 January 1995 to 31 December 2010

Fig. 3
figure3

Absolute values of the differences between the accumulated observed and estimated means for all stations and versions of the Poisson model. Continuous lines with \(\bullet \) correspond to the differences between the observed and the estimated means with no change-points, thick dashed lines are related to the differences considering now the estimated means with the presence of one change-point, thin continuous lines corresponding to the differences using the estimated means using Model 2_1, thick continuous lines when we have the estimated means using Model 2_2, and thin dashed line in the case of estimated means using Model 2_3

Fig. 4
figure4

Estimated rate functions and confidence bands for all stations during the entire observational period when two change-points are allowed and Model 2_2 is used

A.4. Further results

In this section some additional tables and results mentioned in the main text are presented. Hence, Table 3 gives the values of the DIC and log-MLF for all stations and and versions of the model with two change-points.

In Table 4 we have the estimated means of \(\varvec{\mu }^{\alpha }\) and \(\varvec{\mu }^{\beta }\) of the multivariate normal distributions in the cases of \(\log (\varvec{\alpha })\) and \(\log (\varvec{\beta })\) for all stations for the version with two change-points given by Model 2_2.

Table 5 gives the MC errors of the parameters present in the Poisson model when Model 2_2 is used.

Next we present the posterior correlation matrices for the vectors of first and second change-points when Model 2_2 is used. Hence, first we have \(\rho _{\varvec{\tau _{1}}}\) followed by \(\rho _{\varvec{\tau _{2}}}\).

$$\begin{aligned} \rho _{\varvec{\tau _{1}}} = \left( \begin{array}{ccccccccccc} 1.000 &{} 0.771 &{} 0.389 &{} 0.263 &{} 0.548 &{} 0.417 &{} 0.280 &{} 0.537 &{} 0.606 &{} 0.610 \\ &{} 1.000 &{} 0.480 &{} 0.297 &{} 0.580 &{} 0.437 &{} 0.292 &{} 0.476 &{} 0.473 &{} 0.535 \\ &{} &{} 1.000 &{} 0.544 &{} 0.561 &{} 0.515 &{} 0.383 &{} 0.355 &{} 0.285 &{} 0.387 \\ &{} &{} &{} 1.000 &{} 0.389 &{} 0.452 &{} 0.414 &{} 0.273 &{} 0.213 &{} 0.292 \\ &{} &{} &{} &{} 1.000 &{} 0.710 &{} 0.452 &{} 0.585 &{} 0.444 &{} 0.646 \\ &{} &{} &{} &{} &{} 1.000 &{} 0.590 &{} 0.543 &{} 0.375 &{} 0.563 \\ &{} &{} &{} &{} &{} &{} 1.000 &{} 0.409 &{} 0.287 &{} 0.395 \\ &{} &{} &{} &{} &{} &{} &{} 1.000 &{} 0.657 &{} 0.844 \\ &{} &{} &{} &{} &{} &{} &{} &{} 1.000 &{} 0.665 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} 1.000 \\ \end{array} \right) \end{aligned}$$
Table 3 Values of DIC and log-MLF for each station as well as their total values for each model allowing two change-points
Table 4 Estimated means, standard deviations (indicated by SD) and 95% credible intervals for the vector of parameters \(\varvec{\mu }^{\alpha }_{j}\) and \(\varvec{\mu }^{\beta }_{j}\), \(j = 1, 2, 3\), of the multivariate normal distributions of \(\log (\varvec{\alpha })\) and \(\log (\varvec{\beta })\), respectively, for all stations when two change-points are allowed and Model 2_2 is used
Table 5 Estimated MC error for \(\tau _{k}^{(i)}\), \(k = 1, 2\) and the parameters \(\alpha _{j}^{(i)}\), \(\beta _{j}^{(i)}\), \(\varvec{\mu }^{\alpha }_{j}\), and \(\varvec{\mu }^{\beta }_{j}\), \(j = 1, 2, 3\), \(i = 1, 2, \ldots , 10\), for all stations in the case of Model 2_2
$$\begin{aligned} \rho _{\varvec{\tau _{2}}} = \left( \begin{array}{ccccccccccc} 1.000 &{} 0.765 &{} 0.385 &{} 0.223 &{} 0.537 &{} 0.413 &{} 0.284 &{} 0.534 &{} 0.587 &{} 0.608 \\ &{} 1.000 &{} 0.457 &{} 0.262 &{} 0.566 &{} 0.425 &{} 0.284 &{} 0.459 &{} 0.467 &{} 0.529 \\ &{} &{} 1.000 &{} 0.545 &{} 0.565 &{} 0.519 &{} 0.348 &{} 0.338 &{} 0.283 &{} 0.379 \\ &{} &{} &{} 1.000 &{} 0.397 &{} 0.445 &{} 0.382 &{} 0.239 &{} 0.176 &{} 0.249 \\ &{} &{} &{} &{} 1.000 &{} 0.715 &{} 0.433 &{} 0.578 &{} 0.434 &{} 0.643 \\ &{} &{} &{} &{} &{} 1.000 &{} 0.588 &{} 0.541 &{} 0.383 &{} 0.558 \\ &{} &{} &{} &{} &{} &{} 1.000 &{} 0.402 &{} 0.289 &{} 0.385 \\ &{} &{} &{} &{} &{} &{} &{} 1.000 &{} 0.650 &{} 0.840 \\ &{} &{} &{} &{} &{} &{} &{} &{} 1.000 &{} 0.651 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} 1.000 \\ \end{array} \right) \end{aligned}$$

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Rodrigues, E.R., Nicholls, G., Tarumoto, M.H. et al. Using a non-homogeneous Poisson model with spatial anisotropy and change-points to study air pollution data. Environ Ecol Stat 26, 153–184 (2019). https://doi.org/10.1007/s10651-019-00423-6

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Keywords

  • Anisotropic spatial model
  • Bayesian inference
  • Change-points
  • Markov chain Monte Carlo algorithms
  • Non-homogeneous Poisson process