A Bayesian kriged Kalman model for short-term forecasting of air pollution levels

Summary.  Short-term forecasts of air pollution levels in big cities are now reported in news-papers and other media outlets. Studies indicate that even short-term exposure to high levels of an air pollutant called atmospheric particulate matter can lead to long-term health effects. Data are typically observed at fixed monitoring stations throughout a study region of interest at different time points. Statistical spatiotemporal models are appropriate for modelling these data. We consider short-term forecasting of these spatiotemporal processes by using a Bayesian kriged Kalman filtering model. The spatial prediction surface of the model is built by using the well-known method of kriging for optimum spatial prediction and the temporal effects are analysed by using the models underlying the Kalman filtering method. The full Bayesian model is implemented by using Markov chain Monte Carlo techniques which enable us to obtain the optimal Bayesian forecasts in time and space. A new cross-validation method based on the Mahalanobis distance between the forecasts and observed data is also developed to assess the forecasting performance of the model implemented.     

Likelihood Inference for Spatial Generalized Linear Mixed Models

Spatial modeling is widely used in environmental sciences, biology, and epidemiology. Generalized linear mixed models are employed to account for spatial variations of point-referenced data called spatial generalized linear mixed models (SGLMMs). Frequentist analysis of these type of data is computationally difficult. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of SGLMM computationally convenient. Recent introduction of the method of data cloning, which leads to maximum likelihood estimate, has made frequentist analysis of mixed models also equally computationally convenient. Recently, the data cloning was employed to estimate model parameters in SGLMMs, however, the prediction of spatial random effects and kriging are also very important. In this article, we propose a frequentist approach based on data cloning to predict (and provide prediction intervals) spatial random effects and kriging. We illustrate this approach using a real dataset and also by a simulation study.     

Residual Kriging for Functional Spatial Prediction of Salinity Curves

Recently, several methodologies to perform geostatistical analysis of functional data have been proposed. All of them assume that the spatial functional process considered is stationary. However, in practice, we often have nonstationary functional data because there exists an explicit spatial trend in the mean. Here, we propose a methodology to extend kriging predictors for functional data to the case where the mean function is not constant through the region of interest. We consider an approach based on the classical residual kriging method used in univariate geostatistics. We propose a three steps procedure. Initially, a functional regression model is used to detrend the mean. Then we apply kriging methods for functional data to the regression residuals to predict a residual curve at a non-data location. Finally, the prediction curve is obtained as the sum of the trend and the residual prediction. We apply the methodology to salinity data corresponding to 21 salinity curves recorded at the Ciénaga Grande de Santa Marta estuary, located in the Caribbean coast of Colombia. A cross-validation analysis was carried out to track the performance of the proposed methodology.     

Theory & Methods: Krige,Smooth, Both or Neither?

Both kriging and non-parametric regression smoothing can model a non-stationary regression function with spatially correlated errors. However comparisons have mainly been based on ordinary kriging and smoothing with uncorrelated errors. Ordinary kriging attributes smoothness of the response to spatial autocorrelation whereas non-parametric regression attributes trends to a smooth regression function. For spatial processes it is reasonable to suppose that the response is due to both trend and autocorrelation. This paper reviews methodology for non-parametric regression with autocorrelated errors which is a natural compromise between the two methods. Re-analysis of the one-dimensional stationary spatial data of Laslett (1994) and a clearly non-stationary time series demonstrates the rather surprising result that for these data, ordinary kriging outperforms more computationally intensive models including both universal kriging and correlated splines for spatial prediction. For estimating the regression function, non-parametric regression provides adaptive estimation, but the autocorrelation must be accounted for in selecting the smoothing parameter.     

Comparison of predictions by kriging and spatial autoregressive models

In geostatistics, the prediction of unknown quantities at given locations is commonly made by the kriging technique. In addition to the kriging technique for modeling regular lattice spatial data, the spatial autoregressive models can also be used. In this article, the spatial autoregressive model and the kriging technique are introduced. We extend prediction method proposed by Basu and Reinsel for SAR(2,1) model. Then, using a simulation study and real data, we compare prediction accuracy of the spatial autoregressive models with that of the kriging prediction. The results of simulation study show that predictions made by the autoregressive models are good competitor for the kriging method.     

The Bootstrap and Kriging Prediction Intervals

Kriging is a method for spatial prediction that, given observations of a spatial process, gives the optimal linear predictor of the process at a new specified point. The kriging predictor may be used to define a prediction interval for the value of interest. The coverage of the prediction interval will, however, equal the nominal desired coverage only if it is constructed using the correct underlying covariance structure of the process. If this is unknown, it must be estimated from the data. We study the effect on the coverage accuracy of the prediction interval of substituting the true covariance parameters by estimators, and the effect of bootstrap calibration of coverage properties of the resulting 'plugin' interval. We demonstrate that plugin and bootstrap calibrated intervals are asymptotically accurate in some generality and that bootstrap calibration appears to have a significant effect in improving the rate of convergence of coverage error.     

Optimal sampling for spatial prediction of functional data

This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogotá city, Colombia.     

Comparison of designs for generalized linear models under model misspecification

The purpose of this article is to demonstrate the use of the quantile dispersion graphs (QDGs) approach for comparing candidate designs for generalized linear models in the presence of model misspecification in the linear predictor. The proposed design criterion is based on the mean-squared error of prediction which incorporates the prediction variance and the bias caused by fitting the wrong model. The method of kriging is used to estimate the unknown function assumed to be the cause of model misspecification. The QDGs approach is also useful in assessing the robustness of a given design to values of the unknown parameters in the linear predictor. Three numerical examples are presented to illustrate the application of the proposed methodology.     

Spatial modeling using frequentist approach for disease mapping

In this article, a generalized linear mixed model (GLMM) based on a frequentist approach is employed to examine spatial trend of asthma data. However, the frequentist analysis of GLMM is computationally difficult. On the other hand, the Bayesian analysis of GLMM has been computationally convenient due to the advent of Markov chain Monte Carlo algorithms. Recently developed data cloning (DC) method, which yields to maximum likelihood estimate, provides frequentist approach to complex mixed models and equally computationally convenient method. We use DC to conduct frequentist analysis of spatial models. The advantages of the DC approach are that the answers are independent of the choice of the priors, non-estimable parameters are flagged automatically, and the possibility of improper posterior distributions is completely avoided. We illustrate this approach using a real dataset of asthma visits to hospital in the province of Manitoba, Canada, during 2000–2010. The performance of the DC approach in our application is also studied through a simulation study.     

The Effect of Leap Years and Seasonal Trends on the Birthday Problem

Most data have a space and time label associated with them; data that are close together are usually more correlated than those that are far apart. Prediction (or forecasting) of a process at a particular label where there is no datum, from observed nearby data, is the subject of this article. One approach, known as geostatistics, is featured, from which linear methods of spatial prediction (kriging) will be considered. Brief reference is made to other linear/nonlinear, stochastic/deterministic predictors. The (linear) geostatistical method is applied to piezometric-head data around a potential nuclear-waste repository site.     

Mixture Kalman filters

In treating dynamic systems, sequential Monte Carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling, importance sampling and weighted resampling to complete the on-line 'filtering' task. We propose a special sequential Monte Carlo method, the mixture Kalman filter, which uses a random mixture of the Gaussian distributions to approximate a target distribution. It is designed for on-line estimation and prediction of conditional and partial conditional dynamic linear models, which are themselves a class of widely used non-linear systems and also serve to approximate many others. Compared with a few available filtering methods including Monte Carlo methods, the gain in efficiency that is provided by the mixture Kalman filter can be very substantial. Another contribution of the paper is the formulation of many non-linear systems into conditional or partial conditional linear form, to which the mixture Kalman filter can be applied. Examples in target tracking and digital communications are given to demonstrate the procedures proposed.     

Prediction in linear mixed models

Following estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. The process has been well defined for linear models, but the introduction of random effects into the model means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions. This paper discusses the interpretation of predictions formed including or excluding random terms. Four datasets are used to illustrate circumstances where different prediction strategies may be appropriate: in an orthogonal design, an unbalanced nested structure, a model with cubic smoothing spline terms and for kriging after spatial analysis. The examples also show the need for different weighting schemes that recognize nesting and aliasing during prediction, and the necessity of being able to detect inestimable predictions.     

Numerical Comparison Between Different Empirical Prediction Intervals Under the Fay-Herriot Model

Recently, an empirical best linear unbiased predictor is widely used as a practical approach to small area inference. It is also of interest to construct empirical prediction intervals. However, we do not know which method should be used from among the several existing prediction intervals. In this article, we first obtain an empirical prediction interval by using the residual maximum likelihood method for estimating unknown model variance parameters. Then we compare the later with other intervals with the residual maximum likelihood method. Additionally, some different parametric bootstrap methods for constructing empirical prediction intervals are also compared in a simulation study.     

Prediction based upon maximizing squared correlation for stationary processes and simple kriging

A spatial prediction procedure is investigated which is based upon maximizing the squared correlation between the predictor and the value to be estimated. Several properties of this predictor are examined and contrasted with the simple kriging predictor. Some examples are discussed.     

On Optimal Point and Block Prediction in Log-Gaussian Random Fields

Abstract.  This work discusses the problems of point and block prediction in log-Gaussian random fields with unknown mean. New point and block predictors are derived that are optimal in mean squared error sense within certain families of predictors that contain the corresponding lognormal kriging point and block predictors, as well as a block predictor originally motivated under the assumption of 'preservation of lognormality', and hence improve upon them. A comparison between the optimal, lognormal kriging and best linear unbiased predictors is provided, as well as between the two new block predictors. Somewhat surprisingly, it is shown that the corresponding optimal and lognormal kriging predictors are almost identical under most scenarios. It is also shown that one of the new block predictors is uniformly better than the other.     

Nearest Neighbor Adjusted Best Linear Unbiased Prediction

Statistical inference for linear models has classically focused on either estimation or hypothesis testing of linear combinations of fixed effects or of variance components for random effects. A third form of inference—prediction of linear combinations of fixed and random effects—has important advantages over conventional estimators in many applications. None of these approaches will result in accurate inference if the data contain strong, unaccounted for local gradients, such as spatial trends in field-plot data. Nearest neighbor methods to adjust for such trends have been widely discussed in recent literature. So far, however, these methods have been developed exclusively for classical estimation and hypothesis testing. In this article a method of obtaining nearest neighbor adjusted (NNA) predictors, along the lines of “best linear unbiased prediction,” or BLUP, is developed. A simulation study comparing “NNABLUP” to conventional NNA methods and to non-NNA alternatives suggests considerable potential for improved efficiency.     

A gm estimation of the location parameters in a spatial linear model

Universal kriging is a form of interpolation that takes into account the local trends in data when minimizing the error associated with the estimator. Under multivariate normality assumptions, the given predictor is the best linear unbiased predictor. but if the underlying distribution is not normal, the estimator will not be unbiased and will be vulnerable to outliers. With spatial data, it is not only the presence of outliers that may spoil the predictions, but also the boundary sites. usually corners, that tend to have high leverage. As an alternative, a weighted one-step generalized M estimator of the location parameters in a spatial linear model is proposed. It is especially recommended in the case of irregularly spaced data.     

Understanding the Kalman Filter

This is an expository article. Here we show how the successfully used Kalman filter, popular with control engineers and other scientists, can be easily understood by statisticians if we use a Bayesian formulation and some well-known results in multivariate statistics. We also give a simple example illustrating the use of the Kalman filter for quality control work.