Information loss on the dependence parameter due to incomplete data from a one-parameter spatial gaussian process

The Fisher information loss on the dependence parameter is considered for three one-parameter Gaussian processes on a rectangular lattice when some values are not observed. Formulae are given for the general case, and simplified versions for two of the special cases. Approximations are considered, and cases with a small number of missing sites are compared.     

Exact maximum likelihood for incomplete data from a correlated gaussian process

In many situations in which a variable is measured at locations in time or space the observed data can be regarded as incomplete, the missing data sites perhaps completing a regular pattern such as a rectangular grid. In this paper general methods not dependent on the sequential nature of time are considered for estimating the parameters of Gaussian processes. An example is given.     

Information loss on the mean for spatial processes when some values are missing

The loss of information on the mean due to the presence of missing values is discussed for a Gaussian univariate process on a rectangular lattice. The exact as well as the approximate formulae for this loss are given for general conditional autoregressive (CAR) and simultaneous autoregressive (SAR) processes. The formulae are evaluated for some low order CAR and SAR processes. The approximate formula is shown to give a good insight into how the loss varies over the different configurations of missing sites.     

Bias from the use of generalized estimating equations to analyze incomplete longitudinal binary data

Patient dropout is a common problem in studies that collect repeated binary measurements. Generalized estimating equations (GEE) are often used to analyze such data. The dropout mechanism may be plausibly missing at random (MAR), i.e. unrelated to future measurements given covariates and past measurements. In this case, various authors have recommended weighted GEE with weights based on an assumed dropout model, or an imputation approach, or a doubly robust approach based on weighting and imputation. These approaches provide asymptotically unbiased inference, provided the dropout or imputation model (as appropriate) is correctly specified. Other authors have suggested that, provided the working correlation structure is correctly specified, GEE using an improved estimator of the correlation parameters (‘modified GEE’) show minimal bias. These modified GEE have not been thoroughly examined. In this paper, we study the asymptotic bias under MAR dropout of these modified GEE, the standard GEE, and also GEE using the true correlation. We demonstrate that all three methods are biased in general. The modified GEE may be preferred to the standard GEE and are subject to only minimal bias in many MAR scenarios but in others are substantially biased. Hence, we recommend the modified GEE be used with caution.     

Estimating spatial means with an application to remotely sensed data

The paper examines alternative estimators for the mean of a spatial process where observations are not independent. Properties of the sample mean and its standard error are contrasted with those of maximum likelihood estimators derived for three spatial models. The information loss caused by spatial dependency in the data is examined. The distribution theory for the estimators is reviewed and the paper concludes with an empirical example illustrating the properties of the estimators and the practical benefits of the maximum likelihood procedure.     

Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data

In this paper, we propose a hidden Markov model for the analysis of the time series of bivariate circular observations, by assuming that the data are sampled from bivariate circular densities, whose parameters are driven by the evolution of a latent Markov chain. The model segments the data by accounting for redundancies due to correlations along time and across variables. A computationally feasible expectation maximization (EM) algorithm is provided for the maximum likelihood estimation of the model from incomplete data, by treating the missing values and the states of the latent chain as two different sources of incomplete information. Importance-sampling methods facilitate the computation of bootstrap standard errors of the estimates. The methodology is illustrated on a bivariate time series of wind and wave directions and compared with popular segmentation models for bivariate circular data, which ignore correlations across variables and/or along time.     

Binary probability maps using a hidden conditional autoregressive Gaussian process with an application to Finnish common toad data

The Finnish common toad data of Heikkinen and Hogmander are reanalysed using an alternative fully Bayesian model that does not require a pseudolikelihood approximation and an alternative prior distribution for the true presence or absence status of toads in each 10 km×10 km square. Markov chain Monte Carlo methods are used to obtain posterior probability estimates of the square-specific presences of the common toad and these are presented as a map. The results are different from those of Heikkinen and Hogmander and we offer an explanation in terms of the prior used for square-specific presence of the toads. We suggest that our approach is more faithful to the data and avoids unnecessary confounding of effects. We demonstrate how to extend our model efficiently with square-specific covariates and illustrate this by introducing deterministic spatial changes.     

Modelling the location decisions of manufacturing firms with a spatial point process approach

The paper is devoted to explore how the increasing availability of spatial micro-data, jointly with the diffusion of GIS software, allows to exploit micro-econometric methods based on stochastic spatial point processes in order to understand the factors that may influence the location decisions of new firms. By using the knowledge of the geographical coordinates of the newborn firms, their spatial distribution is treated as a realization of an inhomogeneous marked point process in the continuous space and the effect of spatial-varying factors on the location decisions is evaluated by parametrically modelling the intensity of the process. The study is motivated by the real issue of analysing the birth process of small and medium manufacturing firms in Tuscany, an Italian region, and it shows that the location choices of the new Tuscan firms is influenced on the one hand by the availability of infrastructures and the level of accessibility, and on the other by the presence and the characteristics of the existing firms. Moreover, the effect of these factors varies with the size and the level of technology of the new firms. Besides the specific Tuscan result, the study shows the potentiality of the described micro-econometric approach for the analysis of the spatial dynamics of firms.     

Application of a predictive distribution formula to Bayesian computation for incomplete data models

We consider exact and approximate Bayesian computation in the presence of latent variables or missing data. Specifically we explore the application of a posterior predictive distribution formula derived in Sweeting And Kharroubi (2003), which is a particular form of Laplace approximation, both as an importance function and a proposal distribution. We show that this formula provides a stable importance function for use within poor man’s data augmentation schemes and that it can also be used as a proposal distribution within a Metropolis-Hastings algorithm for models that are not analytically tractable. We illustrate both uses in the case of a censored regression model and a normal hierarchical model, with both normal and Student t distributed random effects. Although the predictive distribution formula is motivated by regular asymptotic theory, it is not necessary that the likelihood has a closed form or that it possesses a local maximum.     

Correlating point-referenced radon and areal uranium data arising from a common spatial process

Summary.  Because exposure to radon gas in buildings is a likely risk factor for lung cancer, estimation of residential radon levels is an important public health endeavour. Radon originates from uranium, and therefore data on the geographical distribution of uranium in the Earth's surface may inform about radon levels. We fit a Bayesian geostatistical model that appropriately combines data on uranium with measurements of indoor home radon in the state of Iowa, thereby obtaining more accurate and precise estimation of the geographic distribution of average residential radon levels than would be possible by using radon data alone.     

Kernel partial correlation: a novel approach to capturing conditional independence in graphical models for noisy data

Graphical models capture the conditional independence structure among random variables via existence of edges among vertices. One way of inferring a graph is to identify zero partial correlation coefficients, which is an effective way of finding conditional independence under a multivariate Gaussian setting. For more general settings, we propose kernel partial correlation which extends partial correlation with a combination of two kernel methods. First, a nonparametric function estimation is employed to remove effects from other variables, and then the dependence between remaining random components is assessed through a nonparametric association measure. The proposed approach is not only flexible but also robust under high levels of noise owing to the robustness of the nonparametric approaches.     

Inference of a hidden spatial tessellation from multivariate data: application to the delineation of homogeneous regions in an agricultural field

Summary.  In a precision farming context, differentiated management decisions regarding fertilization, application of lime and other cultivation activities may require the subdivision of the field into homogeneous regions with respect to the soil variables of main agronomic significance. The paper develops an approach that is aimed at delineating homogeneous regions on the basis of measurements of a categorical and quantitative nature, namely soil type and resistivity measurements at different soil layers. We propose a Bayesian multivariate spatial model and embed it in a Markov chain Monte Carlo inference scheme. Implementation is discussed using real data from a 15-ha field. Although applied to soil data, this model could be relevant in areas of spatial modelling as diverse as epidemiology, ecology or meteorology.     

Geometrically Corrected Second Order Analysis of Events on a Linear Network,with Applications to Ecology and Criminology

Abstract. We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. Okabe and Yamada developed a ‘network K function’, analogous to Ripley's K function, for analysis of such data. However, values of the network K‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of K, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network K‐function is the identity. The corrected estimator is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of under complete randomness. We extend these results to an ‘inhomogeneous’ network K‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).     

Case study in data analysis: Variables related to codling moth abundance and the efficacy of the Okanagan Sterile Insect Release Program

The authors consider the effect of orchard attributes and landscape in a heterogeneous area on the efficacy of a control program for the codling moth in apple orchards in British Columbia. The context is first presented, along with a set of questions of importance to the Okanagan Valley Sterile Insect Release program. Two groups of analysts then address a number of these issues using methods for spatial‐temporal data including counts, proportions and Bernoulli variables. The models are then compared and the relevance of the results to this operational program is discussed.