Image warping using radial basis functions

Image warping is the process of deforming an image through a transformation of its domain, which is typically a subset of R². Given the destination of a collection of points, the problem becomes one of finding a suitable smooth interpolation for the destinations of the remaining points of the domain. A common solution is to use the thin plate spline (TPS). We find that the TPS often introduces unintended distortions of image structures. In this paper, we will analyze interpolation by TPS, experiment with other radial basis functions, and suggest two alternative functions that provide better results.     

Nonparametric regression using linear combinations of basis functions

This paper discusses a Bayesian approach to nonparametric regression initially proposed by Smith and Kohn (1996. Journal of Econometrics 75: 317–344). In this approach the regression function is represented as a linear combination of basis terms. The basis terms can be univariate or multivariate functions and can include polynomials, natural splines and radial basis functions. A Bayesian hierarchical model is used such that the coefficient of each basis term can be zero with positive prior probability. The presence of basis terms in the model is determined by latent indicator variables. The posterior mean is estimated by Markov chain Monte Carlo simulation because it is computationally intractable to compute the posterior mean analytically unless a small number of basis terms is used. The present article updates the work of Smith and Kohn (1996. Journal of Econometrics 75: 317–344) to take account of work by us and others over the last three years. A careful discussion is given to all aspects of the model specification, function estimation and the use of sampling schemes. In particular, new sampling schemes are introduced to carry out the variable selection methodology.     

A distance-based diagnostic for trans-dimensional Markov chains

Over the last decade the use of trans-dimensional sampling algorithms has become endemic in the statistical literature. In spite of their application however, there are few reliable methods to assess whether the underlying Markov chains have reached their stationary distribution. In this article we present a distance-based method for the comparison of trans-dimensional Markov chain sample output for a broad class of models. This diagnostic will simultaneously assess deviations between and within chains. Illustration of the analysis of Markov chain sample-paths is presented in simulated examples and in two common modelling situations: a finite mixture analysis and a change-point problem.     

A distance-based rounding strategy for post-imputation ordinal data

Multiple imputation has emerged as a widely used model-based approach in dealing with incomplete data in many application areas. Gaussian and log-linear imputation models are fairly straightforward to implement for continuous and discrete data, respectively. However, in missing data settings which include a mix of continuous and discrete variables, correct specification of the imputation model could be a daunting task owing to the lack of flexible models for the joint distribution of variables of different nature. This complication, along with accessibility to software packages that are capable of carrying out multiple imputation under the assumption of joint multivariate normality, appears to encourage applied researchers for pragmatically treating the discrete variables as continuous for imputation purposes, and subsequently rounding the imputed values to the nearest observed category. In this article, I introduce a distance-based rounding approach for ordinal variables in the presence of continuous ones. The first step of the proposed rounding process is predicated upon creating indicator variables that correspond to the ordinal levels, followed by jointly imputing all variables under the assumption of multivariate normality. The imputed values are then converted to the ordinal scale based on their Euclidean distances to a set of indicators, with minimal distance corresponding to the closest match. I compare the performance of this technique to crude rounding via commonly accepted accuracy and precision measures with simulated data sets.     

Empirical Bayes spatial prediction using a Monte Carlo EM algorithm

This paper deals with an empirical Bayes approach for spatial prediction of a Gaussian random field. In fact, we estimate the hyperparameters of the prior distribution by using the maximum likelihood method. In order to maximize the marginal distribution of the data, the EM algorithm is used. Since this algorithm requires the evaluation of analytically intractable and high dimensionally integrals, a Monte Carlo method based on discretizing parameter space, is proposed to estimate the relevant integrals. Then, the approach is illustrated by its application to a spatial data set. Finally, we compare the predictive performance of this approach with the reference prior method.     

A spatial structural equation model for health outcomes

Population level risk factors in spatial epidemiology (e.g. socioeconomic deprivation) are often not directly available but indirectly measured through census or other sources. This paper considers multiple health outcomes (e.g. mortality, hospital admissions) in relation to unmeasured latent constructs of population morbidity, established as relevant to explaining spatial contrasts in such health outcomes. The constructs are derived using a factor analytic approach in which observed area social indicators are measures of a smaller set of latent constructs. The constructs are allowed to be spatially correlated as well as correlated with one another. The possibility of nonlinear construct effects is considered using a spline regression. A case study considers suicide mortality and self-harm contrasts in 32 London boroughs, in relation to two latent constructs: area deprivation and social fragmentation.     

A spatial mixed-effects regression model for electoral data

Statistical Methods & Applications - On 4th March 2018, elections took place in Italy for the two Chambers of the Parliament. Many newspapers emphasized the victory of the 5 Star Movement (5SM)...     

Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation

Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Second, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi-Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.     

Optimizing minimum information pair-copula using genetic algorithm to select optimal basis functions

Constructing pair-copula using the minimum information approach is an appropriate and flexible way to survey the dependency structure between variables of interest. Minimum information pair-copula method approximates multivariate copula by applying some constraints between desired variables that are elicited from the data itself or experts’ judgment. In minimum information pair-copula, selecting basis constraints is a challenge. In this article, we apply genetic algorithms as a heuristic way to select basis constraints to optimize approximated pair-copula. The results gained show that our method optimizes model selection criteria and lead to better pair-copula approximation. Finally, we apply our proposed method to approximate pair-copula density in real dataset.     

A model for estimating point sources in spatial data

We introduce a point source model which may be useful for estimating point sources in spatial data. It may also be useful for modelling general spatial data, and providing a simple explanatory model for some data, whilst in other cases it may give a parsimonious representation. The model assumes that there are point sources (or sinks), usually at unknown positions, and that the mean value at a site depends on the distance from these sources. We discuss the general form of the model, and some methods for estimating the sources and the regression parameters. We demonstrate the methodology using a simulation study, and apply the model to two real data sets. Some possibilities for further research are outlined.     

Modeling and prediction for multivariate spatial factor analysis

Factor analysis of multivariate spatial data is considered. A systematic approach for modeling the underlying structure of potentially irregularly spaced, geo-referenced vector observations is proposed. Statistical inference procedures for selecting the number of factors and for model building are discussed. We derive a condition under which a simple and practical inference procedure is valid without specifying the form of distributions and factor covariance functions. The multivariate prediction problem is also discussed, and a procedure combining the latent variable modeling and a measurement-error-free kriging technique is introduced. Simulation results and an example using agricultural data are presented.     

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.     

A non-homogeneous skew-Gaussian Bayesian spatial model

In spatial statistics, models are often constructed based on some common, but possible restrictive assumptions for the underlying spatial process, including Gaussianity as well as stationarity and isotropy. However, these assumptions are frequently violated in applied problems. In order to simultaneously handle skewness and non-homogeneity (i.e., non-stationarity and anisotropy), we develop the fixed rank kriging model through the use of skew-normal distribution for its non-spatial latent variables. Our approach to spatial modeling is easy to implement and also provides a great flexibility in adjusting to skewed and large datasets with heterogeneous correlation structures. We adopt a Bayesian framework for our analysis, and describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters and performing spatial prediction. Through a simulation study, we demonstrate that the proposed model could detect departures from normality and, for illustration, we analyze a synthetic dataset of CO\(_2\) measurements. Finally, to deal with multivariate spatial data showing some degree of skewness, a multivariate extension of the model is also provided.     

Maximum likelihood estimation and prediction mean square error in the spatial linear model

This paper is concerned with prediction in the spatial linear model using the maximum likelihood estimation of parameters in this model. In particular, we give some properties of predictors obtained on substituting the maximum likelihood estimators of model parameters into the form of the best-in the sense of minimum mean square prediction error-predictor. Such predictors are not optimal but we show them to be asymptotically equivalent to the optimum. We discuss practical aspects of this work and conclude by considering the connection with other areas.     

A distance based regression model for prediction with mixed data

A multiple regression method based on distance analysis and metric scaling is proposed and studied. This method allow us to predict a continuous response variable from several explanatory variables, is compatible with the general linear model and is found to be useful when the predictor variables are both continuous and categorical. Real data examples are given to illustrate the results obtained.     

Testing for separability of spatial–temporal covariance functions

Most applications in spatial statistics involve modeling of complex spatial–temporal dependency structures, and many of the problems of space and time modeling can be overcome by using separable processes. This subclass of spatial–temporal processes has several advantages, including rapid fitting and simple extensions of many techniques developed and successfully used in time series and classical geostatistics. In particular, a major advantage of these processes is that the covariance matrix for a realization can be expressed as the Kronecker product of two smaller matrices that arise separately from the temporal and purely spatial processes, and hence its determinant and inverse are easily determinable. However, these separable models are not always realistic, and there are no formal tests for separability of general spatial–temporal processes. We present here a formal method to test for separability. Our approach can be also used to test for lack of stationarity of the process. The beauty of our approach is that by using spectral methods the mechanics of the test can be reduced to a simple two-factor analysis of variance (ANOVA) procedure. The approach we propose is based on only one realization of the spatial–temporal process.We apply the statistical methods proposed here to test for separability and stationarity of spatial–temporal ozone fields using data provided by the US Environmental Protection Agency (EPA).     

A note on finite population prediction under asymetric loss functions

The purpose of the present note is to derive optimal population total predictors relative to the Linex (Zellner, 1986) loss function under some well known superpopulation models. The risk function and Bayes risk are derived and compared with those of usual predictors. Minimax and and admissibility properties of some of the derived predictors are also investigated.     

A continuous latent spatial model for crack initiation in bone cement

Summary.  Hip replacements rovide a means of achieving a higher quality of life for individuals who have, through aging or injury, accumulated damage to their natural joints. This is a very common operation, with over a million people a year benefiting from the procedure. The replacements themselves fail mainly as a result of the mechanical loosening of the components of the artificial joint due to damage accumulation. This damage accumulation consists of the initiation and growth of cracks in the bone cement which is used to fixate the replacement in the human body. The data come from laboratory experiments that are designed to assess the effectiveness of the bone cement in resisting damage. We examine the properties of the bone cement, with the aim being to estimate the effect that both observable and unobservable spatially varying factors have on causing crack initiation. To do this, an explicit model for the damage process is constructed taking into account the tension and compression at different locations in the specimens. A gamma random field is used to model any latent spatial factors that may be influential in crack initiation. Bayesian inference is carried out for the parameters of this field and related covariates by using Markov chain Monte Carlo techniques.     

Constructing and fitting models for cokriging and multivariable spatial prediction

We consider best linear unbiased prediction for multivariable data. Minimizing mean-squared-prediction errors leads to prediction equations involving either covariances or variograms. We discuss problems with multivariate extensions that include the construction of valid models and the estimation of their parameters. In this paper, we develop new methods to construct valid crossvariograms, fit them to data, and then use them for multivariable spatial prediction, including cokriging. Crossvariograms are constructed by explicitly modeling spatial data as moving averages over white noise random processes. Parameters of the moving average functions may be inferred from the variogram, and with few additional parameters, crossvariogram models are constructed. Weighted least squares is then used to fit the crossvariogram model to the empirical crossvariogram for the data. We demonstrate the method for simulated data, and show a considerable advantage of cokriging over ordinary kriging.