Alexandra M. Schmidt

Nonstationary Multivariate Process Modeling through Spatially Varying Coregionalization

Models for the analysis of multivariate spatial data are receiving increased attention these days. In many applications it will be preferable to work with multivariate spatial processes to specify such models. A critical specification in providing these models is the cross covariance function. Constructive approaches for developing valid cross-covariance functions offer the most practical strategy for doing this. These approaches include separability, kernel convolution or moving average methods, and convolution of covariance functions. We review these approaches but take as our main focus the computationally manageable class referred to as the linear model of coregionalization (LMC). We introduce a fully Bayesian development of the LMC. We offer clarification of the connection between joint and conditional approaches to fitting such models including prior specifications. However, to substantially enhance the usefulness of such modelling we propose the notion of a spatially varying LMC (SVLMC) providing a very rich class of multivariate nonstationary processes with simple interpretation. We illustrate the use of our proposed SVLMC with application to more than 600 commercial property transactions in three quite different real estate markets, Chicago, Dallas and San Diego. Bivariate nonstationary process inodels are developed for income from and selling price of the property.

A class of covariate-dependent spatiotemporal covariance functions for the analysis of daily ozone concentration

In geostatistics, it is common to model spatially distributed phenomena through an underlying stationary and isotropic spatial process. However, these assumptions are often untenable in practice because of the influence of local effects in the correlation structure. Therefore, it has been of prolonged interest in the literature to provide flexible and effective ways to model non-stationarity in the spatial effects. Arguably, due to the local nature of the problem, we might envision that the correlation structure would be highly dependent on local characteristics of the domain of study, namely the latitude, longitude and altitude of the observation sites, as well as other locally defined covariate information. In this work, we provide a flexible and computationally feasible way for allowing the correlation structure of the underlying processes to depend on local covariate information. We discuss the properties of the induced covariance functions and discuss methods to assess its dependence on local covariate information by means of a simulation study and the analysis of data observed at ozone-monitoring stations in the Southeast United States.

Modelling zero-inflated spatio-temporal processes:

We consider models for spatio-temporal processes which assume either non-negative values, and often are observed as zero, or discrete values and are also inflated by zeros. Typically, in the first case, the spatial observations are obtained at fixed locations (point-referenced data) over a region D; whereas in the second, the region D is divided into a finite number of regular or irregular subregions (areal level), resulting in observations for each subregion. Our main idea is based on those of zeroinflated models, by assuming that the value observed at location s and time t, Yt (s), is a realization of a mixture between a Bernoulli distribution with a probability of success θt (s) and a probability density function or probability function p(yt (s) | .) For both cases, we include spatio-temporal latent processes in the model to account for the possible extra variation present in the mean structure of θt (s) and/or p(yt(s) | .). In the context of point-referenced data, we model the amount of rainfall over the city of Rio de Janeiro during 75 weeks; whereas in the areal data level case, we consider weekly cases of dengue fever in the city of Rio de Janeiro during the years of 2001–02.

Hyperparameter estimation in forecast models

A large number of non-linear time series models can be more easily analyzed using traditional linear methods by considering explicitly the difference between parameters of interest, or just parameters, and hyperparameters. One example is the class of conditionally Gaussian dynamic linear models. Bayesian vector autoregressive models and non-linear transfer function models are also important examples in the literature. Until recently, a two-step procedure was broadly used to estimate such models. In the first step maximum likelihood estimation was used to find the best value of the hyperparameter, which turned to be used in the second step where a conditionally linear model was fitted. The main drawback of such an algorithm is that it does not take into account any kind of uncertainty that might have been brought, and usually was, to the modeling at the first step. In other words and more practically speaking, the variances of the parameters are underestimated. Another problem, more philosophical, is the violation of the likelihood principle by using the sample information twice. In this paper we apply sampling importance resampling (SIR) techniques (Rubin, 1988) to obtain a numerical approximation to the full posterior distribution of the hyperparameters. Then, instead of conditioning in a particular value of that distribution we integrate the hyperparameters out in order to obtain the marginal posterior distributions of the parameters. We used SIR to model a set of Brazilian macroeconomic time-series in three different, but important, contexts. We also compare the forecast performance of our approach with traditional ones.

Spatially Varying Autoregressive Processes

We develop a class of models for processes indexed in time and space that are based on autoregressive (AR) processes at each location. We use a Bayesian hierarchical structure to impose spatial coherence for the coefficients of the AR processes. The priors on such coefficients consist of spatial processes that guarantee time stationarity at each point in the spatial domain. The AR structures are coupled with a dynamic model for the mean of the process, which is expressed as a linear combination of time-varying parameters. We use satellite data on sea surface temperature for the North Pacific to illustrate how the model can be used to separate trends, cycles, and short-term variability for high-frequency environmental data. This article has supplementary material online.

A nonlinear population Monte Carlo scheme for the Bayesian estimation of parameters of α -stable distributions

The class of α -stable distributions enjoys multiple practical applications in signal processing, finance, biology and other areas because it allows to describe interesting and complex data patterns, such as asymmetry or heavy tails, in contrast with the simpler and widely used Gaussian distribution. The density associated with a general α -stable distribution cannot be obtained in closed form, which hinders the process of estimating its parameters. A nonlinear population Monte Carlo (NPMC) scheme is applied in order to approximate the posterior probability distribution of the parameters of an α -stable random variable given a set of random realizations of the latter. The approximate posterior distribution is computed by way of an iterative algorithm and it consists of a collection of samples in the parameter space with associated nonlinearly-transformed importance weights. A numerical comparison of the main existing methods to estimate the α -stable parameters is provided, including the traditional frequentist techniques as well as a Markov chain Monte Carlo (MCMC) and a likelihood-free Bayesian approach. It is shown by means of computer simulations that the NPMC method outperforms the existing techniques in terms of parameter estimation error and failure rate for the whole range of values of α , including the smaller values for which most existing methods fail to work properly. Furthermore, it is shown that accurate parameter estimates can often be computed based on a low number of observations. Additionally, numerical results based on a set of real fish displacement data are provided.

Measuring the vulnerability of the Uruguayan population to vector-borne diseases via spatially hierarchical factor models

We propose a model-based vulnerability index of the population from Uruguay to vector-borne diseases. We have available measurements of a set of variables in the census tract level of the 19 Departmental capitals of Uruguay. In particular, we propose an index that combines different sources of information via a set of micro-environmental indicators and geographical location in the country. Our index is based on a new class of spatially hierarchical factor models that explicitly account for the different levels of hierarchy in the country, such as census tracts within the city level, and cities in the country level. We compare our approach with that obtained when data are aggregated in the city level. We show that our proposal outperforms current and standard approaches, which fail to properly account for discrepancies in the region sizes, for example, number of census tracts. We also show that data aggregation can seriously affect the estimation of the cities vulnerability rankings under benchmark models.

A Hierarchical Model for Aggregated Functional Data

In many areas of science, one aims to estimate latent subpopulation mean curves based only on observations of aggregated population curves. By aggregated curves we mean linear combination of functional data that cannot be observed individually. We assume that several aggregated curves with linearly independent coefficients are available, and each aggregated curve is an independent partial realization of a Gaussian process with mean modeled through a weighted linear combination of the disaggregated curves. The mean of the Gaussian process is modeled using B-splines basis expansion methods. We propose a semiparametric, valid covariance function that is modeled as the product of a nonparametric variance function by a correlation function. The variance function is described as the square of a function that is expanded using B-splines basis functions. This results in a nonstationary covariance function and includes constant variance models as special cases. Inference is performed following the Bayesian paradigm allowing experts’ opinion, when available, to be accounted for. Moreover, it naturally provides the uncertainty associated with the parameters’ estimates and fitted values. We analyze artificial datasets and discuss how to choose among the different covariance models. We focus on two different real examples: a calibration problem for NIR spectroscopy data and an analysis of distribution of energy among different types of consumers. In the latter example, our proposed covariance function captures interesting features of the data. Further analysis of different artificial datasets, as well as computer code and data is available as supplementary material online.

Population counts along elliptical habitat contours: Hierarchical modeling using Poisson-lognormal mixtures with nonstationary spatial structure

Ecologists often interpret variation in the spatial distribution of populations in terms of responses to environmental features, but disentangling the effects of individual variables can be difficult if latent effects and spatial and temporal correlations are not accounted for properly. Here, we use hierarchical models based on a Poisson log-normal mixture to understand the spatial variation in relative abundance (counts per standardized unit of effort) of yellow perch, Perca flavescens, the most abundant fish species in Lake Saint Pierre, Quebec, Canada. The mixture incorporates spatially varying environmental covariates that represent local habitat characteristics, and random temporal and spatial effects that capture the effects of unobserved ecological processes. The sampling design covers the margins but not the central region of the lake. We fit spatial generalized linear mixed models based on three different prior covariance structures for the local latent effects: a single Gaussian process (GP) over the lake, a GP over a circle, and independent GP for each shore. The models allow for independence, isotropy, or nonstationary spatial effects. Nonstationarity is dealt with using two different approaches, geometric anisotropy, and the inclusion of covariates in the correlation structure of the latent spatial process. The proposed approaches for specification of spatial domain and choice of Gaussian process priors may prove useful in other applications that involve spatial correlation along an irregular contour or in discontinous spatial domains.

Laparoscopia no abdome agudo não traumático: estudo retrospectivo

Os autores analisaram, retrospectivamente, 117 pacientes portadores de abdome agudo nao-traumatico, submetidos a laparoscopia diagnostica e/ou terapeutica, na Casa de Saude Santa Martha, em Niteroi. A precisao diagnostica do exame laparoscopico foi de 96,6%. Com relacao a terapeutica, 74,4% dos pacientes foram tratados por laparoscopia, 21,4% por laparotomia e 4,3% foram tratados clinicamente. A precocidade na realizacao da laparoscopia relacionou-se a maior taxa de sucesso com o tratamento laparoscopico (valor p < 0,05). Analisando-se a recuperacao pos-operatoria, os pacientes submetidos a intervencoes laparoscopicas iniciaram a dieta oral e receberam alta mais precocemente que os submetidos a laparotomia (valor p < 0,05 e p < 0,01 respectivamente). A taxa de complicacao foi de 13,7%, com mortalidade de 2,6%. Os autores concluem que a laparoscopia e um excelente metodo diagnostico, permite um manejo terapeutico satisfatorio associado a uma recuperacao pos-operatoria mais precoce.