The statistical analysis of compositional data

Noel Cressie and Christopher WikleHardcover: 624 pagesYear: 2011Publisher: John WileyISBN-13: 978-0471692744Here is the new reference book about spatial and spatio-temporal statistical modeling! No...

Statistics for Spatio-Temporal Data

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Spatial and spatio-temporal models with R-INLA

In this work, we consider a hierarchical spatio-temporal model for particulate matter (PM) concentration in the North-Italian region Piemonte. The model involves a Gaussian Field (GF), affected by a measurement error, and a state process characterized by a first order autoregressive dynamic model and spatially correlated innovations. This kind of model is well discussed and widely used in the air quality literature thanks to its flexibility in modelling the effect of relevant covariates (i.e. meteorological and geographical variables) as well as time and space dependence. However, Bayesian inference—through Markov chain Monte Carlo (MCMC) techniques—can be a challenge due to convergence problems and heavy computational loads. In particular, the computational issue refers to the infeasibility of linear algebra operations involving the big dense covariance matrices which occur when large spatio-temporal datasets are present. The main goal of this work is to present an effective estimating and spatial prediction strategy for the considered spatio-temporal model. This proposal consists in representing a GF with Matern covariance function as a Gaussian Markov Random Field (GMRF) through the Stochastic Partial Differential Equations (SPDE) approach. The main advantage of moving from a GF to a GMRF stems from the good computational properties that the latter enjoys. In fact, GMRFs are defined by sparse matrices that allow for computationally effective numerical methods. Moreover, when dealing with Bayesian inference for GMRFs, it is possible to adopt the Integrated Nested Laplace Approximation (INLA) algorithm as an alternative to MCMC methods giving rise to additional computational advantages. The implementation of the SPDE approach through the R-library INLA (
 www.r-inla.org
 ) is illustrated with reference to the Piemonte PM data. In particular, providing the step-by-step R-code, we show how it is easy to get prediction and probability of exceedance maps in a reasonable computing time.

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Spatio-temporal modeling of particulate matter concentration through the SPDE approach

Sources of high PM2.5 concentrations in Milan, Northern Italy: Molecular marker data and CMB modelling

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A Space-Time Extension of the Lee-Carter Model in a Hierarchical Bayesian Framework: Modelling and Forecasting Provincial Mortality in Italy

Trajectories of compositional data, that is, sequences of composition measurements taken along a domain, can be considered as functional data. The present work centres on a way of clustering compositional data trajectories. Functional cluster analysis has been applied in several fields, but has not been extended to cope with the problem of clustering compositional data trajectories. In this work, we extend functional cluster analysis techniques to the analysis of compositional data using suitable compositional algebra both for smoothing observed trajectories and for building suitable metrics to evaluate dissimilarities between objects. When the focus is on the identification of typical shapes, clustering based on derivatives is the most suitable tool. As a motivating example, we consider clustering particulate matter vertical profiles in the lower troposphere. The impact of the choice of metric on the clustering structure is thoroughly discussed in this case study. Copyright © 2011 John Wiley & Sons, Ltd.

Clustering compositional data trajectories: the case of particulate matter in the lower troposphere

Assessing the evolution of territorial disparities in health

Nowadays, traditional population censuses based on total enumeration of the population are being accompanied by sample surveys. Sampling within censuses allows to reduce costs and workload of authorities involved in censuses operations, along with the statistical burden for the people involved in the enumeration. In this paper, we deal with estimation of multi-way contingency tables involving variables measured both via census and sampling. In this framework, two main issues need to be addressed: first of all, sample size for estimating some of the entries of the contingency tables may be too small, delivering estimates prone to huge sampling variability. On the other hand, since estimates of the joint distribution need to be coherent with the marginal distribution of the variable collected via a census, estimation methods need to be coherent with the constraint imposed by marginal distribution of variables measured via census. The problem is tackled via a model-based approach that allows to comply with all coherence constraints following a fairly simple procedure. The merit of the proposed methodology is illustrated by means of a simulation study.

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Estimation of multi-way tables subject to coherence constraints

Methods to perform regression on compositional covariates have recently been proposed using isometric log-ratios (ilr) representation of compositional parts. This approach consists of first applying standard regression on ilr coordinates (with no need for either constraint on regression coefficients or the use of Moore Penrose generalized inverse) and second, transforming the estimated ilr coefficients into their contrast logratios counterparts which give easy-to-interpret parameters for the linear effect of each compositional part, in relative terms, on the response. In this work we present an extension of this framework, where compositional covariate effects are allowed to be smooth in the ilr domain. This is achieved by fitting a smooth function over the multidimensional ilr space, using a basis of B-splines and a set of associated spline coefficients. Smoothness is achieved by assuming random walk priors on spline coefficients in a hierarchical Bayesian framework. The proposed methodology is illustrated on a dataset from an ecological survey on a gypsum outcrop located in the Emilia Romagna Region, Italy. Data consists of measurements of vegetation cover and substrate compositions in a grid format. The application goal is to estimate substrate suitability for vegetation, regarding four different substrate typologies found in the area. With the proposed method non linear effects of substrate compositions on vegetation cover can be investigated.

Fedele Pasquale Greco

Papers

A Space-Time Extension of the Lee-Carter Model in a Hierarchical Bayesian Framework: Modelling and Forecasting Provincial Mortality in Italy

Clustering compositional data trajectories: the case of particulate matter in the lower troposphere

Assessing the evolution of territorial disparities in health

Estimation of multi-way tables subject to coherence constraints

Non-parametric regression on compositional covariates