Dynamic generalized linear models with application to environmental epidemiology

Summary. We propose modelling short-term pollutant exposure effects on health by using dynamic generalized linear models. The time series of count data are modelled by a Poisson distribution having mean driven by a latent Markov process; estimation is performed by the extended Kalman filter and smoother. This modelling strategy allows us to take into account possible overdispersion and time-varying effects of the covariates. These ideas are illustrated by reanalysing data on the relationship between daily non-accidental deaths and air pollution in the city of Birmingham, Alabama.     

Model-based segmentation of spatial cylindrical data

ABSTRACT

A new hidden Markov random field model is proposed for the analysis of cylindrical spatial series, i.e. bivariate spatial series of intensities and angles. It allows us to segment cylindrical spatial series according to a finite number of latent classes that represent the conditional distributions of the data under specific environmental conditions. The model parsimoniously accommodates circular–linear correlation, multimodality, skewness and spatial autocorrelation. A numerically tractable expectation–maximization algorithm is provided to compute parameter estimates by exploiting a mean-field approximation of the complete-data log-likelihood function. These methods are illustrated on a case study of marine currents in the Adriatic sea.     

Using multivariate time series methods to estimate location and climate change effects on temperature readings employed in electricity demand simulation

Long‐term historical daily temperatures are used in electricity forecasting to simulate the probability distribution of future demand but can be affected by changes in recording site and climate. This paper presents a method of adjusting for the effect of these changes on daily maximum and minimum temperatures. The adjustment technique accommodates the autocorrelated and bivariate nature of the temperature data which has not previously been taken into account. The data are from Perth, Western Australia, the main electricity demand centre for the South‐West of Western Australia. The statistical modelling involves a multivariate extension of the univariate time series ‘interleaving method’, which allows fully efficient simultaneous estimation of the parameters of replicated Vector Autoregressive Moving Average processes. Temperatures at the most recent weather recording location in Perth are shown to be significantly lower compared to previous sites. There is also evidence of long‐term heating due to climate change especially for minimum temperatures.     

Cross-validating fit and predictive accuracy of nonlinear quantile regressions

The paper proposes a cross-validation method to address the question of specification search in a multiple nonlinear quantile regression framework. Linear parametric, spline-based partially linear and kernel-based fully nonparametric specifications are contrasted as competitors using cross-validated weighted L ₁-norm based goodness-of-fit and prediction error criteria. The aim is to provide a fair comparison with respect to estimation accuracy and/or predictive ability for different semi- and nonparametric specification paradigms. This is challenging as the model dimension cannot be estimated for all competitors and the meta-parameters such as kernel bandwidths, spline knot numbers and polynomial degrees are difficult to compare. General issues of specification comparability and automated data-driven meta-parameter selection are discussed. The proposed method further allows us to assess the balance between fit and model complexity. An extensive Monte Carlo study and an application to a well-known data set provide empirical illustration of the method.     

Decline and fall: Unemployment among Britain's non-white ethnic communities 1960–1999

The declining employment fortunes of Britain's non-white communities, relative to whites, are explained. To achieve this comparison, some historical official sources of unemployment data are reviewed. The earliest known official time series on unemployment of non-whites dates back to 1960. This historical context is explored, especially the reluctance to make the early data more widely known. An unemployment series for non-white males and females from 1970 to 1999 is derived in two separate ways by splicing together official sources. These series are compared with unemployment of whites to demonstrate a relative increase in unemployment of non-whites.     

Bayesian parsimonious covariance estimation for hierarchical linear mixed models

We consider a non-centered parameterization of the standard random-effects model, which is based on the Cholesky decomposition of the variance-covariance matrix. The regression type structure of the non-centered parameterization allows us to use Bayesian variable selection methods for covariance selection. We search for a parsimonious variance-covariance matrix by identifying the non-zero elements of the Cholesky factors. With this method we are able to learn from the data for each effect whether it is random or not, and whether covariances among random effects are zero. An application in marketing shows a substantial reduction of the number of free elements in the variance-covariance matrix.     

Extending dual multiple factor analysis to categorical tables

This paper describes a proposal for the extension of the dual multiple factor analysis (DMFA) method developed by Lê and Pagès 15 to the analysis of categorical tables in which the same set of variables is measured on different sets of individuals. The extension of DMFA is based on the transformation of categorical variables into properly weighted indicator variables, in a way analogous to that used in the multiple factor analysis of categorical variables. The DMFA of categorical variables enables visual comparison of the association structures between categories over the sample as a whole and in the various subsamples (sets of individuals). For each category, DMFA allows us to obtain its global (considering all the individuals) and partial (considering each set of individuals) coordinates in a factor space. This visual analysis allows us to compare the set of individuals to identify their similarities and differences. The suitability of the technique is illustrated through two applications: one using simulated data for two groups of individuals with very different association structures and the other using real data from a voting intention survey in which some respondents were interviewed by telephone and others face to face. The results indicate that the two data collection methods, while similar, are not entirely equivalent.     

Improved local polynomial estimation in time series regression

We propose a modification of local polynomial estimation which improves the efficiency of the conventional method when the observation errors are correlated. The procedure is based on a pre-transformation of the data as a generalization of the pre-whitening procedure introduced by Xiao et al. [(2003), ‘More Efficient Local Polynomial Estimation in Nonparametric Regression with Autocorrelated Errors’, Journal of the American Statistical Association, 98, 980–992]. While these authors assumed a linear process representation for the error process, we avoid any structural assumption. We further allow the regressors and the errors to be dependent. More importantly, we show that the inclusion of both leading and lagged variables in the approximation of the error terms outperforms the best approximation based on lagged variables only. Establishing its asymptotic distribution, we show that the proposed estimator is more efficient than the standard local polynomial estimator. As a by-product we prove a suitable version of a central limit theorem which allows us to improve the asymptotic normality result for local polynomial estimators by Masry and Fan [(1997), ‘Local Polynomial Estimation of Regression Functions for Mixing Processes’, Scandinavian Journal of Statistics, 24, 165–179]. A simulation study confirms the efficiency of our estimator on finite samples. An application to climate data also shows that our new method leads to an estimator with decreased variability.     

Fractal analysis of surface roughness by using spatial data

We develop fractal methodology for data taking the form of surfaces. An advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale-free component (fractal dimension) and properties that depend purely on scale. Particular emphasis is given to anisotropy where we show that, for many surfaces, the fractal dimension of line transects across a surface must either be constant in every direction or be constant in each direction except one. This virtual direction invariance of fractal dimension provides another canonical feature of fractal analysis, complementing its scale invariance properties and enhancing its attractiveness as a method for summarizing properties of roughness. The dependence of roughness on direction may be explained in terms of scale rather than dimension and can vary with orientation. Scale may be described by a smooth periodic function and may be estimated nonparametrically. Our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrapping. For the soil data, interest centres on the effect of surface roughness on retention of rain-water, and data are recorded as a series of digital images over time. Our analysis captures the way in which both the fractal dimension and the scale change with rainfall, or equivalently with time. The food wrapping data are on a much finer scale than the soil data and are particularly anisotropic. The analysis allows us to determine the manufacturing process which produces the smoothest wrapping, with least tendency for micro-organisms to adhere.     

Selecting the best geometric distribution based on type-I censored data: a Bayesian approach

This paper considers the statistical reliability on discrete failure data and the selection of the best geometric distribution having the smallest failure probability from among several competitors. Using the Bayesian approach a Bayes selection rule based on type-I censored data is derived and its associated monotonicity is also obtained. An early selection rule which allows us to make a selection possible earlier than the censoring time of the life testing experiment is proposed. This early selection rule can be shown to be equivalent to the Bayes selection rule. An illustrative example is given to demonstrate the use and the performance of the early selection rule.     

Bayesian inference in a matrix normal dynamic linear model with unknown covariance matrices

In this paper, we consider the problem of estimating the parameters of a matrix normal dynamic linear model when the variance and covariance matrices of its error terms are unknown and can be changing over time. Given that the analysis is not conjugate, we use simulation methods based on Monte Carlo Markov chains to estimate the parameters of the model. This analysis allows us to carry out a dynamic principal components analysis in a set of multivariate time series. Furthermore, it permits the treatment of series with different lengths and with missing data. The methodology is illustrated with two empirical examples: the value added distribution of the firms operating in the manufacturing sector of the countries participating in the BACH project, and the joint evolution of a set of international stock-market indices.     

Dynamic changepoint detection in count time series: a particle filter approach

We study Bayesian dynamic models for detecting changepoints in count time series that present structural breaks. As the inferential approach, we develop a parameter learning version of the algorithm proposed by Chopin [Chopin N. Dynamic detection of changepoints in long time series. Annals of the Institute of Statistical Mathematics 2007;59:349–366.], called the Chopin filter with parameter learning, which allows us to estimate the static parameters in the model. In this extension, the static parameters are addressed by using the kernel smoothing approximations proposed by Liu and West [Liu J, West M. Combined parameters and state estimation in simulation-based filtering. In: Doucet A, de Freitas N, Gordon N, editors. Sequential Monte Carlo methods in practice. New York: Springer-Verlag; 2001]. The proposed methodology is then applied to both simulated and real data sets and the time series models include distributions that allow for overdispersion and/or zero inflation. Since our procedure is general, robust and naturally adaptive because the particle filter approach does not require restrictive specifications to ensure its validity and effectiveness, we believe it is a valuable alternative for dealing with the problem of detecting changepoints in count time series. The proposed methodology is also suitable for count time series with no changepoints and for independent count data.     

Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Powerful Trend Function Tests That Are Robust to Strong Serial Correlation,With an Application to the Prebisch–Singer Hypothesis

We propose tests for hypotheses on the parameters of the deterministic trend function of a univariate time series. The tests do not require knowledge of the form of serial correlation in the data, and they are robust to strong serial correlation. The data can contain a unit root and still have the correct size asymptotically. The tests that we analyze are standard heteroscedasticity autocorrelation robust tests based on nonparametric kernel variance estimators. We analyze these tests using the fixed-b asymptotic framework recently proposed by Kiefer and Vogelsang. This analysis allows us to analyze the power properties of the tests with regard to bandwidth and kernel choices. Our analysis shows that among popular kernels, specific kernel and bandwidth choices deliver tests with maximal power within a specific class of tests. Based on the theoretical results, we propose a data-dependent bandwidth rule that maximizes integrated power. Our recommended test is shown to have power that dominates a related test proposed by Vogelsang. We apply the recommended test to the logarithm of a net barter terms of trade series and we find that this series has a statistically significant negative slope. This finding is consistent with the well-known Prebisch–Singer hypothesis.     

Efficient Estimation of Fixed and Time-varying Covariate Effects in Multiplicative Intensity Models

The proportional hazards assumption of the Cox model does sometimes not hold in practise. An example is a treatment effect that decreases with time. We study a general multiplicative intensity model allowing the influence of each covariate to vary non-parametrically with time. An efficient estimation procedure for the cumulative parameter functions is developed. Its properties are studied using the martingale structure of the problem. Furthermore, we introduce a partly parametric version of the general non-parametric model in which the influence of some of the covariates varies with time while the effects of the remaining covariates are constant. This semiparametric model has not been studied in detail before. An efficient procedure for estimating the parametric as well as the non-parametric components of this model is developed. Again the martingale structure of the model allows us to describe the asymptotic properties of the suggested estimators. The approach is applied to two different data sets, and a Monte Carlo simulation is presented.     

Backfitting Random Varying-Coefficient Models with Time-Dependent Smoothing Covariates

Abstract.  In this paper, we propose a random varying-coefficient model for longitudinal data. This model is different from the standard varying-coefficient model in the sense that the time-varying coefficients are assumed to be subject-specific, and can be considered as realizations of stochastic processes. This modelling strategy allows us to employ powerful mixed-effects modelling techniques to efficiently incorporate the within-subject and between-subject variations in the estimators of time-varying coefficients. Thus, the subject-specific feature of longitudinal data is effectively considered in the proposed model. A backfitting algorithm is proposed to estimate the coefficient functions. Simulation studies show that the proposed estimation methods are more efficient in finite-sample performance compared with the standard local least squares method. An application to an AIDS clinical study is presented to illustrate the proposed methodologies.     

The Demand for Lotto: The Role of Conscious Selection

This article presents estimates of the elasticity of demand for lottery tickets using time series data in which there is variation in the expected value of a lottery ticket induced by rollovers. An important feature of our data is that there are far more rollovers than expected given the lottery design. We find strong evidence that individuals do not choose their lottery numbers uniformly from a uniform distribution—that is, conscious selection. We use our estimates to derive the inverse supply function for the industry, and this enables us to identify the demand elasticity. We find the price elasticity to be close to unity, which implies that the operator is revenue maximizing—which is the regulator's objective.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

Finding overlapping components with MML

We use minimum message length (MML) estimation for mixture modelling. MML estimates are derived to choose the number of components in the mixture model to best describe the data and to estimate the parameters of the component densities for Gaussian mixture models. An empirical comparison of criteria prominent in the literature for estimating the number of components in a data set is performed. We have found that MML coding considerations allows the derivation of useful results to guide our implementation of a mixture modelling program. These advantages allow model search to be controlled based on the minimum variance for a component and the amount of data required to distinguish two overlapping components.     

Forecasting time series with missing data using Holt's model

This paper deals with the prediction of time series with missing data using an alternative formulation for Holt's model with additive errors. This formulation simplifies both the calculus of maximum likelihood estimators of all the unknowns in the model and the calculus of point forecasts. In the presence of missing data, the EM algorithm is used to obtain maximum likelihood estimates and point forecasts. Based on this application we propose a leave-one-out algorithm for the data transformation selection problem which allows us to analyse Holt's model with multiplicative errors. Some numerical results show the performance of these procedures for obtaining robust forecasts.