Estimating time to detect time trends in continuous cropping

SUMMARY In long-term field trials comparing different sequences of crops and husbandry practices, the identification and understanding of trends in productivity over time is an important issue of sustainable crop production. This paper presents a statistical technique for the estimation of time trends in yield variables of a seasonal annual crop under continuous cropping. The estimation procedure incorporates the correlation structure, which is assumed to follow first-order autocorrelation in the errors that arise over time on the same plot. Because large differences in annual rainfall have a major effect on crop performance, rainfall has been allowed for in the estimation of the time trends. Expressions for the number of years (time) required to detect statistically significant time trends have been obtained. Illustrations are based on a 7-year data set of grain and straw yields from a trial in northern Syria. Although agronomic interpretation is not intended in this paper, the barley yield data indicated that a significant time trend can apparently be detected even in a suboptimal data set of 7 years' duration.     

Spatio-temporal modeling of infectious disease dynamics

A stochastic model, which is well suited to capture space–time dependence of an infectious disease, was employed in this study to describe the underlying spatial and temporal pattern of measles in Barisal Division, Bangladesh. The model has two components: an endemic component and an epidemic component; weights are used in the epidemic component for better accounting of the disease spread into different geographical regions. We illustrate our findings using a data set of monthly measles counts in the six districts of Barisal, from January 2000 to August 2009, collected from the Expanded Program on Immunization, Bangladesh. The negative binomial model with both the seasonal and autoregressive components was found to be suitable for capturing space–time dependence of measles in Barisal. Analyses were done using general optimization routines, which provided the maximum likelihood estimates with the corresponding standard errors.     

Spatio‐temporal modelling of disease mapping of rates

This paper studies generalized linear mixed models (GLMMs) for the analysis of geographic and temporal variability of disease rates. This class of models adopts spatially correlated random effects and random temporal components. Spatio‐temporal models that use conditional autoregressive smoothing across the spatial dimension and autoregressive smoothing over the temporal dimension are developed. The model also accommodates the interaction between space and time. However, the effect of seasonal factors has not been previously addressed and in some applications (e.g., health conditions), these effects may not be negligible. The authors incorporate the seasonal effects of month and possibly year as part of the proposed model and estimate model parameters through generalized estimating equations. The model provides smoothed maps of disease risk and eliminates the instability of estimates in low‐population areas while maintaining geographic resolution. They illustrate the approach using a monthly data set of the number of asthma presentations made by children to Emergency Departments (EDs) in the province of Alberta, Canada, during the period 2001–2004. The Canadian Journal of Statistics 38: 698–715; 2010 © 2010 Statistical Society of Canada     

Dynamic models for spatiotemporal data

We propose a model for non-stationary spatiotemporal data. To account for spatial variability, we model the mean function at each time period as a locally weighted mixture of linear regressions. To incorporate temporal variation, we allow the regression coefficients to change through time. The model is cast in a Gaussian state space framework, which allows us to include temporal components such as trends, seasonal effects and autoregressions, and permits a fast implementation and full probabilistic inference for the parameters, interpolations and forecasts. To illustrate the model, we apply it to two large environmental data sets: tropical rainfall levels and Atlantic Ocean temperatures.     

Statistical estimation of time-trends in two-course crop rotations

An assessment of time-trends in yield parameters is essential to the utilization of data from long-term field trials for the comparison of different crop rotations and input regimes, and the identification of sustainable production systems. The barley-vetch rotation established at Breda in northern Syria has provided the basis for estimation of the time-trends in yield data from selected treatments in a two-course crop rotation trial. The model used for the estimation accounts for the effect of rainfall, a major determinant of each annual yield value, and the first-order autocorrelation structure in the errors arising from the same plot over time. An expression for the minimum number of cycles required to detect a significant time-trend has been obtained. Results from the barley-vetch rotation under two fertilizer regimes have been discussed.     

Spatio-temporal Trend Analysis of Spring Arrival Data for Migratory Birds

There is increasing interest in spatio-temporal analysis of environmental and ecological responses to changes in the climate due to the recent concerns about climate change. In this work, we propose a spatio-temporal modeling framework for analyzing environmental and ecological data while accounting for spatial and temporal structure, as well as climate effects. As an example, we consider data on bird migration in the United States and analyze the spring arrival dates of Purple Martins between historical data (1905–1940) from the North American Bird Phenology Program and recent data (2001–2010) from the Purple Martin Conservation Association. The proposed approach allows researchers to compare mean arrival dates while accounting for spatial and temporal variability. Our results for Purple Martins showed statistically significant earlier spring arrivals in parts of United States over the recent years. The proposed approach provides a useful tool for statistical analysis of spatio-temporal data related to studies of climate change.     

Spatio-temporal modelling using B-spline for disease mapping: analysis of childhood cancer trends

To examine childhood cancer diagnoses in the province of Alberta, Canada during 1983–2004, we construct a generalized additive mixed model for the analysis of geographic and temporal variability of cancer ratios. In this model, spatially correlated random effects and temporal components are adopted. The interaction between space and time is also accommodated. Spatio-temporal models that use conditional autoregressive smoothing across the spatial dimension and B-spline over the temporal dimension are considered. We study the patterns of incidence ratios over time and identify areas with consistently high ratio estimates as areas for potential further investigation. We apply the method of penalized quasi-likelihood to estimate the model parameters. We illustrate this approach using a yearly data set of childhood cancer diagnoses in the province of Alberta, Canada during 1983–2004.     

Regularized covariance matrix estimation under the common principal components model

The common principal components (CPC) model provides a way to model the population covariance matrices of several groups by assuming a common eigenvector structure. When appropriate, this model can provide covariance matrix estimators of which the elements have smaller standard errors than when using either the pooled covariance matrix or the per group unbiased sample covariance matrix estimators. In this article, a regularized CPC estimator under the assumption of a common (or partially common) eigenvector structure in the populations is proposed. After estimation of the common eigenvectors using the Flury–Gautschi (or other) algorithm, the off-diagonal elements of the nearly diagonalized covariance matrices are shrunk towards zero and multiplied with the orthogonal common eigenvector matrix to obtain the regularized CPC covariance matrix estimates. The optimal shrinkage intensity per group can be estimated using cross-validation. The efficiency of these estimators compared to the pooled and unbiased estimators is investigated in a Monte Carlo simulation study, and the regularized CPC estimator is applied to a real dataset to demonstrate the utility of the method.     

Prediction Performance of Correlated Error Procedures for Spatial Counts

Abstract

Few guidelines exist for the application of geostatistical methods to spatial counts and the prediction to unsampled areas is an important aspect of experimental field research. The prediction performances of kriging and a correlated errors Poisson model are compared through simulation. Counts with a known spatial covariance structure are generated in an investigation involving several factors: area size, overall mean, range of correlation, spatial covariance function, and the presence of trend. The correlated errors Poisson model generally gives superior prediction performance when an exponential covariance structure is used.     

Separability tests for high-dimensional,low-sample size multivariate repeated measures data

Longitudinal imaging studies have moved to the forefront of medical research due to their ability to characterize spatio-temporal features of biological structures across the lifespan. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within-subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our approach provides useful guidance for high dimension, low-sample size data that preclude using standard likelihood-based tests. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrate the approaches appeal.     

Robust estimation of a dynamic spatio-temporal model with structural change

We postulate a dynamic spatio-temporal model with constant covariate effect but with varying spatial effect over time and varying temporal effect across locations. To mitigate the effect of temporary structural change, the model can be estimated using the backfitting algorithm embedded with forward search algorithm and bootstrap. A simulation study is designed to evaluate structural optimality of the model with the estimation procedure. The fitted model exhibit superior predictive ability relative to the linear model. The proposed algorithm also consistently produced lower relative bias and standard errors for the spatial parameter estimates. While additional neighbourhoods do not necessarily improve predictive ability of the model, it trims down relative bias on the parameter estimates, specially for spatial parameter. Location of the temporary structural change along with the degree of structural change contributes to lower relative bias of parameter estimates and in better predictive ability of the model. The estimation procedure is able to produce parameter estimates that are robust to the occurrence of temporary structural change.     

Assessing direct and indirect seasonal decomposition in state space

The problem of whether seasonal decomposition should be used prior to or after aggregation of time series is quite old. We tackle the problem by using a state-space representation and the variance/covariance structure of a simplified one-component model. The variances of the estimated components in a two-series system are compared for direct and indirect approaches and also to a multivariate method. The covariance structure between the two time series is important for the relative efficiency. Indirect estimation is always best when the series are independent, but when the series or the measurement errors are negatively correlated, direct estimation may be much better in the above sense. Some covariance structures indicate that direct estimation should be used while others indicate that an indirect approach is more efficient. Signal-to-noise ratios and relative variances are used for inference.     

A flexible and efficient spatial interpolator for radar rainfall estimation

A key challenge in rainfall estimation is spatio-temporal variablility. Weather radars are used to estimate precipitation with high spatial and temporal resolution. Due to the inherent errors in radar estimates, spatial interpolation has been often employed to calibrate the estimates. Kriging is a simple and popular spatial interpolation method, but the method has several shortcomings. In particular, the prediction is quite unstable and often fails to be performed when sample size is small. In this paper, we proposed a flexible and efficient spatial interpolator for radar rainfall estimation, with several advantages over kriging. The method is illustrated using a real-world data set.     

General class of covariance structures for two or more repeated factors in longitudinal data analysis

The main difficulty in parametric analysis of longitudinal data lies in specifying covariance structure. Several covariance structures, which usually reflect one series of measurements collected over time, have been presented in the literature. However there is a lack of literature on covariance structures designed for repeated measures specified by more than one repeated factor. In this paper a new, general method of modelling covariance structure based on the Kronecker product of underlying factor specific covariance profiles is presented. The method has an attractive interpretation in terms of independent factor specific contribution to overall within subject covariance structure and can be easily adapted to standard software.     

Functional principal component analysis of spatially correlated data

This paper focuses on the analysis of spatially correlated functional data. We propose a parametric model for spatial correlation and the between-curve correlation is modeled by correlating functional principal component scores of the functional data. Additionally, in the sparse observation framework, we propose a novel approach of spatial principal analysis by conditional expectation to explicitly estimate spatial correlations and reconstruct individual curves. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov\((X_i(s),X_i(t))\) and cross-covariance surface Cov\((X_i(s), X_j(t))\) at locations indexed by i and j. Then a anisotropy Matérn spatial correlation model is fitted to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We demonstrate the consistency of our estimates and propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Using simulation studies, we show that these methods have some clear advantages over existing methods of curve reconstruction and estimation of model parameters.     

Bayesian spatio‐temporal models based on discrete convolutions

Abstract: The authors consider a class of models for spatio‐temporal processes based on convolving independent processes with a discrete kernel that is represented by a lower triangular matrix. They study two families of models. In the first one, spatial Gaussian processes with isotropic correlations are convoluted with a kernel that provides temporal dependencies. In the second family, AR(p) processes are convoluted with a kernel providing spatial interactions. The covariance structures associated with these two families are quite rich. Their covariance functions that are stationary and separable in space and time as well as time dependent nonseparable and nonisotropic ones.     

Comparison of estimators for heteroskedastic models

Ordinary least squares (OLS) yield inefficient parameter estimates and inconsistent estimates of the covariance matrix in case of heteroskedastic errors. Robinson's adaptive estimator and the Cragg estimator avoid any explicit parameterization of heteroskedasticity, and reduce the danger of misspecification. A small Monte Carlo experiment is performed to compare the behavior of the adaptive estimator with the performance of the Cragg estimator. The Monte Carlo experiment includes simulations of the Generalized Least Squares (GLS) estimator. Indeed, an interesting question is how more sophisticated techniques, like the adaptive estimator, compare with GLS when the latter relies on an incorrect specification of the heteroskedastic process. It turns out that the regression parameters, when estimated adaptively, display small mean squared errors and great efficiency in case of medium or high heteroskedasticity. The covariance matrix, instead, is better estimated by the Cragg estimator or by GLS based on a misspecified error term, since the adaptive estimator overpredicts the standard errors of the regression parameters.     

The impact of integrated measurement errors on modeling long-run macroeconomic time series

ABSTRACT

Data spanning long time periods, such as that over 1860–2012 for the UK, seem likely to have substantial errors of measurement that may even be integrated of order one, but which are probably cointegrated for cognate variables. We analyze and simulate the impacts of such measurement errors on parameter estimates and tests in a bivariate cointegrated system with trends and location shifts which reflect the many major turbulent events that have occurred historically. When trends or shifts therein are large, cointegration analysis is not much affected by such measurement errors, leading to conventional stationary attenuation biases dependent on the measurement error variance, unlike the outcome when there are no offsetting shifts or trends.     

Statistical Modelling and Deconvolution of Yield Meter Data

Abstract.  This paper considers the problem of mapping spatial variation of yield in a field using data from a yield monitoring system on a combine harvester. The unobserved yield is assumed to be a Gaussian random field and the yield monitoring system data is modelled as a convolution of the yield and an impulse response function. This results in an unusual spatial covariance structure (depending on the driving pattern of the combine harvester) for the yield monitoring system data. Parameters of the impulse response function and the spatial covariance function of the yield are estimated using maximum likelihood methods. The fitted model is assessed using certain empirical directional covariograms and the yield is finally predicted using the inferred statistical model.     

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.