Explore the vast range of titles available.

Spatial statistics for very large spatial data sets is challenging. The size of the data set, n, causes problems in computing optimal spatial predictors such as kriging, since its computational cost is of order [graphic removed] . In addition, a large data set is often defined on a large spatial domain, so the spatial process of interest typically exhibits non-stationary behaviour over that domain. A flexible family of non-stationary covariance functions is defined by using a set of basis functions that is fixed in number, which leads to a spatial prediction method that we call fixed rank kriging. Specifically, fixed rank kriging is kriging within this class of non-stationary covariance functions. It relies on computational simplifications when n is very large, for obtaining the spatial best linear unbiased predictor and its mean-squared prediction error for a hidden spatial process. A method based on minimizing a weighted Frobenius norm yields best estimators of the covariance function parameters, which are then substituted into the fixed rank kriging equations. The new methodology is applied to a very large data set of total column ozone data, observed over the entire globe, where n is of the order of hundreds of thousands.

We propose a simple, unified, Monte-Carlo-assisted approach (called ‘Sumca’) to second-order unbiased estimation of the mean-squared prediction error (MSPE) of a small area predictor.The MSPE estimator proposed is easy to derive, has a simple expression and applies to a broad range of predictors that include the traditional empirical best linear unbiased predictor, empirical best predictor and post-model-selection empirical best linear unbiased predictor and empirical best predictor as special cases. Furthermore, the leading term of the MSPE estimator proposed is guaranteed positive; the lower order term corresponds to a bias correction, which can be evaluated via a Monte Carlo method. The computational burden for the Monte Carlo evaluation is much less, compared with other Monte-Carlo-based methods that have been used for producing second-order unbiased MSPE estimators, such as the double bootstrap and Monte Carlo jackknife. The Sumca estimator also has a nice stability feature. Theoretical and empirical results demonstrate properties and advantages of the Sumca estimator.

A conventional linear model for functional data involves expressing a response variable Y in terms of the explanatory function X(t), via the model Y = a + ∫𝓘b(t)X(t)dt + error, where a is a scalar, b is an unknown function and 𝓘 = [0, α] is a compact interval. However, in some problems the support of b or X, 𝓘1 say, is a proper and unknown subset of 𝓘, and is a quantity of particular practical interest. Motivated by a real data example involving particulate emissions, we develop methods for estimating 𝓘1. We give particular emphasis to the case 𝓘1 = [0, θ], where θ ∈ (0, α], and suggest two methods for estimating a, b and θ jointly; we introduce techniques for selecting tuning parameters; and we explore properties of our methodology by using both simulation and the real data example mentioned above. Additionally, we derive theoretical properties of the methodology and discuss implications of the theory. Our theoretical arguments give particular emphasis to the problem of identifiability.

The nested-error regression model is one of the best-known models in small area estimation. A small area mean is often expressed as a linear combination of fixed effects and realized values of random effects. In such analyses, prediction is made by borrowing strength from other related areas or sources and mean-squared prediction error (MSPE) is often used as a measure of uncertainty. In this article, we propose a bias-corrected analytical estimation of MSPE as well as a moment-match jackknife method to estimate the MSPE without specific assumptions about the distributions of the data. Theoretical and empirical studies are carried out to investigate performance of the proposed methods with comparison to existing procedures. Le modèle de régression à erreur imbriquée est l’un des mieux connus pour l’estimation sur des petits domaines. La moyenne d’un petit domaine est souvent exprimée comme une combinaison linéaire d’effets fixes et de valeurs réalisées d’effets aléatoires. Pour de telles analyses, les prévisions sont effectuées en empruntant de l’information d’autres domaines associés ou d’autres sources, et l’erreur quadratique moyenne de prévision (EQMP) sert souvent à mesurer l’incertitude. Les auteurs proposent une estimation analytique de l’EQMP corrigée pour le biais ainsi qu’une méthode jackknife d’appariement des moments afin d’estimer l’EQMP sans formuler d’hypothèses spécifiques sur la distribution des données. Ils présentent des études théoriques et empiriques comparant la performance des méthodes proposées aux procédures existantes.

We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.

Finding reliable estimates of parameters of subpopulations (areas) in small area estimation is an important problem especially when there are few or no samples in some areas. Clustering small areas on the basis of the Euclidean distance between their corresponding covariates is proposed to obtain smaller mean-squared prediction error (MSPE) for the predicted values of area means by using area level linear mixed models. We first propose a statistical test to investigate the homogeneity of variance components between clusters. Then, we obtain the empirical best linear unbiased predictor of small area means by taking into account the difference between variance components in different clusters. We study the performance of our proposed test as well as the effect of the clustering on the MSPE of small area means by using simulation studies. We also obtain a second-order approximation to the MSPE of small area means and derive a second-order unbiased estimator of the MSPE. The results show that the MSPE of small area means can be improved when the variance components are different. The improvement in the MSPE is significant when the difference between variance components is considerable. Finally, the methodology proposed is applied to a real data set.

Argentine ant, Linepithema humile (Mayr) (Hymenoptera: Formicidae), is a pest in southern California citrus orchards because it protects honeydew-producing hemipteran pests from natural enemies. A major impediment to controlling L. humile is estimating ant densities in orchards. Ants use irrigation lines to travel across orchard floors to reach trees infested with hemipterans. However, for making ant control decisions, it is the number of ants in trees, not on pipes that is critical. Work completed here demonstrates that the number of ants counted on pipes is highly correlated with the number of ants counted on trunks. Densities of ants counted on trunks are correlated with trunk diameter, citrus variety, and time of year and time of day counts. Six regression models, linear regression, zero-inflated Poisson regression, and zero-inflated negative binomial regression models, and each of their mixed model extensions, indicated a strong positive relationship between ant counts on irrigation pipes and ant counts on tree trunks. Mean squared prediction error and 5-fold cross-validation analyses indicated that the best performing of these 6 models was the zero-inflated Poisson mixed regression model. A binary classification model developed using support vector machine learning for ant infestation severity levels, categorized as low (<100 ants counted in 1 min) or high (≥100 ants counted in minutes), predicted ant densities on trunks with 85% accuracy. These models can be used to estimate the number of ants on the trunks of citrus trees by using counts of ants made on irrigation pipes.

The aim of this paper is to analyze the prediction accuracy of multivariate spatiotemporal forecasting models with a confounded covariate versus univariate models without covariates for discrete (count and binary) and continuous response variables by means of theoretical considerations and Monte Carlo simulation. For the simulation, we propose a Bayesian latent Gaussian Markov random fields framework for three types of generalized additive prediction models: (i) a multivariate model with a spatiotemporally confounded covariate only, denoted in the rest of the paper as the multivariate model; (ii) a univariate model with spatiotemporal random effects and their interaction only; (iii) and a full multivariate model consisting of the combination of (i) and (ii), that is, a univariate model combined with a multivariate model. One simulation result is that for all three kinds of response variables, the univariate and the full multivariate model uniformly dominate the multivariate model in terms of prediction accuracy measured by the mean-squared prediction error (MSPE). A second finding is that for discrete variables the univariate model uniformly dominates the full multivariate model. A third result is that for continuous response variables the full multivariate model dominates the univariate model in the case of low confoundedness of the covariate. For high confoundedness, the reverse holds. The results provide important guidelines for practitioners.

Many practical problems are related to prediction, where the main interest is at subject (e.g., personalized medicine) or (small) sub-population (e.g., small community) level. In such cases, it is possible to make substantial gains in prediction accuracy by identifying a class that a new subject belongs to. This way, the new subject is potentially associated with a random effect corresponding to the same class in the training data, so that method of mixed model prediction can be used to make the best prediction. We propose a new method, called classified mixed model prediction (CMMP), to achieve this goal. We develop CMMP for both prediction of mixed effects and prediction of future observations, and consider different scenarios where there may or may not be a \"match\" of the new subject among the training-data subjects. Theoretical and empirical studies are carried out to study the properties of CMMP, including prediction intervals based on CMMP, and its comparison with existing methods. In particular, we show that, even if the actual match does not exist between the class of the new observations and those of the training data, CMMP still helps in improving prediction accuracy. Two real-data examples are considered. Supplementary materials for this article are available online.

We establish negative moment bounds for the minimum eigenvalue of the normalized Fisher information matrix in a stochastic regression model with a deterministic time trend. This result enables us to develop an asymptotic expression for the mean squared prediction error (MSPE) of the least squares predictor of the aforementioned model. Our asymptotic expression not only helps better understand how the MSPE is affected by the deterministic and random components, but also inspires an intriguing proof of the formula for the sum of the elements in the inverse of the Cauchy/Hilbert matrix from a prediction perspective.