Empirical Best Linear Unbiased Prediction Research Articles

Performance of linear mixed models and random forests for spatial prediction of soil pH

Digital soil maps describe the spatial variation of soil and provide important information on spatial variation of soil properties which provides policy makers with a synoptic view of the state of the soil. This paper presents a study to tackle the task of how to map the spatial variation of soil pH across Zambia. This was part of a project to assess suitability for rice production across the country. Legacy data on the target variable were available along with additional exhaustive environmental covariates as potential predictor variables. We had the option of undertaking spatial prediction by geostatistical or machine learning methods. We set out to compare the approaches from the selection of predictor variables through to model validation, and to test the predictors on a set of validation observations. We also addressed the problem of how to robustly validate models from legacy data when these have, as is often the case, a strongly clustered spatial distribution. The validation statistics results showed that the empirical best linear unbiased predictor (EBLUP) with the only fixed effect a constant mean (ordinary kriging) performed better than the other methods. Random forests had the largest model-based estimates of the expected squared errors. We also noticed that the random forest algorithm was prone to select as “important” spatially correlated random variables which we had simulated.

Spatial variability and mapping of soil fertility status in a high-potential smallholder farming area under sub-humid conditions in Zimbabwe

A study was conducted to examine spatial variability of soil properties related to fertility in maize fields across varying soil types in ward 10 of Hurungwe district, Zimbabwe; a smallholder farming area with sub-humid conditions and high yield potential. Purposively collected and geo-referenced soil samples were analyzed for texture, pH, soil organic carbon (OC), mineral N, bicarbonate P, and exchangeable K. Linear mixed model was used to analyze spatial variation of the data. The model allowed prediction of soil properties at unsampled sites by the empirical best linear unbiased predictor (EBLUP). Evidence for spatial dependence in the random component of the model was evaluated by calculating Akaike’s information criterion. Soil pH ranged from 4.0 to 6.9 and showed a strong spatial trend increasing from north to south, strong evidence for a difference between the home and outfields with homefields significantly higher and between soil textural classes with the sand clay loam fraction generally higher. Soil OC ranged from 0.2 to 2.02% and showed no spatial trend, but there was strong evidence for a difference between home and outfields, with mean soil OC in homefields significantly larger, and between soil textural classes, with soil OC largest in the sandy clay loams. Both soil pH and OC showed evidence for spatial dependence in the random effect, providing a basis for spatial prediction by the EBLUP, which was presented as a map. There were significant spatial trends in mineral N, available P and exchangeable K, all increasing from north to south; significant differences between homefields and outfields (larger concentrations in homefields), and differences between the soil textural classes with larger concentrations in the sandy clay loams. However, there was no evidence for spatial dependence in the random component, so no attempt was made to map these variables. These results show how management (home fields vs outfields), basic soil properties (texture) and other factors emerging as spatial trends influence key soil properties that determine soil fertility in these conditions. This implies that the best management practices may vary spatially, and that site-specific management is a desirable goal in conditions such as those which apply in Ward 10 of Hurungwe district in Zimbabwe.

Robust Estimation for Small Domains in Business Surveys

Abstract Small area (or small domain) estimation is still rarely applied in business statistics, because of challenges arising from the skewness and variability of variables such as turnover. We examine a range of small area estimation methods as the basis for estimating the activity of industries within the retail sector in the Netherlands. We use tax register data and a sampling procedure which replicates the sampling for the retail sector of Statistics Netherlands’ Structural Business Survey as a basis for investigating the properties of small area estimators. In particular, we consider the use of the empirical best linear unbiased predictor (EBLUP) under a random effects model and variations of the EBLUP derived under (a) a random effects model that includes a complex specification for the level 1 variance and (b) a random effects model that is fitted by using the survey weights. Although accounting for the survey weights in estimation is important, the impact of influential data points remains the main challenge in this case. The paper further explores the use of outlier robust estimators in business surveys, in particular a robust version of the EBLUP, M-regression-based synthetic estimators and M-quantile small area estimators. The latter family of small area estimators includes robust projective (without and with survey weights) and robust predictive versions. M-quantile methods have the lowest empirical mean squared error and are substantially better than direct estimators, although there is an open question about how to choose the tuning constant for bias adjustment in practice. The paper makes a further contribution by exploring a doubly robust approach comprising the use of survey weights in conjunction with outlier robust methods in small area estimation.

Fault detection and diagnosis in water resource recovery facilities using incremental PCA.

Because of the static nature of conventional principal component analysis (PCA), natural process variations may be interpreted as faults when it is applied to processes with time-varying behavior. In this paper, therefore, we propose a complete adaptive process monitoring framework based on incremental principal component analysis (IPCA). This framework updates the eigenspace by incrementing new data to the PCA at a low computational cost. Moreover, the contribution of variables is recursively provided using complete decomposition contribution (CDC). To impute missing values, the empirical best linear unbiased prediction (EBLUP) method is incorporated into this framework. The effectiveness of this framework is evaluated using benchmark simulation model No. 2 (BSM2). Our simulation results show the ability of the proposed approach to distinguish between time-varying behavior and faulty events while correctly isolating the sensor faults even when these faults are relatively small.

Corrected empirical Bayes confidence region in a multivariate Fay–Herriot model

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) in the linear mixed model is useful because it gives a stable estimate for a mean of a small area. For measuring uncertainty of EBLUP, much of research is focused on second-order unbiased estimation of mean squared prediction errors in the univariate case. In this paper, we consider the multivariate Fay–Herriot model where the covariance matrix of random effects is fully unknown, and obtain a confidence region of the small area mean that is based on the Mahalanobis distance centered around EBLUP and has second-order correction. A positive-definite, consistent and second-order unbiased estimator of the covariance matrix of the random effects is also suggested. The performance is investigated through simulation study.

Comparison of mean squared error estimators under the Fay-Herriot model: application to poverty and percentage of food expenditure data

Small area estimates have received much attention from both private and public sectors due to the growing demand for effective planning of health services, apportioning of government funds and policy and decision making. The uncertainty of empirical best linear unbiased predictor (EBLUP) estimates is widely assessed by mean squared error (MSE). MSEs are criticized as they are not area specific since they do not depend on the direct estimators from the survey. In this paper, we compare the performances of different MSE estimators with respect to the relative bias and relative risk using a Monte Carlo simulation study. Simulation results suggest the superiority of the proposed MSEs over the existing methods in some situations. As a case study, the 2010/11 household consumption expenditure survey (HCES) and the 2007 housing and population census of Ethiopia have been used to study the performances of the MSE estimators.

Estimation of variance of random effect in small area model with Spatial Empirical Best Linear Unbiased Prediction (SEBLUP)

Survey sampling is one of the sampling methods of an object to provide information estimation of population parameters that became the focus of research. One of the methods that used to estimate population parameters is direct estimation method. However, when the direct estimation is used it will cause a large standard error. To handle that problem in small area we add information about the same parameters in other small areas which has similar character, or the value of the variables that are related to the variables being observed, this method is known as the small area estimation (SAE). In this mini thesis, small area method that we use consider spatial correlation between area, spatial Empirical Best Linear Unbiased Prediction (EBLUP). The estimator of spatial EBLUP depends on the variance component and spatial correlation, but in practice they are unknown. Therefore, to get the spatial EBLUP estimator it is necessary to first estimate the variance of random effect and correlations between area. In this mini thesis we use maximum likelihood method and scoring algorithm to estimate the variance of random effect and correlations between area.

ESTIMATION OF INCOME CHARACTERISTICS FOR REGIONS IN POLAND USING SPATIO-TEMPORAL SMALL AREA MODELS

Abstract The paper presents the comparison of estimation results for spatial and spatiotemporal small area models. The study was carried out for income-related variables drawn from the Polish Household Budget Survey and explanatory variables from the Polish Local Data Bank for the years 2003-2013. The properties of EBLUPs (Empirical Best Linear Unbiased Predictors) based on spatiotemporal models, which utilize spatial correlation between neighbouring areas as well as historical data, were compared and contrasted with EBLUPs based on spatial models obtained separately for each year and with EBLUPs based on the Rao-Yu model. The computations were performed using sae, sae2 and spdep packages for R-project environment. In the case of sae package, the eblupFH, eblupSFH and the eblupSTFH functions were used for point estimation along with the mseFH, mseSFH and the pbmseSTFH functions for the MSE estimation, whereas the eblupRY function was applied for the purposes of sae2 package. The precision of direct estimators was guaranteed by the adoption of the Balanced Repeated Replication method. The results of the analysis demonstrate that a visible reduction of the estimation error was achieved for the implemented spatiotemporal small-area models, especially when significant spatial and time autocorrelations were observed. These results are even more valuable than those achieved by the means of the Rao-Yu model. In the computations, three author-defined functions were adopted, which not only enabled the author to perform the extract of random effects for spatial, spatiotemporal and Rao-Yu models, but also made it possible to obtain their decomposition with respect to spatial and temporal parts, thus creating a novel solution. The comparison was carried out using choropleth maps for spatial effects and distributions of temporal random effects for the considered years.

Detecting falsification in a television audience measurement panel survey

Case studies reporting real-world experiences with survey falsification are uncommon. In this article, we document the experience of a panel survey in India that produced TV viewing estimates (“TV ratings”) where external parties were illegitimately trying to influence respondents’ behavior. The usual method to detect possible falsifications was that of analysts poring through data to find suspicious viewing patterns. Here, we develop a method using multilevel models and illustrate its use in the detection of an actual incident. We report how the model-based method was used to direct on-ground investigations that ultimately supported our analytic inferences. The model-based method offers four advantages over the usual method. First, by approximating an interpenetrated sample, the model simultaneously controls for several household characteristics. Second, Empirical Best Linear Unbiased Predictors (EBLUPs) of random effects can be examined separately at both the household and interviewer level, thus suggesting where further investigation efforts should be directed. Third, the method is faster and more objective than the usual method. Fourth, the method is easily implemented and can provide regular quality control for survey organizations.

Empirical best linear unbiased predictors in multivariate nested-error regression models

For analyzing unit-level multivariate data in small area estimation, we consider the multivariate nested error regression model (MNER) and provide the empirical best linear unbiased predictor (EBLUP) of a small area characteristic based on second-order unbiased and consistent estimators of the ‘within’ and ‘between’ multivariate components of variance. The second-order approximation of the mean squared error (MSE) matrix of the EBLUP and its unbiased estimator are derived in closed forms. The confidence interval with second-order accuracy is also provided analytically.

Multivariate Small Area Estimation of Multidimensional Latent Economic Well‐being Indicators

SummaryFactor analysis models are used in data dimensionality reduction problems where the variability among observed variables can be described through a smaller number of unobserved latent variables. This approach is often used to estimate the multidimensionality of well‐being. We employ factor analysis models and use multivariate empirical best linear unbiased predictor (EBLUP) under a unit‐level small area estimation approach to predict a vector of means of factor scores representing well‐being for small areas. We compare this approach with the standard approach whereby we use small area estimation (univariate and multivariate) to estimate a dashboard of EBLUPs of the means of the original variables and then averaged. Our simulation study shows that the use of factor scores provides estimates with lower variability than weighted and simple averages of standardised multivariate EBLUPs and univariate EBLUPs. Moreover, we find that when the correlation in the observed data is taken into account before small area estimates are computed, multivariate modelling does not provide large improvements in the precision of the estimates over the univariate modelling. We close with an application using the European Union Statistics on Income and Living Conditions data.

Estimation of Proportions in Small Areas: Application to the Labour Force Using the Swiss Census Structural Survey

SummaryThe main objectives of this paper are to find efficient but computationally simple estimators for the proportions of people in the labour force (economic activity rates) in Swiss communes and to estimate their mean-squared error (MSE) over the sampling replication mechanism (the design MSE). This will be done by combining survey data with administrative data provided by the Swiss Federal Statistical Office. We find estimators with considerably greater efficiency than currently used direct estimators and that are easy to implement. We show that, under a generalized linear mixed model with logit link, the computationally expensive empirical best predictor does not perform appreciably better than a plug-in estimator. Moreover, for moderate proportions of active workers, the empirical best linear unbiased predictor (EBLUP) based on a much simpler linear mixed model performs similarly to the above estimators. We propose new bootstrap estimators of the design MSE of the EBLUPs, which ‘borrow strength’ similarly to EBLUPs. Realistic simulation studies carried out under both model- and design-based set-ups indicate great gains in efficiency of the selected small area estimators over the traditional direct estimators and acceptable performance of the proposed bootstrap MSE estimators. In the application using the Swiss data, coefficient-of-variation reductions of the estimates obtained for the communes are remarkable.

A Cross‐Validation of Statistical Models for Zoned‐Based Prediction in Cultivar Testing

ABSTRACTThe principal goals of a plant breeding program are to provide breeders with cultivar information for selection purposes and to provide farmers with high‐yielding and stable cultivars. For that reason, multi‐environment trials need to be done to predict future cultivar yield, and a robust statistical procedure is needed to provide reliable information on the tested cultivars. In Sweden, the statistical procedure follows the tradition of modeling cultivar effects as fixed. Moreover, the analysis is performed separately by zone and level of fungicide treatment, and so the factorial information regarding cultivar × zone × fungicide combinations is not explored. Thus, the question arose whether the statistical method could be improved to increase accuracy in zone‐based cultivar prediction, since the cultivar recommendation is zone based. In this paper, the performance of empirical best linear unbiased estimation (E‐BLUE) and empirical best linear unbiased prediction (E‐BLUP) are compared using cross‐validation for winter wheat (Triticum aestivum L.) and spring barley (Hordeum vulgare L.), in single‐year and multiyear series of trials. Data were obtained from three agricultural zones of Sweden. Several linear mixed models were compared, and model performance was evaluated using the mean squared error of prediction criterion. The E‐BLUP method outperformed the E‐BLUE method in both crops and series. The prediction accuracy for zone‐based yield was improved by using E‐BLUP because the random‐effects assumption for cultivar × zone interaction allows information to be borrowed across zones. We conclude that E‐BLUP should replace the currently used E‐BLUE approach to predict zone‐based cultivar yield.

Robust estimation of mean squared error matrix of small area estimators in a multivariate Fay–Herriot model

This paper is concerned with the small area estimation in the multivariate Fay–Herriot model where covariance matrix of random effects are fully unknown without normality assumption. The covariance matrix is estimated by a Prasad–Rao type consistent estimator, and the empirical best linear unbiased predictor (EBLUP) of a vector of small area characteristics is provided. When the EBLUP is measured in terms of a mean squared error matrix (MSEM), a second-order approximation of MSEM of the EBLUP and a second-order unbiased estimator of the MSEM are derived analytically in closed forms. The performance is investigated through simulation and empirical studies.

A comparison of empirical BLUP with different considerations of residual error variance for genotype evaluation of multi-location trials

AbstractThe empirical best linear unbiased prediction (eBLUP) is usually based on the assumption that the residual error variance (REV) is homogenous. This may be unrealistic, and therefore limits the accuracy of genotype evaluations for multi-location trials, where the REV often varies across locations. The objective of this contribution was to investigate the direct implications of the eBLUP with different considerations about REV based on the mixed model for evaluation of genotype simple effects (i.e. genotype effects at individual locations). A series of 14 multi-location trials from a rape-breeding program in the north of China were simultaneously analyzed from 2012 to 2014 using a randomized complete block design at each location. The results showed that the model with heterogeneous REV was more appropriate than the one with homogeneous REV in all of the trials according to model fitting statistics. Whether the REV differences across locations were accounted for in the analysis procedure influenced the variance estimate of related random effects and testing of the variance of genotype-location (G-L) interactions. Ignoring REV differences by use of the eBLUP could result not only in an inflation or deflation of statistical Type I error rates for pair-wise testing but also in an inaccurate ranking of genotype simple effects for these trials. Therefore, it is suggested that in application of the eBLUP for evaluation of genotype simple effects in multi-location trials, the heterogeneity of REV should be accounted for based on mixed model approaches with appropriate variance-covariance structure.

Poverty Mapping of the Coastal Areas Using Spatial Empirical Best Linear Unbiased Prediction Method

This research aims to map out the poverty data in the Coastal Areas of Mukomuko District. The estimation method used is Spatial Empirical Best Linear Unbiased Prediction (EBLUP). So, the poverty estimator in the coastal area of Mukomuko District is obtained. The evaluation of parameter estimator is calculated by the value of MSE (Mean Square Error) using Bootstrap resampling method. Based on the result of the study is seen that the MSE value of Spatial EBLUP estimators is smaller than the MSE value of the direct estimator (Y) in each village. Finally, the estimation is presented as poverty map. The estimation of spatial correlation is positive and strong (0.52).

Nonstationary EBLUP on Prediction of Poverty Rate at Village Level in Lembata Regency

The village development program requires accurate village level data, such as the poverty rate. However data poverty rate in Indonesia can only be obtained at the regency/municipality level. An analysis technique to overcome this problem is Small Area Estimation (SAE). SAE model related to poverty rate must be able to produce an estimated proportion that is in the interval of 0 and 1. One approach that can be done is to use logit transformation. The purpose of this study was to estimate the poverty rate at village level in Lembata Regency, Nusa Tenggara Timur Province. This estimation was done by comparing the Empirical Best Linear Unbiased Prediction (EBLUP), Spatial Empirical Best Linear Unbiased Prediction (SEBLUP), and Nonstationary Empirical Best Linear Unbiased Prediction (NSEBLUP). The results showed that logit transformations produced estimates between 0 and 1. The best method to estimate poverty rate at village level in Lembata Regency was NSEBLUP, which produced estimation that more precise than EBLUP and SEBLUP.

Empirical Best Predictor for Nested Error Regression Small Area Models

Nested error regression models have played an important role in small area estimation (SAE) especially for deriving indirect or model-based estimates of small area parameters. The models are valid to be employed whenever the auxiliary information is available at level units, as well as the random effect is independent from the sampling error. Furthermore, the models also assume normality the random effects and sampling errors. The standard SAE method, specifically the Empirical Best Linear Unbiased Predictor (EBLUP), which derives the estimates for small area parameters under the nested error regression models, certainly have to satisfy the strictly assumptions. In this paper, we study the Empirical Best Predictor (EBP) which can be utilized for deriving estimates of small area parameters whenever the variable of interest has skewed distributions. We apply the EBP to estimate the poverty incidence and poverty gap for the regions (‘kabupaten’ and ‘kota’) in West Java Province – Indonesia.

Estimating variance of random effects to solve multiple problems simultaneously

The two-level normal hierarchical model (NHM) has played a critical role in statistical theory for the last several decades. In this paper, we propose random effects variance estimator that simultaneously (i) improves on the estimation of the related shrinkage factors, (ii) protects empirical best linear unbiased predictors (EBLUP) [same as empirical Bayes (EB)] of the random effects from the common overshrinkage problem, (iii) avoids complex bias correction in generating strictly positive second-order unbiased mean square error (MSE) (same as integrated Bayes risk) estimator either by the Taylor series or single parametric bootstrap method. The idea of achieving multiple desirable properties in an EBLUP or EB method through a suitably devised random effects variance estimator is the first of its kind and holds promise in providing good inferences for random effects under the EBLUP or EB framework. The proposed methodology is also evaluated using a Monte Carlo simulation study and real data analysis.

A class of general adjusted maximum likelihood methods for desirable mean squared error estimation of EBLUP under the Fay–Herriot small area model

The empirical best linear unbiased prediction (EBLUP) estimator is utilized for efficient inference in various research areas, especially for small-area estimation. In order to measure its uncertainty, we generally need to estimate its mean squared prediction error (MSE). Ideally, an EBLUP-based method should not only provide a second-order unbiased estimator of MSE of EBLUP but also maintain strict positivity in estimators of both model variance parameter and MSE of EBLUP. Fortunately, the MSE estimators proposed in Yoshimori and Lahiri (2014) and Hirose and Lahiri (2018) achieve the three desired properties simultaneously. As far as we know, no other MSE estimator does so.In this paper, we therefore seek an adequate class of general adjusted maximum-likelihood methods that simultaneously achieve the three desired properties of MSE estimation. To establish that the investigated class does so, we reveal the relationship between the general adjusted maximum-likelihood method for the model variance parameter and the general functional form of the second-order unbiased MSE estimator, maintaining strict positivity. We also compare the performance of several MSE estimators in our investigated class and others through a Monte Carlo simulation study. The results show that the MSE estimators in our investigated class perform better than those in others.