Comparison of REML and MINQUE for Estimated Variance Components and Predicted Random Effects

Linear mixed model (LMM) approaches have been widely applied in many areas of research data analysis because they offer great flexibility for different data structures and linear model systems. In this study, emphasis is placed on comparing the properties of two LMM approaches: restricted maximum likelihood (REML) and minimum norm quadratic unbiased estimation (MINQUE) with and without resampling techniques being included. Bias, testing power, Type I error, and computing time were compared between REML and MINQUE approaches with and without Jackknife technique based on 500 simulated data sets. Results showed that MINQUE and REML methods performed equally regarding bias, Type I error, and power. Jackknife-based MINQUE and REML greatly improved power compared to non-Jackknife based linear mixed model approaches. Results also showed that MINQUE is more time-saving compared to REML, especially with the use of resampling techniques and large data set analysis. Results from the actual cotton data analysis were in agreement with our simulated results. Therefore, Jackknife-based MINQUE approaches could be recommended to achieve desirable power with reduced time for a large data analysis and model simulations.

Nonlinear total least-squares variance component estimation for GM(1,1)model

The solution of the grey model(GM(1,1)model)generally involves equal-precision observations,and the(co)variance matrix is established from the prior information.However,the data are generally available with unequal-precision measurements in reality.To deal with the errors of all observations for GM(1,1)model with errors-in-variables(EIV)structure,we exploit the total least-squares(TLS)algorithm to estimate the parameters of GM(1,1)model in this paper.Ignoring that the effect of the improper prior stochastic model and the homologous observations may degrade the accuracy of parameter estimation,we further present a nonlinear total least-squares variance component estimation approach for GM(1,1)model,which resorts to the minimum norm quadratic unbiased estimation(MINQUE).The practical and simulative experiments indicate that the presented approach has significant merits in improving the predictive accuracy in comparison with control methods.

Robust Variance Components Estimation in the PERG Mixed Distributions of Empirical Variances—PEROBVC Method

A mixed distribution of empirical variances, composed of two distributions the basic and contaminating ones, and referred to as PERG mixed distribution of empirical variances, is considered. In the paper a robust inverse problem solution is given, namely a (new) robust method for estimation of variances of both distributions—PEROBVC Method, as well as the estimates for the numbers of observations for both distributions and, in this way also the estimate of contamination degree.

The mathematical weighting of GNSS observations based on different types of receivers/antennas and environmental conditions

Stochastic models play an important role in achieving high accuracy in positioning,the ideal estimator in the least-squares(LS)can be obtained only by using the suitable stochastic model.This study investigates the role of variance component estimation(VCE)in the LS method for Precise Point Positioning(PPP).This estimation is performed by considering the ionospheric-free(IF)functional model for code and the phase observation of Global Positioning System(GPS).The strategy for estimating the accuracy of these observations was evaluated to check the effect of the stochastic model in four modes:a)antenna type,b)receiver type,c)the tropospheric effect,and d)the ionosphere effect.The results show that using empirical variance for code and phase observations in some cases caused erroneous estimation of unknown components in the PPP model.This is because a constant empirical variance may not be suitable for various receivers and antennas under different conditions.Coordinates were compared in two cases using the stochastic model of nominal weight and weight estimated by LS-VCE.The position error difference for the east-west,north-south,and height components was 1.5 cm,4 mm,and 1.8 cm,respectively.Therefore,weight estimation with LS-VCE can provide more appropriate results.Eventually,the convergence time based on four elevation-dependent models was evaluated using nominal weight and LS-VCE weight.According to the results,the LS-VCE has a higher convergence rate than the nominal weight.The weight estimation using LS-VCE improves the convergence time in four elevation-dependent models by 11,13,12,and 9 min,respectively.

Recent Advances in the Geodesy Data Processing

Geodetic functional models,stochastic models,and model parameter estimation theory are fundamental for geodetic data processing.In the past five years,through the unremitting efforts of Chinese scholars in the field of geodetic data processing,according to the application and practice of geodesy,they have made significant contributions in the fields of hypothesis testing theory,un-modeled error,outlier detection,and robust estimation,variance component estimation,complex least squares,and ill-posed problems treatment.Many functional models such as the nonlinear adjustment model,EIV model,and mixed additive and multiplicative random error model are also constructed and improved.Geodetic data inversion is an important part of geodetic data processing,and Chinese scholars have done a lot of work in geodetic data inversion in the past five years,such as seismic slide distribution inversion,intelligent inversion algorithm,multi-source data joint inversion,water reserve change and satellite gravity inversion.This paper introduces the achievements of Chinese scholars in the field of geodetic data processing in the past five years,analyzes the methods used by scholars and the problems solved,and looks forward to the unsolved problems in geodetic data processing and the direction that needs further research in the future.

Comparative analysis of recent hydrological models and an attempt to generate new combined products for monitoring terrestrial water storage change

Hydrological models are crucial for characterizing large-scale water quantity variations and correcting GNSS reference station vertical displacements.We evaluated the robustness of multiple models,such as the Global Land Data Assimilation System (GLDAS),the Famine Early Warning System Network Land Data Assimilation System (FLDAS),the National Centers for Environmental Prediction (NCEP),and the WaterGAP Global Hydrology Model (WGHM).Inter-model and outer comparisons with Global Positioning System (GPS) coordinate time series,satellite gravity field Mascon solutions,and Global Precipitation Climatology Centre (GPCC) guide our assessment.Results confirm WGHM's 26% greater effectiveness in correcting nonlinear variations in GPS height time series compared to NCEP.In the Amazon River Basin,a 5-month lag between FLDAS,GLDAS,and satellite gravity results is observed.In eastern Asia and Australia,NCEP's Terrestrial Water Storage Changes (TWSC)-derived surface displacements correlate differently with precipitation compared to other models.Three combined hydrological models (H-VCE,H-EWM,and H-CVM) utilizing Variance Component Estimation (VCE),Entropy Weight Method (EWM),and Coefficient of Variation Method (CVM) are formulated.Correcting nonlinear variations with combined models enhances global GPS height scatter by 15%-17%.Correlation with precipitation increases by 25%-30%,and with satellite gravity,rises from 0.2 to 0.8 at maximum.The combined model eliminates time lag in the Amazon Basin TWSC analysis,exhibiting a four times higher signal-to-noise ratio than single models.H-VCE demonstrates the highest accuracy.In summary,the combined hydrological model minimizes discrepancies among individual models,significantly improving accuracy for monitoring large-scale TWSC.

Horizontal crustal movement in China fitted by adaptive collocation with embedded Euler vector

The crustal movements of the Chinese mainland include an average regional movement trend of the mainland and complex local deformations. Thus, both trends in the crustal movement of the mainland and local distortions should be simultaneously taken into consideration in crustal movement estimations. A combined collocation model based on Euler vector (taken as trend parameters) and local distortions (taken as signals) is proposed in this paper. We assume that prior covariance matrices between signals and observations should be consistent with their uncertainties. Otherwise, the station movement estimates provided by the collocation will be distorted. Thus, an adaptive collocation estimator based on simplified Helmert variance components is applied. This means that the contributions of signals and observations to estimates of crustal movements are balanced and reasonable, and consistent covariance matrices of the signals and observations are achieved through the adjustment of the adaptive factor. The calculation of actual horizontal movements of the Chinese crust shows that the estimates of horizontal crustal movement velocities are made more accurate by the adaptive collocation model.

Combining GPS carrier phase and Doppler observations for precise velocity determination

The truncation error and propagation error are analyzed for velocity determination through differential GPS carrier phase observations,and an approach for the choice of the best number of points for the central difference method is developed.In order to overcome the disadvantages of existing GPS velocity determination methods,a new velocity determination algorithm is presented,based on combining carrier phase and Doppler observations.The basic idea is that two types of observation are combined by adding their normal equations,and their weights are evaluated by strict Helmet variance-components estimation.In order to control the influence of outliers,a bifactor equivalent weights strategy is adopted.To validate this method,GPS data of the airborne gravimetry campaign MEXAGE2001 is tested.The results show that the precision and reliability of velocity determination are obviously improved by using the proposed method.

Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions

The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation（VCE） and minimum standard deviation（MSTD） approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.