Recursive weighted least squares estimation algorithm based on minimum model error principle

Kalman filter is commonly used in data filtering and parameters estimation of nonlinear system,such as projectile's trajectory estimation and control.While there is a drawback that the prior error covariance matrix and filter parameters are difficult to be determined,which may result in filtering divergence.As to the problem that the accuracy of state estimation for nonlinear ballistic model strongly depends on its mathematical model,we improve the weighted least squares method(WLSM)with minimum model error principle.Invariant embedding method is adopted to solve the cost function including the model error.With the knowledge of measurement data and measurement error covariance matrix,we use gradient descent algorithm to determine the weighting matrix of model error.The uncertainty and linearization error of model are recursively estimated by the proposed method,thus achieving an online filtering estimation of the observations.Simulation results indicate that the proposed recursive estimation algorithm is insensitive to initial conditions and of good robustness.

Application of Backward Nonlinear Local Lyapunov Exponent Method to Assessing the Relative Impacts of Initial Condition and Model Errors on Local Backward Predictability

Initial condition and model errors both contribute to the loss of atmospheric predictability.However,it remains debatable which type of error has the larger impact on the prediction lead time of specific states.In this study,we perform a theoretical study to investigate the relative effects of initial condition and model errors on local prediction lead time of given states in the Lorenz model.Using the backward nonlinear local Lyapunov exponent method,the prediction lead time,also called local backward predictability limit(LBPL),of given states induced by the two types of errors can be quantitatively estimated.Results show that the structure of the Lorenz attractor leads to a layered distribution of LBPLs of states.On an individual circular orbit,the LBPLs are roughly the same,whereas they are different on different orbits.The spatial distributions of LBPLs show that the relative effects of initial condition and model errors on local backward predictability depend on the locations of given states on the dynamical trajectory and the error magnitudes.When the error magnitude is fixed,the differences between the LBPLs vary with the locations of given states.The larger differences are mainly located on the inner trajectories of regimes.When the error magnitudes are different,the dissimilarities in LBPLs are diverse for the same given state.

A New Method for Identifying the Model Error of Adjust ment System

Some theory problems affecting parameter estimation are discussed in this paper. Influence and transformation between errors of stochastic and functional models is pointed out as well. For choosing the best adjustment model, a formula, which is different from the literatures existing methods, for estimating and identifying the model error, is proposed. On the basis of the proposed formula, an effective approach of selecting the best model of adjustment system is given.

Approach for wideband direction-of-arrival estimation in the presence of array model errors

The presence of array imperfection and mutual coupling in sensor arrays poses several challenges for development of effective algorithms for the direction-of-arrival (DOA) estimation problem in array processing. A correlation domain wideband DOA estimation algorithm without array calibration is proposed, to deal with these array model errors, using the arbitrary antenna array of omnidirectional elements. By using the matrix operators that have the memory and oblivion characteristics, this algorithm can separate the incident signals effectively. Compared with other typical wideband DOA estimation algorithms based on the subspace theory, this algorithm can get robust DOA estimation with regard to position error, gain-phase error, and mutual coupling, by utilizing a relaxation technique based on signal separation. The signal separation category and the robustness of this algorithm to the array model errors are analyzed and proved. The validity and robustness of this algorithm, in the presence of array model errors, are confirmed by theoretical analysis and simulation results.

An Online Model Correction Method Based on an Inverse Problem:Part I—Model Error Estimation by Iteration

Errors inevitably exist in numerical weather prediction(NWP) due to imperfect numeric and physical parameterizations.To eliminate these errors,by considering NWP as an inverse problem,an unknown term in the prediction equations can be estimated inversely by using the past data,which are presumed to represent the imperfection of the NWP model(model error,denoted as ME). In this first paper of a two-part series,an iteration method for obtaining the MEs in past intervals is presented,and the results from testing its convergence in idealized experiments are reported. Moreover,two batches of iteration tests were applied in the global forecast system of the Global and Regional Assimilation and Prediction System(GRAPES-GFS) for July–August 2009 and January–February 2010. The datasets associated with the initial conditions and sea surface temperature(SST) were both based on NCEP(National Centers for Environmental Prediction) FNL(final) data.The results showed that 6th h forecast errors were reduced to 10% of their original value after a 20-step iteration. Then,off-line forecast error corrections were estimated linearly based on the 2-month mean MEs and compared with forecast errors. The estimated error corrections agreed well with the forecast errors,but the linear growth rate of the estimation was steeper than the forecast error. The advantage of this iteration method is that the MEs can provide the foundation for online correction. A larger proportion of the forecast errors can be expected to be canceled out by properly introducing the model error correction into GRAPES-GFS.

An approach to estimating and extrapolating model error based on inverse problem methods:towards accurate numerical weather prediction

Model error is one of the key factors restricting the accuracy of numerical weather prediction(NWP). Considering the continuous evolution of the atmosphere, the observed data(ignoring the measurement error) can be viewed as a series of solutions of an accurate model governing the actual atmosphere. Model error is represented as an unknown term in the accurate model, thus NWP can be considered as an inverse problem to uncover the unknown error term. The inverse problem models can absorb long periods of observed data to generate model error correction procedures. They thus resolve the deficiency and faultiness of the NWP schemes employing only the initial-time data. In this study we construct two inverse problem models to estimate and extrapolate the time-varying and spatial-varying model errors in both the historical and forecast periods by using recent observations and analogue phenomena of the atmosphere. Numerical experiment on Burgers' equation has illustrated the substantial forecast improvement using inverse problem algorithms. The proposed inverse problem methods of suppressing NWP errors will be useful in future high accuracy applications of NWP.

An Online Model Correction Method Based on an Inverse Problem:PartⅡ——Systematic Model Error Correction

An online systematic error correction is presented and examined as a technique to improve the accuracy of real-time numerical weather prediction, based on the dataset of model errors(MEs) in past intervals. Given the analyses, the ME in each interval(6 h) between two analyses can be iteratively obtained by introducing an unknown tendency term into the prediction equation, shown in Part I of this two-paper series. In this part, after analyzing the 5-year(2001–2005) GRAPESGFS(Global Forecast System of the Global and Regional Assimilation and Prediction System) error patterns and evolution,a systematic model error correction is given based on the least-squares approach by firstly using the past MEs. To test the correction, we applied the approach in GRAPES-GFS for July 2009 and January 2010. The datasets associated with the initial condition and SST used in this study were based on NCEP(National Centers for Environmental Prediction) FNL(final) data.The results indicated that the Northern Hemispheric systematically underestimated equator-to-pole geopotential gradient and westerly wind of GRAPES-GFS were largely enhanced, and the biases of temperature and wind in the tropics were strongly reduced. Therefore, the correction results in a more skillful forecast with lower mean bias and root-mean-square error and higher anomaly correlation coefficient.

The Combined Effect of Initial Error and Model Error on ENSO Prediction Uncertainty Generated by the Zebiak-Cane Model

Initial errors and model errors are the source of prediction errors. In this study, the authors compute the conditional nonlinear optimal perturbation(CNOP)-type initial errors and nonlinear forcing singular vector(NFSV)-type tendency errors of the Zebiak-Cane model with respect to El Nio events and analyze their combined effect on the prediction errors for El Nio events. The CNOPtype initial error(NFSV-type tendency error) represents the initial errors(model errors) that have the largest effect on prediction uncertainties for El Nio events under the perfect model(perfect initial conditions) scenario. However, when the CNOP-type initial errors and the NFSVtype tendency errors are simultaneously considered in the model, the prediction errors caused by them are not amplified as the authors expected. Specifically, the prediction errors caused by the combined mode of CNOP-type initial errors and NFSV-type tendency errors are a little larger than those caused by the NFSV-type tendency errors. This fact emphasizes a need to investigate the optimal combined mode of initial errors and tendency errors that cause the largest prediction error for El Nio events.

FORMING DYNAMIC EQUATIONS OF ELASTIC LINKAGE AND INVESTIGATION OF MODEL ERROR

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R-Factor Analysis of Data Based on Population Models Comprising R- and Q-Factors Leads to Biased Loading Estimates

Effects of performing an R-factor analysis of observed variables based on population models comprising R- and Q-factors were investigated. Although R-factor analysis of data based on a population model comprising R- and Q-factors is possible, this may lead to model error. Accordingly, loading estimates resulting from R-factor analysis of sample data drawn from a population based on a combination of R- and Q-factors will be biased. It was shown in a simulation study that a large amount of Q-factor variance induces an increase in the variation of R-factor loading estimates beyond the chance level. Tests of the multivariate kurtosis of observed variables are proposed as an indicator of possible Q-factor variance in observed variables as a prerequisite for R-factor analysis.

Improved cat swarm optimization for parameter estimation of mixed additive and multiplicative random error model

To estimate the parameters of the mixed additive and multiplicative(MAM)random error model using the weighted least squares iterative algorithm that requires derivation of the complex weight array,we introduce a derivative-free cat swarm optimization for parameter estimation.We embed the Powell method,which uses conjugate direction acceleration and does not need to derive the objective function,into the original cat swarm optimization to accelerate its convergence speed and search accuracy.We use the ordinary least squares,weighted least squares,original cat swarm optimization,particle swarm algorithm and improved cat swarm optimization to estimate the parameters of the straight-line fitting MAM model with lower nonlinearity and the DEM MAM model with higher nonlinearity,respectively.The experimental results show that the improved cat swarm optimization has faster convergence speed,higher search accuracy,and better stability than the original cat swarm optimization and the particle swarm algorithm.At the same time,the improved cat swarm optimization can obtain results consistent with the weighted least squares method based on the objective function only while avoiding multiple complex weight array derivations.The method in this paper provides a new idea for theoretical research on parameter estimation of MAM error models.

Positional Error Model of Line Segments with Modeling and Measuring Errors Using Brownian Bridge

Spatial linear features are often represented as a series of line segments joined by measured endpoints in surveying and geographic information science.There are not only the measuring errors of the endpoints but also the modeling errors between the line segments and the actual geographical features.This paper presents a Brownian bridge error model for line segments combining both the modeling and measuring errors.First,the Brownian bridge is used to establish the position distribution of the actual geographic feature represented by the line segment.Second,an error propagation model with the constraints of the measuring error distribution of the endpoints is proposed.Third,a comprehensive error band of the line segment is constructed,wherein both the modeling and measuring errors are contained.The proposed error model can be used to evaluate line segments’overall accuracy and trustability influenced by modeling and measuring errors,and provides a comprehensive quality indicator for the geospatial data.

A Comparative Study on Kinematic Calibration for a 3-DOF Parallel Manipulator Using the Complete-Minimal,Inverse-Kinematic and Geometric-Constraint Error Models

Kinematic calibration is a reliable way to improve the accuracy of parallel manipulators, while the error model dramatically afects the accuracy, reliability, and stability of identifcation results. In this paper, a comparison study on kinematic calibration for a 3-DOF parallel manipulator with three error models is presented to investigate the relative merits of diferent error modeling methods. The study takes into consideration the inverse-kinematic error model, which ignores all passive joint errors, the geometric-constraint error model, which is derived by special geometric constraints of the studied RPR-equivalent parallel manipulator, and the complete-minimal error model, which meets the complete, minimal, and continuous criteria. This comparison focuses on aspects such as modeling complexity, identifcation accuracy, the impact of noise uncertainty, and parameter identifability. To facilitate a more intuitive comparison, simulations are conducted to draw conclusions in certain aspects, including accuracy, the infuence of the S joint, identifcation with noises, and sensitivity indices. The simulations indicate that the complete-minimal error model exhibits the lowest residual values, and all error models demonstrate stability considering noises. Hereafter, an experiment is conducted on a prototype using a laser tracker, providing further insights into the diferences among the three error models. The results show that the residual errors of this machine tool are signifcantly improved according to the identifed parameters, and the complete-minimal error model can approach the measurements by nearly 90% compared to the inverse-kinematic error model. The fndings pertaining to the model process, complexity, and limitations are also instructive for other parallel manipulators.

BDS satellite clock offset prediction based on a semiparametric adjustment model considering model errors

In view of the influence of model errors in conventional BeiDou prediction models for clock offsets,a semiparametric adjustment model for BeiDou Navigation Satellite System(BDS)clock offset prediction that considers model errors is proposed in this paper.First,the model errors of the conventional BeiDou clock offset prediction model are analyzed.Additionally,the relationship among the polynomial model,polynomial model with additional periodic term correction,and its periodic correction terms is explored in detail.Second,considering the model errors,combined with the physical relationship between phase,frequency,frequency drift,and its period in the clock sequence,the conventional clock offset prediction model is improved.Using kernel estimation and comprehensive least squares,the corresponding parameter solutions of the prediction model and the estimation of its model error are derived,and the dynamic error correction of the clock sequence model is realized.Finally,the BDS satellite precision clock data provided by the IGS Center of Wuhan University with a sampling interval of 5 min are used to compare the proposed prediction method with commonly used methods.Experimental results show that the proposed prediction method can better correct the model errors of BDS satellite clock offsets,and it can effectively overcome the inaccuracies of clock offset correction.The average forecast accuracies of the BeiDou satellites at 6,12,and 24 h are 27.13%,37.71%,and 45.08%higher than those of the conventional BeiDou clock offset forecast models;the average model improvement rates are 16.92%,20.96%,and 28.48%,respectively.In addition,the proposed method enhances the existing BDS satellite prediction method for clock offsets to a certain extent.

Revisiting Total Model Errors and Model Validation

The paper contains a discussion of earlier work on Total Model Errors and Model Validation.It is maintained that the recent change of paradigm to kernel based system identification has also affected the basis for(and interest in)giving bounds for the total model error.

ENSO ensemble prediction:Initial error perturbations vs. model error perturbations

Based on our developed ENSO (El Nio-Southern Oscillation) ensemble prediction system (EPS), the impacts of stochastic initial-error and model-error perturbations on ENSO ensemble predictions are examined and discussed by performing four sets of 14-a retrospective forecast experiments in both a deterministic and probabilistic sense. These forecast schemes are differentiated by whether they considered the initial or model stochastic perturbations. The comparison results suggest that the stochastic model-error perturbations, which are added into the modeled physical fields to mainly represent the uncertainties of the physical model, have significant, positive impacts on improving the ensemble prediction skills during the entire 12-month forecast process. However, the stochastic initial-error perturbations have relatively small impacts on the ensemble prediction system, and its impacts are mainly focusing on the first 3-month predictions.

An Error Equivalent Model of Revolute Joints with Clearances for Antenna Pointing Mechanisms

Joint clearances in antenna pointing mechanisms lead to uncertainty in function deviation. Current studies mainly focus on radial clearance of revolute joints, while axial clearance has rarely been taken into consideration. In fact, own?ing to errors from machining and assembly, thermal deformation and so forth, practically, axial clearance is inevitable in the joint. In this study, an error equivalent model(EEM) of revolute joints is proposed with considering both radial and axial clearances. Compared to the planar model of revolute joints only considering radial clearance, the journal motion inside the bearing is more abundant and matches the reality better in the EEM. The model is also extended for analyzing the error distribution of a spatial dual?axis("X–Y" type) antenna pointing mechanism of Spot?beam antennas which especially demand a high pointing accuracy. Three case studies are performed which illustrates the internal relation between radial clearance and axial clearance. It is found that when the axial clearance is big enough, the physical journal can freely realize both translational motion and rotational motion. While if the axial clearance is limited, the motion of the physical journal will be restricted. Analysis results indicate that the consideration of both radial and axial clearances in the revolute joint describes the journal motion inside the bearing more precise. To further validate the proposed model, a model of the EEM is designed and fabricated. Some suggestions on the design of revolute joints are also provided.

Comparative Study of Response Surface Designs with Errors-in-Variables Model

This paper investigates the scaled prediction variances in the errors-in-variables model and compares the performance with those in classic model of response surface designs for three factors.The ordinary least squares estimators of regression coefficients are derived from a second-order response surface model with errors in variables.Three performance criteria are proposed.The first is the difference between the empirical mean of maximum value of scaled prediction variance with errors and the maximum value of scaled prediction variance without errors.The second is the mean squared deviation from the mean of simulated maximum scaled prediction variance with errors.The last performance measure is the mean squared scaled prediction variance change with and without errors.In the simulations,1 000 random samples were performed following three factors with 20 experimental runs for central composite designs and 15 for Box-Behnken design.The independent variables are coded variables in these designs.Comparative results show that for the low level errors in variables,central composite face-centered design is optimal;otherwise,Box-Behnken design has a relatively better performance.

Pattern-Moving-Based Parameter Identification of Output Error Models with Multi-Threshold Quantized Observations

This paper addresses a modified auxiliary model stochastic gradient recursive parameter identification algorithm(M-AM-SGRPIA)for a class of single input single output(SISO)linear output error models with multi-threshold quantized observations.It proves the convergence of the designed algorithm.A pattern-moving-based system dynamics description method with hybrid metrics is proposed for a kind of practical single input multiple output(SIMO)or SISO nonlinear systems,and a SISO linear output error model with multi-threshold quantized observations is adopted to approximate the unknown system.The system input design is accomplished using the measurement technology of random repeatability test,and the probabilistic characteristic of the explicit metric value is employed to estimate the implicit metric value of the pattern class variable.A modified auxiliary model stochastic gradient recursive algorithm(M-AM-SGRA)is designed to identify the model parameters,and the contraction mapping principle proves its convergence.Two numerical examples are given to demonstrate the feasibility and effectiveness of the achieved identification algorithm.

Error model identification of inertial navigation platform based on errors-in-variables model

Because the real input acceleration cannot be obtained during the error model identification of inertial navigation platform, both the input and output data contain noises. In this case, the conventional regression model and the least squares (LS) method will result in bias. Based on the models of inertial navigation platform error and observation error, the errors-in-variables (EV) model and the total least squares (TLS) method are proposed to identify the error model of the inertial navigation platform. The estimation precision is improved and the result is better than the conventional regression model based LS method. The simulation results illustrate the effectiveness of the proposed method.