The reasonable prior information between the parameters in the adjustment processing can significantly improve the precision of the parameter solution. Based on the principle of equality constraints, we establish the ...The reasonable prior information between the parameters in the adjustment processing can significantly improve the precision of the parameter solution. Based on the principle of equality constraints, we establish the mixed additive and multiplicative random error model with equality constraints and derive the weighted least squares iterative solution of the model. In addition, aiming at the ill-posed problem of the coefficient matrix, we also propose the ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints based on the principle of ridge estimation method and derive the U-curve method to determine the ridge parameter. The experimental results show that the weighted least squares iterative solution can obtain more reasonable parameter estimation and precision information than existing solutions, verifying the feasibility of applying the equality constraints to the mixed additive and multiplicative random error model. Furthermore, the ridge estimation iterative solution can obtain more accurate parameter estimation and precision information than the weighted least squares iterative solution.展开更多
The article deals with optimality of combining ridge and principal component estimator.It is proved that combining ridge and principal componet estimate has several types of minimumvariance properties in the class of ...The article deals with optimality of combining ridge and principal component estimator.It is proved that combining ridge and principal componet estimate has several types of minimumvariance properties in the class of reduced dimension estimates and it also has minimum of orthogo-nal unchanged norm of covariance in the same class.展开更多
Ordinary least squares(OLS) algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or ...Ordinary least squares(OLS) algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or noisy data or multi-collinearity, unbiased OLS leads to large variance. Biased estimators, especially ridge estimator, have been introduced to improve OLS by trading bias for variance. Ridge estimator is feasible as an estimator with smaller variance. At the same confidence level, with additive noise as the normal random variable, the less variance one estimator has, the shorter the two-sided symmetric confidence interval is. However, this finding is limited to the unbiased estimator and few studies analyze and compare the confidence levels between ridge estimator and OLS. This paper derives the matrix of ridge parameters under necessary and sufficient conditions based on which ridge estimator is superior to OLS in terms of mean squares error matrix, rather than mean squares error.Then the confidence levels between ridge estimator and OLS are compared under the condition of OLS fixed symmetric confidence interval, rather than the criteria for evaluating the validity of different unbiased estimators. We conclude that the confidence level of ridge estimator can not be directly compared with that of OLS based on the criteria available for unbiased estimators, which is verified by a simulation and a laboratory scale experiment on a single parameter measurement.展开更多
In this article, the restricted almost unbiased ridge logistic estimator (RAURLE) is proposed to estimate the parameter in a logistic regression model with exact linear re-strictions when there exists multicollinearit...In this article, the restricted almost unbiased ridge logistic estimator (RAURLE) is proposed to estimate the parameter in a logistic regression model with exact linear re-strictions when there exists multicollinearity among explanatory variables. The performance of the proposed estimator over the maximum likelihood estimator (MLE), ridge logistic estimator (RLE), almost unbiased ridge logistic estimator (AURLE), and restricted maximum likelihood estimator (RMLE) with respect to different ridge parameters is investigated through a simulation study in terms of scalar mean square error.展开更多
When multicollinearity is present in a set of the regression variables,the least square estimate of the regression coefficient tends to be unstable and it may lead to erroneous inference.In this paper,generalized ridg...When multicollinearity is present in a set of the regression variables,the least square estimate of the regression coefficient tends to be unstable and it may lead to erroneous inference.In this paper,generalized ridge estimate(K)of the regression coefficient=vec(B)is considered in multivaiale linear regression model.The MSE of the above estimate is less than the MSE of the least square estimate by choosing the ridge parameter matrix K.Moreover,it is pointed out that the Criterion MSE for choosing matrix K of generalized ridge estimate has several weaknesses.In order to overcome these weaknesses,a new family of criteria Q(c)is adpoted which includes the criterion MSE and criterion LS as its special case.The good properties of the criteria Q(c)are proved and discussed from theoretical point of view.The statistical meaning of the scale c is explained and the methods of determining c are also given.展开更多
<div style="text-align:justify;"> As a generalized sensor, the RPC model with its accuracy equally matches the physical sensor model. Moreover, the accurate positioning combining with the flexibility i...<div style="text-align:justify;"> As a generalized sensor, the RPC model with its accuracy equally matches the physical sensor model. Moreover, the accurate positioning combining with the flexibility in application leads the RPC model to be the priority in photogrammetry processing. Generally, the RPC model is calculated through a control grid. Different RPC parameters solving methods and the operation efficiency all serve as variables in the accuracy of the model. In this paper, the ridge estimation iterative method, spectrum correction iteration, and conjugate gradient method are employed to solve RPC parameters;the accuracy and efficiency of three solving methods are analyzed and compared. The results show that ridge estimation iterative method and spectrum correction iteration have obvious advantages in accuracy. The ridge estimation iterative method has fewer iteration times and time con-sumption, and spectrum correction iteration has more stable precision. </div>展开更多
The solution properties of semiparametric model are analyzed, especially that penalized least squares for semiparametric model will be invalid when the matrix B^TPB is ill-posed or singular. According to the principle...The solution properties of semiparametric model are analyzed, especially that penalized least squares for semiparametric model will be invalid when the matrix B^TPB is ill-posed or singular. According to the principle of ridge estimate for linear parametric model, generalized penalized least squares for semiparametric model are put forward, and some formulae and statistical properties of estimates are derived. Finally according to simulation examples some helpful conclusions are drawn.展开更多
Navigation technology,which integrates vision,Inertial Measurement Unit(IMU),and Ultra-Wideband(UWB)sensors in GNSS-denied environments has gained a significant attention.However,inaccurate estimation of UWB anchor po...Navigation technology,which integrates vision,Inertial Measurement Unit(IMU),and Ultra-Wideband(UWB)sensors in GNSS-denied environments has gained a significant attention.However,inaccurate estimation of UWB anchor positions and improper sensor weighting among heterogeneous sensors significantly impairs the positioning accuracy and robustness of Visual-Inertial-UWB(VIU)systems.To accurately and rapidly estimate the UWB anchor positions,we employed the robust ridge nonlinear least-squares method to improve the accuracy and reliability of the estimated UWB anchor position.Additionally,we proposed a simple and effective method to assess the accuracy of the UWB anchor position using the geometric dilution precision principle,which facilitates rapid and accurate estimation of the UWB anchor position.Furthermore,we designed a method to calculate the estimated UWB anchor position error in real-world settings.Finally,we proposed a nonlinear optimization method with dynamically adaptive weighting based on the HELMERT variance component estimation principle,which assigns appropriate weights to heterogeneous sensors.To validate the feasibility and effectiveness of the proposed method,comprehensive simulations and real-world experiments were conducted.First,using Monte Carlo simulation and real-world experiments,we validated the effectiveness of the proposed methods for UWB anchor position and its accuracy estimation.Then,we conducted ablation experiments utilizing the open-source VIRAL and real-world datasets.The experimental results demonstrate that the proposed method exhibits superior positioning accuracy and robustness in contrast to the open-source VINS-MONO and VIR-SLAM methods.展开更多
In this paper,we propose a new biased estimator of the regression parameters,the generalized ridge and principal correlation estimator.We present its some properties and prove that it is superior to LSE(least squares ...In this paper,we propose a new biased estimator of the regression parameters,the generalized ridge and principal correlation estimator.We present its some properties and prove that it is superior to LSE(least squares estimator),principal correlation estimator,ridge and principal correlation estimator under MSE(mean squares error) and PMC(Pitman closeness) criterion,respectively.展开更多
In this paper,we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator(AULE)andridge estimator(RE)in a linear regression model when multicollinearity p...In this paper,we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator(AULE)andridge estimator(RE)in a linear regression model when multicollinearity presents amongthe independent variables.Necessary and sufficient conditions for the proposed estimator over the ordinary least square estimator,RE,AULE and Liu estimator(LE)in the mean squared error matrix sense are derived,and the optimal biasing parameters are obtained.To illustrate the theoretical findings,a Monte Carlo simulation study is carried out and a numerical example is used.展开更多
基金supported by the National Natural Science Foundation of China,Grant Nos.42174011,41874001 and 41664001Innovation Found Designated for Graduate Students of ECUT,Grant No.DHYC-202020。
文摘The reasonable prior information between the parameters in the adjustment processing can significantly improve the precision of the parameter solution. Based on the principle of equality constraints, we establish the mixed additive and multiplicative random error model with equality constraints and derive the weighted least squares iterative solution of the model. In addition, aiming at the ill-posed problem of the coefficient matrix, we also propose the ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints based on the principle of ridge estimation method and derive the U-curve method to determine the ridge parameter. The experimental results show that the weighted least squares iterative solution can obtain more reasonable parameter estimation and precision information than existing solutions, verifying the feasibility of applying the equality constraints to the mixed additive and multiplicative random error model. Furthermore, the ridge estimation iterative solution can obtain more accurate parameter estimation and precision information than the weighted least squares iterative solution.
文摘The article deals with optimality of combining ridge and principal component estimator.It is proved that combining ridge and principal componet estimate has several types of minimumvariance properties in the class of reduced dimension estimates and it also has minimum of orthogo-nal unchanged norm of covariance in the same class.
基金Supported by the National Natural Science Foundation of China （21006127） and the National Basic Research Program of China （2012CB720500）.
文摘Ordinary least squares(OLS) algorithm is widely applied in process measurement, because the sensor model used to estimate unknown parameters can be approximated through multivariate linear model. However, with few or noisy data or multi-collinearity, unbiased OLS leads to large variance. Biased estimators, especially ridge estimator, have been introduced to improve OLS by trading bias for variance. Ridge estimator is feasible as an estimator with smaller variance. At the same confidence level, with additive noise as the normal random variable, the less variance one estimator has, the shorter the two-sided symmetric confidence interval is. However, this finding is limited to the unbiased estimator and few studies analyze and compare the confidence levels between ridge estimator and OLS. This paper derives the matrix of ridge parameters under necessary and sufficient conditions based on which ridge estimator is superior to OLS in terms of mean squares error matrix, rather than mean squares error.Then the confidence levels between ridge estimator and OLS are compared under the condition of OLS fixed symmetric confidence interval, rather than the criteria for evaluating the validity of different unbiased estimators. We conclude that the confidence level of ridge estimator can not be directly compared with that of OLS based on the criteria available for unbiased estimators, which is verified by a simulation and a laboratory scale experiment on a single parameter measurement.
文摘In this article, the restricted almost unbiased ridge logistic estimator (RAURLE) is proposed to estimate the parameter in a logistic regression model with exact linear re-strictions when there exists multicollinearity among explanatory variables. The performance of the proposed estimator over the maximum likelihood estimator (MLE), ridge logistic estimator (RLE), almost unbiased ridge logistic estimator (AURLE), and restricted maximum likelihood estimator (RMLE) with respect to different ridge parameters is investigated through a simulation study in terms of scalar mean square error.
基金The projects Supported by Natural Science Foundation of Fujian Province
文摘When multicollinearity is present in a set of the regression variables,the least square estimate of the regression coefficient tends to be unstable and it may lead to erroneous inference.In this paper,generalized ridge estimate(K)of the regression coefficient=vec(B)is considered in multivaiale linear regression model.The MSE of the above estimate is less than the MSE of the least square estimate by choosing the ridge parameter matrix K.Moreover,it is pointed out that the Criterion MSE for choosing matrix K of generalized ridge estimate has several weaknesses.In order to overcome these weaknesses,a new family of criteria Q(c)is adpoted which includes the criterion MSE and criterion LS as its special case.The good properties of the criteria Q(c)are proved and discussed from theoretical point of view.The statistical meaning of the scale c is explained and the methods of determining c are also given.
文摘<div style="text-align:justify;"> As a generalized sensor, the RPC model with its accuracy equally matches the physical sensor model. Moreover, the accurate positioning combining with the flexibility in application leads the RPC model to be the priority in photogrammetry processing. Generally, the RPC model is calculated through a control grid. Different RPC parameters solving methods and the operation efficiency all serve as variables in the accuracy of the model. In this paper, the ridge estimation iterative method, spectrum correction iteration, and conjugate gradient method are employed to solve RPC parameters;the accuracy and efficiency of three solving methods are analyzed and compared. The results show that ridge estimation iterative method and spectrum correction iteration have obvious advantages in accuracy. The ridge estimation iterative method has fewer iteration times and time con-sumption, and spectrum correction iteration has more stable precision. </div>
基金Funded by the National Nature Science Foundation of China(No.40274005) .
文摘The solution properties of semiparametric model are analyzed, especially that penalized least squares for semiparametric model will be invalid when the matrix B^TPB is ill-posed or singular. According to the principle of ridge estimate for linear parametric model, generalized penalized least squares for semiparametric model are put forward, and some formulae and statistical properties of estimates are derived. Finally according to simulation examples some helpful conclusions are drawn.
基金supported by the National Natural Science Foundation of China(NSFC)under Grant 42071454 and Grant 42371466the State Key Laboratory of Geo-Information Engineering under Grant SKLGIE2023-M-2-2.
文摘Navigation technology,which integrates vision,Inertial Measurement Unit(IMU),and Ultra-Wideband(UWB)sensors in GNSS-denied environments has gained a significant attention.However,inaccurate estimation of UWB anchor positions and improper sensor weighting among heterogeneous sensors significantly impairs the positioning accuracy and robustness of Visual-Inertial-UWB(VIU)systems.To accurately and rapidly estimate the UWB anchor positions,we employed the robust ridge nonlinear least-squares method to improve the accuracy and reliability of the estimated UWB anchor position.Additionally,we proposed a simple and effective method to assess the accuracy of the UWB anchor position using the geometric dilution precision principle,which facilitates rapid and accurate estimation of the UWB anchor position.Furthermore,we designed a method to calculate the estimated UWB anchor position error in real-world settings.Finally,we proposed a nonlinear optimization method with dynamically adaptive weighting based on the HELMERT variance component estimation principle,which assigns appropriate weights to heterogeneous sensors.To validate the feasibility and effectiveness of the proposed method,comprehensive simulations and real-world experiments were conducted.First,using Monte Carlo simulation and real-world experiments,we validated the effectiveness of the proposed methods for UWB anchor position and its accuracy estimation.Then,we conducted ablation experiments utilizing the open-source VIRAL and real-world datasets.The experimental results demonstrate that the proposed method exhibits superior positioning accuracy and robustness in contrast to the open-source VINS-MONO and VIR-SLAM methods.
基金Foundation item： the National Natural Science Foundation of China （Nos. 60736047 10671007+2 种基金 60772036） the Foundation of Beijing Jiaotong University （Nos. 2006XM037 2007XM046）.
文摘In this paper,we propose a new biased estimator of the regression parameters,the generalized ridge and principal correlation estimator.We present its some properties and prove that it is superior to LSE(least squares estimator),principal correlation estimator,ridge and principal correlation estimator under MSE(mean squares error) and PMC(Pitman closeness) criterion,respectively.
文摘In this paper,we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator(AULE)andridge estimator(RE)in a linear regression model when multicollinearity presents amongthe independent variables.Necessary and sufficient conditions for the proposed estimator over the ordinary least square estimator,RE,AULE and Liu estimator(LE)in the mean squared error matrix sense are derived,and the optimal biasing parameters are obtained.To illustrate the theoretical findings,a Monte Carlo simulation study is carried out and a numerical example is used.
