Ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints

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.

Confidence Level Based on Ridge Estimator in Process Measurement and Its Application

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.

On the Restricted Almost Unbiased Ridge Estimator in Logistic Regression

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.

GENERALIZED MULTIVARIATE RIDGE REGRESSION ESTIMATE AND CRITERIA Q(c) FOR CHOOSING MATRIX K

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.

Accuracy and Efficiency: The Comparison of Different RPC Parameters Solving Methods

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.

Generalized Penalized Least Squares and Its Statistical Characteristics

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.

Generalized Ridge and Principal Correlation Estimator of the Regression Parameters and Its Optimality

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.

Modified Almost Unbiased Liu Estimator in Linear Regression Model

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.