Restricted Principal Components Regression Research Articles

Amendments of a Stochastic Restricted Principal Components Regression Estimator in the Linear Model

Principal component Analysis (PCA) is one of the popular methods used to solve the multicollinearity problem. Researchers in 2014 proposed an estimator to solve this problem in the linear model when there were stochastic linear restrictions on the regression coefficients. This estimator was called the stochastic restricted principal components (SRPC) regression estimator. The estimator was constructed by combining the ordinary mixed estimator (OME) and the principal components regression (PCR) estimator. It ignores the number of components (orthogonal matrix Tr) that the researchers choose to solve the multicollinearity problem in the data matrix (X). This paper proposed four different methods (Lagrange function, the same technique, the constrained principal component model, and substitute in model) to modify the (SRPC) estimator to be used in case of multicollinearity. Finally, a numerical example, an application, and simulation study have been introduced to illustrate the performance of the proposed estimator.

Stochastic Restricted Biased Estimators in Misspecified Regression Model with Incomplete Prior Information

The analysis of misspecification was extended to the recently introduced stochastic restricted biased estimators when multicollinearity exists among the explanatory variables. The Stochastic Restricted Ridge Estimator (SRRE), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE), Stochastic Restricted Liu Estimator (SRLE), Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE), Stochastic Restricted Principal Component Regression Estimator (SRPCRE), Stochastic Restricted r-k (SRrk) class estimator, and Stochastic Restricted r-d (SRrd) class estimator were examined in the misspecified regression model due to missing relevant explanatory variables when incomplete prior information of the regression coefficients is available. Further, the superiority conditions between estimators and their respective predictors were obtained in the mean square error matrix (MSEM) sense. Finally, a numerical example and a Monte Carlo simulation study were used to illustrate the theoretical findings.

The small sample properties of the restricted principal component regression estimator in linear regression model

ABSTRACTIn regression analysis, to deal with the problem of multicollinearity, the restricted principal components regression estimator is proposed. In this paper, we compared the restricted principal components regression estimator, the principal components regression estimator, and the ordinary least-squares estimator with each other under the Pitman's closeness criterion. We showed that the restricted principal components regression estimator is always superior to the principal components regression estimator, under certain conditions the restricted principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion and under certain conditions the principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion.

Modified and Restricted r-k Class Estimators

In this article, we introduce the modified r-k class estimator and the restricted r-k class estimator. We compare the performances of the new estimators to the r-k class estimator with respect to the matrix mean square error (MSE) criterion. As a special case of the restricted r-k class estimator, we obtain the restricted principal components regression (RPCR) estimator. Finally, we conduct a Monte Carlo simulation study and a numerical example to investigate the performances of the proposed estimators by the scalar mean square error (mse) criterion.

Efficiency of the restricted [formula omitted] class estimator in linear regression

In this paper, a restricted r-d class estimator is introduced for the vector of parameters in a linear regression model with linear restrictions. The risk function for the restricted r-d class estimator and restricted principal component regression estimator associated with balanced loss function are given and the comparison of the restricted r-d class estimator and restricted principal component regression estimator associated with balanced loss function is also presented. The performance of the proposed estimators is evaluated by a simulated data set.

A Stochastic Restricted Principal Components Regression Estimator in the Linear Model

We propose a new estimator to combat the multicollinearity in the linear model when there are stochastic linear restrictions on the regression coefficients. The new estimator is constructed by combining the ordinary mixed estimator (OME) and the principal components regression (PCR) estimator, which is called the stochastic restricted principal components (SRPC) regression estimator. Necessary and sufficient conditions for the superiority of the SRPC estimator over the OME and the PCR estimator are derived in the sense of the mean squared error matrix criterion. Finally, we give a numerical example and a Monte Carlo study to illustrate the performance of the proposed estimator.

Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression

In this article, we propose two stochastic restricted principal components regression estimator by combining the approach followed in obtaining the ordinary mixed estimator and the principal components regression estimator in linear regression model. The performance of the two new estimators in terms of matrix MSE criterion is studied. We also give an example and a Monte Carlo simulation to show the theoretical results.

Principal components regression estimator and a test for the restrictions

In this article, we introduce restricted principal components regression (RPCR) estimator by combining the approaches followed in obtaining the restricted least squares estimator and the principal components regression estimator. The performance of the RPCR estimator with respect to the matrix and the generalized mean square error are examined. We also suggest a testing procedure for linear restrictions in principal components regression by using singly and doubly non-central F distribution.

A simulation study on comparison of prediction methods when only a few components are relevant

An unconditional expected squared error criterion is used for an overall comparison of 5 different prediction methods: Principal component regression by the size of the eigenvalues (PCR1) and by the size of the t-value (PCR2), partial least squares regression (PLS), restricted principal component regression (RPCR), and modified maximum likelihood regression (MML). Because the distributions of the estimated regression coefficients (looking at the calibration set as random) are unknown or only known asymptotically, a large simulation study is performed. By means of a model based on relevant components and by reducing the number of parameters in the model using the symmetries in the situation, the simulations are designed to cover the major part of the parameter space. The main result is that PCR1, PLS and RPCR are the best prediction methods, the three methods being quite similar, with PLS somewhat better when the irrelevant eigenvalues are large, and PCR1 somewhat better when the irrelevant eigenvalues are small.

Calibration Methods for NIRS Instruments: A Theoretical Evaluation and Comparisons by Data Splitting and Simulations

The properties of the recently proposed calibration method called restricted principal component regression (RPCR) were evaluated and compared with partial least-squares regression (PLSR) and two types of principal component regression (PCR1, selected according to the size of the eigenvalues, and PCR2, selected according to the t-value). RPCR can be considered a compromise between PCR and PLSR, since the first component of RPCR is equivalent to the first component of PLSR, while the rest can be regarded as principal components on a space orthogonal to the first. The methods showed almost the same properties when the irrelevant components had small eigenvalues. The prediction error of RPCR selected according to the size of the eigenvalues was intermediate to those of PCR1 and PLSR when the number of components was low, while RPCR and PCR1 nearly coincided when the number of components exceeded the number of relevant ones. The prediction error minimum was about the same for RPCR, PCR1, and PLSR, but the minimum of PLSR was obtained when a lower number of components were included in the calibration model.