Mean Square Error Matrix Research Articles

Holevo bound independent of weight matrices for estimating two parameters of a qubit

Holevo bound plays an important role in quantum metrology as it sets the ultimate limit for multi-parameter estimations, which can be asymptotically achieved. Except for some trivial cases, the Holevo bound is implicitly defined and formulated with the help of weight matrices. Here we report the first instance of an intrinsic Holevo bound, namely, without any reference to weight matrices, in a nontrivial case. Specifically, we prove that the Holevo bound for estimating two parameters of a qubit is equivalent to the joint constraint imposed by two quantum Cramér–Rao bounds corresponding to symmetric and right logarithmic derivatives. This weightless form of Holevo bound enables us to determine the precise range of independent entries of the mean-square error matrix, i.e., two variances and one covariance that quantify the precisions of the estimation, as illustrated by different estimation models. Our result sheds some new light on the relations between the Holevo bound and quantum Cramér–Rao bounds. Possible generalizations are discussed.

A Background-Impulse Kalman Filter With Non-Gaussian Measurement Noises

In the Kalman filter (KF), the estimated state is a linear combination of the one-step prediction and measurement. The two combination weights depend on the prediction mean-square error matrix and the covariances matrix of measurement noise (CMMN), respectively. When the measurement noise values are small (large), the corresponding measurement is close to (far from) the real value, and the weight should be large (small). If there is non-Gaussian measurement noise, especially heavy-tailed noise, most of the noise values are small, and a few of them are large. The occasional impulses cause the overall noise variance to be much greater than the noise without impulses. Since the overall CMMN is adopted in the KF, its performance will deteriorate. In this article, we divide the measurement noise into two parts: 1) the background noise with small variance and 2) the impulse noise with large variance. Then, we use the expectation–maximization algorithm to dynamically calculate the parameters and propose a new KF algorithm to process them separately, which is called background-impulse KF (BIKF). It can dynamically determine whether there is an impulse and eliminate its impact as much as possible. Compared with the recent maximum correntropy criterion and minimum error entropy criterion-based KFs, simulations show that the proposed BIKF works better than them with lower computational complexity.

Machine learning based renewable energy generation and energy consumption forecasting

ABSTRACT Renewable energy generation sources are the need of the hour to mitigate the challenges of fossil fuels, environmental impacts, large transmission and distribution losses, and variable consumption pattern. Renewable energy such as solar PV generation or wind generation is most unpredicted generation and depends upon weather constraints. A promising prediction of generation is a suitable solution for this problem. A machine learning-based conventional neural network algorithm is proposed to forecast the generation and calculate the errors in generation forecast. The generated energy is supplied to consumers, but the consumers consume electricity by their choice and an energy management model is required at the consumer end to supply the available predicted generation to develop a self-sustainable microgrid. The energy consumption at consumer end is essential to predict, and based on the predicted consumption at hourly bases, the cost model for the consumers is developed. The consumption profile is forecasted using long short-term memory as it is a good tool for time-series forecasting using available sequential load consumption data. The demand is forecasted and analyzed for selected load categories (i.e. fixed, non-shift-able, and shift-able loads). The evaluation matrix mean square error (MSE), mean absolute error, and root mean square error (RMSE) are used for the error calculations. The RMSE for solar and wind power generation forecasting have been 2.472 and 3.034, respectively. The RMSE for fixed, non-shiftable, and shiftable demand forecasting have been 0.027, 0.067, and 0.21, respectively. The results show that the proposed methods for predictions of solar, wind power generation and energy consumption forecast achieve better accuracy. The presented methodology can be implemented with energy storage for renewable energy production and demand management for energy from sustainable energy resources.

On the performance of some new ridge parameter estimators in the Poisson-inverse Gaussian ridge regression

The Poisson Inverse Gaussian Regression model (PIGRM) is used for modeling the count datasets to deal with the issue of over-dispersion. Generally, the maximum likelihood estimator (MLE) is used to estimate the PIGRM estimates. In the PIGRM, when the explanatory variables are correlated, the MLE does not provide efficient results. To overcome this problem, we propose a ridge estimator for the PIGRM. The matrix mean square error (MSE) and the scalar MSE properties are derived and then compared with the MLE. In the ridge estimator, ridge parameter play a significant role, so, this study also proposes different ridge parameter estimators for the PIGRM. The performance of the proposed estimator is evaluated with the help of a simulation study and a real-life application using MSE as a performance evaluation criterion. The simulation study and the real-life application results show the superiority of the proposed parameter estimators as compared to the MLE.

Robust Underwater Direction-of-Arrival Tracking Based on AI-Aided Variational Bayesian Extended Kalman Filter

The AI-aided variational Bayesian extended Kalman filter (AI-VBEKF)-based robust direction-of-arrival (DOA) technique is proposed to make reliable estimations of the bearing angle of an uncooperative underwater target with uncertain environment noise. Considering that the large error of the guess of the initial mean square error matrix (MSEM) will lead to inaccurate DOA tracking results, an attention-based deep convolutional neural network is first proposed to make reliable estimations of the initial MSEM. Then, by utilizing the AI-VBEKF estimating scheme, the uncertain measurement noise caused by the unknown underwater environment along with the bearing angle of the target can be estimated simultaneously to provide reliable results at every DOA tracking step. The proposed technique is demonstrated and verified by both of the simulations and the real sea trial data from the South China Sea in July 2021, and both the robustness and accuracy are proven superior to the traditional DOA-estimating methods.

A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity

<abstract><p>This paper is concerned with proposing a generalized Liu-type estimator (GLTE) to address the multicollinearity problem of explanatory variable of the linear part in the logistic partially linear regression model. Using the profile likelihood method, we propose the GLTE as a general class of Liu-type estimator, which includes the profile likelihood estimator, the ridge estimator, the Liu estimator and the Liu-type estimator as special cases. The conditional superiority of the proposed GLTE over the other estimators is derived under the asymptotic mean square error matrix (MSEM) criterion. Moreover, the optimal choices of biasing parameters and function of biasing parameter are given. Numerical simulations demonstrate that the proposed GLTE performs better than the existing estimators. An application on a set of real data arising from the study of Indian Liver Patient is shown for illustrating our theoretical results.</p></abstract>

Statistical inference of a stochastically restricted linear mixed model

<abstract><p>This article compares a predictor with the best linear unbiased predictor (BLUP) for a unified form of all unknown parameters under a stochastically restricted linear mixed model (SRLMM) in terms of the mean squared error matrix (MSEM) criterion. The methodology of block matrix inertias and ranks is employed to compare the MSEMs of these predictors. The comparison results are also demonstrated for a linear mixed model with and without an exact restriction, as well as special cases of the unified form of all unknown parameters in the SRLMM.</p></abstract>

Estimation of structural parameters in balanced Bühlmann credibility model with correlation risk

In this paper, the longitudinal data analysis is used to interpret the balanced Bühlmann credibility model with correlation risk, and the homogeneous credibility estimator is derived. We obtain the restricted maximum likelihood estimators (RMLE) for the structural parameters involved in the credibility factor and show that they are unbiased. In addition, the linear Bayes method is employed to estimate the structural parameters, and the proposed linear Bayes estimators (LBE) appear to outperform RMLE in terms of the mean squared error matrix (MSEM) criterion. Simulation studies show that the proposed LBE performs well.

Approximate Bayesian estimator for the parameter vector in linear models with multivariate t distribution errors

This article constructs an approximate Bayes estimator for the parameter vector consisted of regression coefficients and variance parameter in the linear model in which the error terms follow multivariate t distribution. Its superiorities over the classical estimators are strictly proved in terms of the mean squared error matrix (MSEM) criterion. Compared with the Bayes estimator computed via the MCMC method, the proposed Bayes estimator is simple and easy to interpret and compute, which only requires relatively little prior designation. The numerical computations further verify that the approximate Bayes estimator performs well. Also, the proposed procedure can be easily extended to other multivariate distribution cases.

Ridge estimation in linear mixed measurement error models using generalized maximum entropy

In this article, we concentrate on the generalized maximum entropy (GME) estimators and their asymptotic properties in linear mixed models (LMMs) with measurement error in the fixed effects (MEFE) variables. Moreover, we obtain the Ridge-GME estimator in these models as a remedy to collinearity. Finally, a simulation study and an example of real data are given to characterize the superiority of the Ridge-GME estimator over the corrected score estimator (CSE) and ridge estimator (RE), using the mean squared error matrix (MSEM).

A new stochastic restricted two-parameter estimator in multiplelinear regression model

In this paper, we proposed a biased estimator, a new stochastic restricted two-parameter estimator (NSRTPE), for the multiple linear regression model to tackle the multicollinearity problem when the stochastic restrictions are available. Necessary and sufficient conditions for the superiority of the proposed estimator over the ordinary least square estimator (OLSE), ridge estimator (RE), Liu estimator (LE), almost unbiased Liu estimator (AULE), modified new two-parameter estimator (MNTPE), mixed estimator (ME), stochastic restricted Liu estimator (SRLE) were derived in the mean square error matrix (MSEM) criterion. Finally, we showed the superiority of the estimator proposed using a simulation study and a real-world example in the scalar mean square error (SMSE) criterion.

A Femto-Satellite Localization Method Based on TDOA and AOA Using Two CubeSats

This article presents a feasibility analysis to remotely estimate the geo-location of a femto-satellite only using two station-CubeSats and the communication link between the femto-satellite and each CubeSat. The presented approach combines the Time Difference Of Arrival (TDOA) and Angle Of Arrival (AOA) methods. We present the motivation, the envisioned solution together with the constraints for reaching it, and the best potential sensitivity of the location precision for different (1) deployment scenarios of the femto-satellite, (2) precisions in the location of the CubeSats, and (3) precisions in each CubeSat’s Attitude Determination and Control Systems (ADCS). We implemented a simulation tool to evaluate the average performance for different random scenarios in space. For the evaluated cases, we found that the Cramér-Rao Bound (CRB) for Gaussian noise over the small error region of the solution is highly dependent on the deployment direction, with differences in the location precision close to three orders of magnitude between the best and worst deployment directions. For the best deployment case, we also studied the best location estimation that might be achieved with the current Global Navigation Satellite System (GNSS) and ADCS commercially available for CubeSats. We found that the mean-square error (MSE) matrix of the proposed solution under the small error condition can attain the CRB for the simulated time, achieving a precision below 30 m when the femto-satellite is separated by around 800 m from the mother-CubeSat.

Experiment design for impulse response identification with signal matrix models

This paper formulates an input design approach for truncated infinite impulse response identification in the context of implicit model representations recently used as basis for data-driven simulation and control approaches. Precisely, the considered model consists of a linear combination of the columns of a data (or signal) matrix. An optimal combination for the case of noisy data was recently proposed using a maximum likelihood approach, and the objective here is to optimize the input entries of the data matrix such that the mean-square error matrix of the estimate is minimized. A least-norm problem is derived in terms of the optimality criteria typically considered in the experiment design literature. Numerical results showcase the improved estimation fit achieved with the optimized input.

Study of Some Kinds of Ridge Regression Estimators in Linear Regression Model

In linear regression model, the biased estimation is one of the most commonly used methods to reduce the effect of the multicollinearity. In this paper, a simulation study is performed to compare the relative efficiency of some kinds of biased estimators as well as for twelve proposed estimated ridge parameter (k) which are given in the literature. We propose some new adjustments to estimate the ridge parameter. Finally, we consider a real data set in economics to illustrate the results based on the estimated mean squared error (MSE) criterion.
 According to the results, all the proposed estimators of (k) are superior to ordinary least squared estimator (OLS), and the superiority among them based on minimum MSE matrix will change according to the sample under consideration.

Ridge estimation in linear mixed measurement error models with stochastic linear mixed restrictions

This article is concentrated on the problem of multicollinearity in linear mixed models (LMMs) with measurement error in the fixed effects variables. After introducing a ridge estimator (RE) in these models, we propose a new estimator called the stochastic restricted ridge estimator (SRRE) by combining the ridge estimator (RE) and the stochastic restricted estimator (SRE). Moreover, asymptotic properties of these estimators will be derived and the necessary and sufficient conditions for the superiority of the SRRE over the RE and SRE are obtained using the mean squared error matrix (MSEM). Finally, the theoretical findings of the proposed estimators are also evaluated with a Monte Carlo simulation study and a numerical example.

Modifying Two-Parameter Ridge Liu Estimator Based on Ridge Estimation

In this paper, we introduce the new biased estimator to deal with the problem of multicollinearity. This estimator is considered a modification of Two-Parameter Ridge-Liu estimator based on ridge estimation. Furthermore, the superiority of the new estimator than Ridge, Liu and Two-Parameter Ridge-Liu estimator were discussed. We used the mean squared error matrix (MSEM) criterion to verify the superiority of the new estimate. In addition to, we illustrated the performance of the new estimator at several factors through the simulation study.

A Low-Complexity Linear Precoding for Secure Transmission Over MIMOME Wiretap Channels With Finite-Alphabet Inputs

In this paper, we investigate the precoding scheme for multiple-input multiple-output multiple antennas eavesdropper (MIMOME) channels under the constraint of finite-alphabet inputs to ensure secure transmission between the source and the destination in the presence of an eavesdropper. With finite-alphabet inputs, there are no closed-form solutions for mutual information (MI) and its gradient-minimum mean-square error (MMSE) matrix. Therefore traditional precoding schemes based on direct calculation of MI and MMSE matrix through numerical integration involve prohibitive computational burden. To address this problem, this paper first derives asymptotically optimal lower bounds of MI and ergodic MI with which the computational complexity can be significantly reduced compared with the direct computation of MI. With the derived lower bounds, a low-complexity precoding scheme is proposed to maximize the secrecy rate, wherein a gradient descent based algorithm is used to obtain the precoding result. In the low and high signal-to-noise ratio (SNR) regions, we demonstrate that the proposed precoding scheme gives the same result as directly maximizing the secrecy rate, which indicates that the proposed low-complexity precoding scheme is asymptotically optimal.

Stochastic restricted Liu estimator in linear mixed measurement error models

This paper is concerned with the Liu estimator, stochastic restricted estimator and stochastic restricted Liu estimator of fixed and random effects in the linear mixed measurement error models. We compare the proposed estimators under the criterion of mean squared error matrix (MSEM). Furthermore, the selection of the Liu biasing parameter is discussed. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of estimators in the linear mixed measurement error model.

Estimation of parameters in linear mixed measurement error models with stochastic linear restrictions

In this paper, we concentrate on the estimation of parameters in the linear mixed models (LMMs) with stochastic linear restrictions on the fixed and random effects, when the fixed effects are measured with error. In addition, the asymptotic properties of the estimators are derived and their performances are compared with the estimators for no restrictions cases, in the sense of mean square error matrix (MSEM). Finally, a simulation study and numerical example have been conducted to show the superiority of the restricted estimator over the unrestricted estimator.

Robust estimation of mean squared error matrix of small area estimators in a multivariate Fay–Herriot model

This paper is concerned with the small area estimation in the multivariate Fay–Herriot model where covariance matrix of random effects are fully unknown without normality assumption. The covariance matrix is estimated by a Prasad–Rao type consistent estimator, and the empirical best linear unbiased predictor (EBLUP) of a vector of small area characteristics is provided. When the EBLUP is measured in terms of a mean squared error matrix (MSEM), a second-order approximation of MSEM of the EBLUP and a second-order unbiased estimator of the MSEM are derived analytically in closed forms. The performance is investigated through simulation and empirical studies.