Measurement Error Covariance Matrix Research Articles

A new kernel two-parameter prediction under multicollinearity in partially linear mixed measurement error model

A Partially linear mixed effects model relating a response Y to predictors ( X , Z , T ) with the mean function X T β + Zb + g ( T ) is considered in this paper. When the parametric parts' variable X are measured with additive error and there is ill-conditioned data suffering from multicollinearity, a new kernel two-parameter prediction method using the kernel ridge and Liu regression approach is suggested. The kernel two parameter estimator of β and the predictor of b are derived by modifying the likelihood and Henderson methods. Matrix mean square error comparisons are calculated. We also demonstrate that under suitable conditions, the resulting estimator of β is asymptotically normal. The situation with an unknown measurement error covariance matrix is handled. A Monte Carlo simulation study, together with an earthquake data example, is compiled to evaluate the effectiveness of the proposed approach at the end of the paper.

Instrumental Variable Method for Regularized Estimation in Generalized Linear Measurement Error Models

Regularized regression methods have attracted much attention in the literature, mainly due to its application in high-dimensional variable selection problems. Most existing regularization methods assume that the predictors are directly observed and precisely measured. It is well known that in a low-dimensional regression model if some covariates are measured with error, then the naive estimators that ignore the measurement error are biased and inconsistent. However, the impact of measurement error in regularized estimation procedures is not clear. For example, it is known that the ordinary least squares estimate of the regression coefficient in a linear model is attenuated towards zero and, on the other hand, the variance of the observed surrogate predictor is inflated. Therefore, it is unclear how the interaction of these two factors affects the selection outcome. To correct for the measurement error effects, some researchers assume that the measurement error covariance matrix is known or can be estimated using external data. In this paper, we propose the regularized instrumental variable method for generalized linear measurement error models. We show that the proposed approach yields a consistent variable selection procedure and root-n consistent parameter estimators. Extensive finite sample simulation studies show that the proposed method performs satisfactorily in both linear and generalized linear models. A real data example is provided to further demonstrate the usage of the method.

Kernel mixed and Kernel stochastic restricted ridge predictions in the partially linear mixed measurement error models: an application to COVID-19

In this article, we define mixed predictor and stochastic restricted ridge predictor of partially linear mixed measurement error models by taking advantage of Kernel approximation. Under matrix mean square error criterion, we make the comparison of the superiorities the linear combinations of the new defined predictors. Then we investigate the asymptotic normality characteristics and the situation of the unknown covariance matrix of measurement errors. Finally, the study is ended with a Monte Carlo simulation study and COVID-19 data application.

Analytical weak-lensing shear responses of galaxy properties and galaxy detection

ABSTRACTShear estimation bias from galaxy detection and blending identification is now recognized as an issue for ongoing and future weak-lensing surveys. Currently, the empirical approach to correcting for this bias involves numerically shearing every observed galaxy and rerunning the detection and selection process. In this work, we provide an analytical correction for this bias that is accurate to sub per cent level and far simpler to use. With the interpretation that smoothed image pixel values and galaxy properties are projections of the image signal onto a set of basis functions, we analytically derive the linear shear responses of both the pixel values and the galaxy properties (i.e. magnitude, size, and shape) using the shear responses of the basis functions. With these derived shear responses, we correct for biases from shear-dependent galaxy detection and galaxy sample selection. With the analytical covariance matrix of measurement errors caused by image noise on pixel values and galaxy properties, we correct for the noise biases in galaxy shape measurement and the detection/selection process to the second-order in noise. The code used for this paper can carry out the detection, selection, and shear measurement for ∼1000 galaxies per CPU second. The algorithm is tested with realistic image simulations, and we find, after the analytical correction (without relying on external image calibration) for the detection/selection bias of about $-4~{{\ \rm per\ cent}}$, the multiplicative shear bias is $-0.12 \pm 0.10~{{\ \rm per\ cent}}$ for isolated galaxies; and about $-0.3 \pm 0.1~{{\ \rm per\ cent}}$ for blended galaxies with Hyper Suprime-Cam observational condition.

Обращение матриц поисковым методом при уравнивании геодезических сетей

The theoretical substantiation of matrix inversion by the search method of nonlinear programming is given for equalizing and evaluating the accuracy of geodetic networks’ elements. A theorem on the uniqueness of the extremum point and its corollary (also a remark to the latter) are formulated and proved with minimizing the objective function by the search method at performing matrix inversion. A step-by-step algorithm for matrix conversion by the search method was compiled. The approbation of the developed algorithm was carried out when equalizing and evaluating the geodetic network elements’ accuracy, taking into account the errors of the source data. The correctness of the proposed solutions is confirmed by coinciding the covariance matrix of measurement errors results reversal and that of unknowns’ coefficients normal equations with the calculations performed by the MOBR function in Microsoft Excel, as well as the coincidence of the equalization results and evaluation of the geodetic network’s elements accuracy by the search method with similar calculations by parametric method in the NW program by Prof. V. A. Kougiya.

A Novel Adaptive Filtering for Cooperative Localization Under Compass Failure and Non-Gaussian Noise

Multi-autonomous underwater vehicles (AUVs) cooperative localization has become a research hotspot in the marine navigation field. In this paper, a filtering algorithm for slave AUV with compass failure is proposed based on ‘two-master-one slave’ cooperative localization model. In this algorithm, the course angle is unknown, and non-Gaussian measurement noise caused by measurement outliers is considered. The cooperative localization model is reconstructed, and a filtering algorithm is derived whose performance will not be affected by the unknown course angle. Moreover, the maximum information potential (MIP) criterion is introduced into a recursive model to deal with the non-Gaussian measurement noise in a complicated underwater environment. Simultaneously, the kernel bandwidth is adaptively adjusted by the innovation matrix and the measurement error covariance matrix. The proposed algorithm can accurately estimate the variation of the unknown course angle without expanding the dimension of state under the condition of non-Gaussian measurement noise. Finally, the proposed algorithm is verified through field experiments. The results show that the proposed algorithm has higher accuracy and robustness than other existing algorithms.

A Numerical Procedure for Multivariate Calibration Using Heteroscedastic Principal Components Regression

Many methods have been developed to allow for consideration of measurement errors during multivariate data analyses. The incorporation of the error structure into the analytical framework, usually described in terms of the covariance matrix of measurement errors, can provide better model estimation and prediction. However, little effort has been made to evaluate the effects of heteroscedastic measurement uncertainties on multivariate analyses when the covariance matrix of measurement errors changes with the measurement conditions. For this reason, the present work describes a new numerical procedure for analyses of heteroscedastic systems (heteroscedastic principal component regression or H-PCR) that takes into consideration the variations of the covariance matrix of measurement fluctuations. In order to illustrate the proposed approach, near infrared (NIR) spectra of xylene and toluene mixtures were measured at different temperatures and stirring velocities and the obtained data were used to build calibration models with different multivariate techniques, including H-PCR. Modeling of available xylene–toluene NIR data revealed that H-PCR can be used successfully for calibration purposes and that the principal directions obtained with the proposed approach can be quite different from the ones calculated through standard PCR, when heteroscedasticity is disregarded explicitly.

Improved Attitude and Heading Accuracy with Double Quaternion Parameters Estimation and Magnetic Disturbance Rejection.

The use of unmanned aerial vehicle (UAV) applications has grown rapidly over the past decade with the introduction of low-cost microelectromechanical system (MEMS)-based sensors that measure angular velocity, gravity, and magnetic field, which are important for an object orientation determination. However, the use of low-cost sensors has also been limited because their readings are easily distorted by unwanted internal and/or external noise signals such as environmental magnetic disturbance, which lead to errors in attitude and heading estimation results. In an extended Kalman filter (EKF) process, this study proposes a method for mitigating the effect of magnetic disturbance on attitude determination by using a double quaternion parameters for representation of orientation states, which decouples the magnetometer from attitude computation. Additionally, an online measurement error covariance matrix tuning system was implemented to reject the impact of magnetic disturbance on the heading estimation. Simulation and experimental tests were conducted to verify the performance of the proposed methods in resolving the magnetic noise effect on attitude and heading. The results showed that the proposed method performed better than complimentary, gradient descent, and single quaternion-based EKF.

Formulating the history matching problem with consistent error statistics

It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

소방관의 실내 측위를 위한 두 발에 장착된 보행항법장치의 입체구적 포인트 기반 위치 영역 융합

The ZUPT (Zero-velocity UPdaTe)-based PDR (Pedestrian Dead Reckoning) mounted on the foot calculates navigation information through the INS (Inertial Navigation System) algorithm, detects the moment when the foot touches the ground through the ZVD (Zero-Velocity Detection) algorithm, and corrects the INS errors through zero-velocity correction filtering. Through this, it is possible to estimate fairly stable position information of a pedestrian. However, the accumulation of errors over time is inevitable. In this paper, we propose a technique to reduce the accumulation of errors through the fusion of PDRs mounted on both feet. After converting the position information of the two PDRs into cubature points, the body position is estimated using the points. The two PDRs are then updated using the body position information as a measurement of the filter. Here, the measurement error covariance matrix is set using the structural constraint of the body and feet to reduce the phenomenon that the position difference between the two PDRs increases over time. The performance of the proposed technique is verified through experiments.

A Rapid and Adaptive Alignment under Mooring Condition Using Adaptive EKF and CNN-Based Learning.

Alignment of the inertial navigation system (INS) in the mooring environment should take into account the movements of the waves or wind. The alignment of the INS is performed through an extended Kalman filter (EKF) using zero velocity as a measurement. However, in the mooring condition, this is not perfect stationary, thus the measurement error covariance matrix should be adjusted. In addition, if the measurement error covariance matrix is fixed to one value, the alignment time may take longer or the performance may be reduced depending on the change in mooring conditions. To solve this problem, we propose an alignment method using adaptive Kalman filter and convolution neural network (CNN)-based learning. The proposed method was verified for the superiority of alignment time and accuracy through Monte Carlo simulation in a mooring environment.

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.

Robust Ensemble Kalman Filter for Medium-Voltage Distribution System State Estimation

This article proposes a forecasting-aided medium-voltage (MV) distribution system state estimation (SE) method using a robust ensemble Kalman filter (REnKF). In the proposed solution, low-voltage (LV) measurements at the LV side of secondary substations are used together with the mathematical model of the MV/LV substations to derive new equivalent MV measurements. This yields the improvement of measurement redundancy and robustness of the REnKF to bad data and unknown system process noise. Specifically, we rely on the temporal correlations of the constructed innovation vector as well as the projection statistics (PS) to detect and modify the measurement error covariance matrix. Furthermore, the system process noise covariance matrix is updated adaptively to mitigate the impacts of uncertainties ON-state forecasting and measurement filtering. Extensive comparisons have been carried out with both the traditional EnKF and other SE formulations under balanced and unbalanced distribution system conditions. The results reveal that our proposed REnKF is able to obtain accurate SE results under various operation conditions, including the presence of intermittent renewable energy sources, bad data, and system unbalance.

Simulation of 1/fα noise for analytical measurements

A simple procedure is described that can be used to generate 1/fα noise, also known as power law noise, in simulated analytical measurement vectors. Certain types of power law noise, such as pink noise (α=1), dominate many types of analytical signals, so its simulation is important in optimizing data processing strategies. In this work, simulated 1/fα error sequences are created directly from white noise via the theoretical measurement error covariance matrix (ECM) by rotation and scaling. The 1/fα ECM is obtained from the coefficients of a finite impulse response filter and is easily adapted to generate multiplicative 1/fα noise that is probably more common for analytical systems exhibiting proportional noise characteristics. Simulating 1/fα noise directly from the ECM offers two main advantages. First, 1/fα noise can be easily simulated in the presence of other common analytical measurement errors by additive combination of the ECMs. Second, the theoretical ECM can be used to model real experimental measurement noise. It is shown that the power spectral density function of measurement error sequences generated by the proposed method closely approximates the theoretical behaviour of 1/fα noise. To demonstrate the utility of this method in evaluating data processing methods, simulated data exhibiting 1/f (pink) noise is analyzed by maximum likelihood principal component analysis (MLPCA) that takes measurement error structure into account, and baseline noise is simulated using brown noise to test baseline fitting by asymmetric least squares (AsLS).

Enhanced History Matching of Gas Reservoirs With an Aquifer Using the Combination of Discrete Cosine Transform and Level Set Method in ES-MDA

Reservoir characterization is a process to make dependable reservoir models using available reservoir information. There are promising ensemble-based methods such as ensemble Kalman filter (EnKF), ensemble smoother (ES), and ensemble smoother with multiple data assimilation (ES-MDA). ES-MDA is an iterative version of ES with inflated covariance matrix of measurement errors. It provides efficient and consistent global updates compared to EnKF and ES. Ensemble-based method might not work properly for channel reservoirs because its parameters are highly non-Gaussian. Thus, various parameterization methods are suggested in previous studies to handle nonlinear and non-Gaussian parameters. Discrete cosine transform (DCT) can figure out essential channel information, whereas level set method (LSM) has advantages on detailed channel border analysis in grid scale transforming parameters into Gaussianity. However, DCT and LSM have weaknesses when they are applied separately on channel reservoirs. Therefore, we propose a properly designed combination algorithm using DCT and LSM in ES-MDA. When DCT and LSM agree with each other on facies update results, a grid has relevant facies naturally. If not, facies is assigned depending on the average facies probability map from DCT and LSM. By doing so, they work in supplementary way preventing from wrong or biased decision on facies. Consequently, the proposed method presents not only stable channel properties such as connectivity and continuity but also similar pattern with the true. It also gives trustworthy future predictions of gas and water productions due to well-matched facies distribution according to the reference.

An Efficient Indoor Target Tracking Algorithm Using TDOA Measurements With Applications to Ultra-Wideband Systems

The ultra-wideband technique has shown its effectiveness for indoor target tracking. Various types of measurements have been applied to ultra-wideband systems for indoor target tracking, and the time difference of arrival (TDOA) measurement-based approaches are the most widely used methods due to their good accuracy and feasibility. Target tracking with the TDOA measurements usually encounters the problem of correlated measurement noises, as one sensor network utilizes the common reference sensor for measurement generation. The off-diagonal entries in the measurement error covariance matrix become non-zero values, which makes the standard target tracking algorithms inconvenient for practical installation of an ultra-wideband system. Another problem in sensor networks is properly exploiting the measurements obtained from multiple sensors considering practical conditions, such as storage limitations or computational resource consumption. The parallel update and the serial update are usually applied for the multi-sensor tracking problem. This paper presents a target tracking algorithm that integrates the Cholesky decomposition to decorrelate the measurement noises for the serial update, thus improving computational efficiency. The proposed algorithm is realized in an ultra-wideband system for real-time target tracking, and an experiment using real data is conducted to validate its practicability.

Power system dynamic state estimation considering correlation of measurement error from PMU and SCADA

SummaryIt is well known that measurements from phasor measurement unit (PMU) or supervisory control and data acquisition (SCADA) are not generally independent. Since the correlation of measurement error is a very representative feature of the actual measurement system, traditional assumptions on error independency are not adequate. In this paper, taking the correlation of measurement error of both PMU and SCADA measurements into consideration, a novel correlated extended Kalman filter (CEKF) is proposed. The actual measurement configurations are analyzed with the consideration of measurement error transfer characteristics. Then, the modified measurement error covariance matrix is calculated by using the point estimation method, which will replace the traditional diagonal variance matrix. At last, IEEE 14‐bus system and 57‐bus system are provided to illustrate the effectiveness and superiority of the method, respectively.

Conditioning reservoir models on rate data using ensemble smoothers

There are several issues to consider when we use ensemble smoothers to condition reservoir models on rate data. The values in a time series of rate data contain redundant information that may lead to poorly conditioned inversions and thereby influence the stability of the numerical computation of the update. A time series of rate data typically has correlated measurement errors in time, and negligence of the correlations leads to a too strong impact from conditioning on the rate data and possible ensemble collapse. The total number of rate data included in the smoother update will typically exceed the ensemble size, and special care needs to be taken to ensure numerically stable results. We force the reservoir model with production rate data derived from the observed production, and the further conditioning on the same rate data implies that we use the data twice. This paper discusses strategies for conditioning reservoir models on rate data using ensemble smoothers. In particular, a significant redundancy in the rate data makes it possible to subsample the rate data. The alternative to subsampling is to model the unknown measurement error correlations and specify the full measurement error covariance matrix. We demonstrate the proposed strategies using different ensemble smoothers with the Norne full-field reservoir model.

A weighted least squares approach to retrieve aerosol layer height over bright surfaces applied to GOME-2 measurements of the oxygen A band for forest fire cases over Europe

Abstract. This paper presents a weighted least squares approach to retrieve aerosol layer height from top-of-atmosphere reflectance measurements in the oxygen A band (758–770 nm) over bright surfaces. A property of the measurement error covariance matrix is discussed, due to which photons travelling from the surface are given a higher preference over photons that scatter back from the aerosol layer. This is a potential source of biases in the estimation of aerosol properties over land, which can be mitigated by revisiting the design of the measurement error covariance matrix. The alternative proposed in this paper, which we call the dynamic scaling method, introduces a scene-dependent and wavelength-dependent modification in the measurement signal-to-noise ratio in order to influence this matrix. This method is generally applicable to other retrieval algorithms using weighted least squares. To test this method, synthetic experiments are done in addition to application to GOME-2A and GOME-2B measurements of the oxygen A band over the August 2010 Russian wildfires and the October 2017 Portugal wildfire plume over western Europe.

Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error

ABSTRACTIn many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p > n setting. We study a correction for generalized linear models, based on Rosenbaum and Tsybakov’s matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements. Supplementary materials for this article are available online.