Measurement Error Covariance Matrices Research Articles

Measurement errors and implications for preprocessing in miniaturised near-infrared spectrometers: Classification of sweet and bitter almonds as a case of study.

Near-infrared (NIR) spectroscopy is a well-established analytical technique that has been used in many applications over the years. Due to the advancements in the semiconductor industry, NIR instruments have evolved from benchtop instruments to miniaturised portable devices. The miniaturised NIR instruments have gained more interest in recent years because of the fast and robust measurements they provide with almost no sample pretreatments. However, due to the very different configurations and characteristics of these instruments, they need a dedicated optimization of the measurement conditions, which is crucial for obtaining reliable results. To comprehensively grasp the capabilities and potentials offered by these sensors, it is imperative to examine errors that can affect the raw data, which is a facet frequently overlooked. In this study, measurement error covariance and correlation matrices were calculated and then visually inspected to gain insight into the error structures associated with the devices, and to find the optimal preprocessing technique that may result in the improvement of the models built. This strategy was applied to the classification of sweet and bitter almonds, which were measured with the three portable low-cost NIR devices (SCiO, FlameNIR+ and NeoSpectra Micro Development Kit) after removing the shelled, since their classification is of utmost importance for the almond industry. The results showed that bitter almonds can be classified from sweet almonds using any of the instruments after selecting the optimal preprocessing, obtained through inspection of covariance and correlation matrices. Measurements obtained with FlameNIR+device provided the best classification models with an accuracy of 98%. The chosen strategy provides new insight into the performance characterization of the fast-growing miniaturised NIR instruments.

Approximate Gaussian variance inference for state‐space models

SummaryState‐space models require an accurate knowledge of the process error () and measurement error () covariance matrices for exact state estimation. Even though the matrix can be, in many situations, considered to be known from the measuring instrument specifications, it is still a challenge to infer the matrix online while providing reliable estimates along with a low computational cost. In this article, we propose an analytically tractable online Bayesian inference method for inferring the matrix in state‐space models. We refer to this method as approximate Gaussian variance inference (AGVI) using which we are able to treat the error variance and covariance terms in the full matrix as Gaussian hidden states and infer them simultaneously with the other hidden states in a closed‐form manner. The two case studies show that the method is able to provide statistically consistent estimates for the mean and uncertainties of the error variance terms for univariate and multivariate cases. The method also exceeds the performance of the existing adaptive Kalman filter methods both in terms of accuracy and computational efficiency.

Smoothed Kalman MRAS observer for speed sensorless indirect vector controlled induction motor drives

This paper proposes a modified Kalman filter algorithm based on fixed lag smoothing which can be used to estimate the rotor speed and flux of three phase induction motor. The behaviour of a typical extended Kalman filter (EKF) algorithm is influenced by the process and measurement error covariance matrices. In EKF, these matrices are chosen by trial and error method and have to be varied according to the varying operating conditions of the motor. In this paper a smoothing-based extended Kalman filter (SKF) algorithm which uses additional data points for estimation is proposed. Since additional data is used, the algorithm is able to give a better estimate, with the same values of process and measurement error covariance matrices. The performance of the proposed algorithm is explored for sensorless indirect vector control application of three-phase induction motor. Finally, in order to validate the superiority of this algorithm, it is compared with the performance of EKF algorithm for various reference speeds in real-time.

Harmonic estimation in power systems using an optimised adaptive Kalman filter based on PSO‐GA

A hybrid particle swarm optimisation (PSO) and genetic algorithm (GA) aided Kalman filter (PSO-GA-KF) method for dynamic harmonic state estimation in the power system is investigated. The initial choice of states and KF's parameters, such as process and measurement error covariance matrices significantly affects the estimation precision. Aiming at this problem, a combination of PSO and GA (PSO-GA) is proposed to optimise KF's parameters. In this hybrid algorithm, by replacing the gene mutation operator of GA, the PS algorithm mutation operator is constructed, which can adjust the evolution direction and range based on historical records and swarm records, leading to avoiding the blindness of GA mutation and reducing the probability of devious mutation and enhancing the velocity of evolution. Thus, the global search ability of the hybrid PSO-GA can ensure KF has more accurate estimation of harmonics. To test the effectiveness of the algorithm, several time-varying signals are simulated with harmonics and decaying DC components in the presence of amplitude drift and white noise. Simulation results show that the proposed hybrid algorithm has more accurate estimation, faster convergence and better robustness against noise in comparison with the conventional Kalman filter (KF), KF based on the maximum likelihood and PSO aided KF (PSO-KF).

ASENKRON MOTORUN İNDİRGENMİŞ-DERECELİ UYARLAMALI GENİŞLETİLMİŞ KALMAN FİLTRESİ İLE DURUM VE PARAMETRE KESTİRİMİ

In this study, the design of reduced-order adaptive extended Kalman filter (EKF) for speed-sensorless control of induction motors (IMs) is performed, and its performance is tested using it in a speed-sensorless direct vector controlled drive system under simulations. The proposed observer estimates the stator stationary axis components of rotor fluxes and rotor mechanical speed required for vector control in addition to disturbance load torque. On the other hand, estimation performance of EKFs depends on the correct selection of system ( ) and measurement ( ) error covariance matrices. In the literature, these matrices are generally assumed as constant and determined by the trial-and-error method. However, those matrices are affected by operating conditions of IM and should be updated according to operating conditions in order to obtain higher performance estimations. Since the simultaneously update of both and may lead to divergence or tracking problems, only is updated considering operating conditions by adaptive EKF (AEKF) algorithm having a forgetting factor. In addition, AEKF has been designed as reduced-order with the aim of reduction of its computational burden for real-time applications.

Adaptive Unscented Kalman Filter with Correntropy Loss for Robust State of Charge Estimation of Lithium-Ion Battery

As an effective computing technique, Kalman filter (KF) currently plays an important role in state of charge (SOC) estimation in battery management systems (BMS). However, the traditional KF with mean square error (MSE) loss faces some difficulties in handling the presence of non-Gaussian noise in the system. To ensure higher estimation accuracy under this condition, a robust SOC approach using correntropy unscented KF (CUKF) filter is proposed in this paper. The new approach was developed by replacing the MSE in traditional UKF with correntropy loss. As a robust estimation method, CUKF enables the estimate process to be achieved with stable and lower estimation error performance. To further improve the performance of CUKF, an adaptive update strategy of the process and measurement error covariance matrices was introduced into CUKF to design an adaptive CUKF (ACUKF). Experiment results showed that the proposed ACUKF-based SOC estimation method could achieve accurate estimate compared to CUKF, UKF, and adaptive UKF on real measurement data in the presence of non-Gaussian system noises.

Algorithm for Estimating the Parameters of Sea Pitching for Increasing the Accuracy of a Ship-Borne Radar Beam Positioning

The sea waves cause the effect of a rotational motion (i.e., pitch, roll and heading angle) of ships. The ship rotational motions result in beam pointing error of the phased array radar aboard the ship. The antenna stabilization aimed to achieve a beam pointing accuracy over a long dwell time is an important problem for the shipborne phased array radars. Due to the dynamic and stochastic nature of the sea environment, the shipborne phased array radar must be able to compensate for the ship's motion adaptively. In this paper, the linear discrete Kalman filter is proposed as a predictor for the shipborne phased array radar, which can compensate for the beam pointing error and track the air and sea surface targets. The pitch, roll and heading angles and the its velocities data are measured on-line from a gyroscope of the sea vehicle and used from the ship motion mathematical models for their prediction. It is proved, that the pitch, roll, and heading motion models may be considered as independent. All these models are presented as the same second order linear differential equations with different parameters. Besides, equivalent linear discrete state space models for the angles' changes are constructed in the paper. The estimation accuracy of the Kalman filter (predictor) depends on the values of different parameters, such as the parameters of the ship's motion mathematical model, measurement error covariance matrices, etc. The sea environments are very dynamic, hence, there is a need for an adaptive system for the controlling and compensating devices, operating regardless of the ship's motion. Continuous monitoring of the environment and adapting filter ensure parameters with less computational burden needed for a real time application. The paper describes a technique for identification of such parameters by the measured correlation functions of the pitch, roll, and heading angles. Finally, it is proved by the computer simulations that the proposed compensations technique is a real time applicable algorithm for a shipborne phased array radar. The computer simulation was performed with the following parameters: the measurement frequency for the gyroscopes was 100 Hz, the prediction times of the Kalman filter were 0.01 s and 0.1 s. The following two cases were considered. In the first case, only the predicted angles were measured with gyroscopes. In the second case, the angles and the rates of their change were measured. The simulations demonstrated that the presented prediction algorithm ensured higher accuracy (less root mean square errors of the predicted angles) than the initial accuracy of the gyroscopes measurements.

Detection of Voltage Sag using An Adaptive Extended Kalman Filter Based on Maximum Likelihood

An adaptive extended Kalman filter based on the maximum likelihood (EKF-ML) is proposed for detecting voltage sag in this paper. Considering that the choice of the process and measurement error covariance matrices affects seriously the performance of the extended Kalman filter (EKF), the EKF-ML method uses the maximum likelihood method to adaptively optimize the error covariance matrices and the initial conditions. This can ensure that the EKF has better accuracy and faster convergence for estimating the voltage amplitude (states). Moreover, without more complexity, the EKF-ML algorithm is almost as simple as the conventional EKF, but it has better anti-disturbance performance and more accuracy in detection of the voltage sag. More importantly, the EKF-ML algorithm is capable of accurately estimating the noise parameters and is robust against various noise levels. Simulation results show that the proposed method performs with a fast dynamic and tracking response, when voltage signals contain harmonics or a pulse and are jointly embedded in an unknown measurement noise.

Retrieval of ash properties from IASI measurements

Abstract. A new optimal estimation algorithm for the retrieval of volcanic ash properties has been developed for use with the Infrared Atmospheric Sounding Interferometer (IASI). The retrieval method uses the wave number range 680–1200 cm−1, which contains window channels, the CO2 ν2 band (used for the height retrieval), and the O3 ν3 band.Assuming a single infinitely (geometrically) thin ash plume and combining this with the output from the radiative transfer model RTTOV, the retrieval algorithm produces the most probable values for the ash optical depth (AOD), particle effective radius, plume top height, and effective radiating temperature. A comprehensive uncertainty budget is obtained for each pixel. Improvements to the algorithm through the use of different measurement error covariance matrices are explored, comparing the results from a sensitivity study of the retrieval process using covariance matrices trained on either clear-sky or cloudy scenes. The result showed that, due to the smaller variance contained within it, the clear-sky covariance matrix is preferable. However, if the retrieval fails to pass the quality control tests, the cloudy covariance matrix is implemented.The retrieval algorithm is applied to scenes from the Eyjafjallajökull eruption in 2010, and the retrieved parameters are compared to ancillary data sources. The ash optical depth gives a root mean square error (RMSE) difference of 0.46 when compared to retrievals from the MODerate-resolution Imaging Spectroradiometer (MODIS) instrument for all pixels and an improved RMSE of 0.2 for low optical depths (AOD < 0.1). Measurements from the Facility for Airborne Atmospheric Measurements (FAAM) and Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR) flight campaigns are used to verify the retrieved particle effective radius, with the retrieved distribution of sizes for the scene showing excellent consistency. Further, the plume top altitudes are compared to derived cloud-top altitudes from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument and show agreement with RMSE values of less than 1 km.

Six Object Tracking Algorithms: A Comparative Study

Objectives: To compare five different objects tracking algorithms performance wise with the proposed algorithm and to find out the best one among them for tracking of an object in occlusion and Background clutter condition i.e. when background contains target features. Methods/Analysis: In the proposed algorithm, the object’s position is obtained by Mean Shift tracking and then the prediction of object’s position is done through Kalman Filter. The Proposed method has Kalman Filter which consists of an adaptive system matrix and adaptive process error covariance and measurement error covariance matrices respectively. Adaptive system matrix of Kalman Filter is getting updated online depending on the quality of observation by Mean Shift algorithm and adaptive process and measurement noise covariance matrices are getting updated according to the variation in Bhattacharya Coefficient respectively. Findings: Proposed algorithm has a maximum value (0.2177 and 0.4821) of tracking efficiency metric i.e. Bhattacharya Coefficient in both video dataset among all the algorithms and it is taking less execution time (0.002761 sec and 0.005431 sec) than other Kalman filter based tracking algorithms. Applications/Improvements: Proposed algorithm is able to track the target in occlusion and background variation condition with more efficiency and less execution time than rest of the algorithms.

Accurate tracking of harmonic signals in VSC-HVDC systems using PSO based unscented transformation

This paper presents the estimation of harmonics in a voltage source converter based HVDC (VSC-HVDC) system for designing AC side filters. The extended Kalman filter (EKF) is well known for estimating amplitude, phase, frequency, and harmonic content of a signal corrupted with noise. However, the EKF algorithm suffers from instability due to linearization and costly calculation of Jacobian matrices, and its performance deteriorates when the signal model is highly nonlinear. This paper, therefore, proposes an unscented Kalman filter (UKF) to overcome these difficulties of linearization and derivative calculations for robust tracking of harmonics in VSC-HVDC system. The model and measurement error covariance matrices Q and R along with the UKF parameters are selected using a modified particle swarm optimization (PSO) algorithm. To circumvent the problem of premature convergence and local minima, a dynamically varying inertia weight based on the variance of the population fitness is used. This results in a better local and global searching ability of the particles, which improves the convergence of the velocity and better accuracy of the UKF parameters. Various simulation results for harmonic signals corrupted with noise obtained from VSC-HVDC system reveal significant improvement in noise rejection and speed of convergence and accuracy.

Fast Tracking of Power Quality Disturbance Signals Using an Optimized Unscented Filter

This paper presents a hybrid approach for tracking the amplitude, phase, frequency, and harmonic content of power quality disturbance signals occurring in power networks using an unscented Kalman filter (UKF) and swarm intelligence. The UKF is a novel extension of the well-known extended Kalman filter (EKF) using an unscented transformation to overcome the difficulties of linearization and derivative calculations of signals with a low signal-to-noise ratio (SNR). Further, the model and measurement error covariance matrices Q and R, along with the UKF parameters, are selected using a modified particle swarm optimization (PSO) algorithm for accurate tracking of signal parameters. To circumvent the problem of premature convergence and local minima in conventional PSO, a dynamically varying inertia weight based on the variance of the population fitness is used. This results in a better local and global searching ability of the particles, which improves the convergence of the velocity, and in a better accuracy of the UKF parameters. Various simulation results for nonstationary sinusoidal signals occurring in power networks with varying amplitudes, phases, and harmonic contents corrupted with noise having a low SNR reveal significant improvements in noise rejection and speed of convergence and accuracy.

Chemometrics description of measurement error structure: Study of an ultrafast absorption spectroscopy experiment

The error structure inherent to a transient absorption spectroscopy kinetic experiment is described through the examination of the related measurement error covariance matrices in the time and the spectral direction. The systematic approach proposed by Wentzell's group to characterize the measurement error covariance matrices is used for this purpose. The goals of this study are identifying and modelling the factors contributing to the error measurement and proposing a weighting error-based scheme that can be generally used when analysing data coming from this experimental setup. From the analysis performed, independent noise structure is associated with the time direction of the data sets, whereas the spectral direction is affected by two sources of correlated errors, which can be related to the probe spectrum and to the mean signal registered.

Methods for systematic investigation of measurement error covariance matrices

Several recently developed methods for multivariate data analysis allow the incorporation of prior information about the measurement error structure into the analysis to permit better model estimation and prediction. This error structure is described in the form of the measurement error covariance matrix, which defines the complex relationship between measurement uncertainty at one channel and those at other channels and/or for other samples. In this work, a systematic approach for characterizing the measurement error covariance matrix for a particular experimental or instrumental environment is presented. This approach involves a number of strategies that include visualization of covariance and correlation matrices, bilinear modelling through principal components analysis (PCA) and target-testing factor analysis, trilinear modelling through PARAFAC, and refinement of models to include interaction terms and independent errors. The primary goals of this characterization are to obtain a better understanding of the factors contributing to measurement error and to develop parametric models for error covariance that do not rely on extensive replication. To illustrate this approach, four experimental data sets are employed: (1) UV-visible absorbance data, (2) near-infrared (NIR) reflectance data, (3) fluorescence emission data, and (4) short-wave NIR (SW-NIR) absorbance data from a kinetics experiment. For both the UV-visible and SW-NIR spectra, the main contribution to the error structure is a constant offset term that appears to have a dependence on the reciprocal of the wavelength. The NIR reflectance spectra are dominated by constant and multiplicative offset noise that have a strong interaction. The fluorescence data is affected by independent shot-noise with a variance proportional to the magnitude of the spectrum, as well as by correlated offset noise with two largely independent terms, one which is fixed in magnitude and the other which depends on the square root of the spectrum. It was also found that the SW-NIR data exhibits a strong correlation among samples.

Academic Insights: An Application of Multiple-Group Causal Models in Assessing Cross-Cultural Measurement Equivalence

In cross-national marketing research, the question whether measurement accuracy, reliability, and validity are achieved across samples traditionally has hampered research efforts. Measurement equivalence across samples has particularly perplexed researchers in their efforts to evaluate responses regarding latent variables. Recently, multiple-group structural equations modeling has been suggested as a reliable method for determining measurement equivalence. To date, however, the use of this approach has remained limited. In this study, the authors apply multiple-group structural equations modeling to assess measurement equivalence in three distinct constructs across U.S. and Korean samples on data derived from cross-cultural advertising research. The authors propose an extension of this method using the measurement error covariance matrices, which will enable researchers to evaluate measurement reliability across samples better and to attempt to disentangle cultural differences in instrument usage from measurement-related differences. The authors evaluate the results and discuss the findings, as well as outline the implications and limitations of this method for further research.

Errors in Variables in Linear Systems

This paper extends the simple errors-in-variable bound to the setting of systems of equations. Both diagonal and nondiagonal measurement error covariance matrices are considered. In the nondiagonal case, the analogue of the simple errors-in-variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two-stage least squares estimate. For the diagonal case, the set of estimates under some conditions must lie within the convex hull of 2k points.