Measurement Error Distribution Research Articles

Clustering Functional Data With Measurement Errors: A Simulation-Based Approach.

Clustering analysis of functional data, which comprises observations that evolve continuously over time or space, has gained increasing attention across various scientific disciplines. Practical applications often involve functional data that are contaminated with measurement errors arising from imprecise instruments, sampling errors, or other sources. These errors can significantly distort the inherent data structure, resulting in erroneous clustering outcomes. In this article, we propose a simulation-based approach designed to mitigate the impact of measurement errors. Our proposed method estimates the distribution of functional measurement errors through repeated measurements. Subsequently, the clustering algorithm is applied to simulated data generated from the conditional distribution of the unobserved true functional data given the observed contaminated functional data, accounting for the adjustments made to rectify measurement errors. We illustrate through simulations show that the proposed method has improved numerical performance than the naive methods that neglect such errors. Our proposed method was applied to a childhood obesity study, giving more reliable clustering results.

Tsunami Data Assimilation Using High‐Frequency Radar‐Derived Surface Currents by Considering Beam Angle‐Dependent Measurement Error Distributions

AbstractThe application of high‐frequency radar as an instrument for assimilating tsunami‐induced current fields is garnering increasing interest. The performance of surface current velocity measurements depends on the azimuthal differences between the crossing radar beams at the measurement points. This study aimed to incorporate the measurement error distributions of the east‐west and north‐south velocity components into tsunami data assimilation based on an optimal interpolation method, assuming Gaussian noise with the time‐invariant and a uniform standard deviation (STD = 5 cm/s) of radial velocity measurements. Through the empirical orthogonal function (EOF) analysis of radar‐derived surface currents in the Kii Channel, Japan, the velocities reconstructed using higher modes (EOFs 16–274) were associated with measurement errors, portraying nonuniform distribution depending on the crossing beam angle of two radar beams. Based on independent fifteen‐time assimilation experiments for two different tsunami scenarios, for a uniform water depth of 500 m, we observed a significant improvement of up to 29% and 0.9% in the assimilation performance (on average) over the along‐coast stations for scenarios with 1‐ and 5‐m maximum initial sea surface heights, respectively. The measurement errors dependent on the crossing beam angle reduced the error‐induced tsunamis, resulting in stable assimilations, with lower STDs in the fifteen‐time assimilation performances. When the STD of Gaussian noise varies with time, it is important to consider the temporal change in the radial velocity measurement errors and/or noise‐filtering techniques, to maintain a certain level of noise intensity.

Optimal designs of accelerated degradation tests with random shock failures and measurement errors

AbstractAccelerated degradation tests (ADTs) are widely used for assessing the reliability of long‐life products. During an ADT, accelerated stresses not only expedite the degradation of test products but also increase the likelihood of encountering traumatic shocks. Moreover, it is important to acknowledge that measurement errors can be inevitable during the observation process of an ADT. Unfortunately, these errors are often overlooked in the optimal design of the ADT, especially when multiple competing failure modes are present. In this article, we propose a new approach to design ADTs when measurement errors exist and test products suffer from degradation failures and random shock failures. We utilize the Wiener process to model the degradation path, incorporating normally distributed measurement errors, and an exponential distribution to fit the time between random shock failures. Given the number of test products and the termination time, we optimize the ADT plans under three common design criteria. The equivalence theorem is used to verify the optimality of the optimal ADT plans. A real‐life example and sensitivity analysis are provided to illustrate our proposed method. The results demonstrate that when competing failure modes are present, the optimal ADT plans, which account for measurement errors, differ significantly from those that do not consider such errors.

Using artificial intelligence methods to recognize radar stations based on the radio technical parameters of signals with filtering signals from radar stations not taken into account in the training set

Currently, the tasks of radio-electronic surveillance using spacecraft are the most relevant in ensuring the defense capability of the Russian Federation. In the course of solving the problem of radio-electronic surveillance, the issue of classifying the detected signals. Classical classification methods, based on comparing the obtained data on detected stations with reference a priori-known station parameters, do not allow obtaining results with a high degree of reliability, so to solve this issue we turned to machine learning technologies. This paper describes the solution to the problem of recognizing radar stations by the radio technical parameters of signals received during radio-electronic monitoring (SEM) of the Earth, using machine learning methods, taking into account the possibility of signals from unknown radars appearing in the test sample. In the process of work, an analysis of existing machine learning methods for processing SEM information was carried out using a developed mathematical model, implemented in program code, using simulated test data. In order to probabilistically evaluate methods for recognizing types of radar stations (radars) as part of the analysis of SEM information in a complex signal-interference environment, modeling of the initial data, consisting of a set of pairs “classification object - class label” was carried out. The radar type number was chosen as the class label, and the radar signal, which can be described using three characteristics: frequency, period, and signal pulse duration, was chosen as the classification object. The mathematical model for generating the initial data includes the ability to add a normally distributed measurement error depending on the specified standard deviation (RMSD) for each of the characteristics. In addition, the possibility of using various machine learning methods to search for outliers in the test sample was considered, and a probabilistic assessment of the accuracy of their work was also carried out. As a result of the research, based on the results obtained, the two most preferred methods of radar recognition were selected, as well as the most promising method of filtering signals from unknown radars. The use of the developed algorithms makes it possible to significantly speed up the recognition of radars based on the radio technical parameters of the signal with a high degree of recognition reliability.

Error Analysis of Normal Surface Measurements Based on Multiple Laser Displacement Sensors.

The robotic drilling of assembly holes is a crucial process in aerospace manufacturing, in which measuring the normal of the workpiece surface is a key step to guide the robot to the correct pose and guarantee the perpendicularity of the hole axis. Multiple laser displacement sensors can be used to satisfy the portable and in-site measurement requirements, but there is still a lack of accurate analysis and layout design. In this paper, a simplified parametric method is proposed for multi-sensor normal measurement devices with a symmetrical layout, using three parameters: the sensor number, the laser beam slant angle, and the laser spot distribution radius. A normal measurement error distribution simulation method considering the random sensor errors is proposed. The measurement error distribution laws at different sensor numbers, the laser beam slant angle, and the laser spot distribution radius are revealed as a pyramid-like region. The influential factors on normal measurement accuracy, such as sensor accuracy, quantity and installation position, are analyzed by a simulation and verified experimentally on a five-axis precision machine tool. The results show that increasing the laser beam slant angle and laser spot distribution radius significantly reduces the normal measurement errors. With the laser beam slant angle ≥15° and the laser spot distribution radius ≥19 mm, the normal measurement error falls below 0.05°, ensuring normal accuracy in robotic drilling.

Reliability between the two-colour chewing gum and the gummy-jelly tests used for the assessment of masticatory performance.

This study aimed to evaluate the reliability of two methods used to assess masticatory performance and attempt to correlate them to achieve interchangeability between the methods. Twelve healthy dentate volunteers (men = 6, women = 6; mean age = 28.3 ± 4.1) with no known dental or medical pathologies were requested to participate in this study. Each participant completed three masticatory performance assessments, including two two-colour mixing-ability tests using chewing-gums(CG: gum#1 and gum#2) and the gummy-jelly (GJ) test. For each method, participants created five samples each (total = 15 measurements per participant, gum#1 = 5, gum#2 = 5, GJ = 5). For the gum#1 and gum#2 methods, the predetermined chewing cycles were fixed at 10, 15, 20, 25 and 30 cycles, and for the GJ method, the time duration was fixed at 10, 15, 20, 25 and 30 s. The parameter measures were submitted to Z-score transformation, and Bland-Altman plots were generated to graphically compare the differences between two techniques against their means. Additionally, mountain plot was used to assess the cumulative distribution of measurement error between the methods. A total of 180 measurements were recorded. There were significant correlations between the number of chewing cycles/chewing time and masticatory performance using the gum#1 (r = -.753; p < .001), gum#2 (r = -.838; p < .001) and GJ (r = .730). When all tests were considered together for each method, significant correlations were found (p < .001). A descriptive range of mean values aiming to produce reference value ranges for predictive purposes was achieved considering the interchangeably among the methods [CG = GJ (VoH-mg = dL): 10 cycle = 10 s: 0.329 = 110; 15 cycles = 15 s: 0.177 = 164; 20 cycles = 20 s: 0.130 = 205; 25 cycles = 25 s: 0.086 = 200; 30 cycles = 30 s: 0.077 = 267]. The strong correlations and high consistency between the two masticatory performance methods found in this study conclude that the two assessment methods are reliable and interchangeable. Further evaluations are warranted to arrive at a conversion formula for translation of the results between the two methods.

Bias Analysis of PMU-Based State Estimation and Its Linear Bayesian Improvement

With the rapidly changing states of modern power systems, real-time state estimation based on phasor measurement units (PMUs) plays an increasingly important role in energy management systems (EMSs). However, due to the complicated distribution of PMU measurement errors, conventional state estimation methods such as the weighted least squares (WLS) estimator suffer from the estimation bias. Consequently, they do not represent the least variance unbiased estimators. To solve the estimation bias, a state estimation method combining the PMU linear measurement model and the linear Bayesian estimation theory is proposed. Specifically, considering the prior information on estimated parameters and the complicated probability distribution of PMU measurement errors, the linear Bayesian theory is applied to PMU-based state estimation to improve the performance of state estimation. To support this method, the correlation between real and imaginary errors is analyzed, and the prior information of estimated parameters is determined by the volatility of system states. When the real states of practical power systems are unavailable, performance indicators for state estimation are proposed based on the estimation residuals. The efficacies of the proposed estimation and evaluation methods are verified using IEEE 14 bus system and a large-scale power grid in China.

Constructions of statistical estimates of digital measurement error in the case of small samples

The paper presents a computational-experimental method for the construction of statistical estimates of the error of digital measurements carried out on complex technical objects with metrological support. The algorithm of construction of the estimation of the distribution density function of random error of measurements on the basis of complex application of methods of the theory of characteristic functions, the theory of Lie operator series and methods of statistical modeling in the processing of a limited amount of statistical information in the conditions of small samples is described. The solution of the practical problem on the construction of the estimation of the measurement error distribution density and the problem on the zero mark of the measuring instrument in the probabilistic formulation is given. The presented computational and experimental method can be used to construct statistical estimates for the probability distribution density of the general type, including those given by implicit functions, characteristic functions and operator series.

A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system’s true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.

A multi-sensor data fusion algorithm based on consistency preprocessing and adaptive weighting

ABSTRACT In the data collection of a multi-sensor system, there are problems with large errors, conflicts, and redundancy. To solve the above problem, a multi-sensor data fusion algorithm based on anomaly data preprocessing and adaptive weighted estimation is proposed. To improve the reliability of the algorithm, first, for a single sensor measurement signal sequence, a consistency preprocessing using the off-centre distance method is performed, and the weighting factor of each measurement data is calculated. Then, the measurement signal sequence is weighted and fused; Secondly, in response to the uneven distribution of measurement errors among multiple sensors in different directions, an adaptive weighted data fusion method based on the principle of optimal weight allocation is proposed. The proposed method was compared with the adaptive weighting method and arithmetic mean method. The simulation results showed that the total mean square error of the data fusion results obtained using the proposed algorithm is smaller. The proposed algorithm can effectively improve the accuracy of data measurement, reduce redundancy, and improve the stability of data measurement.

Estimation of a clustering model for non Gaussian functional data

Model-based clustering analysis of functional data often has normality assumption. This article considers clustering non Gaussian functional data. We propose a novel non Gaussian functional mixed-effects model without the prior information and clustering number. We use transformation functions to accommodate non Gaussian functional data. Smoothing spline ANOVA and cubic B-spline approximate unknown fixed effects and random effects, respectively. A penalized likelihood is used to estimate unknown parameters, and the consistency and asymptotic normality is provided after that. We take simulations for different measurement error distribution assumptions and adopt the air quality of Italian city data. Both simulation and actual data analysis show that the proposed method performs well and has a better clustering effect.

The mean residual life model for the right-censored data in the presence of covariate measurement errors.

In this article, we consider the mean residual life regression model in the presence of covariate measurement errors. In the whole cohort, the surrogate variable of the error-prone covariate is available for each subject, while the instrumental variable (IV), which is related to the underlying true covariates, is measured only for some subjects, the calibration sample. Without specifying distributions of measurement errors but assuming that the IV is missing at random, we develop two estimation methods, the IV calibration and cohort estimators, for the regression parameters by solving estimation equations (EEs) based on the calibration sample and cohort sample, respectively. To improve estimation efficiency, a synthetic estimator is derived by applying the generalized method of moment for all EEs. The large sample properties of the proposed estimators are established and their finite sample performance are evaluated via simulation studies. Simulation results show that the cohort and synthetic estimators outperform the IV calibration estimator and the relative efficiency of the cohort and synthetic estimators mainly depends on the missing rate of IV. In the case of low missing rate, the synthetic estimator is more efficient than the cohort estimator, while the result can be reversed when the missing rate is high. We illustrate the proposed method by application to data from the patients with stage 5 chronic kidney disease in Taiwan.

Data-Driven Event-Triggered Bipartite Consensus for Multi-Agent Systems Preventing DoS Attacks

This paper considers event-triggered bipartite consensus issues for discrete-time nonlinear networked multi-agent systems with antagonistic interactions and denial-of-service (DoS) attacks. Firstly, a pseudo partial derivative technology is applied to obtain an equivalent dynamic linearization model of the controlled system. The signed graph theory is employed to analyze the coopetition relationships among agents. Next, a distributed combined measurement error function is formulated to transform the bipartite consensus issue into a consensus issue. Then, an output predictive compensation scheme is proposed to offset the influence of DoS attacks. Furthermore, a dead-zone operator is designed to improve the flexibility of the proposed event-triggered mechanism. Additionally, a data-driven event-triggered resilient bipartite consensus scheme is formulated. Then, the convergence of the proposed method is strictly proved by using the Lyapunov theory and the contraction mapping principle, which indicates that the bipartite consensus error could be cut to a small region around zero. Finally, hardware tasks are conducted to verify the effectiveness of the proposed method.

Emitter Location Using Frequency Difference of Arrival Measurements Only.

It is desirable to enable emitter location using frequency difference of arrival (FDoA) measurements only, since many signals are characterized by coarse range resolution and fine Doppler resolution. For instance, while using the cross-ambiguity function (CAF) to measure the time difference of arrival (TDoA) and the FDoA of a narrowband signal, it is difficult to obtain accurate TDoA measurements because the Doppler resolution is higher than the range resolution. Grid-based and sample-based algorithms are developed to solve the two-dimensional (2D) emitter location problem, where the solution space is approximated, respectively, by generating deterministic and random emitter location candidates. Simulation results corroborate the viability of both non-iterative algorithms to estimate the emitter location using a single-time snapshot of FDoA measurements only, without any prior location information or any knowledge about the distribution of measurement errors. The achieved accuracies are sufficient for early warning purposes, preparing defenses, and cueing more accurate location sensors by directing additional surveillance resources.

Using shock tube species time-histories in Bayesian parameter estimation: Effective independent-data number and target selection

Species time-histories in shock tube experiments provide rich kinetic information for parameter estimation, but there are two problems in using these data in Bayesian approaches. First, the effective independent-data number is not equal to the number of data points in a curve, so brute multiplication of all data points in likelihood function can weaken the constraints from prior information. Second, taking all points of a curve as targets can lead to results different from that of taking several representative points in the curve. In this paper, we employed maximum a posteriori estimation combined with a neural network response surface to optimize a propane mechanism against multispecies time-histories of propane pyrolysis in a shock tube. Three methods of calculating the likelihood function are used: multiplying all points in a curve (C-160), taking the averaged likelihood in each point (C-1), and taking the likelihood of last points (LastP). The influence of effective independent-data number was studied by comparing C-1 and C-160. It was found that C-160 performed slightly better in fitting experimental data, but brute multiplication overtuned the rate constants beyond a reasonable range. The larger the effective independent-data number, the more severe the overtuning, leading to only a slight improvement of model predictions. The influence of target selection is investigated by comparing LastP and C-1. LastP outperformed C-1 slightly, which can be attributed to the fact that larger discrepancies observed between experimental data and model predictions of the last point can increase the weights of likelihood functions. This further implies that several critical points can represent the entire line for point estimation. This paper can provide a reference both for modelers about reasonable utilization of species time-histories, and for experimentalists about the importance of a detailed probability distribution of measurement error, as well as experiment design with emphasis on critical points.

Nonparametric regression on Lie groups with measurement errors

This paper develops a foundation of methodology and theory for nonparametric regression with Lie group-valued predictors contaminated by measurement errors. Our methodology and theory are based on harmonic analysis on Lie groups, which is largely unknown in statistics. We establish a novel deconvolution regression estimator, and study its rate of convergence and asymptotic distribution. We also provide asymptotic confidence intervals based on the asymptotic distribution of the estimator and on the empirical likelihood technique. Several theoretical properties are also studied for a deconvolution density estimator, which is necessary to construct our regression estimator. The case of unknown measurement error distribution is also covered. We present practical details on implementation as well as the results of simulation studies for several Lie groups. A real data example is also provided.

Behavior and mechanisms of Doppler wind lidar error in varying stability regimes

Abstract. Wind lidars are widespread and important tools in atmospheric observations. An intrinsic part of lidar measurement error is due to atmospheric variability in the remote-sensing scan volume. This study describes and quantifies the distribution of measurement error due to turbulence in varying atmospheric stability. While the lidar error model is general, we demonstrate the approach using large ensembles of virtual WindCube V2 lidar performing a profiling Doppler-beam-swinging scan in quasi-stationary large-eddy simulations (LESs) of convective and stable boundary layers. Error trends vary with the stability regime, time averaging of results, and observation height. A systematic analysis of the observation error explains dominant mechanisms and supports the findings of the empirical results. Treating the error under a random variable framework allows for informed predictions about the effect of different configurations or conditions on lidar performance. Convective conditions are most prone to large errors (up to 1.5 m s−1 in 1 Hz wind speed in strong convection), driven by the large vertical velocity variances in convective conditions and the high elevation angle of the scanning beams (62∘). Range-gate weighting induces a negative bias into the horizontal wind speeds near the surface shear layer (−0.2 m s−1 in the stable test case). Errors in the horizontal wind speed and direction computed from the wind components are sensitive to the background wind speed but have negligible dependence on the relative orientation of the instrument. Especially during low winds and in the presence of large errors in the horizontal velocity estimates, the reported wind speed is subject to a systematic positive bias (up to 0.4 m s−1 in 1 Hz measurements in strong convection). Vector time-averaged measurements can improve the behavior of the error distributions (reducing the 10 min wind speed error standard deviation to &lt;0.3 m s−1 and the bias to &lt;0.1 m s−1 in strong convection) with a predictable effectiveness related to the number of decorrelated samples in the time window. Hybrid schemes weighting the 10 min scalar- and vector-averaged lidar measurements are shown to be effective at reducing the wind speed biases compared to cup measurements in most of the simulated conditions, with time averages longer than 10 min recommended for best use in some unstable conditions. The approach in decomposing the error mechanisms with the help of the LES flow field could be extended to more complex measurement scenarios and scans.

Disentangling the Influence of Data Contamination in Growth Curve Modeling: A Median Based Bayesian Approach

Growth curve models (GCMs), with their ability to directly investigate within-subject change over time and between-subject differences in change for longitudinal data, are widely used in social and behavioral sciences. While GCMs are typically studied with the normal distribution assumption, empirical data often violate the normality assumption in applications. Failure to account for the deviation from normality in data distribution may lead to unreliable model estimation and misleading statistical inferences. A robust GCM based on conditional medians was recently proposed and outperformed traditional growth curve modeling when outliers are present resulting in nonnormality. However, this robust approach was shown to perform less satisfactorily when leverage observations existed. In this work, we propose a robust double medians growth curve modeling approach (DOME GCM) to thoroughly disentangle the influence of data contamination on model estimation and inferences, where two conditional medians are employed for the distributions of the within-subject measurement errors and of random effects, respectively. Model estimation and inferences are conducted in the Bayesian framework, and Laplace distributions are used to convert the optimization problem of median estimation into a problem of obtaining the maximum likelihood estimator for a transformed model. A Monte Carlo simulation study has been conducted to evaluate the numerical performance of the proposed approach, and showed that the proposed approach yields more accurate and efficient parameter estimates when data contain outliers or leverage observations. The application of the developed robust approach is illustrated using a real dataset from the Virginia Cognitive Aging Project to study the change of memory ability.

Layout Design of Strapdown Array Seeker and Extraction Method of Guidance Information

This paper proposed a multi-view surface array to enlarge the field-of-view (FOV) from 45° × 45° to 72° × 75° and improve the estimation precision of guidance information. First, based on circular and rectangular FOV sensors, the superposition characteristics of FOV in the +-shaped layout and the X-shaped layout were explored and analyzed. Secondly, normalization processing was applied to obtain the equivalent measurement value of the central sensor and the corresponding error distribution. Based on such measurement value and error distribution, the filtering model could be constructed, which could effectively solve the problem of observation number variation caused by multiple sensors. Thirdly, a multivariate iterated extended Kalman filter (MIEKF) was proposed to make full use of multiple measurements. By iterating multiple unequal observations and making full use of the known error distribution information, the noise of filtered data was found to be effectively reduced and the estimation precision of guidance information was improved. Finally, based on a 6-DOF trajectory simulation, the correctness and effectiveness of the proposed method were verified. Simulation results show that MIEKF can improve the estimation accuracy of the line-of-sight (LOS) angle by at least 30% and the estimation accuracy of the LOS angle rate by nearly 80% compared with EKF.

Непараметрические критерии согласия при проверке нормальности в условиях округления измерений

When analyzing measurement series in various applications, the verification of whether measurement errors belong to the normal law is considered as a mandatory procedure. For this purpose, various special tests for testing hypotheses about normality can be used; non-parametric tests of goodness or chi-square tests can be used. When using nonparametric goodness-of-fit tests to test normality, it must be taken into account that a complex hypothesis is being tested. When testing a complex hypothesis, the distributions of the statistics of the goodness-of-fit tests differ significantly from the classical ones that occur when testing simple hypotheses. It is known that the presence of rounding errors can significantly change the distribution of test statistics. In such situations, ignoring the fact of influence can lead to incorrect conclusions about the results of the normality test. In metrology, when carrying out high-precision measurements, as a rule, scientists do not even allow thoughts about the possible influence of D rounding errors on the results of statistical analysis. This allows the possibility of incorrect conclusions since there is no influence not only at small D, but at values of D much less than the standard deviation s of the measurement error distribution law and sample sizes n not exceeding some maximum values. For sample sizes larger than these maximum values, the real distributions of the test statistics deviate from the asymptotic ones towards larger statistics values. In this work, based on real and well-known data, using statistical modeling methods, we demonstrate the dependence of the distributions of statistics of nonparametric goodness-of-fit tests when testing normality on the ratio of D and s for specific n. The possibility of correct application of the tests under the influence of rounding errors on the conclusions is shown and implemented.