Estimation Error Variance Research Articles

Bayesian Updating and Model Class Selection for Magnitude Estimation in Earthquake Early Warning: Application to Earthquake Sequences in Sichuan Region, China

ABSTRACT A key component of earthquake early warning system is the development of an accurate and robust predictive model relating the recorded waveform to seismicity and ground-shaking characteristics. This article presents a probabilistic Bayesian inference methodology to address relevant sources of uncertainty in the development of such predictive models. We investigate the calibration of models for the earthquake magnitude based on the maximum predominant period, considering the Sichuan region of Southwestern China as a case study. Established approaches for developing predictive models in this context adopt deterministic tools for some aspects of this development. They consider a linear regression calibrated typically through least squares optimization and frequently utilize the mean observations for each event averaging across the data available from different stations. The proposed Bayesian learning accommodates the following improvements: a model class selection is established, comparing across different candidate models to promote the most appropriate from accuracy and robustness perspectives; the full posterior distribution of the model parameters is identified, quantifying relevant uncertainties in their values; a heteroscedastic model is considered for the estimation error variance; and the observations are separately considered at the calibration stage. Each of these improvements ultimately addresses a different source of uncertainty impacting the predictive model development. We utilize transitional Markov chain Monte Carlo for obtaining samples from the posterior and for calculating the evidence to perform the model class selection. Different regression models are examined, and the Bayesian-based model identification is compared against the common least squares identification approach. Results show the value added by comparing across the different models and by considering a heteroskedastic variance model, offering insights into the advantages of Bayesian-based predictive models in earthquake early warning applications.

A risk-averse learning machine via variance-dependent penalization

Based on Hoeffding’s inequality, many popular regression and classification models in supervised learning relax the expected risk minimization problem to the empirical risk minimization problem. Nevertheless, the recent theoretical results disclose that the bound of Bernstein’s inequality - which includes variance information - is often significantly tighter than the bound of Hoeffding’s inequality. In this paper, based on the empirical Bernstein bound, we proposed a risk-averse learning machine, which can achieve better generalization performance by trading off good loss performance (approximation error) and small variance (estimation error) as well as suitable complexity of the model. We prove that the resulting learning machine is tractable for many popular loss functions. Moreover, to solve our model which is mostly non-convex because of the square root term, we introduce an extra variable to get rid of the square root and obtain an optimization problem that is convex in most cases for two variables respectively. Then Newton’s method can be used to solve it alternately. The experimental results on artificial and benchmark datasets demonstrate that the proposed models can achieve better performance compared to other existing empirical risk minimization models.

Physical Layer Authentication in Spatial Modulation

Spatial Modulation (SM) is a promising low-complexity modulation scheme for Multiple-Input Multiple-Output (MIMO) systems. In this paper, we address the problem of authenticating the transmitter device in the SM. We propose an authentication approach for an SM system by using Physical-Layer Authentication (PLA) mechanisms because the PLA has the following advantages: high security and low complexity. Based on the features of an SM system, we propose two PLA schemes: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PLA with Superimposed Authentication Tag</i> (PLA-SAT) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PLA with Superimposed Imaginary authentication Tag</i> (PLA-SIT). We provide performance analyses of our schemes over fading channels in terms of robustness, compatibility, and security. Moreover, we derive their closed-form expressions under both perfect and imperfect channel estimates, including the Probability of Detection (PD), Probability of False Alarm (PFA), and Average Error Probability (AEP). Although the two proposed schemes have the same robustness and security, the PLA-SIT scheme has better compatibility than the PLA-SAT scheme. Our schemes were implemented and extensive performance comparisons through simulations were conducted. We observe that the simulation results of the two proposed schemes perfectly match their corresponding theoretical analyses. The authentication accuracy of the two proposed schemes is close to one when the received SNR is greater than 20 dB and the security performances of the two proposed schemes improve as the variance of estimation errors increases.

Application of Rasch Analysis to Timed Instrumental Activities of Daily Living in Low Vision: A Validation Study.

This work validates Rasch analysis of a performance-based low vision outcome measure evaluated in patients' own homes to ensure real-world relevance. Inclusion of sources of variance from the patient's home environment in functional outcome measures introduced nonuniform variance in measurements but did not preclude estimation of valid measures. This study aimed to validate Rasch analysis of a performance-based outcome measure with real-world relevance. Low vision patients (N = 161) receiving services from an occupational therapist performed Timed Instrumental Activity of Daily Living (TIADL) tasks in their homes. Rasch analysis was applied to error count and performance time data. Internal validity was assessed with evaluations of the accuracy and precision of estimated measures. External validity was assessed by comparing TIADL measures with measures estimated from the Activity Inventory (i.e., from self-reported difficulty ratings). Task measures were well targeted to person measures estimated from task performance time but were poorly targeted for measures estimated from task performance errors, for which most task trials (72%) were performed without error at baseline. Error-based person measures had larger standard errors with a smaller pseudo- R2 than time-based person or task measures and error-based task measures. Person measure infits for time- and error-based estimates conformed to expected values. The linear regressions between time-based person and task measures and corresponding error-based estimates had slopes of approximately 0.5, an observation consistent with larger estimation error variance for error-based measures than for time-based measures. Time-based TIADL person measures ( x ) and Activity Inventory person measures (estimated from all items, y ) were colinear but weakly correlated ( R = 0.19). Functional ability measures estimated from performance times of instrumental activity of daily living tasks in patients' homes demonstrate good internal and external validity. The ceiling effect from the infrequency of task performance errors in our data set limits use of TIADL error data to measure rehabilitation outcomes.

Robust weighted fusion Kalman estimators for systems with uncertain noise variances, multiplicative noises, missing measurements, packets dropouts and two‐step random measurement delays

AbstractFor multisensor networked systems with uncertain noise variances, multiplicative noises and multiple networked‐induced uncertainties including missing measurements, packets dropouts and two‐step random measurement delays, the robust weighted fusion estimation problem is addressed in this article. More precisely, the system noise variances are assumed to be uncertain but bounded, the other four uncertainties are compensated into fictitious white noise by the proposed model transformation method, which includes the augmented method and extended fictitious noise technique. Then local multi‐model system is obtained, for which robust local Kalman estimator is obtained based on the minimax robust estimation principle and unified estimation method. Based on this, the six robust weighted fusion time‐varying Kalman estimators are presented in a unified form, which include robust weighted fusers weighted by matrices, diagonal, scalars, and a robust covariance intersection (CI) fuser and two fast CI (FCI) fusers. The robustness proving method, including the extended Lyapunov equation approach with two kinds of generalized Lyapunov equations, non‐negative matrix factorization and elementary transformation of matrix, is presented to prove that the actual estimation error variances are guaranteed to have minimal upper bounds for all admissible uncertainties. The accuracy relations are proved. Further, the robust local and fused steady‐state Kalman estimators are presented. Finally, a simulation example applied to Internet‐based three tank water system is given to demonstrate effectiveness of the proposed results.

Explicit construction of the minimum error variance estimator for stochastic LTI-ss systems

We showcase the derivation of the optimal (minimum error variance) estimator, when one part of the stochastic LTI system outputs is not measured but is able to be predicted from the measured system outputs.

Data fusion uncertainty-enabled methods to map street-scale hourly NO2 in Barcelona: a case study with CALIOPE-Urban v1.0

Abstract. Comprehensive monitoring of NO2 exceedances is imperative for protecting human health, especially in urban areas with traffic. However, an accurate spatial characterization of the exceedances is challenging due to the typically low density of air quality monitoring stations and the inherent uncertainties in urban air quality models. We study how observational data from different sources and timescales can be combined with a dispersion air quality model to obtain bias-corrected NO2 hourly maps at the street scale. We present a kriging-based data fusion workflow that merges dispersion model output with continuous hourly observations and uses a machine-learning-based land use regression (LUR) model constrained with past short intensive passive dosimeter campaign measurements. While the hourly observations allow the bias adjustment of the temporal variability in the dispersion model, the microscale LUR model adds information on the NO2 spatial patterns. Our method includes an uncertainty calculation based on the estimated error variance of the universal kriging technique, which is subsequently used to produce urban maps of probability of exceeding the 200 µg m−3 hourly and the 40 µg m−3 annual NO2 average limits. We assess the statistical performance of this approach in the city of Barcelona for the year 2019. Our results show that simply merging the monitoring stations with the model output already significantly increases the correlation coefficient (r) by +29 % and decreases the root mean square error (RMSE) by −32 %. When adding the time-invariant microscale LUR model in the data fusion workflow, the improvement is even more remarkable, with +46 % and −48 % for the r and RMSE, respectively. Our work highlights the usefulness of high-resolution spatial information in data fusion methods to better estimate exceedances at the street scale.

Estimation of Error Variance in Genomic Selection for Ultrahigh Dimensional Data

Estimation of error variance in the case of genomic selection is a necessary step to measure the accuracy of the genomic selection model. For genomic selection, whole-genome high-density marker data is used where the number of markers is always larger than the sample size. This makes it difficult to estimate the error variance because the ordinary least square estimation technique cannot be used in the case of datasets where the number of parameters is greater than the number of individuals (i.e., p &gt; n). In this article, two existing methods, viz. Refitted Cross Validation (RCV) and kfold-RCV, were suggested for such cases. Moreover, by considering the limitations of the above methods, two new methods, viz. Bootstrap-RCV and Ensemble method, have been proposed. Furthermore, an R package “varEst” has been developed, which contains four different functions to implement these error variance estimation methods in the case of Least Absolute Shrinkage and Selection Operator (LASSO), Least Squares Regression (LSR) and Sparse Additive Models (SpAM). The performances of the algorithms have been evaluated using simulated and real datasets.

Adaptive Relay Selection Strategy in Underwater Acoustic Cooperative Networks: A Hierarchical Adversarial Bandit Learning Approach

Relay selection solutions for underwater acoustic cooperative networks suffer significant performance degradation as they fail to adapt to incomplete information, noisy interference and overwhelming dynamics. To address this challenge, a hierarchical adversarial multi-armed bandit learning framework by proposing an online reward estimation layer is designed to improve adaptive relay decision control. In online reward estimation layer, adaptive Kalman filter estimator is developed to properly handle noisy observation to support accurate reward. Meanwhile, an online predict mechanism is projected for all relays to enrich learning information. Furthermore, based on estimate error variance, an adaptive exploration structure is developed to accelerate the balance between exploration and exploitation. All gathered information are exploited to learn relay quality for the decision-making. Accordingly, we present a Hierarchical Adversarial Bandit Learning (HABL) algorithm to fully exploit the heuristic interaction between the hierarchical framework. HABL integrates reward estimation, information prediction, adaptive exploration and decision making carefully in a holistic algorithm to maximize the learning efficiency. Thereby, the HABL-based relay selection algorithm has higher system throughput and lower communication cost. Further, we rigorously analyze the convergence of HABL algorithm and give its upper bound on the cumulative regret. Finally, extensive simulations elucidate the effectiveness of the HABL.

A Kalman-Based Compensation Strategy for Platoons Subject to Data Loss: Numerical and Empirical Study

This article considers a homogeneous platoon with vehicles that communicate through channels prone to data loss. The vehicles use a predecessor-following topology, where each vehicle sends relevant data to the next, and data loss is modeled through a Bernoulli process. To address the lossy communication, we propose a strategy to estimate the missing data based on the Kalman filter with intermittent observations combined with a linear extrapolation stage. This strategy enables the followers to better deal with data dropouts. We compare this approach to one purely based on the linear extrapolation of previous data. The performance of both strategies is analyzed through Monte Carlo simulations and experiments in an ad hoc testbed, considering various data loss and transmission loss probabilities depending on the inter-vehicle distance. The results show that for the considered cases, the proposed strategy outperforms the linear extrapolation approach in terms of tracking and estimation error variances. Our results also show that the proposed strategy can achieve string stability for the mean and variance for both the tracking and estimation errors in scenarios where the basic extrapolation strategy cannot.

Feature Screening and Error Variance Estimation for Ultrahigh-Dimensional Linear Model with Measurement Errors

Feature Screening and Error Variance Estimation for Ultrahigh-Dimensional Linear Model with Measurement Errors

A new non‐parametric control chart for monitoring general linear profiles based on log‐linear modelling

AbstractProfile monitoring has received much attention from the applications in statistical process control. It is a method for monitoring the stability of a functional relationship between a response variable and one or more explanatory variables over the time axis. General linear profiles monitoring is very important since the linear relationship between a response variable and explanatory variables is easy to characterize besides it is simple and flexible. In addition, most of general linear profiles monitoring techniques assume normality of random error variables. However, the normality of random error variables is not satisfied in certain applications. This causes the existing monitoring methods for general linear profiles both inadequate and inefficient. Based on the log‐linear modelling, in this paper, we develop a non‐parametric control chart for Phase II monitoring of general linear profiles where normality of random error variables is not assumed. The proposed charting method applies the CUSUM (cumulative sum) to the Pearson chi‐square test for the Wilcoxon‐type rank‐based estimators of coefficient parameters and an estimator of random error variance. Effectiveness of the developed control chart is assessed and compared with that of two existing control charts based on the criterion of ARL (average run length). An industrial production process example is also applied to illustrate how the proposed control chart can be used in practice.

Constrained radar waveform optimization for a cooperative radar-communication system

The coexistence of radar-sensing and communication systems research has received a surge of interest in recent times to tackle the issue of spectrum inadequacy. Designing an optimized radar waveform for a coexistence scenario has been a challenging task for accomplishing the convergence of radar-sensing and communication functionalities, without degrading the performance at either end. This paper proposes a novel global optimization-based Spatial Branch and Bound (SBnB) approach to optimize the phase coefficients of a Non-Linear Frequency Modulated (NLFM) waveform in a CRCS framework. In addition, the Modified-Power Ratio Constraint-Cramér–Rao Lower Bound (M-PRC-CRLB), a local optimization-based approach is proposed to optimize the phase coefficients of an NLFM waveform. The spectral energy distribution and auto-correlation characteristics of an NLFM waveform are comprehensively investigated for various values of polynomial order (N) and at different threshold Signal-to-Noise-Ratio (SNR) values. To compare the proposed waveform design approaches (M-PRC-CRLB, SBnB) with the existing waveform design approaches namely, Minimum Estimation Error Variance (MEEV) and PRC- CRLB, a Peak-to-Side-Lobe-Ratio (PSLR), and Integrated-Side-Lobe-Ratio (ISLR) are evaluated at various polynomial orders and threshold SNR values. Furthermore, the performance of a CRCS is assessed using the radar estimation rate and communication data rate. The simulation results reveal that the proposed optimized radar waveform design approaches provide improved performance compared to the existing radar waveform design approaches in terms of radar estimation rate. Further, the proposed global optimization-based SBnB approach achieves a comparable performance of the communication data rate. In addition, the proposed approaches accomplish enhanced spectral utilization, controlled side-lobe energy levels, reduced range-domain ambiguities, and a higher information rate in a CRCS.

Observer-based Quasi-Synchronization Control for Master-Slave Neural Networks under Sojourn-Probability-based Stochastic Communication Protocol

This paper investigates the observer-based quasi-synchronization control for a class of master-slave neural networks under sojourn-probability-based (SP-based) stochastic communication protocol (SCP). The novel scheduling protocol, is introduced to determine which sensor node obtains the transmission right, and the transmission probabilities of SCP are determined by the alleged SP. The primary control objectives are threefold: 1) designing proper observer-based controller such that the quasi-synchronization error be bounded in the mean-square sense; 2) letting the estimation error of the master neural network constrained within specified variance bounds; and 3) finding the optimized quasi-synchronization error and estimation error variance upper bounds. To achieve these aims, the recursive matrix inequality approach and stochastic analysis method are utilized, and a quasi-synchronization control algorithm is developed to ensure the pre-specified quasi-synchronization performance under the proposed SP-based SCP. In addition, two simulations are employed to validate the efficiency and practicality of the designed scheme.

Error variance estimation for the partially linear model

To address the &lt;i&gt;curse of dimensionality&lt;/i&gt; problem associated with multivariate nonparametric regression models, we consider partially linear regression models. A contribution to statistical inference in the partially linear regression model is to propose estimators of the error variance with good asymptotic properties. We propose two estimators of the error variance in a partially linear regression model and their respective asymptotic normality. Using a simulation–based study, we will compare the performance of our estimator with the performance of other concurrent estimators existing in the statistical literature

Influence of the spatial configuration of available data on hydraulic conductivity estimates for a geostatistical-Kalman filter method

Abstract Accurate hydraulic conductivity estimates are vital for groundwater evaluation. Usually, interpolations of hydraulic conductivity data are needed to obtain spatial estimates over larger areas, but the results present a high uncertainty which can be reduced by adding a secondary variable in the estimation. In this paper, the influence of the number and spatial configuration of hydraulic conductivity (K) and hydraulic head (HH) data on the estimation of K is evaluated using univariate and bivariate geostatistical-Kalman filter approaches (similar to kriging and cokriging, respectively). A synthetic case based on a transient groundwater flow model is used to generate different numbers, spatial arrays, and data. With these data, variogram models for the univariate and bivariate cases were fitted and used to calculate the corresponding covariance matrices for the Kalman filter. The results show that K estimates are more reliable when HH data is added than when only K is used, independently of the number and distribution of the data, since there is a better agreement between the calculated errors and estimate error variances. HH data provides valuable information only where K is not sampled. This evaluation could support the design of optimal sampling strategies to obtain reliable K estimates.

Robust integrated sequential covariance intersection fusion Kalman filters and their convergence and stability for networked sensor systems with five uncertainties

AbstractFor networked sensor systems (NSSs) with hard and soft sensors including five uncertainties, two universal approaches of solving the robust fusion estimation problems are presented. It includes an integrated sequential covariance intersection (SCI) fusion minimax robust Kalman filtering approach with cross‐covariance information and a generalized Lyapunov equation approach with four pairs of Lyapunov equations. Applying them, the robust local and SCI fused time‐varying and steady‐state Kalman filters are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds. The equivalent batch SCI fusers are also presented. Their robustness and accuracy relations are proved, and the sensitivity of the SCI fuser with respect to the fused orders of sensors is analyzed. Applying the dynamic error system analysis method and the dynamic variance error system analysis method, a new convergence and absolute asymptotic stability theory of robust fusion Kalman filtering is presented. The classical Kalman filtering convergence and stability theory is developed. Compared with the original covariance intersection fuser, they significantly reduced the computational complexity and burden. Compared with the optimal and conservative SCI fusers, they significantly improved the robust accuracies. They are suitable to deal with asynchronous or random delayed data and are suitable for real‐time applications. A simulation applied to the two‐mass spring damper mechanical system shows their effectiveness.

Strong homogeneous clutter and random target scattering analysis for closed-form Cramer Rao bound and optimal power allocations in MIMO radars

This paper considers the effects of strong homogeneous clutter and the target's random scattering on radar measurements. The signal model is improved and closed-form Cramer-Rao lower bounds (CRLBs) are calculated for delay and Doppler frequency estimations in attendance of practical clutter, and target's scatterings models. The calculated closed-form CRLBs are simplified to extract the effect of assigned powers to the transmitters for using power allocation (PA) strategies. The maximum likelihood estimation error variances (MLEEVs) are derived for calculated CRLBs' verifications. Four power allocation strategies are defined as mathematical optimization problems that consider the power budget constraints and measurement error statistics as the objective function or constraints. Simulation results verify the proposed simplified closed-form CRLBs accuracies by comparing with exact CRLBs and MLE estimators' performance. It is shown that proposed power allocations not only preserve required performance but also optimize objectives.

Nonparametric Estimation of Error Variance under Simple Random Sampling without Replacement

This study adopts a nonparametric approach in the estimation of a finite population error variance in the setting where the variance is a constant (homoscedastic) using a model-based technique under simple random sampling without replacement (SRSWOR). A mean square analysis of the estimator has been conducted, including the asymptotic behaviour of the estimator and the results show that the asymptotic distribution in a homoscedastic setting is asymptotically unbiased and consistent. The performance of the developed estimator is compared to that of other existing estimators using real data. R statistical software was utilized to analyze data and numerical results presented graphically for selected models.

Optimal quantised bits for estimation over a capacity and power limited, lossy channel

This letter considers the problem of estimating the state of a scalar dynamical system over a wireless channel that is lossy, capacity, and power limited. The limited power assumption, which is valid for most real problems, links the number of quantisation bits and packet error rate. As the number of quantisation bits increases, the quantisation noise decreases but the packet loss rate increases. It is shown that the estimation error variance (EEV) is minimised for a range of quantisation bits. Further, it is shown that operating beyond the optimal range sharply increases the EEV. Simulation results corroborating the analytical results are also presented.