Estimation Error Variance Research Articles

Framework for the comparison of a priori and a posteriori error variance estimation and tuning schemes

AbstractThe performance of an assimilation system is strongly dependent on the quality of the error statistics used. A number of error statistics estimation and tuning methods have previously been developed to better assess and determine these statistics. Many of these are a posteriori methods which make use of quantities calculated during the assimilation procedure, while other a priori methods do not require information from the assimilation. In this study, we develop a conceptual framework that relates these methods when applied to error variance determination, where each method is associated with the minimization of a particular cost function. The minimization of these cost functions describes a fitting procedure that fits parts of the prescribed modelled innovation covariance to its observed values. Each method must in some way separate the innovation covariance into its contributions from the background and the observations, which are then used in the fitting procedure. It is shown that the examined a posteriori methods use the analysis filter to make this separation and that the minimization of their associated cost functions is done implicitly within the tuning procedure. Analytical expressions for the expectation value and variance of estimates for error variance scaling parameters are determined for each method. The expressions for the expectation values of these estimates show that the accuracy of each method is dependent on its ability to separate the background from the observation contributions to the innovation covariance. This separability is quantified by use of the Frobenius inner product between the background‐ and observation‐error covariances, which additionally allows for geometric interpretations of the covariances to be made. Comparisons between variance parameter estimates from different methods are made for the case of a 1D periodic domain.

Low complexity robust recursive beamforming for two‐way full duplex relay network

In this study, a two-way full-duplex amplify-and-forward relay network is considered in which two multi-antenna fullduplex user nodes manage to exchange their information via a full-duplex multi-antenna relay. The users and relay can concurrently transmit their symbols in all time slots in the same frequency band, which leads to self-interference (SI) at all nodes. The joint beamforming algorithm is used in which the relay and users beamforming matrices are updated in each time slot to suppress the SI. This algorithm needs the channel state information (CSI) of all previous time slots, so the complexity increases as time lapses. To decrease the complexity, an algorithm is proposed in which the beamforming matrix is computed recursively. Moreover, the channel estimation error is considered when the relay and users have partial CSI. Thus, a robust algorithm, i.e. an algorithm that takes the estimation error into account is proposed to decrease the effect of estimation error. The proposed algorithm has been evaluated in terms of mean square error, achievable sum rate and BER for various values of the estimation error variances and compared with the non-robust algorithm. The results show that the superiority of the recursive robust algorithm in terms of SNR and complexity.

Robust integrated covariance intersection fusion Kalman estimators for networked mixed uncertain time-varying systems

Abstract For the multisensor time-varying networked mixed uncertain systems with random one-step sensor delays and uncertain-variance multiplicative and linearly dependent additive white noises, a new augmented state method with fictitious noises is presented, by which the original system is transformed into a standard system without delays and with uncertain-variance fictitious white noises. According to the minimax robust estimation principle and the Kalman filtering theory, based on the worst-case system with the conservative upper bounds of uncertain noise variances, the local and integrated covariance intersection (ICI) fused robust time-varying Kalman estimators (filter, predictor and smoother) are presented respectively in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Their robustness is proved by the extended Lyapunov equation method, and their accuracy relations are compared based on the traces of the variance matrices and the covariance ellipsoids, respectively. Specially, a universal ICI fusion robust Kalman filtering method of integrating the local robust estimators and their conservative cross-covariances is presented. It overcomes the drawbacks of the original covariance intersection (CI) fusion method and improves robust accuracy of the original CI fuser. A simulation example applied to two-mass spring system shows the effectiveness of the proposed methods and results.

Robust fusion Kalman estimators for networked mixed uncertain systems with random one-step measurement delays, missing measurements, multiplicative noises and uncertain noise variances

For mixed uncertain multisensor networked systems simultaneously with four uncertainties including random one-step measurement delays, missing measurements, multiplicative noises and uncertain noise variances, three new approaches of solving robust fusion estimation problem are presented. They include augmented state approach with fictitious white noises, extended Lyapunov equation approach, and universal integrated covariance intersection (ICI) fusion approach. Applying them, the minimax robust local and five fused time-varying Kalman estimators (predictor, filter an smoother) are presented in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. The five robust fusers include centralized fuser, fusers weighted respectively by matrices, diagonal matrices and scalars, and ICI fuser. Their robustness and accuracy relations are proved. The proposed approaches and results constitute an important methodology and a unified robust fusion Kalman filtering theory of solving the robust estimation problem. A simulation example applied to the vehicle suspension system shows their effectiveness and applicability.

Novel error variance estimation rule for nonparametric VSS-NLMS algorithm

This paper presents a robust error variance estimation rule for the nonparametric variable step-size normalized least mean square (NPVSS-NLMS) algorithm. The proposed variance estimation rule accurately estimates the variance of the error signal. This is achieved by the variable exponential windowing parameter depending on the standard deviations of the sequential error signals. The accurate estimation of the error signal variance in the NPVSS-NLMS algorithm considerably improves the performance of the adaptive filter when compared to the classical NPVSS-NLMS algorithm. Moreover, the convergence and steady-state performances of the NPVSS-NLMS based on the proposed rule are analyzed in this study. The performance of the proposed algorithm is evaluated on system identification and acoustic echo canceling experiments and compared with that the classical NPVSS-NLMS algorithm. As a result, simulations show that the proposed algorithm with the help of the novel robust error variance estimation rule not only yields a dramatically reduced steady-state error but also achieves a faster convergence rate as compared with the classical counterparts. Furthermore, the theoretical results of the variable exponential windowing parameter used in the proposed rule are in very good agreement with its simulation results.

Physical Layer Authentication Jointly Utilizing Channel and Phase Noise in MIMO Systems

In this paper, we propose a physical layer authentication scheme in heterogeneous coexist multiple-input-multiple-output (MIMO) systems. This scheme utilizes two physical layer features in terms of location-specific channel gains and transmitter-specific phase noise caused by imperfect oscillators to identify transmitters. Three properties of the proposed scheme: covertness, robustness, and security, are analyzed in detail. By using a maximum-likelihood estimator (MLE) and extended Kalman filter (EKF), we estimate channel gains and phase noise, and formulate variances of estimation errors. We also quantize the temporal variations of channel gains and phase noise through the developed quantizers. Based on quantization results and theories of hypothesis testing and stochastic process, we then derive the closed-form expressions for false alarm and detection probabilities with the consideration of quantization errors. Simulations are carried out to validate the theoretical results of the two probabilities. Based on theoretical models, we further demonstrate that the proposed scheme makes it possible for us to flexibly control authentication performance by adjusting thresholds (for channel gain, phase noise, and decision, respectively) to achieve a required authentication performance in specific MIMO applications.

The shaky ground truth of real-time phase estimation

Instantaneous phase of brain oscillations in electroencephalography (EEG) is a measure of brain state that is relevant to neuronal processing and modulates evoked responses. However, determining phase at the time of a stimulus with standard signal processing methods is not possible due to the stimulus artifact masking the future part of the signal. Here, we quantify the degree to which signal-to-noise ratio and instantaneous amplitude of the signal affect the variance of phase estimation error and the precision with which “ground truth” phase is even defined, using both the variance of equivalent estimators and realistic simulated EEG data with known synthetic phase. Necessary experimental conditions are specified in which pre-stimulus phase estimation is meaningfully possible based on instantaneous amplitude and signal-to-noise ratio of the oscillation of interest. An open source toolbox is made available for causal (using pre-stimulus signal only) phase estimation along with a EEG dataset consisting of recordings from 140 participants and a best practices workflow for algorithm optimization and benchmarking. As an illustration, post-hoc sorting of open-loop transcranial magnetic stimulation (TMS) trials according to pre-stimulus sensorimotor μ-rhythm phase is performed to demonstrate modulation of corticospinal excitability, as indexed by the amplitude of motor evoked potentials.

Least-squares estimators for systems with stochastic sensor gain degradation, correlated measurement noises and delays in transmission modelled by Markov chains

ABSTRACTThis paper addresses the linear least-squares estimation of a signal from measurements subject to stochastic sensor gain degradation and random delays during the transmission. These uncertainty phenomena, common in network systems, have traditionally been described by independent Bernoulli random variables. We propose a model that is more general and therefore has greater applicability to real-life situations. The model has two particular characteristics: firstly, the sensor gain degradation is represented by a white sequence of random variables with values in [0,1]; in addition, the absence or presence of delays in the transmission is described by a homogeneous three-state Markov chain, which reflects a possible correlation of delays at different sampling times. Furthermore, assuming that the measurement noise is one-step correlated, we obtain recursive prediction, filtering and fixed-point smoothing algorithms using the first and second-order moments of the signal and the processes present in the observation model. Simulation results for a scalar signal are provided to illustrate the feasibility of the proposed algorithms, using the estimation error variances as a measure of the quality of the estimators.

Recursive distributed filtering for two-dimensional shift-varying systems over sensor networks under stochastic communication protocols

This paper is concerned with the distributed filtering problem for a class of two-dimensional shift-varying systems over sensor networks subject to stochastic communication protocol (SCP) over a finite horizon. The communication between the sensor nodes and the filters is implemented through shared channels of limited capacity. To avoid data collisions, the SCP is applied to determine the transmission order of the signals/packets for each sensor. The considered scheduling behavior is characterized by mutually uncorrelated random variables with known probability distributions. Recursive distributed filters are proposed to estimate the system state through available information from both individual and neighboring nodes in the sensor network according to a given topology. Attention is focused on the design of distributed filters in order to ensure the locally minimal upper bound on the error variance of the state estimation. Sufficient conditions are first established, via intensive stochastic analysis and mathematical induction, on the existence of an upper bound of the estimation error variance. Then, by means of a matrix simplification technique, the desired filter gains are designed to optimize the obtained upper bound at each shift step. Finally, a practical example is given to verify the effectiveness of the proposed filter strategy.

Estimation of error variance via ridge regression

Summary We propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory. The proposed estimator is valid under both low- and high-dimensional models, and performs well not only in nonsparse cases, but also in sparse ones. The finite-sample performance of the proposed method is assessed through an intensive numerical study, which indicates that the method is promising compared with its competitors in many interesting scenarios.

Choosing various likelihood functions on uncertainty assessment in groundwater simulation-optimization model

Abstract The main goal in this research is study of impacts of various likelihood functions on DREAM(zs) (Differential Evolution Adaptive Metropolis) method results in simulation-optimization model of aquifer. In this study, DREAM(zs) algorithm has been developed to study aquifer simulation-optimization model uncertainties. DREAM(zs) is used to investigate uncertainty of parameters of the simulation-optimization model in Isfahan-Barkhar aquifer, Isfehan province, Iran. This study is carried out on an aquifer simulation model of MODFLOW that is coupled with MOPSO (multi-objective particle swarm optimization) optimization. Three likelihood functions, L1, L2, and L3, are considered as informal and the remaining (L4 and L5) are represented as formal categories. Likelihood function L1 is Nash–Sutcliffe efficiency and L2 is based on minimum mean square error. L3 uses estimation of model error variance and L4 focuses on the relationship between the traditional least squares fitting and the Bayesian inference. In likelihood function L5 the serial dependence of residual errors is calculated using a first-order autoregressive model of the residuals. Results suggested that the parameters sensitivity depend on the likelihood function selection, and sensitivity of all parameters is not similar in different likelihood functions. MOPSO algorithm outputs indicated that likelihood function No. 5 has a higher speed in reaching convergence and this function also showed that objective functions had a better performance compared to the other likelihood functions.

Use Omega Rather than Cronbach’s Alpha for Estimating Reliability. But…

ABSTRACTCronbach’s alpha (α) is a widely-used measure of reliability used to quantify the amount of random measurement error that exists in a sum score or average generated by a multi-item measurement scale. Yet methodologists have warned that α is not an optimal measure of reliability relative to its more general form, McDonald’s omega (). Among other reasons, that the computation of is not available as an option in many popular statistics programs and requires items loadings from a confirmatory factor analysis (CFA) have probably hindered more widespread adoption. After a bit of discussion of α versus , we illustrate the computation of using two structural equation modeling programs (Mplus and AMOS) and the MBESS package for R. We then describe a macro for SPSS and SAS (OMEGA) that calculates in two ways without relying on the estimation of loadings or error variances using CFA. We show that it produces estimates of that are nearly identical to when using CFA-based estimates of item loadings and error variances. We also discuss the use of the OMEGA macro for certain forms of item analysis and brief form construction based on the removal of items from a longer scale.

Robust Finite-Horizon Filtering for 2-D Systems With Randomly Varying Sensor Delays

This paper is concerned with the robust finite-horizon filter design problem for a class of two-dimensional (2-D) time-varying systems with norm-bounded parameter uncertainties and incomplete measurements. The incomplete measurements cover randomly occurring sensor delays and missing measurements that are presented in a unified form by resorting to a stochastic Kronecker delta function. The occurrences of the sensor delays and missing measurements are governed by stochastic variables with known probability distributions. The main aim of the addressed problem is to design a recursive filter with appropriate gain parameters that ensure the local minimum of certain upper bound on the estimation error variance at each time instant. With the aid of the inductive approach and the 2-D Riccati-like difference equations, one of the first few attempts is made to tackle the robust filter design problem for 2-D uncertain systems with random sensor delays over a finite horizon. Sufficient conditions are provided for the existence of an upper bound on the estimation error variance, an algorithm is then developed to derive such an upper bound, and finally the desired filter is designed to minimize the obtained upper bound. The filter design procedure is of a recursive form that facilitates the online calculation. A numerical simulation is carried out to show the effectiveness of the developed filtering scheme.

Intraoperative Brain Shift Estimation Using Atlas of Brain Deformations and Constrained Kalman Filter

Intraoperative brain shift decreases the accuracy of neuronavigation systems based on preoperative images. In this paper, this problem is addressed by calculating an estimation of brain shift which can be employed to update the preoperative brain images. Therefore, the precision of navigation can be improved. In this regard, a brain shift estimation method is proposed using an atlas of brain deformations and constrained Kalman filter (ACKF). In addition, it is proven that the obtained ACKF estimation is the best unbiased minimax estimation when the risk function is the estimation error variance. Furthermore, a comparison is performed between the ACKF and two existing methods, namely, CKF and atlas-based method. The comparison demonstrates that the ACKF results in a more accurate estimation and needs less computation time. Finally, the supremacy of the proposed ACKF method with respect to the CKF and atlas-based method is illustrated through simulation.

Outage Probability of Cooperative NOMA Networks Under Imperfect CSI With User Selection

Non-orthogonal multiple access (NOMA) is a promising spectrally-efficient technology to satisfy the massive data requirement of the next-generation wireless communication networks. In this paper, we consider cooperative non-orthogonal multiple access (CNOMA) networks with relaying user selection, where a two-stage user selection scheme is adopted to optimize the received signal-to-noise ratio (SNR) of the weak user. We derive analytical expressions of the outage probability of the CNOMA users under two different types of channel estimation errors. Afterwards, we investigate the asymptotic characteristics the outage probability of the CNOMA systems in the high SNR regimes. Our results show that for the case of constant variance of the channel estimation error, the outage probability of each user is saturated and cannot be improved by increasing the transmit power of the access point. In contrast, there is no performance bottleneck of the outage probability for all the users when the variance of the channel estimation errors decease with SNR, whilst fully diversity gain is achieved by the weak user.

Frequency-Domain Prony Method for Autoregressive Model Identification and Sinusoidal Parameter Estimation

In this study, the weighted integral method for identifying differential equation models is extended to a discrete-time system with a difference equation (DE) model and a finite-length sampled data sequence, and obtain a frequency-domain algorithm for short-time signal analysis and frequency estimation. The derivation consists of three steps. 1) Provide the DE (autoregressive model) with unknown coefficients, which is satisfied in a finite observation interval. 2) Discrete Fourier transform (DFT) the DE to obtain algebraic equations (AEs) among the Fourier coefficients. Two mathematical techniques are introduced to maintain the circulant nature of time shifts. 3) Simultaneously solve a sufficient number of AEs with least squares criterion to obtain unknowns exactly when the driving term is absent, or to obtain unknowns that minimize the driving power when it is present. The methods developed enable a decomposed processing of identification and estimation in the frequency domain. Thus, they will be suitable for maximizing statistical efficiency (smallness of estimation error variance), reducing the computational cost, and use in a resolution-enhanced time-frequency analysis of real-world signals. The performance of the proposed methods are compared with those of several DFT-based methods and Cramer–Rao lower bound. Also, the interference effect and its reduction in frequency-decomposed processing are examined.

Meta-analysis of the effects of on-farm management strategies on milk yields of dairy cattle on smallholder farms in the Tropics

Although East Africa is home to one of the most advanced dairy industries in Sub-Saharan Africa, regional annual milk production is insufficient to meet the demand. The challenge of increasing milk yields (MYs) among smallholder dairy cattle farmers (SDCFs) has received considerable attention and resulted in the introduction of various dairy management strategies (DMSs). Despite adoption of these DMSs, MYs remain low on-farm and there is a large discrepancy in the efficacy of DMSs across different farms. Therefore, the present study sought to: (1) identify on-farm DMSs employed by East African SDCFs to increase MYs and (2) summarize existing literature to quantify the expected MY changes associated with these identified DMSs. Data were collected through a comprehensive literature review and in-depth semi-structured interviews with 10 experts from the East African dairy sector. Meta-analysis of the literature review data was performed by deriving four multivariate regression models (i.e. models 1 to 4) that related DMSs to expected MYs. Each model differed in the weighting strategy used (e.g. number of observations and inverse of the standard errors) and the preferred model was selected based on the root estimated error variance and concordance correlation coefficient. Nine DMSs were identified, of which only adoption of improved cattle breeds and improved feeding (i.e. increasing diet quality and quantity) consistently and significantly (P < 0.05) increased daily MYs across the available studies. Improved breeds alongside adequate feeding explained ≤50% of the daily MYs observed in the metadata while improved feeding explained ≤30% of the daily MYs observed across the different models. Conversely, calf suckling significantly (P < 0.05) reduced MYs according to model 2. Other variables including days in milk, trial length and maximum ambient temperature (used as a proxy for heat stress) contributed significantly to decreasing MYs. These variables may explain some of the heterogeneity in MY responses to DMSs reported in the literature. Our results suggest that using improved cattle breeds alongside improved feeding is the most reliable strategy to increase MYs on-farm in East Africa. Nevertheless, these DMSs should not be considered as standalone solutions but as a pool of options that should be combined depending on the resources available to the farmer to achieve a balance between using dairy cattle genetics, proper husbandry and feeding to secure higher MYs.

Grade 3D Block Modeling and Reserve Estimation of the C-North Iron Skarn Ore Deposit, Sangan, NE Iran

Estimation of ore grade is a time and cost consuming process that requires laboratory-based and exploratory information to present the shape and the ore grade distribution of ore deposit in three dimensional space. The block size is one of the most important parameters which impacts the quality of grade estimates in a resource model. This study aims at spatial modeling of iron ore deposit using geostatistical estimation methods such as Ordinary Kriging based on error estimation, selection of the appropriate size for mining blocks using Vlse Kriterijumsk Optimizacija Kompromisno Resenje method, and performing a three-dimensional block modeling along a grade estimation study for the resource estimation in the C-North iron ore deposit, NE Iran. The variogram that was used in OK estimation was cross validated. Cross validation results showed that compared with the local model, OK with the global model was the most appropriate model for the ore body. Detailed distribution maps of total iron contents in the C-North ore deposit showed a close relationship between structural features and higher iron contents, relative to other areas of the ore deposit. Structural features included the major faults and fault zones along the axial plane. These structures are interpreted to have played a significant role in (re) mobilisation and concentration of the metals, in agreement with observations made elsewhere in the Sangan iron ore complex. Based on the estimation results, 83 million tons of resource was estimated at an average grade of 41.86 % Fe using OK method. The C-North ore deposit has been classified based on the relative estimation error variance and the Australasian Code for Reporting of Mineral Resources and Ore Reserves. It is hoped that this example, taken from very different application fields, will encourage practitioners in applying an OK method with variety of ore deposits.

An effective approach for low-complexity maximum likelihood based automatic modulation classification of STBC-MIMO systems

A low-complexity likelihood methodology is proposed for automatic modulation classification of orthogonal space-time block code (STBC) based multiple-input multiple-output (MIMO) systems. We exploit the zero-forcing equalization technique to modify the typical average likelihood ratio test (ALRT) function. The proposed ALRT function has a low computational complexity compared to existing ALRT functions for MIMO systems classification. The proposed approach is analyzed for blind channel scenarios when the receiver has imperfect channel state information (CSI). Performance analysis is carried out for scenarios with different numbers of antennas. Alamouti-STBC systems with 2 × 2 and 2 × 1 and space-time transmit diversity with a 4 × 4 transmit and receive antenna configuration are considered to verify the proposed approach. Some popular modulation schemes are used as the modulation test pool. Monte-Carlo simulations are performed to evaluate the proposed methodology, using the probability of correct classification as the criterion. Simulation results show that the proposed approach has high classification accuracy at low signal-to-noise ratios and exhibits robust behavior against high CSI estimation error variance.

Weighted fusion robust steady‐state estimators for multisensor networked systems with one‐step random delay and inconsecutive packet dropouts

SummaryIn this paper, the weighted fusion robust steady‐state Kalman filtering problem is studied for a class of multisensor networked systems with mixed uncertainties. The uncertainties include same multiplicative noises in system parameter matrices, uncertain noise variances, as well as the one‐step random delay and inconsecutive packet dropouts, which modeled by sequences of Bernoulli variables with different probabilities. By defining a new observation vector and applying the augmented method, the system under study is converted into one with only uncertain noise variances. The sufficient conditions for the existence of steady‐state estimators are given. According to the minimax robust estimation principle, based on the worst‐case subsystems with conservative upper bounds of uncertain noise variances, the robust local steady‐state Kalman estimators (predictor, filter, and smoother) are proposed. Applying the optimal fusion algorithm weighted by matrices and the covariance intersection fusion algorithm, the two kinds of robust fusion steady‐state Kalman estimators are derived in a unified framework. The robustness of the proposed fusion estimators is proved by applying the permutation matrices and the global Lyapunov equations method, such that, for all admissible uncertainties, the actual steady‐state estimation error variances of the estimators are guaranteed to have the corresponding minimal upper bounds. The accuracy relations among the robust local and fusion steady‐state Kalman estimators are proved. An example with application to autoregressive moving average signal processing is proposed, which shows that the robust local and fusion signal estimation problems can be solved by the state estimation problems. Simulation example verifies the effectiveness and correctness of the proposed results.