Estimation Theory Research Articles

Impact of composition and knitting methods of fiberglass grid on the progression of reflective crack in asphalt pavement

In the asphalt pavement with semi-rigid base, the shrinkage cracks initiating from base course often propagate upward to surface course to develops into reflective cracks. The fiberglass grid, as a woven geotextile, is widely used to retard the occurrence of reflective cracks in China, but the impacts of material composition and knitting methods of it on reflective cracks are still not clear. In this paper, two kinds of models, including a two-scale model of fiber bundle-fiberglass grids layer and a coupled thermo-mechanical model of pavement structure, are established. In the two-scale model, the elastic constants (ECs) of fiber bundle and fiberglass grids layer are calculated under different compositions and knitting methods and then assigned to the coupled thermo-mechanical model to estimate the fatigue lives at initiation phase (FL-IP) and propagation phase (FL-PP) of reflective cracks based on an estimation theory. Moreover, this study designs the laboratory tests to validate this estimation method of fatigue life. Then, based on the predicted fatigue life at different phases of crack progression, the sensitivity analysis is implemented to analyze the impacts of composition and knitting methods on the fatigue life. As a result, the recommendations on design and preparation of fiberglass grid are proposed. The results show that, in material composition, the glass fibers with high elastic modulus and volume fraction and the coating materials with high elastic modulus are beneficial to retarding the initiation and propagation of reflective cracks and in knitting methods, increasing density of fiber bundles parallel to the traffic direction and reducing braided angle also has positive effect. These results can be used to optimizing design and preparation of fiberglass grid to reduce the occurrence of reflective cracks in asphalt pavement.

Not All Probability Density Functions Are Tomograms.

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and nonparametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".

Stochastic Arbitrage Opportunities: Set Estimation and Statistical Testing

We provide a formal statistical theory of consistent estimation of the set of all arbitrage portfolios that meet the description of being a stochastic arbitrage opportunity. Two empirical likelihood ratio tests are developed: one for the null that a given arbitrage portfolio is qualified, and another for the alternative that the portfolio is not qualified. Apart from considering generalized concepts and hypotheses based on multiple host portfolios, the statistical assumption framework is also more general than in earlier studies that focused on special cases with a single benchmark portfolio. Various extensions and generalizations of the theory are discussed. A Monte Carlo simulation study shows promising statistical size and power properties for testing the null, for representative data dimensions. The results of an empirical application illustrate the importance of selecting a proper blocking structure and moment estimation method.

Robust sequential fusion Kalman estimators with asymptotic equivalence and stability for networked uncertain sensor systems

For networked uncertain sensor systems (NUSSs) with an improved unified sensor model with “hard” and “soft” sensors including five uncertainties plus two-step random measurement delays and colored measurement noises, applying the block matrix inversion formula, a new sequential inverse covariance fusion (SICF) approach is presented. It can sequentially compute the global inverse covariance according to the time order of the received data, but need not wait until all data are received. The corresponding minimax robust time-varying and steady-state sequential fusion (SF) Kalman estimators (predictor, filter, and smoother) are presented in the sense that their actual error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. They have lower computation complexities and higher accuracies equivalent to ones of the robust optimal batch-fusion (BF) Kalman estimators, and are suitable for real-time applications. Their asymptotic equivalence, absolute asymptotic stability, and accuracy relations are proved. Their strict complexity analysis is given. They constitute a novel universal robust SF estimation, convergence and stability theory for NUSSs. One simulation example applied to the uncertain ARMA signal processing verifies the effectiveness of the proposed approach and theory.

Beating the spectroscopic Rayleigh limit via post-processed heterodyne detection.

Quantum-inspired superresolution methods surpass the Rayleigh limit in imaging, or the analogous Fourier limit in spectroscopy. This is achieved by carefully extracting the information carried in the emitted optical field by engineered measurements. An alternative to complex experimental setups is to use simple homodyne detection and customized data analysis. We experimentally investigate this method in the time-frequency domain and demonstrate the spectroscopic superresolution for two distinct types of light sources: thermal and phase-averaged coherent states. The experimental results are backed by theoretical predictions based on estimation theory.

Physics-Informed Neural Network surrogate model for bypassing Blade Element Momentum theory in wind turbine aerodynamic load estimation

This paper proposes the use of Artificial Neural Networks (ANNs), specifically Physics-Informed Neural Networks (PINNs), for dynamic surrogate modelling of wind turbines. PINNs offer the flexibility to model complex relationships while incorporating physics-based constraints, enabling accurate representation of wind turbine dynamics. In this paper, a PINN-based surrogate model is developed for the Blade Element Momentum (BEM) aerodynamic model used in state-of-the-art numerical wind turbine simulations. The PINN model replaces the time-consuming root-finding process in BEM with high-dimensional regression, significantly improving computational efficiency. The PINN model is trained using data generated from a numerical model of the IEA-15MW reference wind turbine, and its performance is compared against conventional data-driven Neural Network (DDNN) models. The proposed surrogate model provides more efficient and accurate evaluations of wind turbine responses compared with traditional surrogate modelling approaches. A significant computational advantage is obtained by using the developed surrogate models with a forty-fold speedup demonstrated compared to the BEM model. Replacing the BEM model with the PINN-based surrogate model for load computation in the numerical model used for dynamic analysis results in an overall reduction of 35% in computational time for a complete dynamic simulation. This is a substantial improvement in efficiency without sacrificing accuracy — the maximum Mean Absolute Error (MAE) values for the surrogate models are of the order of 10−2, which shows that the surrogate models can predict the angle of attack at any blade node with a discrepancy of less than 0.5°. The surrogate models significantly reduce computational time while maintaining high accuracy, making them a promising approach for simulating wind turbine dynamics, especially in fields such as reliability analysis or fatigue estimation where many simulations are necessary.

Synchronous Retrieval of Wheat Cab and LAI from UAV Remote Sensing: Application of the Optimized Estimation Inversion Framework

Chlorophyll a and b content (Cab) and leaf area index (LAI) are two key parameters of crops, and their quantitative inversions are important for growth monitoring and the field management of wheat. However, due to the close correlation between the spectral signals of these two parameters and the effects of soil and atmospheric conditions, as well as modeling errors, synchronous retrieval of LAI and Cab from remote sensing data is still a challenging task. In a previous study, we introduced the optimal estimation theory and established the inversion framework by coupling the PROSAIL (PROSPECT + SAIL) model with the unified linearized vector radiative transfer model (UNL-VRTM). The framework fully utilizes the simulated radiance spectra for synchronous retrieval of Cab and LAI at the UAV observation scale and has good convergence and self-consistency. In this study, based on this inversion framework, synchronized retrieval of Cab and LAI was carried out by real wheat UAV observation data and validated with the ground-measured data. By comparing with the empirical statistical model constructed by the PROSAIL model and coupled model, least squares support vector machine (LSSVM), and random forest (RF), the proposed method has the highest accuracy of Cab and LAI estimated from UAV multispectral data (for Cab, R2 = 0.835, RMSE = 14.357; for LAI, R2 = 0.892, RMSE = 0.564). Our proposed method enables the fast and efficient estimation of Cab and LAI in multispectral data without prior measurements and training.

Estimation theory of photon-magnon coupling strength in a driven-dissipative double-cavity-magnon system

Estimation theory of photon-magnon coupling strength in a driven-dissipative double-cavity-magnon system

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.

Research on Probability Estimation Method of Investment Cost of Water Conservancy Construction Project

The construction period of water conservancy construction project is long, and the project cost is affected by many uncertain factors. There is a great difference between the final account value after the completion of the project and the budget value before the commencement.Therefore, this paper considers the complexity and variability of the project cost, analyzes and constructs the probability estimation theory of investment cost from the perspective of system analysis, and applies it to the estimation of the investment cost of the construction project of Matan River miaotazi hydropower station, and calculates the probability distribution of the total cost change.The research results can provide a new idea for the estimation of investment cost of water conservancy projects, and also provide valuable reference for the estimation of investment cost of Matan River miaotazi hydropower station.

Performance reliability degradation analysis and lifetime prediction based on generalized normal grey cloud Bayesian Wiener process

AbstractThis study addresses the problem of performance reliability degradation and lifetime prediction in complex systems with high reliability but insufficient information. It comprehensively considers the effects of various uncertainties, such as measurement, cognitive, and stochastic uncertainties, and proposes a generalized normal grey cloud Bayesian Wiener process (GNGCBWP) stochastic degradation model. First, based on the historical degradation information of the system, the interval grey number sequence of the degradation increment is constructed, and the performance degradation path of the system is analyzed to determine the degradation model type. Then, combining the normal grey cloud and Bayesian estimation theory, the relevant definitions and theorems of generalized normal grey cloud Bayesian estimation (GNGCBE) are proposed. Finally, the parameter estimation results obtained using the GNGCBE method are substituted into the grey Wiener process degradation model to dynamically analyze performance reliability degradation and predict the remaining useful life (RUL) in real‐time. And the validity and scientificity of the proposed model is verified through a case study of laser equipment. Moreover, the comparison results of the models indicate that the proposed model has advantages in dynamic analysis of performance reliability degradation and real‐time RUL prediction for highly reliable complex systems with limited or poor degradation information. It can significantly reduce the impact of multiple uncertainties on the analysis results and improve the accuracy and reliability of the final prediction results.

Quantitative assessment of the potential of optimal estimation for aerosol retrieval from geostationary weather satellites in the frame of the iAERUS‐GEO algorithm

AbstractSatellite remote sensing enables the study of atmospheric aerosols at large spatial scales, with geostationary platforms making this possible at sub‐daily frequencies. High‐temporal‐resolution aerosol observations can be made from geostationary data by using robust numerical inversion methods such as the widely‐used optimal estimation (OE) theory. This is the case of the instantaneous Aerosol and surfacE Retrieval Using Satellites in GEOstationary orbit (iAERUS‐GEO) algorithm, which successfully retrieves aerosol optical depth (AOD) maps from the Meteosat Second Generation weather satellite based on a simple implementation of the OE approach combined with the Levenberg–Marquardt method. However, the exact gain in inversion performances that can be obtained from the multiple and more advanced possibilities offered by OE is not well documented in the current literature. Against this background, this article presents the quantitative assessment of OE for the future improvement of the iAERUS‐GEO algorithm. To this end, we use a series of comprehensive experiments based on AOD maps retrieved by iAERUS‐GEO using different OE implementations, and ground‐based observations used as reference data. First, we assess the varying importance in the inversion process of satellite observations and a priori information according to the content of satellite aerosol information. Second, we quantify the gain of AOD estimation in log space versus linear space in terms of accuracy, AOD distribution and number of successful retrievals. Finally, we evaluate the accuracy improvement of simultaneous AOD and surface reflectance retrieval as a function of the regions covered by the Meteosat Earth's disk.

Automatic zoning for retinopathy of prematurity with a key area location system.

Retinopathy of prematurity (ROP) usually occurs in premature or low birth weight infants and has been an important cause of childhood blindness worldwide. Diagnosis and treatment of ROP are mainly based on stage, zone and disease, where the zone is more important than the stage for serious ROP. However, due to the great subjectivity and difference of ophthalmologists in the diagnosis of ROP zoning, it is challenging to achieve accurate and objective ROP zoning diagnosis. To address it, we propose a new key area location (KAL) system to achieve automatic and objective ROP zoning based on its definition, which consists of a key point location network and an object detection network. Firstly, to achieve the balance between real-time and high-accuracy, a lightweight residual heatmap network (LRH-Net) is designed to achieve the location of the optic disc (OD) and macular center, which transforms the location problem into a pixel-level regression problem based on the heatmap regression method and maximum likelihood estimation theory. In addition, to meet the needs of clinical accuracy and real-time detection, we use the one-stage object detection framework Yolov3 to achieve ROP lesion location. Finally, the experimental results have demonstrated that the proposed KAL system has achieved better performance on key point location (6.13 and 17.03 pixels error for OD and macular center location) and ROP lesion location (93.05% for AP50), and the ROP zoning results based on it have good consistency with the results manually labeled by clinicians, which can support clinical decision-making and help ophthalmologists correctly interpret ROP zoning, reducing subjective differences of diagnosis and increasing the interpretability of zoning results.

Sparse Grid Imputation Using Unpaired Imprecise Auxiliary Data: Theory and Application to PM2.5 Estimation

Sparse grid imputation (SGI) is a challenging problem, as its goal is to infer the values of the entire grid from a limited number of cells with values. Traditionally, the problem is solved using regression methods such as KNN and kriging, whereas in the real world, there is often extra information—usually imprecise—that can aid inference and yield better performance. In the SGI problem, in addition to the limited number of fixed grid cells with precise target domain values, there are contextual data and imprecise observations over the whole grid. To solve this problem, we propose a distribution estimation theory for the whole grid and realize the theory via the composition architecture of the Target-Embedding and the Contextual CycleGAN trained with contextual information and imprecise observations. Contextual CycleGAN is structured as two generator–discriminator pairs and uses different types of contextual loss to guide the training. We consider the real-world problem of fine-grained PM2.5 inference with realistic settings: a few (less than 1%) grid cells with precise PM2.5 data and all grid cells with contextual information concerning weather and imprecise observations from satellites and microsensors. The task is to infer reasonable values for all grid cells. As there is no ground truth for empty cells, out-of-sample mean squared error and Jensen–Shannon divergence measurements are used in the empirical study. The results show that Contextual CycleGAN supports the proposed theory and outperforms the methods used for comparison.

Multi-tier dynamic sampling weak RF signal estimation theory

This paper presents a theoretical analysis in discrete time for a multi-tier weak radiofrequency (RF) signal estimation process with N simultaneous signals. Discrete time dynamic sampling is introduced and is shown to provide the capability to extract signal parameter values with increased accuracy compared with accuracy of estimates obtained in prior work. This paper advances phase measurement approaches by proposing discrete time dynamic sampling which our paper shows offers the desirable capability for more accurate weak signal parameter estimates. For N=2 simultaneous signals with a strong signal at 850 MHz and a weak signal at 855 MHz, the results show that dynamically sampling the instantaneous frequency at 24 times the Nyquist rate provides weak signal frequency estimates that are within 1.7 times 10^{-5} of the actual weak signal frequency and weak signal amplitude estimates that are within 428 PPM of the actual weak signal amplitude. Results are also presented for situations with N=2 simultaneous 5G signals. In one case, the strong signal is 3950 MHz, and the weak signal is 3955 MHz; in the other case the strong case is 5950 MHz, and the weak signal is 5955 MHz. The results for these cases show that estimates obtained with dynamic sampling are more accurate than estimates provided using a single sample rate of 65 MSPS. This work has promising applications for weak signal parameters estimation using instantaneous frequency measurements.

On the estimation of the core for TU games

The core of a transferable utility (TU) game, if it is not empty, is prescribed by the set of all stable allocations. The exact determination of the core reaches exponential time complexity. Therefore, its exact computation is often avoided as the number of players increases. In this work, we propose an estimator for the core of a TU game based on the statistical theory of set estimation. Concretely, we provide a core reconstruction that is obtained in polynomial time for general dimension. Additionally, convergence rates for the estimation error are derived. Finally, a consistent core-center estimator is established as a geometrical application of this methodology.

A new indicator for estimating the degree of mining-induced land subsidence: the overburden’s average GSI value

Underground coal mining leads to land subsidence, which, in turn, results in damage to buildings and infrastructure, disturbs the original ecological environment, and hinders the sustainable development of coal mining cities. A reasonable estimation of land subsidence, on the other hand, is the foundation for building protection, land reclamation, and ecological environment reconstruction. However, when we applied the existing land subsidence estimation theory to the deep mining areas of the Ordos coalfield in western China, there was a significant deviation between the estimations and the measurements. To explain such unusual case, we propose using the overburden’s average GSI (Geological Strength Index) value instead of the compressive strength (UCS) of rock specimens for a better representation of the overburden’s overall properties. By using on-site subsidence monitoring results and historical data, we provided evidence which supports that the overburden’s average GSI value has a much greater impact on subsidence rates than the UCS. Subsequently, we investigated the relationship between three typical overburden’s GSI values and the subsidence rates via a calibrated numerical model, revealing the variation patterns of maximum surface subsidence when the overburden’s average GSI value is set at 30, 50, and 75, respectively. Finally, on the basis of the measured and simulated results, we discussed a non-conventional strip mining method for mining subsidence control in the deep mining areas of the Ordos coalfield in western China, and explained why it is possible and what are the significant advantages behind. The proposed methods, findings, and suggestions in this paper are therefore quite helpful for researchers and engineers who wish to estimate and control the mining-induced land subsidence, as well as for those who are particularly interested in the study of environment science related to land subsidence.

Algorithm for capacity estimation based on time resolution theory with linear computational complexity for frequency-selective communication channels and PAM-n-signals

Problem statement. At that moment, it is relevant to create algorithms for communication channels capacity estimation within the framework of the Theory of Resolution Time for broadband wired telecommunication systems, which use bipolar PAM-n-signal whose execution time complexity is linear. The creation of such algorithms is of high importance both for wired network technologies (Ethernet, InfiniBand etc), and in the field of creating high-speed interchip connections, high-speed memory controllers and peripheral computer buses, since it will allow to improve channel capacity at low computational costs in real-time regime due to the transition to the “above Nyquist speed” regime. Target. Develop an algorithm which has linear computational complexity in the framework of The Resolution Time Theory for frequency-selective communication channels capacity estimation with a linear receiver and PAM-n-signal. Results. An algorithm for estimating specific channel capacity and resolution time is presented, which has linear computational complexity, depending only on the value of the effective memory, for continuous frequency-selective communication channels with a discrete source, in which information is transmitted utilizing bipolar PAM-n-signal. This algorithm has no restrictions on the pulse shape, and the computational complexity linearly depends on the channel memory. The key features of the algorithm are considered, allowing its application in real time. The developed algorithm allows to user fully estimate all the effects caused by data depended jitter. Practical significance. The proposed algorithm makes it possible in real time to select the shape of a partial pulse and the configuration of the signal constellation that increase the specific capacity of a communication system with serial data transmission, including under conditions of transmission in the “Faster than Nyquist” regime.

Quantum Fisher information in moving reference frame

In the field of quantum metrology, an important application is quantum parameter estimation. As the fundamental theory of quantum parameter estimation, quantum Cramér-Rao inequality shows that the variance of parameter estimation is determined by the inverse of quantum Fisher information. Higher quantum Fisher information corresponds to a lower variance, thereby improving the precision of parameter estimation. Quantum Fisher information has been extensively investigated in many aspects of non-relativistic quantum mechanics, including entanglement structure detection, quantum teleportation, quantum phase transition, quantum chaos, and quantum computation. However, there are few researches considering the influence of relativistic effect on quantum Fisher information, and therefore, we attempt to investigate this topic in this work. The relativistic transformation of particle states is employed, and the quantum Fisher information about amplitude parameter <inline-formula><tex-math id="M3">\begin{document}$ \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.png"/></alternatives></inline-formula> and phase parameter <inline-formula><tex-math id="M4">\begin{document}$\varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.png"/></alternatives></inline-formula> are investigated in moving reference frame. In this work, the parameters to be estimated are encoded into the spin degree of freedom, and the pure single-qubit state and the pure two-qubit state are both considered. The quantum Fisher information about <inline-formula><tex-math id="M5">\begin{document}$ \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$\varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.png"/></alternatives></inline-formula> of single-qubit state and two-qubit state in moving reference frame are numerically calculated, respectively. It can be observed that the quantum Fisher information is associated with rapidity, amplitude parameter, and the ratio of the width to the particle mass <inline-formula><tex-math id="M7">\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.png"/></alternatives></inline-formula>. The quantum Fisher information of the estimated parameters decreases with rapidity increasing for both single-qubit state and two-qubit state. As rapidity approaches infinity, i.e. increases to the speed of light, the quantum Fisher information reaches to a constant which decreases as the ratio <inline-formula><tex-math id="M8">\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.png"/></alternatives></inline-formula> increases. More importantly, for the phase parameter <inline-formula><tex-math id="M9">\begin{document}$ \varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.png"/></alternatives></inline-formula>, it is observed that the quantum Fisher information of two-qubit state reduces more significantly than that of single-qubit state. While, for the amplitude parameter <inline-formula><tex-math id="M10">\begin{document}$\theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.png"/></alternatives></inline-formula>, the quantum Fisher information of two-qubit state is greater than that of single-qubit state. These results are useful and valuable for improving the precision of parameter estimation under the influence of relativistic effect.

Condition numbers of the generalized ridge regression and its statistical estimation

<abstract><p>In this paper, we considered the condition number theory of a new generalized ridge regression model. The explicit expressions of different types of condition numbers were derived to measure the ill-conditionness of the generalized ridge regression problem with respect to different circumstances. To overcome the computational difficulty of computing the exact value of the condition number, we employed the statistical condition estimation theory to design efficient condition number estimators, and the numerical examples were also given to illustrate its efficiency.</p></abstract>