Estimation Problem Research Articles

Comparative Analysis of Quasi-Linear Kalman-Type Algorithms in Estimating a Markov Sequence with Nonlinearities in the System and Measurement Equations

The so-called Kalman type algorithms (KTA) are considered, among them quasi-linear KTAs introduced as a separate class, the features of which are the Gaussian approximation of the a posteriori probability density function (p.d.f.) at each step and the procedure for processing the current measurement based on the ideology of a linear optimal algorithm. The unified structure of such algorithms and their features are discussed. Two groups are distinguished the quasi-linear KTA: the first is algorithms using Taylor series expansion of the nonlinear functions, and the second is the so-called linear regression KTA. The methods of their designing are considered, and the common features are described. Detailed attention is paid to the following KTAs: the extended Kalman filter (EKF), polynomial filters of the second and third order (PF2 and PF3), as representatives of the first group, and the Unscented and Cubature Kalman filters (UKF and CKF), as representatives the second one. Their comparative analysis is carried out using the estimation problem of a scalar Markov sequence in the presence of nonlinearities in the shaping filter and in the measurement equations. For all the studied algorithms, the formulas are given in a form convenient for comparison. Based on these formulas, possible causes of a decrease in accuracy and a violation of the consistency properties are identified. Using the previously proposed procedure based on the method of statistical tests, predictive simulation was carried out, which made it possible to confirm the conclusions obtained previously on the basis of an analysis of the formulas for the algorithms being compared. The simulation also allowed to compare the computational complexity of the compared algorithms. The results of the study may be useful to developers involved in the processing of measurement information when choosing a filtering algorithm for solving specific practical estimation problems.

Finding the optimal probe state for multiparameter quantum metrology using conic programming

The ultimate precision in quantum sensing could be achieved using optimal quantum probe states. However, current quantum sensing protocols do not use probe states optimally. Indeed, the calculation of optimal probe states remains an outstanding challenge. Here, we present an algorithm that efficiently calculates a probe state for correlated and uncorrelated measurement strategies. The algorithm involves a conic program, which minimizes a linear objective function subject to conic constraints on a operator-valued variable. Our algorithm outputs a probe state that is a simple function of the optimal variable. We prove that our algorithm finds the optimal probe state for channel estimation problems, even in the multiparameter setting. For many noiseless quantum sensing problems, we prove the optimality of maximally entangled probe states. We also analyze the performance of 3D-field sensing using various probe states. Our work opens the door for a plethora of applications in quantum metrology.

Portfolio selection revisited

AbstractIn 1952, Harry Markowitz formulated portfolio selection as a trade-off between expected, or mean, return and variance. This launched a massive research effort devoted to finding suitable inputs to mean-variance optimization. The estimation problem is high dimensional and a factor model is at the core of many attempts. A factor model can reduce the number of parameters that need to be estimated to a manageable size, but these parameters may incorporate substantial, hidden estimation error. Recent analysis elucidates the nature of this error, identifies a mechanism by which it can corrupt optimization and provides a method for its mitigation. We explore this analysis here by illustrating how to improve the volatility ratio of large optimized portfolios, leading to superior portfolio selection.$$^{*}$$ ∗

Estimating Asset Pricing Models in the Presence of Cross-Sectionally Correlated Pricing Errors

In this study, we propose an adversarial learning approach to the asset pricing model estimation problem which aims to find estimates of factors and loadings that capture time-series covariations while minimizing the worst-case cross-sectional pricing errors. The proposed estimator is defined by a novel min-max optimization problem in which finding a solution is known to be difficult. This contrasts with other related estimators that admit a well-defined analytic solution but do not effectively account for correlations among the pricing errors. To this end, we propose an approximate algorithm based on the alternating optimization procedure and empirically demonstrate that our proposed adversarial estimation framework outperforms other existing factor models, especially when the explanatory power of the pricing model is limited.

Some Properties for Certain Subclasses of Spiral-Like and Robertson Analytic Functions

Making use of the definition of subordination, we introduce certain subclasses of spiral-like and Robertson functions in the open unit disk and study some important results such as convolution results, coefficients estimate, subordination properties and Fekete-Szego problems for these subclasses. Further, some known and new outcomes which follow as special cases of our outcomes are also mentioned.

A Closed-Form Solution to Observer Design Problem for Ostensible Metzler Takagi-Sugeno Systems

This paper addresses the state estimation problems related to the generalized fuzzy observer design for ostensible Metzler Takagi-Sugeno (T-S) systems. Attention is focused on design constraints for the concept of diagonal stabilization and positivity of observer gain matrices. On the basis of some new interpretations, the parameterizations of ostensible Metzler T-S fuzzy systems is presented, which opens the way to the solution of the design problem using only the principle of linear matrix inequalities. The same approach is intended to ensure the stability of the dynamics of the estimation error. The presented method extends and generalizes the results that have been presented in the literature so far.

Mitigating the Impact of Lognormal Atmospheric Turbulence Channel Estimation on FSO Communication Systems Using Advanced Deep Learning Modules

ABSTRACTCurrently, researchers and commercial entities worldwide are highly interested in optical wireless communication links operating in terrestrial environments. These links offer the distinct advantage of enabling high‐speed data transmission while keeping operational and installation costs low, all without requiring licensing. Free‐space optical (FSO) communication has established itself as a reliable method for delivering ultra‐high data‐rate services. However, the presence of atmospheric turbulence‐induced fading poses a significant challenge for FSO communication systems, particularly when operating over channels affected by time‐varying turbulence. The effectiveness of FSO systems is greatly influenced by atmospheric conditions in a specific area because the laser beam propagates through the atmosphere. As a consequence, this can lead to a significant degradation such as high complexity, high estimation error, and high bit error rate (BER) in the performance of FSO systems. This paper presents a stacked dual attention LSTM network with layered neural Turing machine (SDALSTM‐LaNTM) for FSO channel estimation over lognormal turbulence‐induced fading channels. In this approach, the stacked structure ensures a robust feature extraction capacity of SDALSTM and enhances its ability to capture intricate relationships among multiple atmospheric variables. After employing the stacked LSTM networks, the extracted features are combined with the augmented features obtained through the dual attention (DAttn) mechanism by concatenation. Then the LaNTM architecture used external memory to solve the channel parameter estimation problem and update new set of hidden features for estimation task. The ability to access and update memory enables the model to capture long‐term dependencies and context, making it beneficial for channel prediction in FSO systems. Presently, we are in the process of verifying the channel estimation model using measured FSO channel data. Simulation results clearly demonstrate the outstanding estimation performance of the SDALSTM‐LaNTM model in FSO systems, specifically when dealing with lognormal turbulence‐induced fading channels. Moreover, we highlight the effectiveness of the SDALSTM‐LaNTM model by conducting a comprehensive comparison with existing models, considering factors such as average capacity performance, outage probability estimation, and BER.

Analytical Second-Order Extended Kalman Filter for Satellite Relative Orbit Estimation

This study considers a relative orbit estimation problem wherein an observing spacecraft navigates with respect to a target space object at a large separation distance (several kilometers) using only the bearing angles obtained by a single onboard camera. Generally, the extended Kalman filter (EKF), which is based on linear relative motion equations such as the Clohessy–Wiltshire equation, is used for the relative navigation of satellites. The EKF linearizes the estimation error around the current estimate and applies the Kalman filter equations to this linearized system. However, it has been shown that nonlinearities of the orbit determination problem can make the linearization assumption insufficient to represent the actual uncertainty. Therefore, an analytical second-order extended Kalman filter (ASEKF) for relative orbit estimation is proposed in this study. The ASEKF, to sequentially estimate the relative states of satellites and their associated uncertainties, is formulated based on a second-order analytic relative-motion equation under J2-perturbtation, which can overcome the deficiencies of existing approaches that mainly focus on applications in two-body, near-circular, and linearized orbit dynamics. Numerical results show that the proposed method provides superior robustness and mean-square error performance compared to linear estimators under the conditions considered.

Consensus-Based Power System State Estimation Algorithm Under Collaborative Attack

Due to its vulnerability to a variety of cyber attacks, research on cyber security for power systems has become especially crucial. In order to maintain the safe and stable operation of power systems, it is worthwhile to gain insight into the complex characteristics and behaviors of cyber attacks from the attacker’s perspective. The consensus-based distributed state estimation problem is investigated for power systems subject to collaborative attacks. In order to describe such attack behaviors, the denial of service (DoS) attack model for hybrid remote terminal unit (RTU) and phasor measurement unit (PMU) measurements, and the false data injection (FDI) attack model for neighboring estimation information, are constructed. By integrating these two types of attack models, a different consensus-based distributed estimator is designed to accurately estimate the state of the power system under collaborative attacks. Then, through Lyapunov stability analysis theory, a sufficient condition is provided to ensure that the proposed distributed estimator is stable, and a suitable consensus gain matrix is devised. Finally, to confirm the viability and efficacy of the suggested algorithm, a simulation experiment on an IEEE benchmark 14-bus power system is carried out.

Reachable Set Estimation for Singular Discrete‐Time Nonlinear Systems With Time‐Varying Delays

ABSTRACTIn this article, the reachable set estimation problems for a class of singular discrete‐time nonlinear systems (SDNS) with time‐varying delay (TVD) are presented. First, we employ a Lyapunov function scheme and utilize zero‐type free matrix equations. Based on this, a criterion is derived in the form of linear matrix inequalities (LMIs) to guarantee that the considered system to be regular, causal, and the state trajectory of the system is bounded. Second, by developing the singular systems decomposition technique into slow and fast subsystems and using the introduced auxiliary lemma, we establish that the state trajectory of the fast subsystem is also bounded, and the reachable set of singular nonlinear system is bounded within the ellipsoid. Finally, Numerical examples are presented to illustrate the effectiveness of the proposed method.

Identification of a Non‐Commensurate Fractional‐Order Nonlinear System Based on the Separation Scheme

ABSTRACTThis article is aimed to study the parameter estimation problems of a non‐commensurate fractional‐order system with saturation and dead‐zone nonlinearity. In order to reduce the structural complexity of the system, the model separation scheme is used to decompose the fractional‐order nonlinear system into two subsystems, one includes the parameters of the linear part and the other includes the parameters of the nonlinear part. Then, we derive an auxiliary model separable gradient‐based iterative algorithm with the help of the model separation scheme. In addition, to improve the utilization of the real time information, an auxiliary model separable multi‐innovation gradient‐based iterative algorithm is presented based on the sliding measurement window. Finally, the feasibility of the presented algorithms is validated by numerical simulations.

COMPARATOR REGRESSION ANALYSIS IN CONDITIONS OF FUZZY INITIAL DATA

Abstract. The canonical task of regression analysis consists in studying the problem of estimation and mathematical description of the influence caused by one or more independent variables (predictors, explanatory variables) on the dependent (explained) variable. The subject of comparator identification problem is a specific modification of the standard regression analysis problem in the case when, for some reason, it is not possible to measure values of the dependent, explained variable. However, the problem of reconstructing a regression polynomial can be solved if it is possible to qualitatively compare the results of experiments and rank them, for example, in the order of increasing. Taking into account the results of ranking the values of the explained variable in different experiments, a corresponding system of inequalities is formed. The purpose of the research consists in developing a method for mathematical processing of the resulting system of inequalities for analytical assessment of the level of influence of explanatory variables on the explained variable. The method for solving the problem is based on transforming a system of inequalities into a system of linear algebraic equations. A significant drawback of the known approach to solving this problem consists in the fact that the resulting system of linear algebraic equations is an underdetermined one (the number of unknowns in this system exceeds the number of equations). At the same time, the system has an infinite number of solutions, among which there may be an uncontrollable number of unacceptable ones. The exhaustive search method for finding acceptable solutions is futile. In this regard, the problem of developing a correct method of comparator identification remains relevant. Another difficulty that accompanies the actual procedure of comparator identification when solving practical problems consists in the fact that measurements of values of the controlled parameters are not accurate. The uncertainty that arises in the conditions of a small sample of initial data can be removed using the tools of fuzzy mathematics. It is assumed that, based on preliminary research, a mathematical description of the corresponding membership function can be obtained for each of the controlled parameters. As a result, effective approaches to solving emerging problems have been proposed, based on the use of developed optimization procedures in the conditions of a small sample of fuzzy initial data.

Optimisation models for estimating public transport OD matrices using different data types

Efficient public transport systems rely on origin–destination matrices (ODMs) estimation to accurately capture passenger travel patterns, enabling adjustments to frequencies and lines as needed. In this study, we address the ODM estimation problem by employing multiple bi-level programs that consider an outdated ODM and observed passenger flows on specific transit line arcs. Additionally, we consider various optional data types, including boarding and alighting data, as well as the structure of the outdated ODM and passenger flows, either all, separately, in combination, or none. In our study, we reformulate these bi-level programs into single-level models, and we use a commercial solver to address the problem in benchmark instances. In our analysis, we focus on the impact of incorporating different types of information into the estimation process, leading to valuable insights. We find that considering all the data types leads to a higher accuracy than only a subset of these data types. In particular, focussing only on boarding and alighting data leads to improvements in the estimation process, whereas considering only the structure of the outdated ODM and passenger flows leads to reduced accuracy compared to not incorporating either. The latter highlights the significance of data selection in ODM estimations.

Statistical and computational efficiency for smooth tensor estimation with unknown permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to ( m − 2 ) ( m + 1 ) / 2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis.

Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis.

Deep optimal experimental design for parameter estimation problems

Abstract Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter estimation techniques are changing rapidly with the introduction of deep learning techniques to replace traditional estimation methods. This in turn requires the adaptation of optimal experimental design that is associated with these new techniques. In this paper we investigate a new experimental design methodology that uses deep learning. We show that the training of a network as a Likelihood Free Estimator can be used to significantly simplify the design process and circumvent the need for the computationally expensive bi-level optimization problem that is inherent in optimal experimental design for non-linear systems. Furthermore, deep design improves the quality of the recovery process for parameter estimation problems. As proof of concept we apply our methodology to two different systems of Ordinary Differential Equations.

Sensitivity fields and parameter estimation from dielectric objects

The quantitative phase image formation process is posed as a problem of parameter estimation from intensity measurements. This approach is inclusive of traditional pixel-oriented imaging, where the sought parameters are the pixel values. The resulting optimization process to find the parameters is then seen to depend on the sensitivity field: this is the gradient of the scattered field with respect to the parameters, and it turns out to obey a scattering relationship that is analogous to that of the original scattered field. Examples are given from several regimes of scattering strength.

Variational Inference based on a Subclass of Closed Skew Normals

Gaussian distributions are widely used in Bayesian variational inference to approximate intractable posterior densities, but the ability to accommodate skewness can improve approximation accuracy significantly, when data or prior information is scarce. We study the properties of a subclass of closed skew normals constructed using affine transformation of independent standardized univariate skew normals as the variational density, and illustrate how it provides increased flexibility and accuracy in approximating the joint posterior in various applications, by overcoming limitations in existing skew normal variational approximations. The evidence lower bound is optimized using stochastic gradient ascent, where analytic natural gradient updates are derived. We also demonstrate how problems in maximum likelihood estimation of skew normal parameters occur similarly in stochastic variational inference, and can be resolved using the centered parameterization. Supplementary materials for this article are available online.

James-Stein estimator improves accuracy and sample efficiency in human kinematic and metabolic data.

Human biomechanical data are often accompanied with measurement noise and behavioral variability. Errors due to such noise and variability are usually exaggerated by fewer trials or shorter trial durations, and could be reduced using more trials or longer trial durations. Speeding up such data collection by lowering number of trials or trial duration, while improving the accuracy of statistical estimates, would be of particular interest in wearable robotics applications and when the human population studied is vulnerable (e.g., the elderly). Here, we propose the use of the James-Stein estimator (JSE) to improve statistical estimates with a given amount of data, or reduce the amount of data needed for a given accuracy. The JSE is a shrinkage estimator that produces a uniform reduction in the summed squared errors when compared to the more familiar maximum likelihood estimator (MLE), simple averages, or other least squares regressions. When data from multiple human participants are available, an individual participant's JSE can improve upon MLE by incorporating information from all participants, improving overall estimation accuracy on average. Here, we apply the JSE to multiple time-series of kinematic and metabolic data from the following parameter estimation problems: foot placement control during level walking, energy expenditure during circle walking, and energy expenditure during resting. We show that the resulting estimates improve accuracy - that is, the James-Stein estimates have lower summed squared error from the 'true' value compared to more conventional estimates.

Bayesian inference for link travel time correlation of a bus route

Estimation of link travel time correlation of a bus route is essential to many bus operation applications. Link travel time on a bus route could exhibit complex correlation structures, such as long-range correlations, negative correlations, and time-varying correlations. This paper develops a Bayesian Gaussian model to estimate the link travel time correlation matrix of a bus route using smart-card-like data. Our method overcomes the small-sample-size problem in correlation matrix estimation by borrowing/integrating those incomplete observations from other bus routes. We first conduct a synthetic experiment and results show that the proposed method produces an accurate estimation for correlations with credible intervals. Next, we perform experiments on a real-world bus route with in-out-stop record data; results show that both local and long-range correlations exist on this bus route. Finally, we demonstrate an application of using the estimated covariance matrix to make probabilistic forecasting of link and trip travel time.