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This paper investigates a time-varying version of weak-form market efficiency in the BRICS countries. A moving window test for sample autocorrelations is applied alongside a Kalman filter approach to recover the hidden dynamics of the market efficiency process through appropriate time-varying autoregressive models with both homoscedastic and heteroscedastic conditional variance. Monthly data covers the period from January 1995 to December 2020, which includes the 2008-2009 global financial crisis and the recent COVID-19 recession. The results reveal that all the BRICS stock markets were affected during both periods, but generally remained weak-form efficient, with the exception of China., Comment: arXiv admin note: text overlap with arXiv:2310.04125

In this paper, we continue to study the derivative-free extended Kalman filtering (DF-EKF) framework for state estimation of continuous-discrete nonlinear stochastic systems. Having considered the Euler-Maruyama and It\^{o}-Taylor discretization schemes for solving stochastic differential equations, we derive the related filters' moment equations based on the derivative-free EKF principal. In contrast to the recently derived MATLAB-based continuous-discrete DF-EKF techniques, the novel DF-EKF methods preserve an information about the underlying stochastic process and provide the estimation procedure for a fixed number of iterates at the propagation steps. Additionally, the DF-EKF approach is particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, but the price to be paid is its degraded numerical stability (to roundoff) compared to the standard EKF framework. To eliminate the mentioned pitfall of the derivative-free EKF methodology, we develop the conventional algorithms together with their stable square-root implementation methods. In contrast to the published DF-EKF results, the new square-root techniques are derived within both the Cholesky and singular value decompositions. A performance of the novel filters is demonstrated on a number of numerical tests including well- and ill-conditioned scenarios.

One of the modern research lines in econometrics studies focuses on translating a wide variety of structural econometric models into their state-space form, which allows for efficient unknown dynamic system state and parameter estimations by the Kalman filtering scheme. The mentioned trend yields advanced state-space model structures, which demand innovative estimation techniques driven by application requirements to be devised. This paper explores both the linear time-invariant multiple-input, multiple-output system (LTI MIMO) and the pairwise Markov model (PMM) with the related pairwise Kalman filter (PKF). In particular, we design robust gradient-based adaptive Kalman-like filtering methods for the simultaneous state and parameter estimation in the outlined model structures. Our methods are fast and accurate because their analytically computed gradient is utilized in the calculation instead of its numerical approximation. Also, these employ the numerically robust $UDU^\top$-factorization-based Kalman filter implementation, which is reliable in practice. Our novel techniques are examined on numerical examples and used for treating one stochastic model in econometrics.

In this paper, a singular value decomposition (SVD) approach is developed for implementing the cubature Kalman filter. The discussed estimator is one of the most popular and widely used method for solving nonlinear Bayesian filtering problem in practice. To improve its numerical stability (with respect to roundoff errors) and practical reliability of computations, the SVD-based methodology recently proposed for the classical Kalman filter is generalized on the nonlinear filtering problem. More precisely, we suggest the SVD-based solution for the continuous-discrete cubature Kalman filter and design two estimators: (i) the filter based on the traditionally used Euler-Maruyama discretization scheme; (ii) the estimator based on advanced It\^{o}-Taylor expansion for discretizing the underlying stochastic differential equations. Both estimators are formulated in terms of SVD factors of the filter error covariance matrix and belong to the class of stable factored-form (square-root) algorithms. The new methods are tested on a radar tracking problem.

In mathematical neuroscience, a special interest is paid to a working memory mechanism in the neural tissue modeled by the Dynamic Neural Field (DNF) in the presence of model uncertainties. The working memory facility implies that the neurons' activity remains self-sustained after the external stimulus removal due to the recurrent interactions in the networks and allows the system to cope with missing sensors' information. In our previous works, we have developed two reconstruction methods of the neural membrane potential from {\it incomplete} data available from the sensors. The methods are derived within the Extended Kalman filtering approach by using the Euler-Maruyama method and the It\^{o}-Taylor expansion of order 1.5. It was shown that the It\^{o}-Taylor EKF-based restoration process is more accurate than the Euler-Maruyama-based alternative. It improves the membrane potential reconstruction quality in case of incomplete sensors information. The aim of this paper is to investigate their pattern recognition facilities, i.e. the quality of the pattern formation reconstruction in case of model uncertainties and incomplete information. The numerical experiments are provided for an example of the stochastic DNF with multiple active zones arisen in a neural tissue.

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

A stable square-root approach has been recently proposed for the unscented Kalman filter (UKF) and fifth-degree cubature Kalman filter (5D-CKF) as well as for the mixed-type methods consisting of the extended Kalman filter (EKF) time update and the UKF/5D-CKF measurement update steps. The mixed-type estimators provide a good balance in trading between estimation accuracy and computational demand because of the EKF moment differential equations involved. The key benefit is a consolidation of reliable state mean and error covariance propagation by using delicate discretization error control while solving the EKF moment differential equations and an accurate measurement update according to the advanced UKF and/or 5D-CKF filtering strategies. Meanwhile the drawback of the previously proposed estimators is an utilization of sophisticated numerical integration scheme with the built-in discretization error control that is, in fact, a complicated and computationally costly tool. In contrast, we design here the mixed-type methods that keep the same estimation quality but reduce a computational time significantly. The novel estimators elegantly utilize any MATLAB-based numerical integration scheme developed for solving ordinary differential equations (ODEs) with the required accuracy tolerance pre-defined by users. In summary, a simplicity of the suggested estimators, their numerical robustness with respect to roundoff due to the square-root form utilized as well as their estimation accuracy due to the MATLAB ODEs solvers with discretization error control involved are the attractive features of the novel estimators. The numerical experiments are provided for illustrating a performance of the suggested methods in comparison with the existing ones.

This paper continues our research devoted to an accurate nonlinear Bayesian filters' design. Our solution implies numerical methods for solving ordinary differential equations (ODE) when propagating the mean and error covariance of the dynamic state. The key idea is that an accurate implementation strategy implies the methods with a discretization error control involved. This means that the filters' moment differential equations are to be solved accurately, i.e. with negligible error. In this paper, we explore the continuous-discrete extended-cubature Kalman filter that is a hybrid method between Extended and Cubature Kalman filters (CKF). Motivated by recent results obtained for the continuous-discrete CKF in Bayesian filtering realm, we propose the numerically stable (to roundoff) square-root approach within a singular value decomposition (SVD) for the hybrid filter. The new method is extensively tested on a few application examples including stiff systems.

The maximum correntropy criterion (MCC) methodology is recognized to be a robust filtering strategy with respect to outliers and shown to outperform the classical Kalman filter (KF) for estimation accuracy in the presence of non-Gaussian noise. However, the numerical stability of the newly proposed MCC-KF estimators in finite precision arithmetic is seldom addressed. In this paper, a family of {\it factored-form} (square-root) algorithms is derived for the MCC-KF and its improved variant, respectively. The family traditionally consists of three factored-form implementations: (i) Cholesky factorization-based algorithms, (ii) modified Cholesky, i.e. UD-based methods, and (iii) the recently established SVD-based filtering. All these strategies are commonly recognized to enhance the numerical robustness of conventional filtering with respect to roundoff errors and, hence, they are the preferred implementations when solving applications with high reliability requirements. Previously, only Cholesky-based IMCC-KF algorithms have been designed. This paper enriches a factored-form family by introducing the UD- and SVD-based methods as well. A special attention is paid to {\it array} algorithms that are proved to be the most numerically stable and, additionally, suitable for parallel implementations. The theoretical properties are discussed and numerical comparison is presented for determining the most reliable implementations.

This paper continues the research devoted to the design of numerically stable square-root implementations for the maximum correntropy criterion Kalman filtering (MCC-KF). In contrast to the previously obtained results, here we reveal the first robust (with respect to round-off errors) method within the Cholesky factorization-based approach. The method is formulated in terms of square-root factors of the {\it covariance} matrices, i.e. it belongs to the covariance-type filtering methodology. Additionally, a numerically stable orthogonal transformation is utilized at each iterate of the algorithm for accurate propagation of the Cholesky factors involved. The results of numerical experiments illustrate a superior performance of the novel MCC-KF implementation compared to both the conventional algorithm and its previously published Cholesky-based variant.

Recursive adaptive filtering methods are often used for solving the problem of simultaneous state and parameters estimation arising in many areas of research. The gradient-based schemes for adaptive Kalman filtering (KF) require the corresponding filter sensitivity computations. The standard approach is based on the direct differentiation of the KF equations. The shortcoming of this strategy is a numerical instability of the conventional KF (and its derivatives) with respect to roundoff errors. For decades, special attention has been paid in the KF community for designing efficient filter implementations that improve robustness of the estimator against roundoff. The most popular and beneficial techniques are found in the class of square-root (SR) or UD factorization-based methods. They imply the Cholesky decomposition of the corresponding error covariance matrix. Another important matrix factorization method is the singular value decomposition (SVD) and, hence, further encouraging KF algorithms might be found under this approach. Meanwhile, the filter sensitivity computation heavily relies on the use of matrix differential calculus. Previous works on the robust KF derivative computation have produced the SR- and UD-based methodologies. Alternatively, in this paper we design the SVD-based approach. The solution is expressed in terms of the SVD-based KF covariance quantities and their derivatives (with respect to unknown system parameters). The results of numerical experiments illustrate that although the newly-developed SDV-based method is algebraically equivalent to the conventional approach and the previously derived SR- and UD-based strategies, it outperforms the mentioned techniques for estimation accuracy in ill-conditioned situations., Comment: 6 pages, 2 tables. Accepted to IEEE Transactions on Automatic Control

This technical note addresses the UD factorization based Kalman filtering (KF) algorithms. Using this important class of numerically stable KF schemes, we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications. For instance, it can be used for efficient calculations in sensitivity analysis and in gradient-search optimization algorithms for the maximum likelihood estimation. The new theory presented in this technical note is a solution to the problem formulated by Bierman in , which has been open since 1990s. As in the cited paper, our method avoids the standard approach based on the conventional KF (and its derivatives with respect to unknown system parameters) with its inherent numerical instabilities and, hence, improves the robustness of computations against roundoff errors., Comment: The revised version of this preprint has been accepted for publication in IEEE Transactions on Automatic Control. arXiv admin note: text overlap with arXiv:1303.4622

The paper presents a new Kalman filter (KF) implementation useful in applications where the accuracy of numerical solution of the associated Riccati equation might be crucially reduced by influence of roundoff errors. Since the appearance of the KF in 1960s, it has been recognized that the factored-form of the KF is preferable for practical implementation. The most popular and beneficial techniques are found in the class of square-root algorithms based on the Cholesky decomposition of error covariance matrix. Another important matrix factorization method is the singular value decomposition (SVD) and, hence, further encouraging implementations might be found under this approach. The analysis presented here exposes that the previously proposed SVD-based KF variant is still sensitive to roundoff errors and poorly treats ill-conditioned situations, although the SVD-based strategy is inherently more stable than the conventional KF approach. In this paper we design a new SVD-based KF implementation for enhancing the robustness against roundoff errors, provide its detailed derivation, and discuss the numerical stability issues. A set of numerical experiments are performed for comparative study. The obtained results illustrate that the new SVD-based method is algebraically equivalent to the conventional KF and to the previously proposed SVD-based method, but it outperforms the mentioned techniques for estimation accuracy in ill-conditioned situations., Comment: Revised version. Following the strong recommendation of the referee, only one SVD-based method is left. 7 pages, 2 tables

Recent developments in the realm of state estimation of stochastic dynamic systems in the presence of non-Gaussian noise have induced a new methodology called the maximum correntropy filtering. The filters designed under the maximum correntropy criterion (MCC) utilize a similarity measure (or correntropy) between two random variables as a cost function. They are shown to improve the estimators' robustness against outliers or impulsive noises. In this paper we explore the numerical stability of linear filtering technique proposed recently under the MCC approach. The resulted estimator is called the maximum correntropy criterion Kalman filter (MCC-KF). The purpose of this study is two-fold. First, the previously derived MCC-KF equations are revise and the related Kalman-like equality conditions are proved. Based on this theoretical finding, we improve the MCC-KF technique in the sense that the new method possesses a better estimation quality with the reduced computational cost compared with the previously proposed MCC-KF variant. Second, we devise some square-root implementations for the newly-designed improved estimator. The square-root algorithms are well known to be inherently more stable than the conventional Kalman-like implementations, which process the full error covariance matrix in each iteration step of the filter. Additionally, following the latest achievements in the KF community, all square-root algorithms are formulated here in the so-called array form. All the MCC-KF variants developed in this paper are demonstrated to outperform the previously proposed MCC-KF version in two numerical examples., Comment: The paper is accepted for publication in Systems & Control Letters, 2017

Using the array form of numerically stable square-root implementation methods for Kalman filtering formulas, we construct a new square-root algorithm for the log-likelihood gradient (score) evaluation. This avoids the use of the conventional Kalman filter with its inherent numerical instabilities and improves the robustness of computations against roundoff errors. The new algorithm is developed in terms of covariance quantities and based on the "condensed form" of the array square-root filter.

This paper addresses the numerical aspects of adaptive filtering (AF) techniques for simultaneous state and parameters estimation arising in the design of dynamic positioning systems in many areas of research. The AF schemes consist of a recursive optimization procedure to identify the uncertain system parameters by minimizing an appropriate defined performance index and the application of the Kalman filter (KF) for dynamic positioning purpose. The use of gradient-based optimization methods in the AF computational schemes yields to a set of the filter sensitivity equations and a set of matrix Riccati-type sensitivity equations. The filter sensitivities evaluation is usually done by the conventional KF, which is known to be numerically unstable, and its derivatives with respect to unknown system parameters. Recently, a novel square-root approach for the gradient-based AF by the method of the maximum likelihood has been proposed. In this paper, we show that various square-root AF schemes can be derived from only two main theoretical results. This elegant and simple computational technique replaces the standard methodology based on direct differentiation of the conventional KF equations (with their inherent numerical instability) by advanced square-root filters (and its derivatives as well). As a result, it improves the robustness of the computations against roundoff errors and leads to accurate variants of the gradient-based AFs. Additionally, such methods are ideal for simultaneous state estimation and parameter identification since all values are computed in parallel. The numerical experiments are given., Comment: The revised version of this preprint has been accepted for publication in International Journal of Adaptive Control and Signal Processing. 15 pages