Filter Recursions Research Articles

Multi-Hypothesis Marginal Multi-Target Bayes Filter for a Heavy-Tailed Observation Noise

A multi-hypothesis marginal multi-target Bayes filter for heavy-tailed observation noise is proposed to track multiple targets in the presence of clutter, missed detection, and target appearing and disappearing. The proposed filter propagates the existence probabilities and probability density functions (PDFs) of targets in the filter recursion. It uses the Student’s t distribution to model the heavy-tailed non-Gaussian observation noise, and employs the variational Bayes technique to acquire the approximate distributions of individual targets. K-best hypotheses, obtained by minimizing the negative log-generalized-likelihood ratio, are used to establish the existence probabilities and PDFs of targets in the filter recursion. Experimental results indicate that the proposed filter achieves better tracking performance than other filters.

SINS/GNSS-Integrated Navigation Based on Group Affine SINS Mechanization in Local-Level Frame

In this article, a group affine strapdown inertial navigation system (SINS) mechanization in local-level frame is investigated for SINS and global navigation satellite system (GNSS) integrated navigation. Traditionally, the SINS mechanization in local-level frame is strong coupling and, therefore, is not group affine. In order to make it group affine, two main procedures are presented. One is making use of damping calculation for SINS, that is, the Coriolis and centrifugal force are ingeniously calculated outside the SINS calculation and filtering recursion using known velocity and position information provided by GNSS. The other is introducing an auxiliary velocity. Then, the trajectory-independent left-invariant error state-space model is derived corresponding to the group affine model. The derived group affine model and its corresponding error state-space model can be used in the SINS/GNSS indirect integration. Simulations and field tests are conducted to evaluate the actual performance of the investigated method. The results show that the proposed method has a much superior performance for SINS/GNSS with large initial attitude error. In contrast, the proposed method has a similar performance with existing state-of-the-art-integrated navigation method when the initial attitude error is small.

The Multiple Model Poisson Multi-Bernoulli Mixture Filter for Extended Target Tracking

In an intelligent transportation system (ITS), the single motion model is not enough to meet the needs of simulation systems (or practical applications) and guarantee good tracking performance for multiple maneuvering extended target tracking (ETT). This paper presents multiple models Poisson multi-Bernoulli mixture (MM-PMBM) filter for extended target tracking to address this problem. To make multiple models transform in an accurate manner, the jump Markov system (JMS) is timely introduced in filtering recursion. After that, the model probability of target is recursively updated and used to estimate the target position. In addition, we derive the closed-form solution to the proposed filter based on the Gamma Gaussian inverse-Wishart (GGIW) implementation. The approach to handling the merging problem of GGIW components via Kullback-Leibler divergence (KLD) minimization is also presented, resulting in better utilization of available components. Simulation results demonstrate that the proposed filter has superior performance in comparison to several state-of-the-art multi-target filters.

Accuracy Evaluation of Marginalized Unscented Kalman Filter

Compared with a conventional unscented Kalman filter (UKF), the recently proposed marginalized unscented Kalman filter (MUKF) uses a partially sampling strategy to achieve similar filter accuracy with fewer sigma points, demonstrating its powerful ability to deal with state estimation with mixed linearity and nonlinearity. However, the hypothesis that the accuracy of MUKF is equivalent to that of UKF is not true for all systems. In this paper, it is proved that when the state equation is expressed as a differential equation, the accuracy of MUKF is equivalent to that of UKF only when the propagations of the states to be sampled and the states not to be sampled in MUKF are independent of each other. The above condition corresponds to a common problem in engineering, which is a system with measurement biases. Through theoretical proof and numerical simulation, it is verified in this paper that for the systems with measurement biases, the differences of the computed means and covariances in filtering recursions between MUKF and UKF are restricted to the same order infinitesimal as the square of the scaling parameter. To sum up, this paper evaluates the accuracy of MUKF, providing references for engineers to choose MUKF or UKF in different systems.

Poisson Multi-Bernoulli Approximations for Multiple Extended Object Filtering

The Poisson multi-Bernoulli mixture (PMBM) is a multiobject conjugate prior for the closed-form Bayes random finite set filter. The extended object PMBM filter provides a closed-form solution for multiple extended object filtering with standard models. This article considers computationally lighter alternatives to the extended object PMBM filter by propagating a Poisson multi-Bernoulli (PMB) density through the filtering recursion. A new local hypothesis representation is presented, where each measurement creates a new Bernoulli component. This facilitates the developments of methods for efficiently approximating the PMBM posterior density after the update step as a PMB. Based on the new hypothesis representation, two approximation methods are presented: one is based on the track-oriented multi-Bernoulli (MB) approximation, and the other is based on the variational MB approximation via Kullback–Leibler divergence minimization. The performance of the proposed PMB filters with gamma Gaussian inverse-Wishart implementations are evaluated in a simulation study.

An enabling framework for automated extraction of signals from market information in real time

This paper proposes a framework for a financial technology that characteristically embeds an automated parameter estimation scheme. Such a scheme ensures that market signals from the most recent observations are reflected readily in trading and decision making. This capitalises on filtering recursions that support a data-analytics scheme in the predictive domain using observed prices revealed in real time. The resulting technology facilitates and strengthens a futures trading environment that evolves dynamically and enhances market liquidity. In the implementation of our framework’s algorithms, we choose the fish futures market data in the context of providing the seafood market stability and sustainability. We develop a knowledge-based related mechanism, via a higher-order Markov chain and online filtering, that brings about price discovery and serves an important risk management tool for fish producers and buyers. Our results demonstrate the computational efficiency, flexibility in capturing price dynamics with complex features and adaptability to other financial markets of our proposed automation infrastructure.

Motion Modeling and State Estimation in Range-Squared Coordinate

In some target tracking applications, angle measurements are absent due to sensor characteristics, which makes Cartesian states unobservable for a single stationary sensor. When multi-sensor target tracking techniques with only range or Doppler measurements are applied to produce Cartesian state estimates, their performance and efficiency can be benefited from accurate range and Doppler estimates obtained from single sensors. This paper investigates the range and Doppler estimation problem based on motion modeling in range-squared coordinate considering range-squared as a generic coordinate. A pseudostate vector consisting of range squared and its derivatives is defined to completely summarize the dynamic of the Cartesian nearly constant velocity (NCV) motion and it is found time evolution of the pseudostate is linear based on the methodology of state space representation. To estimate the pseudostate from range-Doppler or range-only measurements, a recursive filter is developed using the decorrelated unbiased converted measurement (DUCM) method. Finally, the pseudostate estimates in the range-squared coordinate are converted into the range coordinate using the scaled unscented transformation (SUT) method to produce range and Doppler estimates. Although the conversion is nonlinear, it is performed outside the filtering recursions without propagating nonlinearity approximation errors in the filtering recursions. Numerical experiments with various conditions of measurements demonstrate the superior performance benefits from the proposed accurate linear state equation and estimation method.

Computation of the Exact Fisher Information Matrix of a Multiple Input Single Output Time Series Models

Klein, Mélard, and Zahaf (1998) have proposed the computation of the exact Fisher information matrix of a large class of Gaussian time series models called the single-input-single-output (SISO) model, includes dynamic regression with autocorrelated errors and the transfer function model, with autoregressive moving average errors. For computing the Fisher information matrix of a SISO model, they introduced an algorithm based on a combination of two computational procedures: recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters and the fast Kalman filter recursions used in the evaluation of the likelihood function. In this paper, we propose a generalization of this method for computing the Fisher information matrix of a MISO model.

Deception attack monitoring in vulnerable hydroelectric generator system

AbstractThis article deals with unknown deception attacks detection of hydroelectric generator. Such cyber attacks are switching disturbances that affect the input information transmitted from the power system stabilizer to the plant through a constrained communication network. The generator dynamics can be modeled as an augmented state model, whereas the unknown input reconstruction is obtained from a modified Kalman filter recursions under communication constraints. The obtained design ensures the minimization of disturbances with accurate cyberattack monitoring.

Signaling Equilibria for Dynamic LQG Games With Asymmetric Information

In this article, we consider a finite-horizon dynamic game with two players who observe their types privately and take actions that are publicly observed. Players' types evolve as conditionally independent, controlled linear Gaussian processes, and players incur quadratic instantaneous costs. This forms a dynamic linear quadratic Gaussian game with asymmetric information. We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs, form a perfect Bayesian equilibrium (PBE) of the game. This is a signaling equilibrium due to the fact that future beliefs on players' types are affected by the equilibrium strategies. We provide a backward-forward algorithm to find the PBE. Each step of the backward algorithm reduces to solving an algebraic matrix equation for every possible realization of the state estimate covariance matrix. The forward algorithm consists of Kalman filter recursions, where state estimate covariance matrices depend on equilibrium strategies. We provide necessary and sufficient conditions for the existence of the solution of the algebraic matrix equation for scalar actions. Finally, we present numerical examples.

Multiple Model Poisson Multi-Bernoulli Mixture Filter for Maneuvering Targets

The Poisson multi-Bernoulli mixture (PMBM) filter is conjugate prior composed of the union of a Poisson point process (PPP) and a multi-Bernoulli mixture (MBM). Considering that the single model is not enough to guarantee stable tracking performance for maneuvering targets, in this article, a multiple model PMBM (MM-PMBM) filter is proposed to cope with this problem. The proposed MM-PMBM filter extends the single-model PMBM filter recursion to multiple motion models by exploiting the jump Markov system (JMS). The performance of the proposed algorithm is examined from two scenarios with different detection probabilities. Moreover, the robustness of Markovian model transition probability matrices (TPMs) for the proposed MM-PMBM filter is also explored. The simulation results demonstrate that the proposed MM-PMBM filter performs well in terms of the tracking accuracy, including the target states and cardinality estimates, and also has good tolerance with respect to different TPMs.

Online estimation for a predictive analytics platform with a financial-stability-analysis application

An online parameter estimation via filtering recursions is constructed to support a data-analytics scheme in the predictive domain. Multivariate financial market indices or signals revealed in real time are used in our numerical implementation. This work contributes to the analysis and forecasting of financial crises in an environment that evolves dynamically. In particular, we capture the regime-switching characteristics of the Financial Stress Index (FSI) and Industrial Production Index (IPI), designed to detect periods of financial crisis. We integrate two different stochastic models for FSI and IPI deemed to mirror the systemic financial stress levels in the financial and business cycles, respectively. The joint dynamics of the FSI and IPI, exhibit stochasticity, mean reversion, seasonality, and occasional jumps are identified in the most efficient way. All parameters are modulated by a discrete-time hidden Markov chain that switches between economic regimes reflecting various interacting economic forces. Through change of reference probability technique, adaptive multivariate filters are derived which in turn provides online optimal parameter estimates. Historical Canadian economic-based FSI and IPI are examined and an early-warning signal extraction method is put forward to generate alerts at the early stage of some crisis events. Our modelling approach captures the empirical characteristics of FSI and IPI as well as provides auspiciously early warnings for episodes of systemic financial crisis.

Dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series.

Forecasting with the damped trend model using the structural approach

The damped trend model is a strong benchmark for time series forecasting. This model is usually estimated by adopting the innovations approach rather than the structural one, since the latter is more complex, requiring the use of the Kalman filter. In this paper, we introduce a simple method for estimating the damped trend using the structural approach. The proposed method relies on the analytical solution to the algebraic Riccati equation for the covariance matrix of the state vector’s estimation error. The solution fully simplifies both the Kalman filter recursions and the likelihood evaluation. The likelihood evaluation using the proposed method actually becomes very similar to that of the innovations approach. Moreover, the solution facilitates the smoothing of the state vector, which is crucial for signal extraction. A Monte Carlo simulation shows that both innovations and structural approaches have a similar out-of-sample forecasting performance. This is also confirmed empirically by working with the annual time series from the M3-competition database and with quarterly time series on total credit to the non-financial sector relative to GDP published by the Bank for International Settlements.

Differential flatness theory‐based approach to the control of gas‐turbine electric power generation units

A differential flatness theory-based control and state estimation method has been developed for electric power units that consist of synchronous generators connected to gas turbines. The dynamic model of the power unit satisfies the properties of differential flatness and this allows for its transformation into an input–output linearized form. Moreover, it is shown that the state-space description of the power system can be written in the canonical (Brunvsky) form. Using this, a solution for the power unit's control and state estimation problem is given. First, a stabilizing feedback controller is designed. Moreover, with the use of differential flatness theory-based implementation of the Kalman Filter it becomes possible to solve the state and disturbances estimation problem of the gas-turbine power unit. The considered filtering method, under the name of Derivative-free nonlinear Kalman Filter, consists of application of the Kalman Filter's recursion on the linearized equivalent model of the power system, and of an inverse transformation that allows for computing estimates for the state variables of the initial nonlinear system. By redesigning the aforementioned Kalman Filter as a disturbance observer one can also identify and annihilate in real time exogenous perturbations.

Multidimensional Information Fusion in Active Sonar via the Generalized Labeled Multi-Bernoulli Filter

Under the conditions of low detection probabilities, high clutter rates, low data-sampling rates, large measurement errors, and unknown prior information of the target position, multi-object tracking is difficult. This paper proposes a multidimensional information fusion method in active sonar via the generalized labeled multi-Bernoulli (GLMB) filter. After modeling the position measurement, radial velocity measurement and amplitude measurement of the target and clutter, new target births are adaptively generated by the measurement-driven model, predictions are made by the target motion model and updates are performed via multidimensional measurements with the generalized likelihood in the GLMB filter, which enables the measurement information of different dimensions to be elegantly applied to information fusion and significantly improves the filter performance. The contribution of specific dimension measurement to fusion can be evaluated by the Kullback-Leibler (KL) divergence. In the efficient implementation, we propose flat Gibbs sampling to realize multiple hypothesis optimization. Moreover, the filtering recursion is derived from Gaussian mixtures. Simulations are presented to verify the proposed method.

Trajectory Poisson Multi-Bernoulli Filters

This paper presents two trajectory Poisson multi-Bernoulli (TPMB) filters for multi-target tracking: one to estimate the set of alive trajectories at each time step and another to estimate the set of all trajectories, which includes alive and dead trajectories, at each time step. The filters are based on propagating a Poisson multi-Bernoulli (PMB) density on the corresponding set of trajectories through the filtering recursion. After the update step, the posterior is a PMB mixture (PMBM) so, in order to obtain a PMB density, a Kullback-Leibler divergence minimisation on an augmented space is performed. The developed filters are computationally lighter alternatives to the trajectory PMBM filters, which provide the closed-form recursion for sets of trajectories with Poisson birth model, and are shown to outperform previous multi-target tracking algorithms.

SVD-based state and parameter estimation approach for generalized Kalman filtering with application to GARCH-in-Mean estimation

In this paper, a gradient-based adaptive filtering technique for unknown state and parameter estimation is proposed for some extensions of classical state-space models: (i) linear time-invariant multiple-input, multiple-output (LTI MIMO) systems, and (ii) linear pairwise Markov models (PMMs) with the related pairwise Kalman filter (PKF). The solution to robust adaptive filtering problem is grounded in the singular value decomposition (SVD) recently adopted for the classical Kalman filter. In contrast to our previous result, the new methods are designed for the case of thin matrices (pre-arrays) involved in the filtering recursions and sensitivities calculation. Additionally, the paper discusses a time-varying variance models’ calibration by the proposed innovative adaptive filters. In particular, the state-space approach for estimating GARCH-in-Mean(1,1) is explored.

Fault diagnosis of gas-turbine power units with the derivative-free nonlinear Kalman Filter

A method is developed for diagnosing faults and cyberattacks in electric power generation units that consist of a gas-turbine and of a synchronous generator. By proving that such a power generation unit is differentially flat its transformation into an input–output linearized form becomes possible. Moreover, by applying the Derivative-free nonlinear Kalman Filter, state estimation for the power unit is performed. The latter filtering method, consists of the Kalman Filter's recursion on the linearized equivalent model of the power unit, as well as of an inverse transformation providing estimates of the initial nonlinear system. By subtracting the estimated outputs of the Kalman Filter from the measured outputs of the power unit the residuals’ sequence is generated. The residuals undergo statistical processing. It is shown that the sum of the squares of the residuals’ vectors, weighted by the associated covariance matrix, forms a stochastic variable that follows the χ2 distribution. By exploiting the statistical properties of this distribution, confidence intervals are defined, which allow for detecting the power unit's malfunctioning. As long the aforementioned stochastic variable remains within the previous confidence intervals the normal functioning of the power unit is inferred. Otherwise, a fault or cyber-attack is detected. It is also shown that by applying the statistical method into subspaces of the system's state-space model, fault or cyber-attack isolation can be also performed.

A Parametric Factor Model of the Term Structure of Mortality

The prototypical Lee–Carter mortality model is characterized by a single common time factor that loads differently across age groups. In this paper, we propose a parametric factor model for the term structure of mortality where multiple factors are designed to influence the age groups differently via parametric loading functions. We identify four different factors: a factor common for all age groups, factors for infant and adult mortality, and a factor for the “accident hump” that primarily affects mortality of relatively young adults and late teenagers. Since the factors are identified via restrictions on the loading functions, the factors are not designed to be orthogonal but can be dependent and can possibly cointegrate when the factors have unit roots. We suggest two estimation procedures similar to the estimation of the dynamic Nelson–Siegel term structure model. First, a two-step nonlinear least squares procedure based on cross-section regressions together with a separate model to estimate the dynamics of the factors. Second, we suggest a fully specified model estimated by maximum likelihood via the Kalman filter recursions after the model is put on state space form. We demonstrate the methodology for US and French mortality data. We find that the model provides a good fit of the relevant factors and, in a forecast comparison with a range of benchmark models, it is found that, especially for longer horizons, variants of the parametric factor model have excellent forecast performance.