Batch Estimation Research Articles

MAP estimation of target manoeuvre sequence with the expectation-maximization algorithm

Two algorithms are derived for the problem of tracking a manoeuvring target based on a sequence of noisy measurements of the state. Manoeuvres are modeled as unknown input (acceleration) terms entering linearly into the state equation and chosen from a discrete set. The expectation maximization (EM) algorithm is first applied, resulting in a multi-pass estimator of the MAP sequence of inputs. The expectation step for each pass involves computation of state estimates in a bank of Kalman smoothers tuned to the possible manoeuvre sequences. The maximization computation is efficiently implemented using the Viterbi algorithm. A second, recursive estimator is then derived using a modified EM-type cost function. To obtain a dynamic programming recursion, the target state is assumed to satisfy a Markov property with respect to the manoeuvre sequence. This results in a recursive but suboptimal estimator implementable on a Viterbi trellis. The transition costs of the latter algorithm, which depend on filtered estimates of the state, are compared with the costs arising in a Viterbi-based manoeuvre estimator due to Averbuch, et al. (1991). It is shown that the two criteria differ only in the weighting matrix of the quadratic part of the cost function. Simulations are provided to demonstrate the performance of both the batch and recursive estimators compared with Averbuch's method and the interacting multiple model filter.

Semiparametric model and superefficiency in blind deconvolution

In this paper, we study convergence and efficiency of the batch estimator and natural gradient algorithm for blind deconvolution. First, the blind deconvolution problem is formulated in the framework of a semiparametric model, and a family of estimating functions is derived for blind deconvolution. To improve the learning efficiency of the online algorithm, explicit standardized estimating functions are given and within this framework the superefficiency of batch learning and online natural gradient learning is proven.

Bearings-only tracking of maneuvering targets using a batch-recursive estimator

We present a new batch-recursive estimator for tracking maneuvering targets from bearings-only measurements in clutter (i.e., for low signal-to-noise ratio (SNR) targets), Standard recursive estimators like the extended Kalman Iter (EKF) suffer from poor convergence and erratic behavior due to the lack of initial target range information, On the other hand, batch estimators cannot handle target maneuvers. In order to rectify these shortcomings, we combine the batch maximum likelihood-probabilistic data association (ML-PDA) estimator with the recursive interacting multiple model (IMM) estimator with probabilistic data association (PDA) to result in better track initialization as well as track maintenance results in the presence of clutter. It is also demonstrated how the batch-recursive estimator can be used for adaptive decisions for ownship maneuvers based on the target state estimation to enhance the target observability. The tracking algorithm is shown to be effective for targets with 8 dB SNR.

Efficient identification of non-Gaussian mixtures

Non-Gaussian mixtures are commonly used as parametric models for impulsive interference superimposed on a Gaussian background. In this study, the problem of efficient batch and recursive estimation of the parameters of such mixtures via minimization of the Kullback-Leibler distance is considered. The maximum-likelihood estimator (MLE) provides the starting point for the development of these estimators. First, it is shown that the MLE yields consistent estimates of the parameters, despite the existence of multiple roots in the Kullback-Leibler distance function. Since direct implementation of the MLE is difficult, an alternative estimator, designed with the objective of maximizing the likelihood function, is proposed. A simulation study of the proposed estimator reveals that it performs very well in terms of attaining the Cramer-Rao lower bound. Stochastic approximation is also considered in this study. A modification to the standard recursion is presented, which greatly facilitates its implementation. Three initiating estimators are developed for these recursions: an iterative likelihood-based scheme, a batch estimator that uses a histogram of the data, and an iterative scheme that integrates the concept of complete data into the method-of-moments estimator. Upon initiating the modified stochastic approximation recursion with these estimators, it is seen that excellent estimates of the parameters can be obtained.

On eigenvalue decomposition estimators of centro-symmetric covariance matrices

This paper is focused on estimators, both batch and adaptive, of the eigenvalue decomposition (EVD) of centro-symmetric (CS) covariance matrices. These estimators make use of the property that eigenvectors and eigenvalues of such structured matrices can be estimated via two decoupled eigensystems. As a result, the number of operations is roughly halved, and moreover, the statistical properties of the estimators are improved. After deriving the asymptotic distribution of the EVD estimators, the closed-form expressions of the asymptotic bias and covariance of the EVD estimators are compared to those obtained when the CS structure is not taken into account. As a by-product, we show that the closed-form expressions of the asymptotic bias and covariance of the batch and adaptive EVD estimators are very similar provided that the number of samples is replaced by the inverse of the step size. Finally, the accuracy of our asymptotic analysis is checked by numerical simulations, and it is found that the convergence speed is also improved thanks to the use of the CS structure.

On adaptive EVD asymptotic distribution of centro-symmetric covariance matrices

This article investigates the gain in statistical performance/complexity of the adaptive estimation of the eigenvalue decomposition (EVD) of covariance matrices when the centro-symmetric (CS) structure of such matrices is utilized. After deriving the asymptotic distribution of the EVD estimators, it is shown, in particular, that the closed-form expressions for the asymptotic covariance of batch and adaptive EVD estimators are very similar, provided that the number of samples is replaced by the inverse of the step size.

Superefficiency in blind source separation

Blind source separation is the problem of extracting independent signals from their mixtures without knowing the mixing coefficients nor the probability distributions of source signals and may be applied to EEG and MEG imaging of the brain. It is already known that certain algorithms work well for the extraction of independent components. The present paper is concerned with superefficiency of these based on the statistical and dynamical analysis. In a statistical estimation using t examples, the covariance of any two extracted independent signals converges to 0 of the order of 1/t. On-line dynamics shows that the covariance is of the order of /spl eta/ when the learning rate /spl eta/ is fixed to a small constant. In contrast with the above general properties, a surprising superefficiency holds in blind source separation under certain conditions where superefficiency implies that covariance decreases in the order of 1/t/sup 2/ or of /spl eta//sup 2/. The paper uses the natural gradient learning algorithm and method of estimating functions to obtain superefficient procedures for both batch estimation and on-line learning. A standardized estimating function is introduced to this end. Superefficiency does not imply that the error variances of the extracted signals decrease in the order of 1/t/sup 2/ or /spl eta//sup 2/ but implies that their covariances (and independencies) do.

REAL-TIME MODAL PARAMETER ESTIMATION USING SUBSPACE METHODS: THEORY

This article describes the underlying theory of a newly developed algorithm for online modal parameter identification. These online subspace estimation methods use eigenanalysis for data filtering, and are derived from a recent multi-input, multi-output batch algorithm. One method is obtained by deriving a new efficient data update expression combined with a recently developed modified singular value decomposition known as the URV method. The second method combines an existing data update expression with the URV method. The URV method enables recursive update of the signal subspace. The close relationship of a modified form of the batch estimation approach to the Eigensystem Realization Algorithm (ERA) is also shown through the introduction of an extended ERA method.

Natural Gradient Works Efficiently in Learning

When a parameter space has a certain underlying structure, the ordinary gradient of a function does not represent its steepest direction, but the natural gradient does. Information geometry is used for calculating the natural gradients in the parameter space of perceptrons, the space of matrices (for blind source separation), and the space of linear dynamical systems (for blind source deconvolution). The dynamical behavior of natural gradient online learning is analyzed and is proved to be Fisher efficient, implying that it has asymptotically the same performance as the optimal batch estimation of parameters. This suggests that the plateau phenomenon, which appears in the backpropagation learning algorithm of multilayer perceptrons, might disappear or might not be so serious when the natural gradient is used. An adaptive method of updating the learning rate is proposed and analyzed.

Asymptotic performance analysis of subspace adaptive algorithms introduced in the neural network literature

In the neural network literature, many algorithms have been proposed for estimating the eigenstructure of covariance matrices. We first show that many of these algorithms, when presented in a common framework, show great similitudes with the gradient-like stochastic algorithms usually encountered in the signal processing literature. We derive the asymptotic distribution of these different recursive subspace estimators. A closed-form expression of the covariances in distribution of eigenvectors and associated projection matrix estimators are given and analyzed. In particular, closed-form expressions of the mean square error of these estimators are given. It is found that these covariance matrices have a structure very similar to those describing batch estimation techniques. The accuracy of our asymptotic analysis is checked by numerical simulations, and it is found to be valid not only for a small step size but in a very large domain. Finally, convergence speed and deviation from orthonormality of the different algorithms are compared, and several tradeoffs are analyzed.

Asymptotic distributions associated to Oja's learning equation for neural networks

In this paper, we perform a complete asymptotic performance analysis of the stochastic approximation algorithm (denoted subspace network learning algorithm) derived from Oja's learning equation, in the case where the learning rate is constant and a large number of patterns is available. This algorithm drives the connection weight matrix W to an orthonormal basis of a dominant invariant subspace of a covariance matrix. Our approach consists in associating to this algorithm a second stochastic approximation algorithm that governs the evolution of WWT to the projection matrix onto this dominant invariant subspace. Then, using a general result of Gaussian approximation theory, we derive the asymptotic distribution of the estimated projection matrix. Closed form expressions of the asymptotic covariance of the projection matrix estimated by the SNL algorithm, and by the smoothed SNL algorithm that we introduce, are given in case of independent or correlated learning patterns and are further analyzed. It is found that the structures of these asymptotic covariance matrices are similar to those describing batch estimation techniques. The accuracy or our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a "small" learning rate but in a very large domain. Finally, improvements brought by our smoothed SNL algorithm are shown, such as the learning speed/misadjustment tradeoff and the deviation from orthonormality.

Performance analysis of an adaptive algorithm for tracking dominant subspaces

This paper provides a performance analysis of a least mean square (LMS) dominant invariant subspace algorithm. Based on an unconstrained minimization problem, this algorithm is a stochastic gradient algorithm driving the columns of a matrix W to an orthonormal basis of a dominant invariant subspace of a correlation matrix. We consider the stochastic algorithm governing the evolution of WW/sup H/ to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution. A closed-form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this LMS type algorithm. In particular, it is shown that even though the algorithm does not constrain W to have orthonormal columns, there is deviation from orthonormality of the first order. We also give a closed-form expression of the asymptotic covariance of the DOAs estimated by the MUSIC algorithm applied to the adaptive estimate of the projector. It is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion. Finally, the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a small step size but in a very large domain.

Minimum Model Error Approach for Attitude Estimation

In this paper, an optimal batch estimator and smoother based on the Minimum Model Error (MME) approach is developed for three-axis stabilized spacecraft. The formulation described in this paper is shown using only attitude sensors (e.g., three-axis magnetometers, sun sensors, star trackers, etc). This algorithm accurately estimates the attitude of a spacecraft, and substantially smoothes noise associated with attitude sensor measurements. The general functional form of the optimal estimation approach involves the solution of a nonlinear two-point-boundary-valueproblem that can only be solved using computational intense methods. A linearized solution is also shown that is computationally more efficient than methods which solve the general form. The linearized solution is useful when an a priori estimate of the angular velocity is already known, which may be obtained from a finite difference of a determined attitude, or from propagation of a dynamics model. Results using this new algorithm indicate that an MME-based approach accurately estimates the attitude of an actual spacecraft with the sole use of magnetometer sensor measurements. 1 Assistant Professor, Catholic University of America, Dept. of Mech. Eng., Washington, D.C. 20064. Member AIAA. 2 Staff Engineer, Goddard Space Flight Center, Code 712, Greenbelt, MD 20771. Associate Fellow AIAA

Maximum likelihood estimation of fractional Brownian motion and Markov noise parameters

Abstract : Maximum likelihood estimation and power spectral density analysis are developed as tools for the analysis of stochastic processes. Some useful results from the theory of Markov stochastic processes are then presented followed by the introduction of fractional Brownian motion and fractional Gaussian noise as non-Markov models for systems with power spectral density proportional to fBeta, where -3 < Beta < -1 and -1 < Beta < 1 over all frequencies. Maximum likelihood system identification is applied to estimating the unknown parameters in a Markov model which approximates fractional Brownian motion. The algorithm runs a Kalman filter on states and a maximum likelihood estimator on parameters. Results are presented from estimating trend, white noise, random walk, and exponentially correlated noise parameters from fits to simulated and real test data. Maximum likelihood estimation is applied to the batch estimation of parameters in the non-Markov fractional Brownian motion model. New in this thesis is the use of partial derivatives to minimize the resulting likelihood function, and the capability to estimate the unknown parameters of additional trend and Markov noise processes. Results are presented from fits to computer simulated sample paths.

Modelling of batch fermentation and estimation of state variables using self-organizing feature maps

A self-organizing feature map was used for modelling of batch yeast cultures. The model was constructed by training the neural network with experimental data of the specific rates. Estimates of state variables were obtained from the neural network model and differential mass balance equations via integration. They were compared with the experimental data. The neural network model showed a good modelling accuracy and interpolation capability.

Parameter estimation for trace element sorption on a new granular iron oxide

AbstractIn order to acquire information on sorptive systems suitable for predictive and design applications, it is necessary to gather and analyze data in bench or prototype reactors, the most common types being batch and packed column. In this paper, data from both types of reactor are presented and contrasted for the case of two trace elements, cadmium and chromate, sorbing to a granular form of iron oxide. The granular oxide was developed in conjunction with the ALCOA Laboratories specifically for use in column reactors.Data are analyzed using a surface coordination chemical approach and film transfer, surface and pore diffusion transport models. Governing coefficients are extracted using nonlinear optimization techniques which provide an objective interpretation of the data without the errors introduced by various linear transformations or analysis bias.Results indicate that both experimental techniques are capable of accurately measuring the equilibrium coordination constant for sorption reactions, however batch estimates for the total sorptive capacities are consistently lower than column estimates, which range from 1.1 mg/g for Cd to 3.3 mg/g for CrO4. In addition, when column adsorption and desorption data are taken together, it is possible to obtain reliable estimates of rate‐controlling parameters. This is not generally possible for batch systems under conditions representative of those encountered during environmental control.

Bias and Optimal Linear Estimation in Comparison Calibrations

Comparison calibration designs are rank insufficient to permit a purely batch estimation of parameters, and require the input of some a priori knowledge. The estimator commonly used, which is based on Restrained Least Squares, is shown to be inappropriate because of its inability to take account of uncertainty in the a priori knowledge. A model in which the parameters represent the state of a dynamic stochastic system is proposed together with the appropriate recursive estimator, namely the Kalman filter-predictor. This estimator is less affected than Restrained Least Squares by errors in the a priori knowledge. Its application in mass comparisons in conjunction with a fully recursive approach, i.e. with use of all the available a priori knowledge, is discussed.

Generalized covariance analysis for partially autonomous deep space missions

A new covariance analysis method is presented that is suitable for the evaluation of multiple impulsive controllers acting on some stochastic process x. The method accommodates batch and sequential estimators with equal ease and accounts for time-delay effects in a natural manner. The formalism is developed in terms of a generalized state vector that is formed from the system state vector x, augmented by various fixed epoch estimates, and a data vector formed from discrete time observations of the system. Recursions are developed for time transition, measurement incorporation, and impulsive control updating of the generalized covariance matrix. Means of limiting the dimensional growth of the generalized state vector via the processes of estimator epoch adjustment and measurement vector deflation are described and the application of numerically stable matrix factorization methods to the generalized covariance recursions is outlined. The method is applied to the Magellan spacecraft to demonstrate the capability of ground-based optimal estimation and control of gyro/star scanner misalignment.

Parameter identification algorithms in max-algebra

This paper studies the parameter identification problems in Max-algebra sense. For deterministic regressive equation in Max-algebra, batch and recursive parameter estimate algorithms are studied and given, respectively. A brief discussion about the solution properties and both of convergence and consistence of these parameter estimate algorithms are carried out. For stochastic regressive equation, recursive parameter estimate algorithm is given and its convergence is discussed briefly. Two computer simulations for deterministic and stochastic systems, respectively, are carried out and these good simulation results show us the effectiveness of these parameter algorithms proposed in this paper.

Recursive 3-D motion estimation from a monocular image sequence

Consideration is given to the design and application of a recursive algorithm to a sequence of images of a moving object to estimate both its structure and kinematics. The object is assumed to be rigid, and its motion is assumed to be smooth in the sense that it can be modeled by retaining an arbitrary number of terms in the appropriate Taylor series expansions. Translational motion involves a standard rectilinear model, while rotational motion is described with quaternions. Neglected terms of the Taylor series are modeled as process noise. A state-space model is constructed, incorporating both kinematic and structural states, and recursive techniques are used to estimate the state vector as a function of time. A set of object match points is assumed to be available. The problem is formulated as a parameter estimation and tracking problem which can use an arbitrarily large number of images in a sequence. The recursive estimation is done using an iterated extended Kalman filter (IEKF), initialized with the output of a batch algorithm run on the first few frames. Approximate Cramer-Rao lower bounds on the error covariance of the batch estimate are used as the initial state estimate error covariance of the IEKF. The performance of the recursive estimator is illustrated using both real and synthetic image sequences. >