Showing papers on "Kalman filter published in 1981"

A One-Factor Multivariate Time Series Model of Metropolitan Wage Rates

Abstract: The paper formulates and estimates a single-factor multivariate time series model. The model is a dynamic generalization of the multiple indicator (or factor analysis) model. It is shown to be a special case of the general state space model and can be estimated by maximum likelihood methods using the Kalman filter algorithm. The model is used to obtain estimates of the unobserved metropolitan wage rate for Los Angeles, based on observations of sectoral wages within the Standard Metropolitan Statistical Area. Hypothesis tests, model diagnostics, and out-of-sample forecasts are used to evaluate the model.

Fitting a continuous time autoregression to discrete data

Abstract: A brief review of time series analysis with missing observations and unequally spaced data in both time and frequency domain is presented. The exact likelihood for Gaussian ARMA processes can be calculated using Kalman recursive estimation, and non-linear optimization programs used to calculate the maximum likelihood estimates of the parameters including the observational error variance. This algorithm readily handles missing observations. For unequally spaced data it is necessary to consider continuous time models. This paper considers the problem of fitting a continuous time autoregression (all pole spectrum) with observational error. If sampled at equally spaced time points this process would be ARMA (p, p) so for p > 1 this gives the possibility of a more parsimonious representation. Using a state space representation, the exact likelihood function can be calculated by first performing an orthogonal rotation on the state vector producing an uncoupled equation of state with a diagonal state transition matrix. This greatly simplifies the recursive calculation of the likelihood for equally or unequally spaced data. Non-linear optimization involving constraints to ensure a stationary solution produces maximum likelihood estimates of the parameters and observational error variance. Numerical examples are presented of fitting continuous time models to yearly sunspot numbers and producing forecasts and of fitting models to unequally spaced respiration data.

The Kalman filter - Its recognition and development for aerospace applications

Abstract: HE Kalman filter is widely used today for state estimation in aerospace systems. This well-known filter made the transition from a relatively abstract theory to application in many aerospace systems within a very short period during the early 1960's. This paper describes the recognition of its utility and its subsequent development. The first publicly known Kalman filter application was made during the feasibility studies for the Apollo space program, This paper begins with a review of the events which led to defining the requirements for that filter. Then, a discussion is presented of how the theory came to the author's attention and some of the problems involved in interpreting the theory for potential application to the on-board navigation system for the Apollo mission. Next, a brief review of the dissemination of the findings to the general scientific community is covered. Subsequent sections outline the author's contributions to Kalman filter improvements. The paper thus presents a chronological account of how a mathematical filtering theory came of age and achieved its current wide engineering acceptance and practical utilization.

Application of Kalman Filtering in Computer Relaying

Abstract: During the first cycle following a power system fault, a high speed computer relay has to make a decision usually based on the 60 Hz information, which is badly corrupted by noise The noise in this case is the nonfundamental frequency components in the transient current or voltage, as the case may be For research and development purposes of computer relaying techniques, the precise nature of the noise signal is required The autocorrelation function and variance of the noise signal was obtained based on the frequency of occurrence of the different types of faults, and the probability distribution of fault location A new technique for modelling the signal and the measurements is developed based on Kalman Filtering theory for the optimal estimation of the 60 Hz information The results indicate that the technique converges to the true 60 Hz quanitities faster than other algorithms that have been used The new technique also has the lowest computer burden among recently published algorithms and appears to be within the state of the art of current microcomputer technology

A nonstationary time series model and its fitting by a recursive filter

Abstract: . The use of the state space representation for the analysis of nonstationary time series is proposed. For the fitting of the models, the use of a modified AIC based on the likelihood of the innovation process is proposed. A square root filter/smoother algorithm for the evaluation of the likelihood and state estimation is discussed.

Estimation and identification of two-dimensional images

Abstract: Estimation of image pixel density can be performed using a reduced update Kalman filter provided that a mathematical model for the image generating process is available. To this effect various algorithms suitable for identifying the parameters of autoregressive image models are discussed and evaluated in terms of the mean-squared error between the true and filtered images. Algorithms considered include general and bias-compensated least-square procedures, a correlation-based algorithm, and procedures involving the simultaneous estimation of both the image model coefficient vector and pixel estimates. Experiments using two real images and two random fields indicate that bias-compensated least squares and correlation-based procedures might be most useful for image identification and adaptive filtering.

Simple Tracking Filters: Closed-Form Solutions

Abstract: A closed-form steady-state solution is presented for a three-state tracking filter with continuous position measurements and exponentially correlated target acceleration. Some other related closed-form solutions are discussed, and some comparative performance data is presented for the discrete measurement case.

Robust Sequential Approximate Bayesian Estimation

Abstract: SUMMARY An approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non-normal observations. The behaviour of the resulting non-linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non-normal error distributions. Illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned. WE consider here the problem of sequential estimation of the location vector of a linear time series model, termed the Dynamic Linear Model by Harrison and Stevens (1976). The straightforward, exact analysis obtained by assuming normal error and prior structure is generally lost when alternative error distributions are adopted, and yet considerations of realism or robustness may strongly suggest such non-normal assumptions. The multi-state model of Harrison and Stevens provides an approximate analysis based on a discrete variance mixture of normal distributions, an approach which has been extensively investigated in the engineering literature under the name of Gaussian Sum approximations; see, for example, Alspach and Sorenson (1971). Our aim in this paper is to provide an approximate, tractible, recursive updating procedure for the location parameters, which is applicable to a wide range of symmetric, unimodal error distributions and is both more realistic and more elegant than the discrete Gaussian Sum approach. In particular, for heavy-tailed distributions our procedures provide approximate Bayesian methods for time series analysis which extend considerably the work of Masreliez and Martin (1977) and have close connections with classical robustness ideas such as M-estimation and influence functions.

Adaptive nonlinear image restoration by a modified Kalman filtering approach

Abstract: An adaptive nonlinear Kalman-type filter is presented for the restoration of two-dimensional images degraded by general image formation system degradations and additive white noise. A vector difference equation model is used to model the degradation process. The object plane distribution function is partitioned into disjoint regions based on the amount of spatial activity in the image, and difference equation models are used to characterize this nonstationary object plane distribution function. Features of the restoration filter include the ability to account for the response of the human visual system to additive noise in an image; a two-dimensional interpolation scheme to improve the estimates of the initial states in each region; and a nearest neighbor algorithm to choose the previous state of vector for the state of pixel (i,j).

Least-squares algorithms for adaptive equalizers

Abstract: Least-squares algorithms are the fastest converging algorithms for adaptive signal processors, such as adaptive equalizers. The Kalman, fast Kalman, and adaptive lattice algorithms using a least-squares cost function are investigated and extended to complex, fractionally spaced equalizers. It is shown that, for a typical telephone channel, these algorithms converge roughly three times as fast as the conventional stochastic-gradient technique. We analyze and compute the computational complexities and demonstrate that the fast Kalman algorithm is the most efficient in terms of overall performance.

Edge detection using sequential methods for change in level--Part I: A sequential edge detection algorithm

Abstract: A sequential algorithm for edge detection using a line-byline detector of edge elements connected to a recursive edge-following scheme is presented. On each line, edge elements are detected by means of a filtering operation in order to follow the slow variations of the gray level and some sequential and recursive estimators for locating jumps in this level. The edge-following problem is solved by a Kalman filter, the state model corresponding to a noisy straight line. In this first part, the complete edge detection algorithm is presented after a brief survey of edge detection methods available in the literature. Two main examples of applications are given: detection of white and black targets in the landscape in order to perform automatic driving of vehicles and detection of blood vessels in stereographic images of the brain. In the second part, a detailed study of the sequential estimators for change in mean, which are used in the line-by-line detection, will be found.

Best Linear Recursive Estimation for Mixed Linear Models

Abstract: Recursive estimation techniques for fixed and completely random models are extended to mixed linear models The Kalman filter is used to obtain recursive estimators for a two-part random model where the second random factor obeys a generalized autoregressive process By passing to the limit in an appropriate way, recursions for the mixed model are derived

Time Series Analysis in 1880: A Discussion of Contributions Made by T.N. Thiele

Abstract: Summary We describe and discuss a paper of T.N. Thiele from 1880 where he formulates and analyses a model for a time series consisting of a sum of a regression component, a Brownian motion and a white noise. He derives a recursive procedure for estimating the regression component and predicting the Brownian motion. The procedure is now known as Kalman filtering. He estimates the unknown variances of the Brownian motion and the white noise by an iterative procedure that essentially is the EM-algorithm. We finally give a short account of an application of Thiele's model and method to the description of hormone production during normal pregnancy.

New results in two-dimensional Kalman filtering with applications to image restoration

Abstract: This paper describes the application of 2-D Kalman filtering to the restoration of images degraded by linear space invariant (LSI) blur and additive white Gaussian noise (WGN). The image restoration problem is formulated in the framework of the well-known Kalman strip filter. However, the Kalman filtering scheme assumes the availability of a statespace dynamic model for the image process as well as the blur. In the past, most researchers have sought to track the problem of image modeling by making the sometimes unrealistic assumption of separability of the image correlation. A new technique for image modeling which does not make this assumption is proposed. On-line, recursive methods for implementing the modeling algorithm are also presented. We then introduce a novel technique for the state-space realization of separable blurs, since separable 2-D blurs are often encountered in practice and constitute an important subset of 2-D blurs. This state-space model is rendered compatible with strip filtering by using a new recursive scheme which we call as pseudorecursion. An extension for estimating the blur when it is unknown, but can be parameterized, has also been indicated. Simulated experimental results using natural scenery are presented.

Estimation and tests of hypotheses for the initial mean and covariance in the kalman filter model

Abstract: Kalman filtering techniques are widely used by engineers to recursively estimate random signal parameters which are essentially coefficients in a large-scale time series regression model. These Bayesian estimators depend on the values assumed for the mean and covariance parameters associated with the initial state of the random signal. This paper considers a likelihood approach to estimation and tests of hypotheses involving the critical initial means and covariances. A computationally simple convergent iterative algorithm is used to generate estimators which depend only on standard Kalman filter outputs at each successive stage. Conditions are given under which the maximum likelihood estimators are consistent and asymptotically normal. The procedure is illustrated using a typical large-scale data set involving 10-dimensional signal vectors.

Signal processing in evoked potential research: applications of filtering and pattern recognition.

Abstract: The separate but closely related topics of waveform estimation by filtering and information extraction by pattern recognition are covered in this review. Because of the low signal-to-noise ratio generally encountered in evoked potential research, a variety of filtering methods have been employed for improving waveform estimation. Initially the filtering was done with analog devices but with the availability of high performance minicomputers virtually all filtering is now done digitally. Filters of various types are considered. Among them are single and multiple channel Wiener filtering, Kalman filtering, minimum mean square error filtering, maximum signal-to-noise filtering, and several types of nonlinear filters. The application of adaptive filtering techniques is also considered. In recent years there has been a continual increase in the application of pattern recognition techniques to the processing of evoked potentials. The techniques are based on statistical decision theory and the underlying basis of these procedures is reviewed. The technique of linear stepwise discriminant analysis is considered as well as the use of general discriminant functions of linear and quadratic types. Applications of these procedures to psychophysiological testing are discussed with particular emphasis on auditory and visual event-related potentials.

Idenntification of Parameters for Power System Stability Analysis Using Kalman Filter

Abstract: We propose that parameters on generators and the control units, used for power stability analysis, should be identified on the field data for reliable analysis in planning and operation of power supply systems. We apply Kalman filter technique in estimating the parameters. The identified data are compared with the design data and are well verified by analyzing the dynamic stability under line switching condition. Which deviation of designed values from identified values has the largest influence on the analysis, is also investigated.

Discrete-time complementary models and smoothing algorithms: The correlated noise case

Abstract: The concept of complementary models for discrete-time linear finite dimensional systems with correlated observation and process noise is developed. Using this concept a new algorithm for the fixed interval smoothing problem is obtained. The new algorithm offers great flexibility with respect to changes in the initial state variance ?o. Next, using the framework developed in Sections II and III, a new and a simple derivation of the two-filter smoother is presented. Furthermore the relationship between the new smoothing algorithm, the two-filter smoother and the reversed-time Kalman filter is explored. It is shown that a similarity transformation on the Hamiltonian system simultaneously produces the new smoothing algorithm as well as the reversed-time Kalman filter.

Tracking Accuracies with Position and Rate Measurements

Abstract: Tracking accuracies for the radial component of motion are computed for a track-while-scan radar system which obtains position and rate data during the dwell time on a target These results will be of interest to persons developing trackers for pulse Doppler surveillance radars. The normalized accuracies, computed for a two state Kalman tracking filter with white noise maneuver capability, are shown to depend upon two parameters, r = 4?0/?aT2 and s = ?dT/?0. The symbols ?0 and ?d are the position and rate measurement accuracies, respectively, ?a is the standard deviation of the white noise maneuver process and T is the antenna scan time. The scalar tracking filter equations are derived and numerical results are presented. Lower steady state tracking errors plus the earlier attainment of steady state accuracies are the direct consequence of incorporating the rate measurements into the tracking filter.

Statistically Linearized Estimation of Reentry Trajectories

Abstract: Several filters are applied to the problem of state estimation from inertial measurements of reentry drag. This is a highly nonlinear problem of practical significance. It is found that a filter based on the technique of statistical linearization performs better than the extended Kalman in this application. This is believed to be the first application of the statistically linearized filter to a practical dynamics problem. A sensitivity analysis is performed to demonstrate the relative insensitivity of this filter to modeling errors and approximations.

Finite sample prediction and overdifferencing

Abstract: . If the process generating a time series contains a deterministic component the differencing operations carried out to achieve stationarity may lead to an ARMA model which is strictly noninvertible. This is known as overdifferencing but it is shown here that overdifferencing need not have serious implications for prediction provided that a finite sample prediction procedure is used. The proposed method is based on the Kalman filter and it allows both the optimal predictors and their mean square errors to be computed.

Real Time Estimation of the Heaving and Pitching Motions of a Ship, Using a Kalman Filter

Abstract: Real time ship motion estimation is of interest to naval ship operators for, vertical aircraft landing, as well as to the offshore industry for installing rigs and transferring equipment in rough seas. In the present study the estimation of the heave and pitch motion of a ship is considered, using Kalman filtering techniques. A significant part of the study is devoted to constructing appropriate models for the sea and the ship. The governing equations are obtained from hydrodynamic considerations in the form of linear differential equations with frequency dependent coefficients. In addition, non-minimum phase characteristics are obtained due to the spatial integration of the water wave forces. The resulting transfer matrix function is irrational and non-minimum phase. The conditions for a finite-dimensional approximation are considered and the impact of the various parameters is assessed. A numerical application is considered for a DD-963 destroyer.

Piece-wise EEG analysis: an objective evaluation.

Abstract: In this paper, a method is described to evaluate EEGs by means of a piece-wise analysis. The procedure involves the recursive computation of a 5th-order autoregressive model by means of a Kalman filter. As an illustration the result of applying this method to sleep recordings is described in this paper. Also, an objective comparison of this method with a more conventional approach (based on analyzing 30-s intervals), and with adaptive segmentation (Praetorius et al., 1977) was carried out using the same data. The results obtained indicate that our method is useful in extracting elementary patterns from an EEG and that the piece-wise analysis approach is to be favored over more conventional techniques.

Two New Algorithms for On-Line Modelling and Forecasting of the Load Demand of a Multinode Power System

Abstract: Two on-line algorithms are proposed for modelling and forecasting short-term multiple load demand. First a multivariable time series model is presented with a systematic method for determining its order and estimating its parameters. Another model based on the state variable form is then considered. Two decoupled algorithms, recursive least-squares and adaptive Kalman filtering, are combined in a bootstrap manner to estimate the model parameters and states. The performance of the two methods is compared using data provided by the Ontario Hydro for four loading nodes.

Computing the controllability - observability decomposition of a linear time-invariant dynamic system, a numerical approach

Abstract: We examine various numerical properties involved in computing the complete Controllability-Observability (Kalman) Decomposition for a linear time-invariant dynamic system. We discuss the numerical stability, the cost, and the particular advantages of several algorithms. We also examine several ways to measure ill-conditioning in the data. The problem is to decompose the system into four parts: controllable and observable, not controllable and observable, controllable and not observable, not controllable and not observable. We examine four algorithms, two that compute the controllable part, or equivalently the observable part, and two that combine these parts to form the complete Kalman Decomposition. We describe the algorithms, analyze their numerical properties and develop several a-posteriori measures of ill-conditioning in the original data. Numerical examples are used to illustrate the results.

Restoration of images degraded by motion blur and noise

Abstract: This paper considers the restoration of images degraded by a motion blur in the presence of noise. Based on a two-dimensional separable autoregressive image model, a one-dimensional horizontally causal vector state space model with multiple delays is derived. By the discrete sine transform, the one-dimensional vector state space model is decomposed into a set of nearly uncorrelated scalar subsystems, to which the Kalman filter is applied to obtain an approximate recursive computationally efficient restoration algorithm for motion degraded images. The same technique is also applied to a semicausal minimum variance image model in order to derive a related recursive restoration algorithm. The computational efficiency is accomplished by the discrete sine transform and the transform data compression technique. Numerical results are presented to show the applicability of the algorithms developed. Finally, the possible extension of the present method to the case of general blur is suggested.

Inversion of ground‐based radiometric data by Kalman filtering

Abstract: Retrieval of atmospheric vertical temperature profiles from ground-based radiometric observations requires shrewdness and judicious choice of parameters to surmount the noxious effects of the ill-posed nature of the inversion. Kalman linear estimation, already successfully used in satellite microwave sounding of atmospheric temperature, can be also applied to infer the thermal state of the lower troposphere from ground-based infrared measurements. After a short survey of the relevant formalism the implementation of the Kalman filter is discussed with regard to the transition operator and to the error covariances. In particular, the measurement errors are considered in some detail, and their dependence on the atmospheric dynamics is pointed out. The attainable spatial resolution is compared with that of another commonly used inversion technique, and, finally, a set of temperature profiles estimated by the Kalman algorithm from a sequence of successive radiometric measurements is reported.

Performance Degradation in Digitally Implemented Kalman Filters

Abstract: A technique is presented for analyzing expected degradations in the performance of a fixed-point arithmetic implementation of a Kalman filter with precomputed gains. A quantitative approach is provided for comparing the relative degradations associated with different mechanizations of the same Kalman filter. The causes of divergence in digitally implemented filters are investigated. Finally, simulation studies are utilized to show how closely the analytical predictions agree with actual results.