Showing papers on "Recursive least squares filter published in 1974"

Stability of the solutions of linear least squares problems

Abstract: In 1966 Golub and Wilkinson gave upper bounds for the errors of least squares solutions based on orthogonal transformations, in which the square of the condition number of the matrix occurs. In the present paper a geometrical explanation and lower bounds are given. The upper bounds will be shown to be realistic for some classes of problems and types of perturbations, and to be unrealistic for others.

Efficient implementation of a variable projection algorithm for nonlinear least squares problems

Abstract: Nonlinear least squares problems frequently arise for which the variables to be solved for can be separated into a linear and a nonlinear part. A variable projection algorithm has been developed recently which is designed to take advantage of the structure of a problem whose variables separate in this way. This paper gives a slightly more efficient and slightly more general version of this algorithm than has appeared earlier.

Comments on "Identification of optimum filter steady-state gain for systems with unknown noise covariances"

Abstract: The approach to the estimation of the optimum Kalman filter steady-state gain proposed by Mehra and modified by Carew and Belanger can be improved by noting the true statistical nature of the problem. A new approach based on both simple or weighted least squares is outlined and tested by Monte Carlo simulation.

An application of least squares to one-dimensional transient problems

Abstract: The method of ‘least squares’, which falls under the category of weighted residual processes, is applied as a time-stepping algorithm to one-dimensional transient problems including the heat conduction equation, diffusion-convection equation, and a non-linear unsaturated flow equation. Comparison is made with other time-stepping algorithms, and the least squares method is seen to offer definite advantages.

A class of bootstrap estimators and their relationship to the generalized two stage least squares estimators

Abstract: This paper deals with the identification of a process modeled by a stable, linear difference equation of known order. Its output is subject to additive observation noise that is identically and independently distributed with zero mean and a constant variance. On-line estimators in which the process parameters as well as the process outputs are estimated simultaneously in real time are considered. For improving the stability of such on-line algorithms, a simple adaptive filter for the reference model is proposed. Further, it is shown that inclusion of such a filter relates the resulting bootstrap algorithms to the more general forms of the two stage least squares estimators viz. the k -class, h -class and the double k -class estimators. Effectiveness of the filter in stabilizing the on-line algorithms is demonstrated by using data generated by a fourth-order model.

Initial-condition robustness of linear least squares filtering algorithms

Abstract: We give some sufficient conditions under which the mean-square error of linear least squares (lls) estimates converges to its true steady-state value despite perturbations due to uncertainties in initial conditions, round-off errors in calculation, etc. For state-variable estimators, this property, called initial-condition robustness, is implied by the exponential asymptotic stability of the estimating filter, but this latter property though desirable is of course far from necessary for the more basic (since mean-square error is the ultimate criterion) property of robustness. We present a general sufficient condition for such robustness of lls predictors of stochastic processes. This condition is then specialized to lls estimators for processes described by state-variable models and by autoregressive-moving agerage difference equation models. It is shown that our conditions can establish robustness in cases where previous criteria either fail or are inconclusive.

Fixed-memory recursive filters (Corresp.)

Abstract: This correspondence presents three algorithms for computing least-squares estimates based on fixed amounts of data.

A direct error analysis for least squares

Abstract: A direct error analysis is given for orthogonal factorization methods for calculating the least squares solution of an overdetermined system of linear equations. The direct method has the interesting advantage in that it permits the separation of errors occurring in the transformation and back-substitution phases of solution. This shows the partial elimination of potentially significant terms occurring in different stages of the algorithm. Presumably it is prudent to minimize the error at each stage of the algorithm, so it is significant that numerical evidence shows column pivoting can reduce the magnitude of these terms. This is offered as an explanation for the common observation that column pivoting is beneficial in least squares calculations. We also summarize the corresponding error analysis for the calculation of the minimum norm solution of an underdetermined system using orthogonal transformations.

On the solution of the linear least squares problems and pseudo-inverses

Abstract: It is known that the computed least squares solutionx ofAx=b, in the presence of the round-off error, satisfies the perturbed equation(A+E)(x+h)=b+f. The practical considerations of computing the solution are discussed and it is found that rank(A+E)=rank (A). A general analysis of the condition of the linear least squares problems and pseudo-inverses is then presented using this assumption. Norms of relevant round-off error perturbations are estimated for two known methods of solution. Comparison between different algorithms is given by numerical examples.

Comparison of four methods for the identification of linear systems with a time delay

Abstract: Four methods for identification of the parameters and time delay in a linear time-invariant system are compared. The four techniques are (1) least squares, (2) "Extended" Kalman Filter, (3) filtered maximum likelihood, and (4) Fast Fourier Transform. The input and output data are obtained by simulating a second order system, and noise is added to the output data. Five parameters in the system transfer function are identified: the gain constant, two pole locations, one zero location, and the time delay. The results show that the filtered maximum likelihood technique is superior with noisy measurements. All the methods identified the parameters and time delay with noiseless measurements.