Iterative Reweighted Least Squares Algorithm Research Articles

L/sub 1/ and L/sub infinity / minimization via a variant of Karmarkar's algorithm

Simple iterative algorithms are presented for L/sub 1/ and L/sub infinity / minimization (regression) based on a variant of Karmarkar's linear programming algorithm. Although these algorithms are based on entirely different theoretical principles to the popular IRLS (iteratively reweighted least squares) algorithm, they have almost identical matrix operations. Also presented are the results of a Monte Carlo study comparing the numerical convergence properties of the Karmarkar algorithm for L/sub 1/ minimization to those of an IRLS and a simplex algorithm. The test problem involves L/sub 1/ estimation of AR (autoregressive) model parameters. The Karmarkar algorithm outperformed IRLS by achieving higher numerical accuracy in fewer iterations. Techniques for reducing the computational cost per iteration of the Karmarkar L/sub 1/ algorithm are discussed. >

L<inf>1</inf>deconvolution and its application to seismic signal processing

This correspondence reviews fast iterative reweighted least squares (IRLS) and residual steepest descent (RSD) algorithms for L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1 \leq p \leq 2</tex> , deconvolution. The timing aspects and the implementation of the IRLS algorithm on an array processor are discussed. The effectiveness of L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> deconvolution and its insensitivity to noise bursts are illustrated using simple synthetic as well as complex seismic data. Finally, it is shown that the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> prediction filters in general need not be stable, and that L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> solutions predict the possibility of nonminimum phase aspects of a given set of data.

The Analysis of Rates Using Poisson Regression Models

Models are considered in which the underlying rate at which events occur can be represented by a regression function that describes the relation between the predictor variables and the unknown parameters. Estimates of the parameters can be obtained by means of iteratively reweighted least squares (IRLS). When the events of interest follow the Poisson distribution, the IRLS algorithm is equivalent to using the method of scoring to obtain maximum likelihood (ML) estimates. The general Poisson regression models include log-linear, quasilinear and intrinsically nonlinear models. The approach considered enables one to concentrate on describing the relation between the dependent variable and the predictor variables through the regression model. Standard statistical packages that support IRLS can then be used to obtain ML estimates, their asymptotic covariance matrix, and diagnostic measures that can be used to aid the analyst in detecting outlying responses and extreme points in the model space. Applications of these methods to epidemiologic follow-up studies with the data organized into a life-table type of format are discussed. The method is illustrated by using a nonlinear model, derived from the multistage theory of carcinogenesis, to analyze lung cancer death rates among British physicians who were regular cigarette smokers.