Least squares is probably the best known method for fitting linear models and by far the most widely used. Surprisingly, the discrete L 1 analogue, least absolute deviations (LAD) seems to have been considered first. Possibly the LAD criterion was forced into the background because of the com putational difficulties associated with it. Recently there has been a resurgence of interest in LAD. It was spurred on by work that has resulted in efficient al gorithms for obtaining LAD fits. Another stimulus came from robust statistics. LAD estimates resist undue effects from a feyv, large errors. Therefore. in addition to being robust, they also make good starting points for other iterative, robust procedures. The LAD criterion has great utility. LAD fits are optimal for linear regressions where the errors are double exponential. However they also have excellent properties well outside this narrow context. In addition they are useful in other linear situations such as time series and multivariate data analysis. Finally, LAD fitting embodies a set of ideas that is important in linear optimization theory and numerical analysis. viii PREFACE In this monograph we will present a unified treatment of the role of LAD techniques in several domains. Some of the material has appeared in recent journal papers and some of it is new. This presentation is organized in the following way. There are three parts, one for Theory, one for Applicatior.s and one for Algorithms.

The least absolute deviations criterion, or the l1 norm, is frequently used for approximation where the data may contain outliers or wild points'. One of the most popular methods for solving the least absolute deviations data fitting problem is the Barrodale and Roberts (BR) algorithm (1973), which is based on linear programming techniques and the use of a modified simplex method. This algorithm is particularly efficient. However, since it is based upon the simplex method it can be susceptible to the accumulation of unrecoverable rounding errors caused by using an inappropriate pivot. In this paper we shall show how we can extend a numerically stable form of the simplex method to the special case of l1 approximation whilst still maintaining the efficiency of the Barrodale and Roberts algorithm. This extension is achieved by using the l1 characterization to rebuild the relevant parts of the simplex tableau at each iteration. The advantage of this approach is demonstrated most effectively when the observation matrix of the approximation problem is sparse, as in the case when using compactly supported basis functions such as B-splines. Under these circumstances the new method is considerably more efficient than the Barrodale and Roberts algorithm as well as being more robust.

Least squares is probably the best known method for fitting linear models and by far the most widely used. Surprisingly, the discrete L 1 analogue, least absolute deviations (LAD) seems to have been considered first. Possibly the LAD criterion was forced into the background because of the com putational difficulties associated with it. Recently there has been a resurgence of interest in LAD. It was spurred on by work that has resulted in efficient al gorithms for obtaining LAD fits. Another stimulus came from robust statistics. LAD estimates resist undue effects from a feyv, large errors. Therefore. in addition to being robust, they also make good starting points for other iterative, robust procedures. The LAD criterion has great utility. LAD fits are optimal for linear regressions where the errors are double exponential. However they also have excellent properties well outside this narrow context. In addition they are useful in other linear situations such as time series and multivariate data analysis. Finally, LAD fitting embodies a set of ideas that is important in linear optimization theory and numerical analysis. viii PREFACE In this monograph we will present a unified treatment of the role of LAD techniques in several domains. Some of the material has appeared in recent journal papers and some of it is new. This presentation is organized in the following way. There are three parts, one for Theory, one for Applicatior.s and one for Algorithms.

A comparison is made between linear programming and weighted least squares approach to absolute deviation curve fitting. The computational results are consistent in showing linear programming substantially reduces solution times while requiring only a relatively small amount of additional storage.

Mathematical Algorithms for Linear Regression discusses numerous fitting principles related to discrete linear approximations, corresponding numerical methods, and FORTRAN 77 subroutines. The book explains linear Lp regression, method of the lease squares, the Gaussian elimination method, the modified Gram-Schmidt method, the method of least absolute deviations, and the method of least maximum absolute deviation. The investigator can determine which observations can be classified as outliers (those with large errors) and which are not by using the fitting principle. The text describes the elimination of outliers and the selection of variables if too many or all of them are given by values. The clusterwise linear regression accounts if only a few of the relevant variables have been collected or are collectible, assuming that their number is small in relation to the number of observations. The book also examines linear Lp regression with nonnegative parameters, the Kuhn-Tucker conditions, the Householder transformations, and the branch-and-bound method. The text points out the method of least squares is mainly used for models with nonlinear parameters or for orthogonal distances. The book can serve and benefit mathematicians, students, and professor of calculus, statistics, or advanced mathematics.

This book describes the development of statistics, which for more than a century was called "the calculus of observations." The approach will help readers gain a clearer understanding of the historical development as well as the essential nature of some of the commonly used statistical estimation procedures. Detailed descriptions of the fitting of linear relationships by the method of least squares and the closely related least absolute deviations and minimax absolute deviations procedures are presented, along with some of the important work by Laplace, Gauss, and Adrain.