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Theory of the Combination of Observations Least Subject to Errors, Part One, Part Two, SupplementTheory of the Combination of Observations Least Subject to Errors, Part One, Part Two, Supplement

Carl Friedrich Gauss


Translated by G. W. Stewart

Classics in Applied Mathematics 11

In the 1820s Gauss published two memoirs on least squares, which contain his final, definitive treatment of the area along with a wealth of material on probability, statistics, numerical analysis, and geodesy. These memoirs, originally published in Latin with German Notices, have been inaccessible to the English-speaking community. Here for the first time they are collected in an English translation. For scholars interested in comparisons the book includes the original text and the English translation on facing pages. More generally the book will be of interest to statisticians, numerical analysts, and other scientists who are interested in what Gauss did and how he set about doing it. An Afterword by the translator, G. W. Stewart, places Gauss's contributions in historical perspective.

Audience

Numerical analysts, statisticians, computer scientists, engineers, historians, and mathematicians with sufficient background in regression theory, numerical linear algebra, and least squares theory will find this volume of great interest.

Contents

Translator's Introduction; Pars Prior/Part One: Random and regular errors in observations; Regular errors excluded; their treatment; General properties of random errors; The distribution of the error; The constant part or mean value of the error; The mean square error as a measure of uncertainty; Mean error, weight and precision; Effect of removing the constant part; Interpercentile ranges and probable error; properties of the uniform, triangular, and normal distribution; Inequalities relating the mean error and interpercentile ranges; The fourth moments of the uniform, triangular, and normal distributions; The distribution of a function of several errors; The mean value of a function of several errors; Some special cases; Convergence of the estimate of the mean error; the mean error of the estimate itself; the mean error of the estimate for the mean value; Combining errors with different weights; Overdetermined systems of equations; the problem of obtaining the unknowns as combinations of observations; the principle of least squares; The mean error of a function of quantities with errors; The regression model; The best combination for estimating the first unknown; The weight of the estimate; estimates of the remaining unknowns and their weights; justification of the principle of least squares; The case of a single unknown; the arithmetic mean. Pars Posterior/Part Two: Existence of the least squares estimates; Relation between combinations for different unknowns; A formula for the residual sum of squares; Another formula for the residual sum of squares; Four formulas for the residual sum of squares as a function of the unknowns; Errors in the least squares estimates as functions of the errors in the observations; mean errors and correlations; Linear functions of the unknowns; Least squares with a linear cons traint; Review of Gaussian elimination; Abbreviated computation of the weights of the unknowns; Computational details; A bbreviated computation of the weight of a linear function of the unknowns; Updating the unknowns and their weights when a new observation is added to the system; Updating the unknowns and their weights when the weight of an observation change s; A bad formula for estimating the errors in the observations from the residual sum of squares; The correct formula; The mean error of the residual sum of squares; Inequalities for the mean error of the residual sum of squares; the case of the normal distribution. Supplementum/Supplement: Problems having constraints on the observations; reduction to an ordinary least squares problem; Functions of the observations; their mean errors; Estimating a function of observations that are subject to constraints; Characterization of permissible estimates; The function that gives the most reliable estimate; The value of the most reliable estimate; Four formulas for the weight of the value of the estimate; The case of more than one function; The most reliable adjustments of the observations and their use in estimation; Least squares characterization of the most reliable adjustment; Difficulties in determining weights; A better method; Computational details; Existence of the estimates; Estimating the mean error in the observations; Estimating the mean error in the observations, continued; The mean error in the estimate; Incomplete adjustment of observations; Relation between complete and incomplete adjustments; A block iterative method for adjusting observations; The inverse of a symetric system is a symmetric; Fundamentals of geodesy; De Krayenhof's triangulation; A triangulation from Hannover; Determining weights in the Hannover triangulation. Anzeigen/Notices. Part One. Part Two. Supplement. After word. Gauss's Schooldays; Legendre and the Priority Controversy; Beginnings: Mayer, Boscovich and Laplace; Gauss and Laplace; The Theoria Motus; Laplace and the Central Limit Theorem; The Theoria Combinationis Observationum; The Precision of Observations; The Combination of Observations; The Inversion of Linear Systems; Gaussian Elimination and Numerical Linear Algebra; The Generalized Minimum Variance Theorem; References.

Royalties from the sale of this book are contributed to the SIAM Student Travel Fund.

1995 / xi + 241 pages / Softcover / ISBN-13: 978-0-898713-47-3 / ISBN-10: 0-89871-347-1 /
List Price $55.50 / SIAM Member Price $38.85 / Order Code CL11
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