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1999 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Fitting Linear Relationships

A History of the Calculus of Observations 1750–1900

verfasst von: Richard William Farebrother

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

This book is intended for students of mathematical statistics who are interested in the early history of their subject. It gives detailed algebraic descriptions of the fitting of linear relationships by the method of least squares (L ) and the related least absolute 2 deviations (L ) and minimax absolute deviations (Loo) procedures. These traditional line J fitting procedures are, of course, also addressed in conventional statistical textbooks, but the discussion of their historical background is usually extremely slight, if not entirely absent. The present book complements the analysis of these procedures given in S.M. Stigler'S excellent work The History of Statistics: The Quantification of Uncertainty before 1900. However, the present book gives a more detailed account of the algebraic structure underlying these traditional fitting procedures. It is anticipated that readers of the present book will obtain a clear understanding of the historical background to these and other commonly used statistical procedures. Further, a careful consideration of the wide variety of distinct approaches to a particular topic, such as the method of least squares, will give the reader valuable insights into the essential nature of the selected topic.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
A fundamental problem in the theory of errors which seems first to have attracted the attention of leading practical scientists in the middle of the eighteenth century, when expressed in the terminology of modern mathematical statistics, was that of estimating the unknown slope parameters of the standard linear statistical model.
Richard William Farebrother
Chapter 2. The Methods of Boscovich and Mayer
Abstract
On 1 August 1714, Queen Anne died. As all nineteen of her children had either been born dead or had died in childhood, she was succeeded on the throne of Great Britain and Ireland1 by her distant relative2 George Louis (1660-1727), the Elector of Hanover3 (Germany), as King George I, and he by his son George (1683-1763) as King George II. In 1748, King GeorgeIImade a tour of his Hanoverian possessions, and, as Forbes (1980, p. 84) notes
“It was no coincidence that the formal decision of the Hanoverian government to establish an observatory in Göttingen was made only a few weeks after George II’s visit to the town on 1 August 1748. The King and his advisors well knew that precise astronomical observations were a necessary prerequisite for improvements in cartography and oceanic navigation. The Paris and Greenwich Observatories were institutional reminders of the links between theoretical and practical science; and the military and naval applications of accurate longitude and latitude determinations were not difficult to see.4[Indeed the Prussian] Field Marshal von Schmettau in Berlin had already been trying over a period of several years to encourage the Paris cartographers César-Francois Cassini de Thury and Joseph De L’Isle and the inheritors of the Homann Cartographic Bureau to participate in extending the French trigonometrical survey into and across Germany ”
Howerver George II had a different scheme in hand which crucially involved a cartographer by the name of Tobias Mayer.
Richard William Farebrother
Chapter 3. Laplace’s Work on the Methods of Boscovich and Mayer
Abstract
Boscovich’s method for determining the true values of y and z from n equations of the form
$$ {a_i} - z - {p_i}y = {v_i}{\text{ }}i = 0,1, \ldots ,n - 1 $$
was drawn to Laplace’s attention by a review of the 1770 French edition of Maire and Boscovich’s book which Jean Bernoulli III’ (1744-1807) had published in 1772 in the Receuil pour les Astronomes. Further details of this review will be given in Chapter 6, where we will address its more immediate impact on Laplace’s work.
Richard William Farebrother
Chapter 4. Laplace’s Minimax Procedure
Abstract
In Chapter 3 we have outlined Laplace’s discussion of his variants of Boscovich’s procedure. However, it should be noted that in each case his presentation (1793, §§ 11–12; 1799, book 3, § 40) of this procedure is immediately preceded (1793, §§ 9–10; 1799, book 3, § 39) by a discussion of an alternative fitting procedure which he had himself proposed some years earlier (1786, § 3).1This alternative procedure chooses y and z to minimise the maximum absolute error in the equations
$$ {a_i} - z - {p_i}y = {v_i}{\text{ }}i = 0,1, \ldots ,n - 1 $$
and is thus now known as the minimax absolute error or minimax procedure.2
Richard William Farebrother
Chapter 5. The Method of Least Squares
Abstract
Following the successful completion of the Revolution in France, the French government sought to export their revolutionary philosophy to neighbouring countries by despatching military expeditions. Thus, in 1797 Napoléon Buonaparte (1769-1821) led successful military expeditions to Austria and Lombardy (Northern Italy) and in 1798 he led a further expedition to Egypt and the Levant. In the following year he returned to France to lead the coup d’état which installed him as First Consul.
Richard William Farebrother
Chapter 6. Statistical Foundations of the Method of Least Squares
Abstract
In Chapter 8, we will discuss Gauss’s 1809 derivation of the method of least squares. In order to trace the intellectual origins of this method, we must return to 1772. In that year, Jean Bernoulli III published a review of the 1770 French translation of Maire and Boscovich’s geodetic survey. Attached to this review is a footnote which may be translated as follows:
“The problem of finding the true mean of a certain number of observations (which is rarely the arithmetic mean), is of great interest to astronomers; it is to be hoped that someone will present the closely related essence of the different methods given for this purpose by Fr Boscovich, by Mr Lambert [1770] in the work cited on page 157 of my first volume, by Mr Daniel Bernoulli in a short memoir which has not yet been published, and finally by Mr de la Grange in a beautiful treatise which he has recently made the subject of several lectures at our academic meetings.” [Author’s translation from Stigler’s (1978b) transcription]
Richard William Farebrother
Chapter 7. Adrain’s Work on the Normal Law
Abstract
Before resuming our discussion of Gauss’s work in Chapter 8, we must examine the contemporaneous work on the normal law of errors by Robert Adrain. This work was first published in 1809 but was lost to the literature until rediscovered and republished by Abbe (1871) and Merriman (1877b).
Richard William Farebrother
Chapter 8. Gauss’s Most Probable Values
Abstract
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Germany). He was the son of a bricklayer and gardener who also kept the accounts of a local insurance company. Gauss received his elementary education at St. Katharine’s School in Brunswick. In 1791, his evident talent for mathematics was brought to the attention of the Duke of Brunswick, who undertook to support his continuing education, at first at the Collegium Carolinum in Brunswick and later at the University of Göttingen1. Gauss was awarded his doctor’s degree in absentia by the University of Helmstedt in 1801.
Richard William Farebrother
Chapter 9. Laplace’s Most Advantageous Method
Abstract
Gauss’s (1809) work on the method of least squares soon came to the attention of Laplace. In 1810, he published a brief supplement to an earlier paper on limit distributions pointing out their relevance in this context, and in 1811, he published a full paper deriving the method of least squares from the limit distribution of a weighted sum of observed errors. Laplace subsequently presented most of this material in a more accessible form in his Théorie Analytique des Probabilités (1812, book 2, §§ 18–21).1 Thus, although Laplace published his book in the year in which Napoléon began to suffer from the military reversals that eventually led to his downfall, it was written and compiled during the years of his greatness.
Richard William Farebrother
Chapter 10. Gauss’s Most Plausible Values
Abstract
In a letter to his friend Heinrich Wilhelm Matthaus Olbers (1758–1840) dated 22 February 1819 [see Plackett (1972, p. 245)], Gauss announced that he had discarded his earlier (Bayesian) approach to the fitting problem and had substituted a new principle, namely the minimum mean squared error criterion, in place of Bernoulli’s principle of maximum probability.
Richard William Farebrother
Chapter 11. Gauss’s Method of Adjustment by Correlates
Abstract
Five years after the publication of the papers discussed in Chapter 10, Gauss (1828) published a supplement in which he considered a closely related problem that arises naturally in the context of geodetic triangulation studies.1 As an indication of the type of problem involved in this field, we consider Benoît’s (1924) simple worked example In this problem, we are given the angular bearings (in centesimal degrees relative to local origins2) of each of the four corners A, B, C, and D of a quadrilateral figure from each of the other three corners.
Richard William Farebrother
Chapter 12. Mechanical Analogies for the Method of Least Squares
Abstract
Following the publication of Gauss’s Theoria Motus in 1809, there were, as we have noted in Section 6.5, several attempts to derive the Principle of the Arithmetic Mean from more elementary systems of axioms. In a similar way, following the publication of Laplace’s Théorie Analytique des Probabilités in 1812 and Gauss’s Theoria Combinationis in 1823, there were numerous attempts to derive the method of least squares without recourse to the calculus of probabilities. In this chapter, we shall examine one of the most significant of these studies, namely the one published by Donkin in 1844 and again in an abridged form in 1850. As the mechanical model discussed in this paper and in our Section 12.2 is now only of pedagogical interest, we shall treat this topic relatively briefly. However, we shall take this opportunity to relate Donkin’s discussion to the contemporary paper by Bravais (1846) and to the later work of Newcomb (1873a, 1873b), Adcock (1877, 1878b), Kummell (1879), and Pearson (1901). For details of other nonstatistical derivations of the method of least squares see Glaisher (1872), Merriman (1877a) and Harter (1974–76).
Richard William Farebrother
Chapter 13. Orthogonalisation Procedures
Abstract
In Chapter 8, we have seen that Gauss (1809, §§ 181-183) had related the partial derivatives of the adjusted quadratic forms,½Ω, ½Ω1, ½Ω2 etc and
$$\begin{array}{*{20}{r}} {{\xi _0} = }&{{u_{00}}x + {u_{01}}y + {u_{02}}z + etc + {s_0}} \\ {{\eta _1} = }&{{u_{11}}y + {u_{12}}z + etc + {s_1}} \\ {{\zeta _2} = }&{{u_{22}}z + etc + {s_2}} \\ {etc}&{} \end{array}$$
to the linear combinations of the errors
$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {\xi = }&{\sum {{a_i}{v_i}} } \end{array}} \\ {\begin{array}{*{20}{c}} {\eta = }&{\sum {{b_i}{v_i}} } \end{array}} \\ {\begin{array}{*{20}{c}} {\zeta = }&{\sum {{c_i}{v_i}} } \end{array}} \\ {etc} \end{array}$$
by means of the inverse relationships
$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {\xi = }&{{\xi _0}} \end{array}} \\ {\begin{array}{*{20}{c}} {\eta = }&{\frac{{{u_{01}}}}{{{u_{00}}}}{\xi _0} + {\eta _1}} \end{array}} \\ {\begin{array}{*{20}{c}} {\zeta = }&{\frac{{{u_{02}}}}{{{u_{00}}}}{\xi _0} + \frac{{{u_{12}}}}{{{u_{11}}}}{\eta _1} + {\zeta _2}} \end{array}} \\ {etc} \end{array}$$
and thus directly to each other by means of the equations
$$\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {{\xi _0} = }&\xi \end{array}} \\ {\begin{array}{*{20}{c}} {{\eta _1} = }&{{g_{10}}\xi + \eta } \end{array}} \\ {\begin{array}{*{20}{c}} {{\zeta _2} = }&{{g_{20}}\xi + {g_{21}}\eta + \zeta } \end{array}} \\ {etc} \end{array}$$
where the gij coefficients are implicitly defined in terms of the uij.
Richard William Farebrother
Chapter 14. Thiele’s Derivation of the Method of Least Squares
Abstract
Thiele is best known to statisticians for his work on cumulants, which he called half-invariants (Danish halvinvarianter). But in this chapter, we will concentrate on his derivation of the method of least squares as a generalisation of the natural estimates of the unknowns in the canonical form of the linear model.
Richard William Farebrother
Chapter 15. Later Work on the Method of Situation
Abstract
The problem of choosing values for the unknowns x,y, z,etc to minimise the unconstrained sum of the absolute errors in the n linear equations
$$ {a_i}x + {b_i}y + {c_i}z + etc - {l_i} = 0,{\text{ i = 1,2,}} \ldots {\text{,n}} $$
had been proposed by Gauss in 1809 and named the Method of Situation by Laplace in 1818. Although Gauss had characterised the solution of this problem in a form which could have been developed as a simple algorithm, no further work on the unconstrained problem seems to have been published between 1818 and 1887, with the exception of the single sentence in the paper by Fourier (1824) noted in Chapter 4.
Richard William Farebrother
Chapter 16. Concluding Remarks
Abstract
In view of the widespread use of the method of least squares for linear curve fitting problems, and in view of the prominent publication of the corresponding iterative procedure for nonlinear problems in Gauss’s (1809) Theoria Motus, it seems reasonable to anticipate that this iterative procedure would have been widely used well before the centenary of Gauss’s death in 1955. However, this was not the case, as practitioners seem to have preferred to transform their nonlinear equations to a linear or a more linear form before applying the method of selected points, the method of averages, or the method of least squares to the transformed relationships.
Richard William Farebrother
Backmatter
Metadaten
Titel
Fitting Linear Relationships
verfasst von
Richard William Farebrother
Copyright-Jahr
1999
Verlag
Springer New York
Electronic ISBN
978-1-4612-0545-6
Print ISBN
978-1-4612-6812-3
DOI
https://doi.org/10.1007/978-1-4612-0545-6