Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-07T22:54:18.239Z Has data issue: false hasContentIssue false
  • English
  • Français

XVI.—Studies in Practical Mathematics. IV. On Linear Approximation by Least Squares

Published online by Cambridge University Press:  14 February 2012

A. C. Aitken
A. C. Aitken
Affiliation:
Mathematical Institute, University of Edinburgh
Get access Rights & Permissions [Opens in a new window]

Extract

R. Frisch, in a paper (Frisch, 1928) on correlation and scatter in statistical variables, made an extensive use of matrices, and in particular of the moment matrix, as he called it, of a set of variables. The matrices were square arrays, with an equal number of rows and columns. This paper of Frisch pointed the way to an even more extensive use of the algebra of matrices in problems of statistics.

What Frisch called the moment matrix may perhaps be more suitably called, nowadays, the variance matrix of a set or vector of variates, since the moments in question are all variances or covariances. In the present paper, which is illustrative of matrix methods, we explore the familiar ground of linear approximation by Least Squares, making full use of the properties of the variance matrix. We also study the linear transformations that convert crude data into smoothed or graduated values, or into residuals, or into coefficients in a linear representation by chosen functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 2 , 1946 , pp. 138 - 146
Copyright
Copyright © Royal Society of Edinburgh 1946

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES TO LITERATURE

Aitken, A. C., 1933. “On Fitting Polynomials to Data with Weighted and Correlated Errors,” Proc. Roy. Soc. Edin., Liv, 12–16.Google Scholar
Aitken, A. C., 1934. “On Least Squares and Linear Combination of Observations,” Proc. Roy. Soc. Edin., LV, 42–48.Google Scholar
Aitken, A. C., 1942. Determinants and Matrices, Oliver & Boyd, Edinburgh, 2nd ed., pp. 107–109.Google Scholar
Brunt, D., 1931. The Combination of Observations, Cambridge, 2nd ed., pp. 75–128.Google Scholar
Buchan, A. F., 1939. Linear Combination of Data with Least Error, Thesis for Ph.D., University of Edinburgh.Google Scholar
Fisher, R. A., and Yates, F., 1943. Statistical Tables for Biological, Agricultural, and Medical Research, Oliver & Boyd, Edinburgh, 2nd ed., pp. 20, 62–68.Google Scholar
Frisch, Ragnar, 1928. “Correlation and Scatter in Statistical Variables,” Nordisk Statistisk Tidskr., VIII, 36–102, esp. 48-50.Google Scholar
Harding, F. M., 1934. Least Square Smoothing by Linear Combination, Thesis for Ph.D., University of Edinburgh.Google Scholar
Tchebychef, P. L., 1858. Journ. de Math., 2nd ser., III, 289, 320.Google Scholar