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IV.—On Least Squares and Linear Combination of Observations

Published online by Cambridge University Press:  15 September 2014

A. C. Aitken
A. C. Aitken
Affiliation:
Mathematical Institute, University of Edinburgh
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Extract

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–

(i) The data being u1, u2, …, un, the representation is to be given by linear combinations

(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than the kth.

(iii) The sum of squared coefficients which measures the mean square error of yi, is to be a minimum for each value of i.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 55 , 1936 , pp. 42 - 48
Copyright
Copyright © Royal Society of Edinburgh 1936

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References

References to Literature

Henderson, R., 1932. “A Postulate for Observations,” Ann. Math. Statistics, vol. iii, pp. 32–37.CrossRefGoogle Scholar
Lidstone, G. J., 1933. “Notes on Orthogonal Polynomials,” Journ. Inst. Act., vol. lxiv, pp. 153–159.Google Scholar
Sheppard, W. F., 1912. “Reduction of Errors by Negligible Differences,” Proc. Fifth Internat. Congr. Math. (Cambridge), vol. ii, pp. 348–384.Google Scholar
Sheppard, W. F., 1914. “Fitting of Polynomials by Method of Least Squares,” Proc. Lond. Math. Soc. (2), vol. xiii, pp. 97–108.CrossRefGoogle Scholar
Sheppard, W. F., 1914. “Graduation by Reduction of Mean Square of Error,” Journ. Inst. Act., vol. xlviii, pp. 171–185, 390–412; vol. xlix, pp. 148–157.Google Scholar