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VI.—On the Graduation of Data by the Orthogonal Polynomials of Least Squares

Published online by Cambridge University Press:  15 September 2014

A. C. Aitken
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Extract

The problem of fitting a polynomial to data by Least Squares has engaged the attention of many writers. The methods of approach have been many and various. Continued fractions, determinants, the calculus of finite differences and sums, the method of moments, the linear combination of data, the use of the orthogonal polynomials of Legendre and Tchebychef, these and doubtless other instruments of analysis have been pressed into service. At the end of the present paper is given a selective bibliography, which we hope on a future occasion to complete and to supplement by adding brief indications of the standpoint and achievement of each investigator.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 53 , 1934 , pp. 54 - 78
Copyright
Copyright © Royal Society of Edinburgh 1934

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References

page 55 note * List A.

page 56 note * We would have used the notation x[r] for this, but Steffensen has adopted that notation for what he defines as a central factorial, but what we should prefer to regard as a mean central factorial. Reference A (5).

page 59 note * Adopted by Tchebychef, Fisher, Jordan, Chotimsky, and Allan. References B (1), (2), (6), (11), (18).

page 65 note * According to precepts well established and justified. Cf. B 6 (a), (b), (c).

page 66 note * This example was constructed from a curve with inflexions, in order to bring out such a feature ; but the fact that the reduction in S2 produced by the parabolic term a2T2 is almost zero was an unanticipated accident.

page 68 note * The expressions in central, not mean central factorials, were found in a different manner by Miss F. E. Allan, and more recently by P. Lorenz. References B, 1, 6 (c), and 12.

page 72 note * Unless one possesses a very efficient machine for ordinary repeated summation. By either mode of summation, ordinary or central, it is easy on a machine with a capacious register to sum two columns at once, and thus save the time and trouble of copying

page 75 note * Annals of Math., (2), vol. xxxii, 1931, pp. 461–462.

page 76 note * That it should not vanish is indeed the condition for the existence of the Legendre polynomials.

page 77 note * I am indebted to Professor J. F. Steffensen for the privilege of perusing his copy of this monograph.