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IV.—On Least Squares and Linear Combination of Observations

  • A. C. Aitken (a1)
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.

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References
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Henderson, R., 1932. “A Postulate for Observations,” Ann. Math. Statistics, vol. iii, pp. 3237.
Lidstone, G. J., 1933. “Notes on Orthogonal Polynomials,” Journ. Inst. Act., vol. lxiv, pp. 153159.
Sheppard, W. F., 1912. “Reduction of Errors by Negligible Differences,” Proc. Fifth Internat. Congr. Math. (Cambridge), vol. ii, pp. 348384.
Sheppard, W. F., 1914. “Fitting of Polynomials by Method of Least Squares,” Proc. Lond. Math. Soc. (2), vol. xiii, pp. 97108.
Sheppard, W. F., 1914. “Graduation by Reduction of Mean Square of Error,” Journ. Inst. Act., vol. xlviii, pp. 171185, 390–412; vol. xlix, pp. 148–157.
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Proceedings of the Royal Society of Edinburgh
  • ISSN: 0370-1646
  • EISSN: 2059-9153
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh
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