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The Computer Journal 1964 7(3):220-221; doi:10.1093/comjnl/7.3.220
© 1964 by British Computer Society
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Error estimates for smoothing and extrapolation formulae

L. de Witte *

Aerospace Corporation, PO Box 1308, San Bernadino, California, USA

A standard smoothing procedure consists of least squares fitting a polynomial Qm(x) of degree m to 2n + 1 observed points and replacing the midpoint (nth point) by the value of the polynomial. In many applications of trajectory extrapolation, or acquisition and reacquistion in radar tracking, it is desirable to smooth at or near the end-point of the observed data. Considering Qm(x) as a linear combination of a set of orthogonal polynomials Pg,2n(x) of degree g < m the error reduction factor for smoothing at an arbitrary jth point is found to be given by

[equation: see PDF]

where

[equation: see PDF]

and the total number of observed points, N + 1, used in the smoothing may be even or odd. If the jth point belongs to the set of N + 1 points, then fs(j) = aj,j, that is, the smoothing factor is equal to the j coefficient of the jth point smoothing formula. Milne's expression for the smoothing factor in midpoint smoothing forms a special case of the above results.


* Aerospace Corporation, P.O. Box 1308, San Bernadino, California, U.S.A.


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