There are two obvious differences from the classical running mean. At
first, one approximates data within the interval [
] by
a parabola instead of a constant (mean), thus parabolic and cubic
variations would be fit exactly for observations equidistantly
distributed in time. This method is efficient also for data with gaps,
but in this case only a parabolic signal will be fit exactly.
At second, by using the coefficients pi, one would obtain a smoothing
function which is continuous at all t as well as its first derivative.
Second and higher-order derivatives are discontinuous at 2n points
. At these points a running mean function is
discontinuous itself, as well as all derivatives. Thus one may say
that a running mean fit is a spline of order 1 and defect 1, a
running parabola fit is a spline of order 6 and defect 5.
For evenly spaced data one often uses a local approximation at times of observations (cf. Whittaker & Robinson 1928) and neglecting intermediate arguments. In this case, discontinuity of the smoothing function is not important and thus unweighted parabolae may be preferred to make smaller accuracy estimates. However, discontinuous smoothing curve may not allow determination of a true extremum, being highly affected by statistical errors of the signal. Moreover, the line interpolating of the smoothed values may not coincide with the smoothing function at the intermediate arguments.
For fixed filter half-width 
, the accuracy estimate is the best for
"um". However, this fit has the worst spectral characteristics and the
largest systematic differences from smooth curves. Thus one has to
determine a value of optimizing the balance between the systematic
and statistical errors of the fits.
If the signal values are evenly distributed in time and their number per
period is large enough, one may use the "continuous limit" discussed
here in detail. For arbitrary distribution of observations in time one
may use the derived precise analytic expressions and to determine
parameter optimizing the preferred papameter(s) of the fit.
Acknowledgements
The author is grateful to G.Foster, V.Yu.Terebizh, B.E.Zhilyaev and P.A.Mason for helpful discussions. The work was supported by the ESO C&EE Grant A4-018.