Figure 6: Dependence of the mean characteristics of the "wp" fit on
for model harmonic wave with noise (left, n=300, P=30) and visual
observations of RT Cyg (right, n=7154, 
:
1
2
3
4-R;
5
6
7
8
The characteristics of the fit are strongly dependent on which is a
free parameter. Its determination for concrete data set is a separate
problem likewise in the case of determination of the degree of
polynomial or the number of harmonics for global approximations.
However, in our case the free parameter is continuous and one may
not apply the Fischer's statistics to estimate statistical significance
of the fit with given 
We propose to use the value of which corresponds to the maximum of
the ratio "signal/noise". This procedure may be illustrated by Fig. 6 (click here).
For numerical study we have used two data sets. The first one is an
artificial one defined at times 
with signal values
being a superposition of pure sine of unit amplitude and period P=30
with normally distributed noise with rms deviation 0.2. The second
set contained n=7154 visual observations of the Mira-type star RT Cyg
obtained by the members of AFOEV and photographic data from the Odessa
plate collection (Marsakova et al. 1997). Both sets were reduced by
using the same program.
With increasing 
the values of 
and 
remain nearly the
same until some value when systematic differences of the fit from the
true shape become significant. One may note that becomes
significantly larger than 
This may be interpreted by the fact
that one uses the sum 
instead of 
to estimate
the mean value of 
whereas the deviation of the central point
of the local fit 
from the true shape is smaller
than of the whole fit. For larger these both estimates coincide
at the higher level as the fit does not response to periodic variations.
The parameter 
is smaller than because it does not take
into account the expression in brackets in the right side of Eq. (22)
and thus is biased. This difference is significant for small
when the number of the data inside the subinterval is small and
decreases with increasing
The parameter 
(Eq. 21)
is equal to R00 for the "wp" fit. Its mean (over all data) value
R2 is shown by line "4" in Fig. 6 (click here). The parameter R
decreases with nearly proportionally to 
because the
number of the data in the subinterval increases proportionally to
For large all data are involved in the local fit, thus R is not
dependent on and only may see a standstill. Accuracy estimates of
the fit 
and 
behave in a more complex way. At first they
decrease with as remain constant and R
decreases. Then their increase becomes more significant than decrease
of R and the product increases, reaches its maximum and
continues to decrease because of the next standstill of and
decrease of R. The standstill of occurs when R has its
standstill. Thus one may conclude that the minimum value
corresponds to 
i.e. the error estimate is the best if we
use the global fit instead of local and approximate the signal
by polynomial of order m. This trivial situation needs no local fits
for different at all. However, if we are interested in the cyclic
variations, we may choose corresponding to local minimum of
Similarly behaves 
but this value is overestimated
in the interval of we are interested in and thus has no practical
meaning.
As the characteristic of the amplitude of the fit one may choose
the rms deviation 
of the smoothed values from the mean.
Its dependence on is shown by line "7" in Fig. 6 (click here). For small
when systematic differences are small, it has a standstill followed by
an abrupt decrease. From and we may combine a parameter
which we call "signal/noise" (S/N) ratio. The position of its
maximum may be used for determination of the optimal value of
It is slightly smaller than that 
obtained from the minimum of
because of decrease of R, but practically this difference does not
exceed 10 per cent and may be used for control of the value 
The position of the maximum of S/N for model harmonic signal is in
good agreement with that obtained for continuous approximation (Eq. (84)
and the following paragraph). For RT Cyg the value 
is smaller than
the expected one 0.5450P because the shape of the light curve is not
sine-like and may be described by 3-harmonic fit (Marsakova et al.
1997).
Determination of the optimal value of needs more computational time
than for the fit with fixed For each data set one will
obtain different values. However, one should recommend to use the same
for all runs not to change spectral properties of the fit (e.g.
Tremko et al. 1996). For this purpose one may extend the summation
from one run to all runs or to use some value close to the mean for
different runs.