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bins. The
sensitivity of these statistics to sharp signals (such as strongly
pulsed variations or light curves of very wide eclipsing binaries) is
poor. For the detection of such signals better use AOV/TSA with
a bin width matching the width of these features.
The command SINEFIT/TSA serves two purposes: a) least squares
estimation of the parameters of a detected signal and b) filtering the
data for a given frequency (so-called prewhitening). The trend
removal (zero frequency) constitutes a special case of this filtering.
For a pure sinusoid model, the 
statistic used in
SINEFIT/TSA is related to that used in SCARGLE/TSA (Lomb, 1976,
Scargle, 1982).
. The phase origins of the
sinusoids are for each frequency chosen in such a way that the sine and cosine
components of 
become independent. Hence for white noise
(
hypothesis) S is the ratio of 
and 
.
For large numbers of observations 
, numerator and denominator
become uncorrelated so that S has a Fisher-Snedecor distribution
approaching an exponential distribution in the asymptotic limit:
for 
.
We recommend this statistic for larger data sets and for the detection
of smooth, nearly sinusoidal signals, since then its test power is
large and the statistical properties are known. In particular the
expected value is 1. For observations correlated in groups of size
, divide the value of the Scargle statistics by 
(Sect. 
). The slow algorithm implemented here is
suitable for modest numbers of observations. For a faster, FFT based
version see Press and Rybicki (1991).
multiply the
errors by 
(Sect. ). With the latter
correction and for purely sinusoidal variations SINEFIT/TSA
computes the frequency with an accuracy comparable to the one of the
power spectrum (Lomb, 1976, Schwarzenberg-Czerny, 1991). Additionally,
the command displays the parameters of the fitted base sinusoid, i.e.
of the first Fourier term.
SINEFIT/TSA returns also the table of the residuals
(i.e. of the observations with the fitted oscillation
subtracted) in a format suitable for further analysis by any method
supported by the TSA package. In this way, the command can be used to
perform a CLEAN-like analysis manually by removing individual
oscillations one by one in the time domain (see Roberts et al.,
1987, Gray & Desikhary, 1973). Since in most astronomical time
series the number of different sinusoids present is quite small, we
recommend this manual procedure rather than its automated
implementation in frequency space by the CLEAN algorithm.
Alternatively, the command can be used to remove a trend from data.
In order to use SINEFIT/TSA for a fixed frequency, specify one
iteration only. The corresponding value of 
may in principle
be recovered from the standard deviation
, where 
and
and 
are the number of observations and the number
of Fourier coefficients (including the mean value), respectively.
However, the computation of the 
periodogram with
SINEFIT/TSA is very cumbersome while the results should correspond
exactly to the Scargle periodogram (Scargle, 1982, Lomb, 1976).
. The distribution of S for white noise (
hypothesis) and 
order bins is the Fisher-Snedecor
distribution 
. The expected value of the AOV statistics
for pure noise is 1 for uncorrelated observations and 
for
observations correlated in groups of size 
.
Among all statistics named in this chapter, AOV is the only one with
exactly known statistical properties even for small samples. On large
samples, AOV is not less sensitive than other statistics using phase
binning, i.e. the step function model: 
, Whittaker & Roberts
and PDM. Therefore we recommend the AOV statistics for samples of all
sizes and particularly for signals with narrow sharp features (pulses,
eclipses). If on the average 
consecutive observations are
correlated, divide the value of the periodogram by 
and
use the 
distribution (Sect. ).
For smooth light curves use low order, e.g. 4 or 3, for optimal
sensitivity. For numerous observations and sharp light curves use phase
bins of width comparable to that of the narrow features (e.g. pulses,
eclipses). Note that phase coverage and consequently quality of the
statistics near 0 frequency are notoriously poor for most observations.