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<H2><A NAME="SECTION001746000000000000000"> </A><A NAME="s:freq"> </A>
<BR>
Time series analysis in the frequency
domain
</H2>
For the detection of smooth signals, e.g. sinusoids, use either
<TT>ORT/TSA with 'order'=1 or 2 harmonics, SCARGLE/TSA</TT> or <TT>AOV/TSA</TT>
with 'order'=3 or 4 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 <TT>ORT/TSA or AOV/TSA</TT> with
the width of these features matched by the width of the of the top
harmonics or the width of a phase bin, respectively.
<P>
The command <TT>SINEFIT/TSA</TT> 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 <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img530.gif"
ALT="$\chi^2$">
statistic used in <TT>SINEFIT/TSA</TT> is related to that used in <TT>SCARGLE/TSA</TT> (Lomb, 1976,
Scargle, 1982).
<P>
<DL>
<DT><STRONG><TT>ORT/TSA</TT> -</STRONG>
<DD><B>Multiharmonic analysis of variance periodogram:</B>
The command computes the analysis of variance (AOV) periodogram
for fitting data with a (multiharmonic) Fourier series.
The fit of the Fourier series is done by a new efficient algorithm, employing projection
onto orthogonal trigonometric polynomials. The results of the fit are evaluated using
the AOV statistics, a powerful method newly adapted for the time series analysis (Schwarzenberg-Czerny, 1996, 1989). The model used in this method is the Fourier
series of <I>n</I> harmonics. The resolution of the method may be tuned
by change of <I>n</I>. Hence it is the method of choice for
both smooth and sharp signals. The AOV statistic is the ratio
<!-- MATH: $S(\nu) = Var_m/Var_r$ -->
<IMG
WIDTH="148" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img531.gif"
ALT="$S(\nu) = Var_m/Var_r$">.
The distribution of <I>S</I> for white noise (<I>H</I><SUB><I>o</I></SUB>hypothesis) and <IMG
WIDTH="66" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img532.gif"
ALT="$n\equiv$">
<TT>order</TT> bins is the Fisher-Snedecor
distribution
<!-- MATH: $F(2n+1,n_o-2n-1)$ -->
<I>F</I>(2<I>n</I>+1,<I>n</I><SUB><I>o</I></SUB>-2<I>n</I>-1). The expected value of the AOV statistics
for pure noise is 1 for uncorrelated observations and <I>n</I><SUB><I>corr</I></SUB> for
observations correlated in groups of size <I>n</I><SUB><I>corr</I></SUB>.
<DT><STRONG><TT>SCARGLE/TSA</TT> -</STRONG>
<DD><B>Scargle sine model:</B>
This command computes Scargle's (1982) periodogram for unevenly
spaced observations <I>x</I>. The Scargle statistic uses a pure sine model
and is a special case of the power spectrum statistic normalized to
the variance of the raw data,
<BR>
<!-- MATH: $|{\cal F}X^{(m)}|^2/Var[X^{(o)}]$ -->
<IMG
WIDTH="154" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img533.gif"
ALT="$\vert{\cal F}X^{(m)}\vert^2/Var[X^{(o)}]$">.
The phase origins of the
sinusoids are for each frequency chosen in such a way that the sine and cosine
components of <IMG
WIDTH="1" HEIGHT="25" ALIGN="BOTTOM" BORDER="0"
SRC="img534.gif"
ALT="${\cal F}X$">
become independent. Hence for white noise
(<I>H</I><SUB><I>o</I></SUB> hypothesis) <I>S</I> is the ratio of <IMG
WIDTH="12" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
SRC="img535.gif"
ALT="$\chi^2(2)$">
and
<!-- MATH: $\chi^2(n_o)$ -->
<IMG
WIDTH="22" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img536.gif"
ALT="$\chi^2(n_o)$">.
For large numbers of observations <I>n</I><SUB><I>o</I></SUB>, numerator and denominator
become uncorrelated so that <I>S</I> has a Fisher-Snedecor distribution
approaching an exponential distribution in the asymptotic limit:
<!-- MATH: $F(2,n_o-1)
\rightarrow \chi^2(2)/2 = e^{-S}$ -->
<IMG
WIDTH="247" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img537.gif"
ALT="$F(2,n_o-1)
\rightarrow \chi^2(2)/2 = e^{-S}$">
for
<!-- MATH: $n \rightarrow \infty$ -->
<IMG
WIDTH="28" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img538.gif"
ALT="$n \rightarrow \infty$">.
<P>
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
<I>n</I><SUB><I>corr</I></SUB>, divide the value of the Scargle statistics by <I>n</I><SUB><I>corr</I></SUB>(Sect. <A HREF="node226.html#s:corr">12.2.4</A>). The slow algorithm implemented here is
suitable for modest numbers of observations. For a faster, FFT based
version see Press and Rybicki (1991).
<DT><STRONG><TT>SINEFIT/TSA</TT> -</STRONG>
<DD><B>Least-squares sinewave fitting:</B>
This command fits sine (Fourier) series by nonlinear least squares
iterations with simultaneous correction of the frequency. Its main
applications are the evaluation of the significance of a detection,
parameter estimation, and massaging of data. The values fitted for
frequency and Fourier coefficients are displayed on the terminal. For
observations correlated in groups of size <I>n</I><SUB><I>corr</I></SUB> multiply the
errors by
<!-- MATH: $\sqrt{n_{corr}}$ -->
<IMG
WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img539.gif"
ALT="$\sqrt{n_{corr}}$">
(Sect. <A HREF="node226.html#s:corr">12.2.4</A>). With the latter
correction and for purely sinusoidal variations <TT>SINEFIT/TSA</TT>
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.
<P>
<TT>SINEFIT/TSA</TT> returns also the table of the residuals
<I>X</I><SUP>(<I>r</I>)</SUP> (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 <EM>et al.</EM>,
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.
<P>
Alternatively, the command can be used to remove a trend from data.
In order to use <TT>SINEFIT/TSA</TT> for a fixed frequency, specify one
iteration only. The corresponding value of <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img540.gif"
ALT="$\chi^2$">
may in principle
be recovered from the standard deviation
<!-- MATH: $\sigma_o=\sqrt{\chi^2(df)/df}$ -->
<IMG
WIDTH="124" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
SRC="img541.gif"
ALT="$\sigma_o=\sqrt{\chi^2(df)/df}$">,
where
<!-- MATH: $df=n_{obs}-n_{parm}$ -->
<I>df</I>=<I>n</I><SUB><I>obs</I></SUB>-<I>n</I><SUB><I>parm</I></SUB> and
<I>n</I><SUB><I>obs</I></SUB> and <I>n</I><SUB><I>parm</I></SUB> are the number of observations and the number
of Fourier coefficients (including the mean value), respectively.
However, the computation of the <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img542.gif"
ALT="$\chi^2$">
periodogram with <TT>SINEFIT/TSA</TT> is very cumbersome while the results should correspond
exactly to the Scargle periodogram (Scargle, 1982, Lomb, 1976).
<DT><STRONG><TT>AOV/TSA</TT> -</STRONG>
<DD><B>Analysis of variance for phase bins:</B>
The command computes the analysis of variance (AOV) periodogram
for phase folded and binned data.
The AOV statistics is a new and powerful method especially suitable for the
detection of nonsinusoidal signals (Schwarzenberg-Czerny, 1989). It
uses the step function model, i.e. phase binning. Its statistic is
<!-- MATH: $S(\nu) = Var_m/Var_r$ -->
<IMG
WIDTH="148" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img543.gif"
ALT="$S(\nu) = Var_m/Var_r$">.
The distribution of <I>S</I> for white noise (<I>H</I><SUB><I>o</I></SUB>hypothesis) and <IMG
WIDTH="66" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img544.gif"
ALT="$n\equiv$">
<TT>order</TT> bins is the Fisher-Snedecor
distribution
<!-- MATH: $F(n-1,n_o-n)$ -->
<I>F</I>(<I>n</I>-1,<I>n</I><SUB><I>o</I></SUB>-<I>n</I>), where <I>n</I> is number of bins. The expected value of the AOV statistics
for pure noise is 1 for uncorrelated observations and <I>n</I><SUB><I>corr</I></SUB> for
observations correlated in groups of size <I>n</I><SUB><I>corr</I></SUB>.
<P>
Among all statistics named in this chapter, AOV used by ORT/TSA
and AOV/TSA 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: <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img545.gif"
ALT="$\chi^2$">,
Whittaker & Roberts
and PDM. Therefore we recommend the ORT/TSA and AOV/TSA
commands for samples of all sizes and particularly for
signals with narrow sharp features (pulses,
eclipses). If on the average <I>n</I><SUB><I>corr</I></SUB> consecutive observations are
correlated, divide the value of the periodogram by <I>n</I><SUB><I>corr</I></SUB> and
use the
<!-- MATH: $F(n-1,n_o/n_{corr}-n)$ -->
<I>F</I>(<I>n</I>-1,<I>n</I><SUB><I>o</I></SUB>/<I>n</I><SUB><I>corr</I></SUB>-<I>n</I>) distribution (Sect. <A HREF="node226.html#s:corr">12.2.4</A>).
For smooth light curves use low <TT>order</TT>, 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.
</DL>
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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