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<H2><A NAME="SECTION001724000000000000000"> </A><A NAME="s:corr"> </A>
<BR>
Corrections to the probability distribution
</H2>
<P>
In principle, it is possible to compute a value of the statistic
<IMG
WIDTH="13" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img461.gif"
ALT="$S(\nu_1)$">
for a single frequency <IMG
WIDTH="50" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
SRC="img462.gif"
ALT="$\nu_1$">
and to test its consistency
with a random signal (<I>H</I><SUB><I>o</I></SUB>). The common procedure of inspecting the
whole periodogram for a detected signal corresponds to the <I>N</I>-fold
repetition of the single test for a set of trial frequencies,
<!-- MATH: $\nu_n,
n=1,\dots,N$ -->
<IMG
WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img463.gif"
ALT="$\nu_n,
n=1,\dots,N$">.
The probability of the whole periodogram being
consistent with <I>H</I><SUB><I>o</I></SUB> is
<!-- MATH: $1-(1-p)^N \rightarrow Np$ -->
<IMG
WIDTH="144" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
SRC="img464.gif"
ALT="$1-(1-p)^N \rightarrow Np$">
for
<!-- MATH: $p\rightarrow
0$ -->
<IMG
WIDTH="17" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img465.gif"
ALT="$p\rightarrow
0$">.
The factor <I>N</I> means that there is an increased probability of
accepting a given value of the statistic as consistent with a random
signal. Therefore, increasing the number of trial frequencies
decreases the sensitivity for the detection of a significant signal
and accordingly is called the penalty factor for multiple trials or
for the frequency bandwidth used. The true number of independent
frequencies, <I>N</I><SUB><I>t</I></SUB>, remains generally unknown. It is usually less than the
number of resolved frequencies
<!-- MATH: $N_r=\Delta\nu \Delta t$ -->
<IMG
WIDTH="72" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img466.gif"
ALT="$N_r=\Delta\nu \Delta t$">
(Sect.
<A HREF="node232.html#s:iftr">12.3.1</A>) because of aliasing and still less than the number of
computed frequencies <I>N</I><SUB><I>c</I></SUB>, because of oversampling:
<!-- MATH: $N_t \le N_r \le
N_c$ -->
<IMG
WIDTH="96" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img467.gif"
ALT="$N_t \le N_r \le
N_c$">.
For a practical and conservative estimate, we recommend to use
<I>N</I><SUB><I>r</I></SUB> as the number of trial frequencies, <I>N</I>.
<P>
According to the standard null hypothesis, <I>H</I><SUB><I>o</I></SUB>, the noise is white
noise. This is not the case in many practical cases. For instance,
often the noise is a stochastic process with a certain correlation
length
<!-- MATH: $l_{corr}>0$ -->
<I>l</I><SUB><I>corr</I></SUB>>0, so that on average <I>n</I><SUB><I>corr</I></SUB> consecutive
observations are correlated. Such noise corresponds to white noise
passed through a low pass filter which cuts off all frequencies above
<!-- MATH: $1/l_{corr}$ -->
1/<I>l</I><SUB><I>corr</I></SUB>. Such correlation is not usually taken into account by
standard test statistics. The effect of this correlation is to reduce
the effective number of observations by a factor <I>n</I><SUB><I>corr</I></SUB>(Schwarzenberg-Czerny, 1989). This has to be accounted for by scaling
both the statistics <I>S</I> and the number of its degrees of freedom <I>n</I><SUB><I>j</I></SUB>by factors depending on <I>n</I><SUB><I>corr</I></SUB>.
<P>
In the test statistic, a continuum level which is inconsistent with
the expected value of the statistic <IMG
WIDTH="14" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img468.gif"
ALT="$E\{S\}$">
may indicate the presence
of such a correlation between consecutive data points. A practical
recipe to measure the correlation is to compute the residual time
series (e.g. with the <TT>SINEFIT/TSA</TT> command) and to look for its
correlation length with <TT>COVAR/TSA</TT> command. The effect of the
correlation in the parameter estimation is an underestimation of the
uncertainties of the parameters; the true variances of the parameters
are a factor <I>n</I><SUB><I>corr</I></SUB> larger than computed.
<P>
In the command individual descriptions, we often refer to probability
distributions of specific statistics. For the properties of these
individual distributions see e.g. Eadie <EM>et. al.</EM> (1971), Brandt
(1970), and Abramovitz & Stegun (1972). The two latter references
contain tables. For a computer code for the computation of the
cumulative probabilities see Press <EM>et. al.</EM> (1986).
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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