There is also another important method for analyzing the data from a network of detectors – the search for coincidences of events among detectors. This analysis is particularly important when we search for supernova bursts the waveforms of which are not very well known. Such signals can be easily mimicked by nonGaussian behavior of the detector noise. The idea is to filter the data optimally in each of the detector and obtain candidate events. Then one compares parameters of candidate events, like for example times of arrivals of the bursts, among the detectors in the network. This method is widely used in the search for supernovae by networks of bar detectors [
11]
.
4.11 Nonstationary, nonGaussian, and nonlinear data
Equations ( 32 ) and ( 33 ) provide maximum likelihood estimators only when the noise in which the signal is buried is Gaussian. There are general theorems in statistics indicating that the Gaussian noise is ubiquitous. One is the central limit theorem which states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution.
The other comes from the information theory and says that the probability distribution of a random variable with a given mean and variance which has the maximum entropy (minimum information) is the Gaussian distribution. Nevertheless, analysis of the data from gravitationalwave detectors shows that the noise in the detector may be nonGaussian (see, e.g., Figure 6 in [
10]
). The noise in the detector may also be a nonlinear and a nonstationary random process.
The maximum likelihood method does not require that the noise in the detector is Gaussian or stationary. However, in order to derive the optimum statistic and calculate the Fisher matrix we need to know the statistical properties of the data. The probability distribution of the data may be complicated, and the derivation of the optimum statistic, the calculation of the Fisher matrix components and the false alarm probabilities may be impractical. There is however one important result that we have already mentioned. The matchedfilter which is optimal for the Gaussian case is also a linear filter that gives maximum signaltonoise ratio no matter what is the distribution of the data. Monte Carlo simulations performed by Finn [
32]
for the case of a network of detectors indicate that the performance of matchedfiltering (i.e., the maximum likelihood method for Gaussian noise) is satisfactory for the case of nonGaussian and stationary noise.
In the remaining part of this section we review some statistical tests and methods to detect nonGaussianity, nonstationarity, and nonlinearity in the data. A classical test for a sequence of data to be Gaussian is the Kolmogorov–Smirnov test [
23]
. It calculates the maximum distance between the cumulative distribution of the data and that of a normal distribution, and assesses the significance of the distance. A similar test is the Lillifors test [
23]
, but it adjusts for the fact that the parameters of the normal distribution are estimated from the data rather than specified in advance. Another test is the Jarque–Bera test [
43]
which determines whether sample skewness and kurtosis are unusually different from their Gaussian values.
Let
${x}_{k}$
and
${u}_{l}$
be two discrete in time random processes (
$\infty <k,l<\infty $
) and let
${u}_{l}$
be independent and identically distributed (i.i.d.). We call the process
${x}_{k}$
linear if it can be represented by
$$\begin{array}{c}{x}_{k}={\sum}_{l=0}^{N}{a}_{l}{u}_{kl},\end{array}$$ 
(88)

where
${a}_{l}$
are constant coefficients. If
${u}_{l}$
is Gaussian (nonGaussian), we say that
${x}_{l}$
is linear Gaussian (nonGaussian). In order to test for linearity and Gaussianity we examine the thirdorder cumulants of the data. The thirdorder cumulant
${C}_{kl}$
of a zero mean stationary process is defined by
$$\begin{array}{c}{C}_{kl}:=E\left[{x}_{m}{x}_{m+k}{x}_{m+l}\right].\end{array}$$ 
(89)

The bispectrum
${S}_{2}({f}_{1},{f}_{2})$
is the twodimensional Fourier transform of
${C}_{kl}$
. The bicoherence is defined as
$$\begin{array}{c}B({f}_{1},{f}_{2}):=\frac{{S}_{2}({f}_{1},{f}_{2})}{S({f}_{1}+{f}_{2})S\left({f}_{1}\right)S\left({f}_{2}\right)},\end{array}$$ 
(90)

where
$S\left(f\right)$
is the spectral density of the process
${x}_{k}$
. If the process is Gaussian then its bispectrum and consequently its bicoherence is zero. One can easily show that if the process is linear then its bicoherence is constant. Thus if the bispectrum is not zero, then the process is nonGaussian; if the bicoherence is not constant then the process is also nonlinear. Consequently we have the following hypothesis testing problems:

1.
${H}_{1}$
: The bispectrum of
${x}_{k}$
is nonzero.

2.
${H}_{0}$
: The bispectrum of
${x}_{k}$
is zero.
If Hypothesis 1 holds, we can test for linearity, that is, we have a second hypothesis testing problem:

3.
${H}_{1}^{\prime}$
: The bicoherence of
${x}_{k}$
is not constant.

4.
${H}_{1}^{\prime \prime}$
: The bicoherence of
${x}_{k}$
is a constant.
If Hypothesis 4 holds, the process is linear.
Using the above tests we can
detect nonGaussianity and, if the process is nonGaussian, nonlinearity of the process. The distribution of the test statistic
$B({f}_{1},{f}_{2})$
, Equation ( 90 ), can be calculated in terms of
${\chi}^{2}$
distributions. For more details see [
38]
.
It is not difficult to examine nonstationarity of the data. One can divide the data into short segments and for each segment calculate the mean, standard deviation and estimate the spectrum.
One can then investigate the variation of these quantities from one segment of the data to the other.
This simple analysis can be useful in identifying and eliminating bad data. Another quantity to examine is the autocorrelation function of the data. For a stationary process the autocorrelation function should decay to zero. A test to detect certain nonstationarities used for analysis of econometric time series is the Dickey–Fuller test [
21]
. It models the data by an autoregressive process and it tests whether values of the parameters of the process deviate from those allowed by a stationary model. A robust test for detection nonstationarity in data from gravitationalwave detectors has been developed by Mohanty [
60]
. The test involves applying Student's ttest to Fourier coefficients of segments of the data.
5 Acknowledgments
One of us (AK) acknowledges support from the National Research Council under the Resident Research Associateship program at the Jet Propulsion Laboratory, California Institute of Technology and from MaxPlanckInstitut für Gravitationsphysik. We would like to thank Dr. Michele Vallisneri for preparing the figure. This research was in part funded by the Polish KBN Grant No. 1 P03B 029 27.
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