6 Non-stationary, Non-Gaussian, and Non-linear Data

Eqs. (61View Equation) and (62View Equation) 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 variables 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 gravitational-wave detectors shows that the noise in the detector may be non-Gaussian (see, e.g., Figure 6 in [22]). The noise in the detector may also be a non-linear and a non-stationary random process.

The maximum likelihood method does not require that the noise in the detector be 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. However, there is one important result that we have already mentioned. The matched-filter, which is optimal for the Gaussian case is also a linear filter that gives maximum signal-to-noise ratio no matter what the distribution of the data. Monte Carlo simulations performed by Finn [49] for the case of a network of detectors indicate that the performance of matched-filtering (i.e., the maximum likelihood method for Gaussian noise) is satisfactory for the case of non-Gaussian and stationary noise.

Allen et al. [10, 11] derived an optimal (in the Neyman–Pearson sense, for weak signals) signal processing strategy, when the detector noise is non-Gaussian and exhibits tail terms. This strategy is robust, meaning that it is close to optimal for Gaussian noise but far less sensitive than conventional methods to the excess large events that form the tail of the distribution. This strategy is based on a locally optimal test [71] that amounts to comparing a first non-zero derivative

| dn Λ[x|𝜖]| Λn [x] = ---d𝜖n-- || (133 ) 𝜖=0
of the likelihood ratio with respect to the amplitude of the signal with a threshold instead of the likelihood ratio itself.

The non-stationarity in the case of Gaussian and uncorrelated noise can be easily incorporated into matched filtering (see Appendix C of [1Jump To The Next Citation Point]). Let us assume that a noise sample nl in the data has a Gaussian pdf with a variance σ2l and zero mean (l = 1, ...,N, where N is the number of data points). Different noise samples may have distributions with different variances. We also assume that the noise samples are uncorrelated, then the autocorrelation function K (l,l′) of the noise is given by [see Eq. (39View Equation)]

K (l,l′) = σ2δ ′, (134 ) l ll
where δll′ is the Kronecker delta function. In the case of a known signal hl and additive noise the optimal filter ql is the solution of the following equation [which is a discrete version of the integral Eq. (41View Equation)]:
∑N hl = K (l,l′)ql′. (135 ) l′=1
Thus, we have the following equation for the filter ql:
q = hl, (136 ) l σ2l
and the following expression for the log likelihood ratio:
N N 2 ∑ hlxl 1∑ h-l ln Λ[x] = σ2l − 2 σ2l . (137 ) l=1 l=1
Thus, we see that for non-stationary, uncorrelated Gaussian noise the optimal processing is identical to matched filtering for a known signal in stationary Gaussian noise, except that we divide both the data xl and the signal hl by time-varying standard deviation of the noise. This may be thought of as a special case of whitening the data and then correlating it using a whitened filter.

In the remaining part of this section we review some statistical tests and methods to detect non-Gaussianity, non-stationarity, and non-linearity in the data. A classical test for a sequence of data to be Gaussian is the Kolmogorov–Smirnov test [37Jump To The Next Citation Point]. 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 [37], 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 [69], which determines whether sample skewness and kurtosis are unusually different from their Gaussian values.

A useful test to detect outliers in the data is Grubbs’ test [55]. This test assumes that the data has an underlying Gaussian probability distribution but it is corrupted by some disturbances. Grubbs’ test detects outliers iteratively. Outliers are removed one by one and the test is iterated until no outliers are detected. Grubbs’ test is a test of the null hypothesis:

  1. H 0: There are no outliers in the data set x k.

against the alternate hypothesis:

  1. H1: There is at least one outlier in the data set xk.

The Grubbs’ test statistic is the largest absolute deviation from the sample mean in units of the sample standard deviation, so it is defined as

G = maxk--|xk-−--μ|, (138 ) σ
where μ and σ denote the sample mean and the sample standard deviation, respectively. The hypothesis of no outliers is rejected if
┌ ------------------ │ 2 n-−-1-│∘ ----tα∕(2n),n−-2---- G > √ n n − 2 + t2 , (139 ) α∕(2n),n− 2
where tα∕(2n),n−2 denotes the critical value of the t-distribution with n − 2 degrees of freedom and a significance level of α∕(2n ).

Grubbs’ test has been used to identify outliers in the search of Virgo data for gravitational-wave signals from the Vela pulsar [1]. A test to discriminate spurious events due to non-stationarity and non-Gaussianity of the data from genuine gravitational-wave signals has been developed by Allen [9]. This test, called the χ2 time-frequency discriminator, is applicable to the case of broadband signals, such as those coming from compact coalescing binaries.

Let now xk and ul be two discrete-in-time random processes (− ∞ < k,l < ∞) and let ul be independent and identically distributed (i.i.d.) random variables. We call the process x k linear if it can be represented by

∑N xk = aluk−l, (140 ) l=0
where al are constant coefficients. If ul is Gaussian (non-Gaussian), we say that xl is linear Gaussian (non-Gaussian). In order to test for linearity and Gaussianity we examine the third-order cumulants of the data. The third-order cumulant Ckl of a zero mean stationary process is defined by
Ckl := E [xmxm+kxm+l ]. (141 )
The bispectrum S2 (f1,f2) is the two-dimensional Fourier transform of Ckl. The bicoherence is defined as
S2(f1,f2) B (f1,f2) := S-(f-+-f-)S-(f-)S(f-), (142 ) 1 2 1 2
where S(f) is the spectral density of the process xk. 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 non-Gaussian; if the bicoherence is not constant then the process is also non-linear. Consequently we have the following hypothesis testing problems:
  1. H1: The bispectrum of xk is nonzero.
  2. H0: The bispectrum of xk is zero.

If Hypothesis 1 holds, we can test for linearity, that is, we have a second hypothesis testing problem:

  1. ′ H 1: The bicoherence of xk is not constant.
  2. ′′ H 1: The bicoherence of xk is a constant.

If Hypothesis 4 holds, the process is linear.

Using the above tests we can detect non-Gaussianity and, if the process is non-Gaussian, non-linearity of the process. The distribution of the test statistic B (f1,f2), Eq. (142View Equation), can be calculated in terms of χ2 distributions. For more details see [59].

It is not difficult to examine non-stationarity 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 non-stationarities used for analysis of econometric time series is the Dickey–Fuller test [34]. 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 detecting non-stationarity in data from gravitational-wave detectors has been developed by Mohanty [93]. The test involves applying Student’s t-test to Fourier coefficients of segments of the data. Still another block-normal approach has been studied by McNabb et al. [88]. It identifies places in the data stream where the characteristic statistics of the data change. These change points divide the data into blocks in characteristics are stationary.

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