Page 256 - Special Topic Session (STS) - Volume 1

STS426 Guillaume B.
phenomena, independently of timescale, share in common that they appear
as a sharp change.
We here address the task of identifying transients—any feature or change
that can be identified in the data as statistically distinct from the underlying
process. It should be understood that if a feature cannot be distinguished by
statistical means, it cannot be detected and identified, whether this is because
the transient is too weak or too long-lived. The limitations of a detection
procedure can always be accurately established before applying it.
The power spectrum refers to the power spectral density distribution of a
physical process, whereas the periodogram refers to an estimate of the power
spectrum. The most common choice of a periodogram statistic is the Discrete
Fourier Transform, and it is generally used in the form of a computationally
fast algorithm called Fast Fourier Transform (FFT; see Press et al., 2002)
applicable only to grouped data. More sensitive periodogram statistics include
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the Rayleigh or R -test (Leahy et al., 1983), the Z -test (Buccheri et al., 1983),
and the H-test (de Jager et al., 1989). Two important features that make these
tests more powerful than the standard FFT periodogram are: (1) they can be
applied directly to event arrival times, and thus access all variability timescales
present in the data, and (2) they impose no constraints on the frequencies that
can be tested, and are thus said to oversample the periodogram by testing
timescales other than those corresponding to independent frequencies.
But oversampling without taking into account that the variables computed
to estimate the power are correlated within an independent Fourier spacing
(IFS) leads to frequency-dependent artifacts that distort the periodogram and
that can be interpreted as signatures of coherent periodic modulations. Each
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of the above R , Z and H tests suffers from this.
Although it is powerful—the most powerful according to Leahy et al.
(1983)—in detecting sinusoidal modulations in event data, the R achieves this
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by estimating the power using the fundamental harmonic only. This advantage
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for sinusoidal signals is a limitation for non-sinusoidal pulses. The Z statistic
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was devised for this purpose as a generalization of the R that combines any
number of harmonics. Accessing the power in higher harmonics confers the
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Z an important advantage over the R , and explains why it is the statistic of
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choice for event data where pulses are peaked or irregular in shape. A powerful
and reliable periodogram statistic must fulfil three conditions: it must (1) be
able to use event arrival times in order to access all variability timescales, (2)
allow for oversampling in order to explore frequency space without
restrictions, and (3) take into account the oscillation in the mean, variance, and
covariance of the Fourier components as a function of frequency. These are
met by the modified Rayleigh statistic.

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