When and how should the $Z_n^2$ statistic be used?
I was reading a paper (Kuechel et al. 2020, since retracted; see original version) claiming a detection of a high-frequency periodic signal coming from a known pulsar.[1] The authors used something called the $Z_n^2$ statistic, a test I'm not familiar with. Assuming we have a time series of $N+1$ photons with arrival times $t_0,t_1,t_2,\dots,t_N$, we write the phase of the $i$th photon after the first as $$\phi_i=\nu(t_i-t_0)+\dot{\nu}(t_i-t_0)^2/2+\cdots$$ with $\nu,\dot{\nu},\dots$ being the frequency and frequency time derivatives of the signal. The $Z_n^2$ statistic is then defined as $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$ Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal. In this case, the authors picked $n=1$ and used the $Z_1^2$ statistic.
I've never encountered the $Z_n^2$ statistic before, and I've had a hard time tracking down and/or accessing references, so my question has to do with its applicability. I assume it should be used only for periodic signals - is that right? If so, what values of $n$ are appropriate for these signals? Is $n=1$ usually adequate?
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As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually coming from the system. While the source has a known frequency of $\sim567$ Hz, the frequency Kuechel et al. report detecting is $\sim890$ Hz, which corresponds to a well-known instrumental frequency of the telescope, NuSTAR. Therefore, while the signal is real, it's likely not astronomical in origin. As such, the preprint has been removed. ↩︎
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By choosing $n=1$, Kuechel et al. actually use the special case of the Rayleigh statistic $R^2$, which only looks at the fundamental harmonic. In general, however, the $Z_n^2$ statistic is ideal for searching for non-sinusoidal periodic signals, which requires looking at contributions from higher harmonics and hence typically choosing $n>1$ (Belanger 2017, Buccheri 1983). The ability to look beyond the fundamental can be quite handy when looking at, among other things, the distinctly non-sinusoidal x-ray or gamma ray pulses from pulsars.
If you know a priori what your signal should look like, you maybe able to determine how many harmonics to use without much trouble. In this context within x-ray astronomy, $n=2$ may actually maximize the signal-to-noise ratio of an observation, as it provides sensitivity to unknown pulses with a range of different broadnesses. An optimal number of harmonics, however, may be determined by applying the $H$-test of de Jager et al. 1989. Here, we define the $H$-statistic by $$H\equiv\max_{1\leq n\leq20}(Z_n^2-4n+4)$$ Realistically, an infinite number of harmonics could be searched, but the authors argue that $n\leq20$ is typically an adequate truncation, at least for these purposes. $H$ itself can also in fact be used as a statistical test in place of $Z_n^2$.
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