Post History
#5: Post edited
by
HDE 226868
·
2020-11-02T02:28:21Z (over 3 years ago)
I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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?
[^1]: 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.
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638), since retracted; see [original version](https://arxiv.org/abs/2010.06638v1)) 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?
- [^1]: 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.
#4: Post edited
by
HDE 226868
·
2020-10-17T16:36:57Z (over 3 years ago)
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
- $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?- [^1]: 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.
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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?
- [^1]: 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.
#3: Post edited
by
HDE 226868
·
2020-10-16T20:16:10Z (over 3 years ago)
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
- $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
- In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.
- 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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?
[^1]: As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually real. 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.
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
- $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
- In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.
- 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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?
- [^1]: 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.
#2: Post edited
by
HDE 226868
·
2020-10-16T20:11:30Z (over 3 years ago)
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
- $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
- In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.
- 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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?
[^1] As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually real. 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.
- I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
- $$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
- In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.
- 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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?
- [^1]: As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually real. 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.
#1: Initial revision
by
HDE 226868
·
2020-10-16T20:10:54Z (over 3 years ago)
When and how should the $Z_n^2$ statistic be used?
I was reading a paper ([Kuechel et al. 2020](https://arxiv.org/abs/2010.06638)) 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
$$Z_n^2=\frac{2}{N}\sum_{k=1}^n\left[\left(\sum_{i=1}^N\cos(2\pi k\phi_i)\right)^2+\left(\sum_{i=1}^N\sin(2\pi k\phi_i)\right)^2\right]$$
In particular, the authors picked $n=1$ and used the $Z_1^2$ statistic. Higher values of $Z_n^2$ appear to correspond to a strong statistical significance of a signal.
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. Should it be used only for periodic signals? If so, what values of $n$ are appropriate for these signals - is $n=1$ usually enough?
[^1] As a side note, folks who know more about the instrument used than I do are fairly confident that this detection isn't actually real. 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.