<p>You have it the wrong way around. You have to establish a periodicity before doing the folding.</p>

<p>There are various ways to do this, but the most frequently (pun intended) used is the <a href="https://en.wikipedia.org/wiki/Least-squares_spectral_analysis#The_Lomb.E2.80.93Scargle_periodogram" rel="nofollow">Lomb-Scargle</a> periodogram. </p>

<p>Start with a datset consisting of a set of points $(t_i, m_i\pm \sigma_i )$, where $t_i$ are times of observation and $m_i$ and $\sigma_i$ are the fluxes and their uncertainties at those times.</p>

<p>You then calculate an array of chi-squared statistics, by fitting a mean level $\bar{m}$ and sinusoidal functions over a range of discrete frequencies $\omega$. The aim is to <em>maximise</em> the statistic
$$p(\omega) = \frac{\chi^2_{0} - \chi^2(\omega)}{\chi^2_{0}},$$
where 
$$\chi^{2}_{0} = \sum (m_i - \bar{m})^2/\sigma_i^{2}$$ and 
$$\chi^{2}(\omega) = \sum  (y_i - \bar{m})^2/\sigma_i^{2},$$ with $y_i = A\sin(\omega t_i + \phi) + A_0$, where $A$, $A_0$ and $\phi$ are free parameters that are optimised for each frequency (i.e. you find the minimum possible $\chi^2(\omega)$ at each frequency).</p>

<p>Once a period is obtained from a peak in the power spectrum, $p_{best}$, <em>then</em> it must be established whether this peak is "real" or significant.</p>

<p>Theoretically, the probability that $p&gt;p_{best}$ is $(1-p_{best})^{(N-3)/2}$ for $N$ independent measurements (see <a href="http://arxiv.org/abs/astro-ph/9906466" rel="nofollow">Cumming, Marcy &amp; Butler 1999</a>). This is the "false alarm probability" - the chance that you would get $p&gt;p_{best}$ from random, aperiodic data with those uncertainties.</p>

<p>However, real data has correlated noise and the points are not independent. So what you do instead (see <a href="http://arxiv.org/abs/0908.0189" rel="nofollow">Collier Cameron et al. 2009</a>) is shuffle all the times of the data points (i.e. randomly reallocate the times of each observation) and go through the process again. Do this a squillion times with different random reallocations and see how many times you end up with $p&gt;p_{best}$. This gives you some idea about how confident you can be that your detected periodic signal is real. This is called a "bootstrap Monte-Carlo method".</p>

<p>There are other ways to do this; my answer is not complete; but it's the one I'm familiar with.</p>

CC BY-SA 3.0
2015-08-25T00:21:05.353Z
it is said that we can calculate the dispersion and chi2 directly?
questionhang
CC BY-SA 3.0
2015-08-25T07:14:24.153Z
@questionhang I don't understand your follow-up question. I have given you a prescription to find the most likely period from a set of data $t_i, m_i\pm \sigma_i$, and then assess how confident you can be that the periodic signal did not arise by chance.
ProfRob
CC BY-SA 3.0
2015-08-25T07:36:56.053Z
my point is how to assess the significance of a periodical signal. Is there a more popular way than CC's?
questionhang
CC BY-SA 3.0
2015-08-25T07:49:57.023Z
@questionhang This is how (or minor variants thereof) periodic signals are found and their significance assessed in Kepler, SuperWASP, HATnet and the various Doppler shift surveys for exoplanets.
ProfRob
CC BY-SA 3.0
2015-08-25T08:53:18.803Z
I remember there is a method which can give significance level directly. Let me search for that first.
questionhang
CC BY-SA 3.0
2015-08-24T16:07:16.673Z
2015-08-25T07:25:04.307Z
2015-08-25T07:25:04.307Z
ProfRob
<p>After folding a light curve, we may  get some kind of shape.<br/>
How to tell whether there is a real periodical signal quantifiably?</p>

<p>The fluctuation may be small, and the folded shape may not be perfect. </p>

<light>
how to assess a periodical signal?
questionhang
# how to assess a periodical signal?

(Asked by: username_0 on 2015-08-24)

After folding a light curve, we may get some kind of shape.  
How to tell whether there is a real periodical signal quantifiably?

The fluctuation may be small, and the folded shape may not be perfect.

**Tags:** light


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## 1 answers

### Answer by username_1 on 2015-08-24. Score: 2

You have it the wrong way around. You have to establish a periodicity before doing the folding.

There are various ways to do this, but the most frequently (pun intended) used is the [Lomb-Scargle](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis#The_Lomb.E2.80.93Scargle_periodogram) periodogram.

Start with a datset consisting of a set of points $(t\_i, m\_i\pm \sigma\_i )$, where $t\_i$ are times of observation and $m\_i$ and $\sigma\_i$ are the fluxes and their uncertainties at those times.

You then calculate an array of chi-squared statistics, by fitting a mean level $\bar{m}$ and sinusoidal functions over a range of discrete frequencies $\omega$. The aim is to *maximise* the statistic
$$p(\omega) = \frac{\chi^2\_{0} - \chi^2(\omega)}{\chi^2\_{0}},$$
where
$$\chi^{2}\_{0} = \sum (m\_i - \bar{m})^2/\sigma\_i^{2}$$ and
$$\chi^{2}(\omega) = \sum (y\_i - \bar{m})^2/\sigma\_i^{2},$$ with $y\_i = A\sin(\omega t\_i + \phi) + A\_0$, where $A$, $A\_0$ and $\phi$ are free parameters that are optimised for each frequency (i.e. you find the minimum possible $\chi^2(\omega)$ at each frequency).

Once a period is obtained from a peak in the power spectrum, $p\_{best}$, *then* it must be established whether this peak is "real" or significant.

Theoretically, the probability that $p>p\_{best}$ is $(1-p\_{best})^{(N-3)/2}$ for $N$ independent measurements (see [Cumming, Marcy & Butler 1999](http://arxiv.org/abs/astro-ph/9906466)). This is the "false alarm probability" - the chance that you would get $p>p\_{best}$ from random, aperiodic data with those uncertainties.

However, real data has correlated noise and the points are not independent. So what you do instead (see [Collier Cameron et al. 2009](http://arxiv.org/abs/0908.0189)) is shuffle all the times of the data points (i.e. randomly reallocate the times of each observation) and go through the process again. Do this a squillion times with different random reallocations and see how many times you end up with $p>p\_{best}$. This gives you some idea about how confident you can be that your detected periodic signal is real. This is called a "bootstrap Monte-Carlo method".

There are other ways to do this; my answer is not complete; but it's the one I'm familiar with.

> **Comments:**
> **username_0**: it is said that we can calculate the dispersion and chi2 directly?  
> **username_1**: @username_0 I don't understand your follow-up question. I have given you a prescription to find the most likely period from a set of data $t_i, m_i\pm \sigma_i$, and then assess how confident you can be that the periodic signal did not arise by chance.  
> **username_0**: my point is how to assess the significance of a periodical signal. Is there a more popular way than CC's?  
> **username_1**: @username_0 This is how (or minor variants thereof) periodic signals are found and their significance assessed in Kepler, SuperWASP, HATnet and the various Doppler shift surveys for exoplanets.  
> **username_0**: I remember there is a method which can give significance level directly. Let me search for that first.  

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