<blockquote>
<p>What does &quot;RV model including a GP with a quasi-periodic covariance structure&quot; mean?</p>
</blockquote>
<p>RV stands for radial velocity, which is an often used quantity in exoplanet science - <a href="https://www.universetoday.com/138014/radial-velocity-method/" rel="nofollow noreferrer">it is a usual method for detecting exoplanets</a>, for example. The mass of an exoplanet <a href="https://www.rmg.co.uk/schools-communities/teacher-resources/mass-exoplanet-radial-velocity" rel="nofollow noreferrer">can be constrained</a> using it as well. So, you measure some radial velocity profile and want to model it to calculate things about the system. The &quot;GP with a quasi-periodic covariance structure&quot; is the type of model that is employed by these authors.</p>
<p>GP stands for Gaussian Process, which basically means that if you have a random (stochastic) time series, i.e. a variable <span class="math-container">$x$</span> that depends on a set of times <span class="math-container">$t$</span>, then the process is &quot;Gaussian&quot; if and only if the variable over those times is Gaussian distributed. One <a href="https://en.wikipedia.org/wiki/Gaussian_process#Definition" rel="nofollow noreferrer">can formulate this definition</a> in terms of the means and <a href="https://en.wikipedia.org/wiki/Covariance" rel="nofollow noreferrer">covariances</a> of the variables.</p>
<blockquote>
<p>if one is looking for something periodic, why do we start with something stochastic? And for that matter, why look for something with only &quot;quasi-periodic covariance&quot;?</p>
</blockquote>
<p>To put it simply, you gotta start simple! Walk before you can run, sort of thing. In astronomy, processes can often be modeled as periodic processes, such as the time series of a spectrum of light from a star with a planet passing between the star and our detector. But the process can have deviations from periodicity that would make use of periodic covariance function not so great. So, a quasi-periodic covariance function can be used instead to account for those deviations from perfect (e.g., sinusoidal) periodicity.</p>
<p>Other examples of quasi-periodic time series in nature <a href="https://academic.oup.com/gji/article/226/3/1705/6253216" rel="nofollow noreferrer">can come from geology/geophysics</a>, due to the deviations from perfect periodicity that arise from a dynamical source.</p>
<p>To put it more mathematically, I borrow from <a href="https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0550" rel="nofollow noreferrer">Gaussian processes for time-series modelling</a> by Roberts <em>et al.</em>:</p>
<blockquote>
<p>For any function <span class="math-container">$u: x \to u(x)$</span>, a covariance function <span class="math-container">$k()$</span> defined over the range of <span class="math-container">$x$</span> gives rise to a valid covariance k′() over the domain of u. Hence, we can use simple, stationary covariances in order to construct more complex (possibly non-stationary) covariances.... As described in Rasmussen &amp; Williams, valid covariance functions can be constructed by adding or multiplying simpler covariance functions. Thus, we can obtain a quasi-periodic kernel simply by multiplying a periodic kernel with one of the basic stationary kernels described earlier. The latter then specifies the rate of evolution of the periodic signal.</p>
</blockquote>
<p>(See Eqtn 3.6 for the definition of the covariance function, and the above quote is taken from above Eqtn. 21.)</p>
<p>This means that you can take a stationary covariance function and make a periodic one, and then obtain a &quot;quasi-periodic&quot; covariance function from a stationary one and a periodic one.</p>
<p>(see <a href="https://en.wikipedia.org/wiki/Gaussian_process#Covariance_functions" rel="nofollow noreferrer">here</a> for meaning of stationarity in this context).</p>
<p>A <a href="https://www.eso.org/public/archives/releases/sciencepapers/eso2202/eso2202a.pdf" rel="nofollow noreferrer">link</a> to the paper you're referencing. The authors also state, &quot;Regardless of the number of planets modelled, without the inclusion of this GP the residuals always display correlated behaviour,&quot; which can be interpreted as a justification that there is some randomness in the process so the covariance <em>shouldn't</em> be perfectly periodic anyway.</p>
<p>Lastly, residuals that are uncorrelated and flat is statistical evidence that the model you're fitting to the data does not suffer from systematics, i.e., if there are correlations between residuals, then there is information left in the residuals which should be used in the computation and you haven't captured all of the physics, and if the residuals are not flat (i.e. have a mean other than zero) then the model predictions are biased.</p>
<p>I highly recommend <a href="https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0550#d3e499" rel="nofollow noreferrer">the section on Gaussian Processes</a> in the Roberts <em>et al.</em> paper as they explain things quite well. Hope this helps!</p>

CC BY-SA 4.0
2022-02-11T09:14:31.390Z
2022-02-11T09:14:31.390Z
Daddy Kropotkin
<p>A candidate short-period sub-Earth orbiting Proxima Centauri says:</p>
<blockquote>
<p>4.1 Simple periodogram analysis</p>
<p>]We first consider a very simple pre-whitening procedure applied to the CCF RVs in order to determine the main periodicities present in the data. The observed RVs are initially modelled with a Gaussian process (GP) using a standard quasi-periodic (QP) kernel (<a href="https://www.newton.ac.uk/files/seminar/20070809140015001-150844.pdf" rel="nofollow noreferrer">Rasmussen &amp; Williams 2006</a>; <a href="https://academic.oup.com/mnras/article/443/3/2517/1750443#sec4" rel="nofollow noreferrer">Haywood et al. 2014</a>). For simplicity, the hyperparameters of the kernel are fixed to the values determined in SM2020 from a combined fit (their Table 3). Two RV offsets are adjusted between the three data subsets. The resulting fit is shown in the top panel of Fig. 2 and the periodogram of the residuals is shown in panel a) of the same figure. The highest peak in the periodogram is at 11.19 days with a false alarm probability well below 1%.</p>
</blockquote>
<p>and</p>
<blockquote>
<p>5.1 Justification for the use of a GP</p>
<p><strong>We found that an RV model including a GP with a quasi-periodic covariance structure was the only model that would yield uncorrelated, flat residuals.</strong> Regardless of the number of planets modelled, without the inclusion of this GP the residuals always display correlated behaviour.</p>
</blockquote>
<p><strong>Question:</strong> What does &quot;RV model including a GP with a quasi-periodic covariance structure&quot; mean?</p>
<p>Is it possible to write a short &quot;general idea&quot; explanation of what this is? The title of Rasmussen &amp; Williams 2006 is &quot;Gaussian Processes for Machine Learning&quot; and on slide 17 it says</p>
<blockquote>
<p>A Gaussian process is a stochastic process specified by its mean and covariance functions</p>
</blockquote>
<p>so I'm already feeling dizzy; if one is looking for something periodic, why do we start with something stochastic? And for that matter, why look for something with only &quot;quasi-periodic covariance&quot;?</p>

<exoplanet><data-analysis><mathematics><modeling>
What does "RV model including a GP with a quasi-periodic covariance structure" mean?
uhoh
# What does "RV model including a GP with a quasi-periodic covariance structure" mean?

(Asked by: username_0 on 2022-02-10)

A candidate short-period sub-Earth orbiting Proxima Centauri says:

> 4.1 Simple periodogram analysis
>
> ]We first consider a very simple pre-whitening procedure applied to the CCF RVs in order to determine the main periodicities present in the data. The observed RVs are initially modelled with a Gaussian process (GP) using a standard quasi-periodic (QP) kernel ([Rasmussen & Williams 2006](https://www.newton.ac.uk/files/seminar/20070809140015001-150844.pdf); [Haywood et al. 2014](https://academic.oup.com/mnras/article/443/3/2517/1750443#sec4)). For simplicity, the hyperparameters of the kernel are fixed to the values determined in SM2020 from a combined fit (their Table 3). Two RV offsets are adjusted between the three data subsets. The resulting fit is shown in the top panel of Fig. 2 and the periodogram of the residuals is shown in panel a) of the same figure. The highest peak in the periodogram is at 11.19 days with a false alarm probability well below 1%.

and

> 5.1 Justification for the use of a GP
>
> **We found that an RV model including a GP with a quasi-periodic covariance structure was the only model that would yield uncorrelated, flat residuals.** Regardless of the number of planets modelled, without the inclusion of this GP the residuals always display correlated behaviour.

**Question:** What does "RV model including a GP with a quasi-periodic covariance structure" mean?

Is it possible to write a short "general idea" explanation of what this is? The title of Rasmussen & Williams 2006 is "Gaussian Processes for Machine Learning" and on slide 17 it says

> A Gaussian process is a stochastic process specified by its mean and covariance functions

so I'm already feeling dizzy; if one is looking for something periodic, why do we start with something stochastic? And for that matter, why look for something with only "quasi-periodic covariance"?

**Tags:** exoplanet, data-analysis, mathematics, modeling

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## 1 answers

### Answer by username_1 on 2022-02-11. Score: 3 | Accepted answer

> What does "RV model including a GP with a quasi-periodic covariance structure" mean?

RV stands for radial velocity, which is an often used quantity in exoplanet science - [it is a usual method for detecting exoplanets](https://www.universetoday.com/138014/radial-velocity-method/), for example. The mass of an exoplanet [can be constrained](https://www.rmg.co.uk/schools-communities/teacher-resources/mass-exoplanet-radial-velocity) using it as well. So, you measure some radial velocity profile and want to model it to calculate things about the system. The "GP with a quasi-periodic covariance structure" is the type of model that is employed by these authors.

GP stands for Gaussian Process, which basically means that if you have a random (stochastic) time series, i.e. a variable $x$ that depends on a set of times $t$, then the process is "Gaussian" if and only if the variable over those times is Gaussian distributed. One [can formulate this definition](https://en.wikipedia.org/wiki/Gaussian_process#Definition) in terms of the means and [covariances](https://en.wikipedia.org/wiki/Covariance) of the variables.

> if one is looking for something periodic, why do we start with something stochastic? And for that matter, why look for something with only "quasi-periodic covariance"?

To put it simply, you gotta start simple! Walk before you can run, sort of thing. In astronomy, processes can often be modeled as periodic processes, such as the time series of a spectrum of light from a star with a planet passing between the star and our detector. But the process can have deviations from periodicity that would make use of periodic covariance function not so great. So, a quasi-periodic covariance function can be used instead to account for those deviations from perfect (e.g., sinusoidal) periodicity.

Other examples of quasi-periodic time series in nature [can come from geology/geophysics](https://academic.oup.com/gji/article/226/3/1705/6253216), due to the deviations from perfect periodicity that arise from a dynamical source.

To put it more mathematically, I borrow from [Gaussian processes for time-series modelling](https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0550) by Roberts *et al.*:

> For any function $u: x \to u(x)$, a covariance function $k()$ defined over the range of $x$ gives rise to a valid covariance k′() over the domain of u. Hence, we can use simple, stationary covariances in order to construct more complex (possibly non-stationary) covariances.... As described in Rasmussen & Williams, valid covariance functions can be constructed by adding or multiplying simpler covariance functions. Thus, we can obtain a quasi-periodic kernel simply by multiplying a periodic kernel with one of the basic stationary kernels described earlier. The latter then specifies the rate of evolution of the periodic signal.

(See Eqtn 3.6 for the definition of the covariance function, and the above quote is taken from above Eqtn. 21.)

This means that you can take a stationary covariance function and make a periodic one, and then obtain a "quasi-periodic" covariance function from a stationary one and a periodic one.

(see [here](https://en.wikipedia.org/wiki/Gaussian_process#Covariance_functions) for meaning of stationarity in this context).

A [link](https://www.eso.org/public/archives/releases/sciencepapers/eso2202/eso2202a.pdf) to the paper you're referencing. The authors also state, "Regardless of the number of planets modelled, without the inclusion of this GP the residuals always display correlated behaviour," which can be interpreted as a justification that there is some randomness in the process so the covariance *shouldn't* be perfectly periodic anyway.

Lastly, residuals that are uncorrelated and flat is statistical evidence that the model you're fitting to the data does not suffer from systematics, i.e., if there are correlations between residuals, then there is information left in the residuals which should be used in the computation and you haven't captured all of the physics, and if the residuals are not flat (i.e. have a mean other than zero) then the model predictions are biased.

I highly recommend [the section on Gaussian Processes](https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0550#d3e499) in the Roberts *et al.* paper as they explain things quite well. Hope this helps!


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