Selection Function Basics*�HWRix, D.Hogg, D.Boubert, A.Brown, R.Drimmel, A.Everall, M. Fousneau, A. Price-Whelan�2021, AJ, 162, 142

• What is a selection function? (SF)
• When do I need it?
• How do I construct it (well)?
• What do I do with it?
• … a worked example ..
• When is it simple, when tricky?

* n the context of Gaia data

What is a selection function, S(d_obs)?�

• S(d_obs) is the probability of object with properties d_obs being in catalog

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• S(d_obs) is multiplicative link between model predictions for quantities d_obs and the catalog-incidence of objects with these d_obs

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When do I need an SF?

• Fit a model to an ensemble of catalog data …

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• Understand the incidence of even one single object …

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• Make predictions for a specific experiment …

A selection function should be a function of …?

• the minimal set of d_obs needed to predict sample membership probability
• for any object with actual (or counter-factual) properties q!!
• if the selection depends on (many) other objects‘ properties: it gets complicated!!
• d_obs‘s for which
• model predictions can be made (cf. selection of YSOs/QSOs by variability?)
• S(d_obs) can be practically quantified
• d_obs‘s should usually be „observables“
• position, magnitude, parallax, T_eff, [X/H], .. (observed vs “modeled“ quantities is a blurry division)
• ..what about selecting on S/N cuts and error flags?

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Fundamental Catalog SF vs Sample Cut

Basic & Common Issue:��The selection function depends on d_obs that are not immediately relevant to the model‘s physics� e.g. d_obs=(actual sky position, parallax, etc..)��⇒ integrate/marginalize out these d_obs��

… for the case of modelling spatial densities …

Example: space density of WD‘s, given a catalog

this calculates an effective survey volume!

That‘s the upshot of this worked example

To recapitulate: simple operative SF procedure

* traditionally often done by Monte-Carlo simulations (=integration)

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Some Selection Function Discussion Topics

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• How to combine different selection functions? (incl. different catalogs)
• a boolean AND becomes a multiplication of SF‘s

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• What if the SF of an object is a function of other objects‘ properties?

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• What are the consequences of making sample cuts on noisy data?
• (When) is it sensible to make S/N sample cuts?

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• How well do we have to know the selection function?

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• What about „contamination“? Is that a selection function issue?

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Selection function (in SDSS-V)

• Basically all SDSS-V (MOS) studies are population studies (incl. discovery!)

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• Population studies require a selection function, S(q) ( 0<S(q)<1 )
• minimize # of q needed to quantify sample membership
• q should be „observables“
• the model you want to fit must be able to predict the q !

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• A simple and stringent , S(q)
• may give you fewer of the objects you love
• but will allow you to do more with them

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• For „but what if…“, have a look at Rix, Hogg et al 2021arXiv210607653

scale length = f(𝛕 ,[Fe/H])

scale height = f(𝛕 ,[Fe/H])

𝛒_*(𝛕 ,[Fe/H]) at Sun

𝛒_*(𝛕 ) at Sun „SFR“

linear scale

density increases with R

velocity shift matrix for X_A and X_B

X_A

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X_B

multi-epoch �obs. data

epoch 1

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epoch 2

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epoch 3

disentangled �spectra X_A, X_B

• Selection „cuts“, say in m and 𝜛, do not need to be rectangular!
• E.g. you can mimic S/N cuts

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Initial candidate sample

• „simple“ cuts:
• 𝝕 > 3 mas; G < 20 mag; G+5log10(𝝕/100 ) „well below MS“
• Gaia query yields

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A simple model for the luminosity-color function of WD‘s: their space density = f(M_G,c) near the Sun

For the total number of WD‘s in the catalog at (M,c) the actual spatial positions are nuisance parameters, to be marginalized out.

• This becomes: for uniform density

S/N cuts: e.g. 𝛡 / 𝛅 𝛡 > 20

• Why? E.g. to get good `resolution‘ in M_G
• How does that affect the SF? … a lot!

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Initial data-quality cut:

• Rybizki, Green et al 2021
• astrometric fidelity > 0.9 🡪 75% of sources were bad measurements
• How does that affect the SF? Basically not at all!
• eliminates only 2% of spectroscopically confirmed WD G<19.5

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before after

Eliminate astrophysical interlopers

• Binaries, Galaxies, ???
• Exploit that ‚objects of interest‘, WDs, have very tight color-color locus
• How does that affect the SF? … not at all, as Model(q)≡0 there

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A closer look at S/N cuts:

• If we choose SF(q), we must ‚model‘ q‘s 🡪 should be physical observables
• Can we avoid making the SF an explicit function of the uncertainties?

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Note: the SF is (can be) defined everywhere�..also for counter-factual objects 🡪 „upper limits“

Issues to be mentioned, but not pursued

• How do I actually determine the selection function
• Gaia Unlimited!!

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• What if I can‘t determine the selection function?
• or How well do I need to know the selection function?

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• What if my selection is on very noisy q?

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• How to combine surveys

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• By what mechanism do we all come to a mutually agreeable scope definition & terminology/formulation?

Quality cuts?

• Cuts designed to eliminate contamination/garbage
• If they don‘t cut „objects of desire“ (OoD):
• No change of the selection function
• But the model got simpler/better (don‘t need to model CV,binary contamination)
• If they cut out a fraction of OoDs, this has do be incorporated in S(o)

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• What about 𝝕/𝜹𝝕 > X ?
• signal to noise cuts can often be expressed in terms of observables
• thereby becoming model-predictable
• Here: 𝝕/𝜹𝝕 ∼ 𝝕/flux^0.5 (+position etc.) 🡪 S(m_G, 𝝕 | M)

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What does one get for different 𝝕/d 𝝕 cuts?

𝝕 cut

pseudo-𝝕/𝜹𝝕 cut

m_G cut

A worked example: �a stellar census of the solar neighbourhood

• Result: the space density of objects, as a function of M and color

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with units:

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How to devise a „good“ selection function?

• Minimal set of „observables“ that
• allow us to determine membership
• yields sample good for constraining „model“, here φ₀(M,c)
• Conjecture: S(m_G,𝟉) „magnitude and parallax“
• Why m_G? … max .. Obvious: „spectral S/N“, catalog availability
• Why 𝟉 ? … min .. Obvious: „solar neighbourhood“
• What about „parallax S/N“?
• S/N _𝟉,min yield data that are good to model (small M_G uncertainties etc..)
• <S/N _𝟉,min > can be expressed as f(m_G,𝟉)
• Avoids taking many spectra at needlessly low S/N_spec, just to be volume-complete

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A closer look at S/N cuts:

• If we choose SF(q), we must ‚model‘ q‘s 🡪 should be physical observables
• Can we avoid making the SF an explicit function of the uncertainties?

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A simple model for the luminosity-color function of object‘s space density = f(M_G,B-R) near the Sun

• This becomes
• for the isotropic case and n(x)=constant

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That‘s what happens for WDs

What about the SDSS-V 100pc or 200pc sample?

What would that mean for the 200pc sample?