Beta model?

[Erik Bray]

Hi all,

In the Astropy modeling package there are two models called Beta1D and
Beta2D, respectively. In the process of going through the
documentation and trying to improve things I looked at these more
closely than I had before and am now very confused by them.

http://docs.astropy.org/en/stable/api/astropy.modeling.functional_models.Beta1D.html#astropy.modeling.functional_models.Beta1D

http://docs.astropy.org/en/stable/api/astropy.modeling.functional_models.Beta2D.html#astropy.modeling.functional_models.Beta2D

They seem to have nothing to do with the Beta function with which I'm
familiar, or the beta distribution. Apparently it originally came
from Sherpa which describes it as a "Lorentz model with a varying
power law" which certainly tells us what it is. But this seems really
obscure--I can scarcely find any other good reference to this
nomenclature outside of Sherpa. MathWorld has nothing.

So does anyone know where this definition comes from, and how this
might be used in practice?

Thanks,

Erik

[John ZuHone]

It looks like the "beta model" fitting function for galaxy cluster surface brightness distributions, but with non-standard symbols for the parameters:

http://www.aanda.org/articles/aa/full_html/2009/22/aa12150-09/aa12150-09.html

It does have the same functional form, however.

John ZuHone

Kavli Center for Astrophysics and Space Research
Massachusetts Institute of Technology
77 Massachusetts Ave., 37-582G
Cambridge, MA 02139
(w) 617-253-2354
(m) 781-708-5004
jzu...@space.mit.edu
jzu...@gmail.com
http://www.jzuhone.com

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[Marten van Kerkwijk]

According to wikipedia, the lorentz distribution is indeed what we have (if normally called Cauchy) with a power of 1 [1].

The old fitting package in MIDAS that I used has CAUCHY for the above and defines LORENTZ to be the general one with a power.

Wikipedia lists the general case as "Pearson type VII" [3], but the two-dimensional version of the general case is also called a Moffat distribution [4] -- Tony Moffat found it is a good description for seeing profiles.

Overall, I think this function should *not* be called `Beta` -- either PearsonVII or Moffat or GeneralizedLorentz?

-- Marten

[1] http://en.wikipedia.org/wiki/Cauchy_distribution
[2] http://www.eso.org/sci/software/esomidas/doc/user/98NOV/vola/node144.html
[3] http://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VII_distribution
[4] http://en.wikipedia.org/wiki/Moffat_distribution

[Erik Bray]

On Tue, Oct 14, 2014 at 10:20 AM, John ZuHone <jzu...@gmail.com> wrote:
> It looks like the "beta model" fitting function for galaxy cluster surface
> brightness distributions, but with non-standard symbols for the parameters:
>
> http://www.aanda.org/articles/aa/full_html/2009/22/aa12150-09/aa12150-09.html
>
> It does have the same functional form, however.

Thank you! That must be it. I figured it had to be something fairly
domain-specific. This also explains why it would show up in CIAO
without requiring any further explanation, I guess.

The way it's been defined here is significantly more generalized
though, and the name "beta model" seems to be a semi-arbitrary choice
that came from one specific paper.

Is there another name for this functional form (other than
Lorentz-with-some-exponent)? Or should it be redefined to be closer
to its actual usage with some something like (-3 beta / 2) in the
exponent and renamed something more specific like "ICMBetaModel"? I
don't know.

[Erik Bray]

On Tue, Oct 14, 2014 at 11:00 AM, Marten van Kerkwijk
<m.h.van...@gmail.com> wrote:
> According to wikipedia, the lorentz distribution is indeed what we have (if
> normally called Cauchy) with a power of 1 [1].
>
> The old fitting package in MIDAS that I used has CAUCHY for the above and
> defines LORENTZ to be the general one with a power.
>
> Wikipedia lists the general case as "Pearson type VII" [3], but the
> two-dimensional version of the general case is also called a Moffat
> distribution [4] -- Tony Moffat found it is a good description for seeing
> profiles.
>
> Overall, I think this function should *not* be called `Beta` -- either
> PearsonVII or Moffat or GeneralizedLorentz?

Moffat sounds good.

> [1] http://en.wikipedia.org/wiki/Cauchy_distribution
> [2]
> http://www.eso.org/sci/software/esomidas/doc/user/98NOV/vola/node144.html
> [3]
> http://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VII_distribution
> [4] http://en.wikipedia.org/wiki/Moffat_distribution
>
>
> On Tuesday, October 14, 2014 10:10:33 AM UTC-4, Erik Bray wrote:
>>
>> Hi all,
>>
>> In the Astropy modeling package there are two models called Beta1D and
>> Beta2D, respectively. In the process of going through the
>> documentation and trying to improve things I looked at these more
>> closely than I had before and am now very confused by them.
>>
>>
>> http://docs.astropy.org/en/stable/api/astropy.modeling.functional_models.Beta1D.html#astropy.modeling.functional_models.Beta1D
>>
>>
>> http://docs.astropy.org/en/stable/api/astropy.modeling.functional_models.Beta2D.html#astropy.modeling.functional_models.Beta2D
>>
>> They seem to have nothing to do with the Beta function with which I'm
>> familiar, or the beta distribution. Apparently it originally came
>> from Sherpa which describes it as a "Lorentz model with a varying
>> power law" which certainly tells us what it is. But this seems really
>> obscure--I can scarcely find any other good reference to this
>> nomenclature outside of Sherpa. MathWorld has nothing.
>>
>> So does anyone know where this definition comes from, and how this
>> might be used in practice?
>>
>> Thanks,
>>
>> Erik
>

[Alex Conley]

On Oct 14, 2014, at 9:07 AM, Erik Bray <erik....@gmail.com> wrote:

> On Tue, Oct 14, 2014 at 11:00 AM, Marten van Kerkwijk
>>

>> Overall, I think this function should *not* be called `Beta` -- either
>> PearsonVII or Moffat or GeneralizedLorentz?
>
> Moffat sounds good.

Moffat is definitely the name I’ve heard used the most often
in astronomy contexts.

Alex

[Erik Tollerud]

+1 to Moffat - that is *the* standard used for point spread functions for ground-based imaging.

This *may* be too domain-specific, but I would also think it makes sense to set a default beta to something reasonably "standard".  The one I'm most familiar with is 4.765, from Trujillo et al. 2001.

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