Lorentz, Cauchy, Distribution fitting...

[Allen]

Hi everyone,

I've got some crazy questions for ya. :)

1) is the "Lorentz" distribution generally just a name for the
Cauchy-Lorentz distribution?

2) I know that matlab can fit using gaussians (there's a double gaussian
peak with a background shown in the help files)... is it possible to do
the same for a Lorentz distribution instead of a Gaussian?

3) I have three peaks, with some tail overlap which I'm trying to sort
of deconvolute from my data... to give me approx. areas for each of the
three peaks.. this will give me information about my material's
makeup/atomic structure... I know the position of the three peaks, but I
need to fit their breadth and height to produce the final data form.
[think of an M with shallow middle, and a tall upside down V to the
right of it (some small tail overlap)].

Any and all thoughts greatly appreciated. [Is this a bit insane to do?
There may not be a single solution would be my guess with three
distributions?]

Thanks!!
-Allen

[Peter Perkins]

Post by Allen
Hi everyone,
I've got some crazy questions for ya. :)
1) is the "Lorentz" distribution generally just a name for the
Cauchy-Lorentz distribution?

Allen, this is kind of like botany: the common names do not always mean the
same thing to different people, so you should cite the PDF or CDF when in doubt.

Wikipedia has this to say:

"The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous
probability distribution with probability density function

f(x; x_0,\gamma) = \frac{1}{\pi\gamma \left[1 +
\left(\frac{x-x_0}{\gamma}\right)^2\right]} \!

where x0 is the location parameter, specifying the location of the peak of the
distribution, and gamma is the scale parameter which specifies the half-width at
half-maximum (HWHM). As a probability distribution, it is known as the Cauchy
distribution while among physicists it is known as the Lorentz distribution or
the Breit-Wigner distribution."

Statisticians would probably call it a Cauchy or a t-location-scale with one
degree of freedom.

Post by Allen
2) I know that matlab can fit using gaussians (there's a double gaussian
peak with a background shown in the help files)... is it possible to do
the same for a Lorentz distribution instead of a Gaussian?
3) I have three peaks, with some tail overlap which I'm trying to sort
of deconvolute from my data... to give me approx. areas for each of the
three peaks.. this will give me information about my material's
makeup/atomic structure... I know the position of the three peaks, but I
need to fit their breadth and height to produce the final data form.
[think of an M with shallow middle, and a tall upside down V to the
right of it (some small tail overlap)].

If you mean this demo:

<<http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/customdist1demo.html>>,


that describes fitting a mixture of two univariate Gaussians by maximum
likelihood, then yes, in theory you should be able to the same thing. But even
for a single Cauchy distribution, the log-likelihood function typically has lots
of local maxima, and is difficult to fit by maximum likelihood. I imagine it
will only be more difficult with several of them.

Hope this helps.

- Peter Perkins
The MathWorks, Inc.

[Allen]

Post by Peter Perkins
<<http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/customdist1demo.html>>,
that describes fitting a mixture of two univariate Gaussians by maximum
likelihood, then yes, in theory you should be able to the same thing.
But even for a single Cauchy distribution, the log-likelihood function
typically has lots of local maxima, and is difficult to fit by maximum
likelihood. I imagine it will only be more difficult with several of them.
Hope this helps.
- Peter Perkins
The MathWorks, Inc.

Peter,

Yes, this does help quite a bit... I guess I didn't get far enough down
the wiki to see that comment about Lorentz. ;) That definitely
explains it as my Prof. is a Physicist.

Actually, I meant the Guassian Fit with Exponential Background as seen
on this page:

http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/ch_fit11.html#69867

It appears at least that it should be possible to use a lorentz equation
here, and fit it in the same manner as the gaussian used for the doublet
peak? This data is fairly similar to the data I have, in various
respects; however, I my peaks are significantly higher, as visual
similarity is to be had on a log-scale. Which is why I believe the
Lorentz is chosen (due to peak shape).

Any and all thoughts are greatly appreciated, Peter! If I get this
working I'll try and make a little webpage regarding what I did, as it
may prove useful for others who do diffractometer/xray work on materials.

My best!
-Allen

[Peter Perkins]

Post by Allen
Actually, I meant the Guassian Fit with Exponential Background as seen
http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/ch_fit11.html#69867

Hi Allen -

That demo is for fitting a curve, as opposed to fitting a distribution: see

<<http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/cfitdfitdemo.html>>

My comments, and the demo I pointed to, were about distribution fitting. The
former should be much easier, if you're doing a least-squares curve fit,
especiually if you have decent starting guesses for the coefficients.

- Peter

[Allen]

Post by Peter Perkins
Hi Allen -
see
<<http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/cfitdfitdemo.html>>
My comments, and the demo I pointed to, were about distribution
fitting. The former should be much easier, if you're doing a
least-squares curve fit, especiually if you have decent starting guesses
for the coefficients.
- Peter

Oh, wonderful! Sorry, I guess I wasn't very clear... I tend to call the
curves from gauss/cauchy etc., distribution curves... :) Sorry, I
confused things here... yes, I want to do curve-fitting with the Lorentz
distribution *function*. :)

Easier is always a good thing in the land of the grad-student! ;)

Thanks so much for your help and thoughts and clearing up these questions!!

My best!
-Allen