numpy-discussion

[Numpy-discussion] Model and experiment fitting.

[<se...@pi...>]

Hi!

This is probably a silly question but I'm getting confused with a 
certain problem: a comparison between experimental data points (2D 
points set) and a model (2D points set - no analytical form).

The physical model produces (by a sophisticated simulations done by an 
external program) some 2D points data and  one of my task is to compare 
those calculated data with an experimental one.

The experimental and modeled data have form of 2D curves, build of n 
2D-points, i.e.:

expDat=[[x1,x2,x3,..xn],[y1,y2,y3,...,yn]]
simDat=[[X1,X2,X3,...,Xn],[Y1,Y2,Y3,...,Yn]]

The task of determining, let's say, a root mean squarred error (RMSe)
is trivial if x1==X1, x2==X2, etc.

In general, which is a common situation xk differs from Xk (k=0..n) and 
one may not simply compare succeeding Yk and yk (k=0..n) to determine 
the goodness-of-fit. The distance h=Xk-X(k-1) is constant, but similar
distance m(k)=xk-x(k-1) depends on k-th point and is not a constant 
value, although the data array lengths for simulation and experiment are 
the same.

My first idea was to do some interpolations to obtain the missing 
points, but I held it 'by a hand' (which, BTW gave quite rewarding 
results)  and I suppose, there's some i.g. numpy method to do it for me, 
isn't it?

I suppose to do something like:

gfit(expDat,simDat,'measure_type')

which I hope will return the number determining the goodness-of-fit
(mean squarred error, root mean squarred error,...) of two sets of 
discrete 2D data points.

Is there something like that in any numerical python modules (numpy, 
pylab) I could use?


I can imagine, I can fit the data with some polynomial or whatever,
and than compare the fitted data, but my goal is to operate on
as raw data as it's possible.

Thanks for your comments!

Sebastian

Re: [Numpy-discussion] Model and experiment fitting.

[Robert K. <rob...@gm...>]

Sebastian Żurek wrote:
> Hi!
> 
> This is probably a silly question but I'm getting confused with a 
> certain problem: a comparison between experimental data points (2D 
> points set) and a model (2D points set - no analytical form).
> 
> The physical model produces (by a sophisticated simulations done by an 
> external program) some 2D points data and  one of my task is to compare 
> those calculated data with an experimental one.
> 
> The experimental and modeled data have form of 2D curves, build of n 
> 2D-points, i.e.:
> 
> expDat=[[x1,x2,x3,..xn],[y1,y2,y3,...,yn]]
> simDat=[[X1,X2,X3,...,Xn],[Y1,Y2,Y3,...,Yn]]
> 
> The task of determining, let's say, a root mean squarred error (RMSe)
> is trivial if x1==X1, x2==X2, etc.
> 
> In general, which is a common situation xk differs from Xk (k=0..n) and 
> one may not simply compare succeeding Yk and yk (k=0..n) to determine 
> the goodness-of-fit. The distance h=Xk-X(k-1) is constant, but similar
> distance m(k)=xk-x(k-1) depends on k-th point and is not a constant 
> value, although the data array lengths for simulation and experiment are 
> the same.

Your description is a bit vague. Do you mean that you have some model function f 
that maps X values to Y values?

   f(x) -> y

If that is the case, is there some reason that you cannot run your simulation 
using the same X points as your experimental data?

OTOH, is there some other independent variable (say Z) that *is* common between 
your experimental and simulated data?

   f(z) -> (x, y)

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless enigma
  that is made terrible by our own mad attempt to interpret it as though it had
  an underlying truth."
   -- Umberto Eco

Re: [Numpy-discussion] Model and experiment fitting.

[<se...@pi...>]

Robert Kern napisał(a):

> Your description is a bit vague. 

Possibly by my weak English... I'll try to make myself clearer now.

Do you mean that you have some model function f
> that maps X values to Y values?
> 
>    f(x) -> y
> 

My model is quantum energy operator - spin hamiltonian (SH) with some
additional assumption about so called 'line shape', 'line widths',etc.

   It describes various electron interactions, visible in electron 
paramagnetic resonance (EPR, ESR) experiment. The simplest SH can
be written in a form:
                        H = m B g S                   (1)
where m is a constant (bohr magneton), B is magnetic field (my 
x-variable), g is so called 'zeeman matrix' and S is total spin angular
momentum operator.

Summing it all together: the simple model is parametrized by:
  - line shape,
  - line width,
  - zeeman matrix (3x3 diagonal matrix - the spatial dependence),
  - total spin S.

After SH (1) diagonalization one can obtain so called 'resonance fields' 
and  'resonance intensities'. After a convolution with appropriate  line 
shape function which is parametrized by the line width one can finally
get the simulated EPR spectrum (simDat=[[X1,...,Xn],[Y1,...,Yn]]).
This  is a roughly, schematic description, appropriate to EPR spectra of
monocrystals.

In my situation the problem is more sophisticated - I have 
polycrystaline (powders) data, and to obtain a simulated EPR powder 
spectrum I need to sum up the EPR spectra of monocrystals that come from 
many possible spatial orientations, and the resultant spectrum is an 
envelope of all the monocrystals spectra.

There's no simple model function that maps X -> Y.


> If that is the case, is there some reason that you cannot run your simulation 
> using the same X points as your experimental data?
> 

I can only demand a X range and number of X values within the range, 
there's no possibility to find the Y(X) for a specified X. These 
limitations on one hand come from  the external program I'm using to 
simulate the EPR spectra, on the other are a result of spatial averaging 
of EPR data for powders, where a lot of interpolations are involved.


> OTOH, is there some other independent variable (say Z) that *is* common between 
> your experimental and simulated data?
> 
>    f(z) -> (x, y)
> 

This is probably the situation I'm in. These other variables are my 
model parameters, namely: line shape-width, zeeman matrix... and they're
commen between the experiment and the simulation.


To make it clear.

I've already solved the problem by a simple linear interpolation of 
simulated points within the narrow neighborhood of experimental data 
point. The simulation points are uniformly distributed along the 
X-range, with a density I'm able to tune. It all works quite well but 
I'm founding it as a 'brute-force' method and I wonder, if there's any 
more sophisticated and maybe already incorporated into any Python module 
method?

Anyway, it looks like it's impossible to compare two discrete 2D data 
sets without any interpolations included... :]


A. M. Archibald has proposed spline fitting, which I'll try. I'll also 
look at the Numerical Recipes discussion he has proposed.


Sebastian

Re: [Numpy-discussion] Model and experiment fitting.

[Charles R H. <cha...@gm...>]

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Re: [Numpy-discussion] Model and experiment fitting.

[<se...@pi...>]

The problem seemed to be solved, by the A. M. Archibald clue.

I've used splines to fit the simulation data. After that, I can easily
find any Y(X) point, for all X in range (x_min,x_max) where x_min and 
x_max are the experiment independent variable. The experimental data 
stay untouched.

Sorry for all the confusion I've made.

Thanks a lot to all of You!

Sebastian

Re: [Numpy-discussion] Model and experiment fitting.

[A. M. A. <per...@gm...>]

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Re: [Numpy-discussion] Model and experiment fitting.

[<se...@pi...>]

A. M. Archibald napisał(a):
> 
> In scipy there are some very convenient spline fitting tools which
> will allow you to fit a nice smooth spline through the simulation data
> points (or near, if they have some uncertainty); you can then easily
> look at the RMS difference in the y values. You can also, less easily,
> look at the distance from the curve allowing for some uncertainty in
> the x values.
> 

I'll try a spline fitting. I've already made some linear interpolations 
(see Robert Kern answer) which works well enough to use it. I'm working 
on a genetic algorithms application to the model parameters 
optimalization problem and this RMSe comparison serves me as 'fitness 
function'. This 'fitness function' is important element in whole 
procedure, so I'm trying to found the best solution to obtain it.


> I suppose you could also fit a curve through the experimental points
> and compare the two curves in some way.
> 

Well, I can do it, indeed. But every single fitting procedure implicate 
some additional error, so when it comes to fit, I must use it very 
cautiously. The simulated data-points fitting should be the only 
acceptable fitting procedure, I guess.

> If you want to avoid using an a priori model, Numerical Recipes
> discuss some possible approaches ("Do two-dimensional distributions
> differ?" at http://www.nrbook.com/a/bookcpdf.html is one) but it's not
> clear how to turn the problem you describe into a solvable one - some
> assumption about how the models vary between sampled x values appears
> to be necessary, and that amounts to interpolation.
> 

I'll look to this NR discussion.



Thank You for these comments!

Sebastian