Model and experiment fitting.

Nadav Horesh nadavh at visionsense.com
Sat Oct 21 14:05:21 CDT 2006


1. If at least one of your data sets to be interpulated is on a grid, you can use numpy.ndimage.map function for fast interpolation for 2d (in fact for any dimensional) dataset.

2. Isn't there an analytic expression to average the expectration values of SH over all possible orientations between B and the crystal axis? My experience shows that some analytic work can save 99% of simulation time.

  Nadav

-----Original Message-----
From:	numpy-discussion-bounces at lists.sourceforge.net on behalf of Sebastian Zurek
Sent:	Sat 21-Oct-06 15:41
To:	numpy-discussion at lists.sourceforge.net
Cc:	
Subject:	Re: [Numpy-discussion] Model and experiment fitting.

Robert Kern napisal(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


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