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: num...@li... on behalf of =
Sebastian Zurek
Sent: Sat 21-Oct-06 15:41
To: num...@li...
Cc:=09
Subject: Re: [Numpy-discussion] Model and experiment fitting.
Robert Kern napisal(a):
> Your description is a bit vague.=20
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?
>=20
> f(x) -> y
>=20
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=20
paramagnetic resonance (EPR, ESR) experiment. The simplest SH can
be written in a form:
H =3D m B g S (1)
where m is a constant (bohr magneton), B is magnetic field (my=20
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=3D[[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=20
polycrystaline (powders) data, and to obtain a simulated EPR powder=20
spectrum I need to sum up the EPR spectra of monocrystals that come from =
many possible spatial orientations, and the resultant spectrum is an=20
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=20
> using the same X points as your experimental data?
>=20
I can only demand a X range and number of X values within the range,=20
there's no possibility to find the Y(X) for a specified X. These=20
limitations on one hand come from the external program I'm using to=20
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=20
> your experimental and simulated data?
>=20
> f(z) -> (x, y)
>=20
This is probably the situation I'm in. These other variables are my=20
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=20
simulated points within the narrow neighborhood of experimental data=20
point. The simulation points are uniformly distributed along the=20
X-range, with a density I'm able to tune. It all works quite well but=20
I'm founding it as a 'brute-force' method and I wonder, if there's any=20
more sophisticated and maybe already incorporated into any Python module =
method?
Anyway, it looks like it's impossible to compare two discrete 2D data=20
sets without any interpolations included... :]
A. M. Archibald has proposed spline fitting, which I'll try. I'll also=20
look at the Numerical Recipes discussion he has proposed.
Sebastian
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