Re: [Ifeffit] Purple & Red E Derivatives

22 Feb 2011

      Should the line
	out[i] = (x[i+1] - x[i])/2.0
be
	out[i] = (x[i+1] - x[i-1])/2.0  ?

That would account for the 2.0.  That is, in fact, exactly what I proposed was meant by 'numerical differential'.

I don't actually resample.  I do a cubic spline interpolation onto the given grid and pull the derivatives out that way.  Of course,
you have to test for
grids that aren't strictly monotonic.  The smooth() function as given would definitely cause problems with non-uniform tabulation
unless resampled.
My program (2-column Editor on the ALS Beamline 10.3.2 website) offers resampling and tensioned spline fitting but does not smooth
the derivative
before or after - the user has to take control of that using the tensioned spline fit.  I thought about smoothing but never got
around to figuring out what makes
a decent smooth algorithm for general data.  A possibility would be a slow version of Savitzky-Golay in which you do the polynomial
fits explicitly instead of
being able to use pre-computed coefficients as in the actual S-G method.  Similarly, a VSFT (Very Slow Fourier Transform) in which
you fit the data to
a sum of sines and cosines (linear fit - the frequencies are fixed), filter using your favorite low-pass filter, then
inverse-transform, might be good.  Or, there's
always reading the literature to see what actual numerical mathematicians have done :-)

There are two obvious ways to get non-uniform tabulation in XAS data.  One is that it's taken on a varying grid, either
constant-k-step or varying E step.  The
other is that pieces have been chopped out due to glitches, Bragg peaks or some other problem.  Also, the 2-column Editor program
was designed to be general,
not restricted to XAS-like data.
    mam
----- Original Message -----
From: "Matt Newville" 
To: "XAFS Analysis using Ifeffit" 
Sent: Tuesday, February 22, 2011 5:28 PM
Subject: Re: [Ifeffit] Purple & Red E Derivatives

Hi Bruce, Matthew,

Oh, deriv() (and smooth()) are not nearly as sophisticated as what
Matthew suggests.  They are simply applied to a single array,
sequentially, without regard to "the x axis"  (as, in fact, it does
not know what x might be):

Deriv would translate (sorry for the python) to
def deriv(x):
    npts = len(x)
    out  = array(npts)
    out[0] = x[1] - x[0]
    out[npts-1] = x[npts-1] - x[npts-2]
    for i in range(1, npts-2):  # (1, 2, 3, ... npts-2)
	out[i] = (x[i+1] - x[i])/2.0

    return out

and Smooth:
def smooth(x):
    npts = len(x)
    out  = array(npts)
    out[0] = 0.75*x[0] + 0.25 * x[1]
    out[npts-1] = 0.75*x[npts-1] + 0.25 * x[npts-2]
    for i in range(1, npts-2): # (1, 2, 3, ... npts-2)
	out[i] = 0.5*x[i] + 0.25*x[i+1] + 0.25*x[i-1]

    return out

Like Matthew suggests, interpolating onto a finer, uniform x (energy)
grid and then taking the derivative might be a better approach, at
least for data which needs smoothing.

--Matt

On Wed, Feb 23, 2011 at 8:46 AM, Bruce Ravel  wrote:
...
On Tuesday, February 22, 2011 06:19:16 pm Matthew Marcus wrote:
...
...
The numerical derivative
is computed as the numerical differential of the mu(E) spectrum
divided by the numerical differential of the energy array. Ã,Â Smoothing
happens as explained above. Ã,Â Justw atch the lines that get printed to
the buffer when you click the plot buttons. - B. Ravel
I wonder if that's the right way to go for data that, for instance, might
have gaps or changes in step size. Ã,Â What I do in my code is run a cubic
spline through the data, with the abscissa as is,
then take the derivative of that spline. Ã,Â Thus, if the data is smooth
enough to be locally approximated by a cubic, the derivative will come out
smooth regardless of how it's tabulated. Ã,Â I'm not
sure that's the case with Bruce's algorithm. Ã,Â Is the 'numerical
differential' the difference E(i+1)-E(i) or (1/2)*(E(i+1)-E(i-1)) or some
form that's intended to approximate dE(i)/di (i= point index)?
It's not clear to me whether it's any more correct to do smoothing before
or after differentiating. Ã,Â You can take the red pill or the blue pill :-)
Ã,Â Ã,Â mam
Well ... red pill or purple pill. Ã,Â They both have side effects that 4
out of 5 doctors advise against :)
So, I do what's easy using Ifeffit as Ifeffit is implemented. Ã,Â I would
have to code dive (or ask Matt) what the deriv() function actually
does.
A big gap in the data might result in a weird derivative spectrum, but
changes in step size isn't so much of a problem because I ask Ifeffit
to compute d(mu)/d(E).
I agree that "smooth then differentiate" is not obviously better than
"differentiate then smooth". Ã,Â What is certainly wrong is to do this
inconsistently.
B
--
Ã,Â Bruce Ravel Ã,Â ------------------------------------ bravel@bnl.gov
Ã,Â National Institute of Standards and Technology
Ã,Â Synchrotron Methods Group at NSLS --- Beamlines U7A, X24A, X23A2
Ã,Â Building 535A
Ã,Â Upton NY, 11973
Ã,Â My homepage: Ã,Â Ã,Â http://xafs.org/BruceRavel
Ã,Â EXAFS software: http://cars9.uchicago.edu/~ravel/software/exafs/
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