[Ifeffit] Question about transform windows and statistical parameters

[Scott Calvin]

Matt,

On May 13, 2011, at 8:39 AM, Matt Newville wrote:
>>
>>  After all, the epsilon should be different for different k-ranges,  
>> as your signal to noise ratio probably changes as a function of k.  
>> Using the same epsilon doesn't reflect that.
>
> Without seeing the data in question, this seems like speculation to  
> me.  I'm not at all sure why epsilon (the variation in chi(k))  
> should depend strongly on the k-range.  In my experience, it usually  
> does not.  The S/N ratio will surely change with k, but that would  
> surely be dominated by the rapid decay in |chi(k)|, rather than a  
> change in epsilon.

I'm confused. We Fourier transform k-weighted data. Since Ifeffit uses  
the high-R amplitude to estimate uncertainty, it seems to me that what  
matters is signal-to-noise, not just noise in the original unweighted  
chi(k). Am I wrong in that? I may be misunderstanding how epsilon_r is  
calculated. And epsilon_r is the relevant epsilon for a fit in R  
space, right?

>
> I think your assumption that epsilon will depend strongly on k may  
> not correct.  Do you have evidence for this?   I would say that it  
> is not strongly dependent on k, and that reduced chi-square is  
> useful in comparing fits with different k-ranges.

I just tried it on the FeC2O4 chi(k) attached to this post. It's a  
good example of data where it's not immediately clear to me what the  
"best" value for kmax is, so it would be tempting to use RCS to  
compare fits over different k-ranges. I used k-weight 3, and Hanning  
windows with dk = 1. I chose kmin as 2 and stepped kmax by 0.5,  
recording epsilon_r for each:

kmax         epsilon_r
7           0.034840105
7.5       0.041843848
8          0.082627337
8.5       0.087550367
9          0.086032007
9.5       0.085996216
10        0.088679339
10.5     0.090364699
11        0.092509939
11.5     0.108103081

There's a general trend of increasing epsilon_r with an increase in k.  
There's also a jump of a factor of 2 between 7.5 and 8. Why? Because  
there's a glitch there, and the glitch adds high-R structure.

To make sure there wasn't something odd about this particular chi(k),  
I took one of the data sets included with the horae distribution: the  
file y300.chi in the ybco folder.

I followed the same procedure as before, except I stepped by 1 inverse  
angstrom each time, because of the greater data range.

kmax         epsilon_r

7         0.012866125
8         0.073383695
9         0.078255772
10       0.080016040
11       0.091634572
12       0.105419473
13       0.164341701
14       0.195266957
15       0.224727593
16       0.411139882
17       0.480293296

If anything, the trend is more clear here.

>
> I find it confusing that you expect  the noise in the data to depend  
> (strongly, even) on k, but not on R.    The general wisdom is the  
> estimate of epsilon from the high-R components is too low,  
> suggesting that the R dependence is significant.    Every time I've  
> looked, I come to the conclusion that noise in data is either  
> consistent with "white" or so small as to be difficult to measure.   
> I believe Corwin Booth's recent work supports the conventional  
> wisdom that  epsilon decreases with R, but I don't recall it  
> suggesting a significant k dependence.

I'm not making any claims as to whether, in general, the noise in the  
data depends on R. I can speculate about circumstances where low R  
noise is greater (due, for instance, to temperature fluctuations in  
cooling water, which are likely to be fairly slow), or where high R  
noise is greater (an example here would be if whatever system is  
keeping the beam on the sample vertically as the mono scans is tending  
to overshoot).

But Ifeffit's estimation of epsilon_r demonstrably does not depend on  
the R-range used for fitting, regardless of the distribution of noise  
in R. That's a very different thing. Thus, changing the R-range of a  
fit is completely safe as far as comparing RCS goes.

--Scott Calvin
Sarah Lawrence College

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