Hi Alan,
Thanks for the response.
I'm not sure what the cleanest way to type math in this forum is, so I'll just write them with latex markup (e.g. could be rendered here for clarity:
https://latex.codecogs.com/eqneditor/editor.php)
You're correct-- here y refers to two distinct profiles being compared, i.e. y_obs and y_calc for the Rietveld application. In the reference [1] the similarity measure is given in context of three correlation functions written as integrals:
c_{12}(r) = \int{ y_1(\theta) y_2(\theta + r) d\theta }
where
\theta is the x-coordinate of the powder pattern (or PDF) and
r is "the distance between two points in the powder diagrams."
The similarity measure
S_{12} is given in terms of these auto- and cross-correlation functions by
S_{12} =
\frac{ \int{w(r) c_{12}(r) dr} }
{ [\int{w(r) c_{11}(r) dr} \int{w(r) c_{22}(r) dr)}]^{1/2} }
where the weight function
w(r) is the piecewise triangle with width
2l
w(r) = \left\{\begin{matrix}
1 - |r| / l, & |r| < l \\
0, & |r| \ge l \\
\end{matrix}\right.
which defines the window in which to compare patterns
y_1 and
y_2. The usual Rietveld case corresponds with l==0 (pointwise comparison). For l > 0, the measure will have some sensitivity e.g. to lattice parameters that are slightly ajar.
The application to the PDF uses an additional rescaling of G(r) to account for the changing sign.
The motivation is hinted at in the abstract of [1], "The procedure is also suitable for unindexed powder data, powder diagrams of very low quality, ..." I've been looking at powder diffraction and PDF data for a material that is disordered and doesn't have a powder pattern suitable for indexing. I'm trying to work out a plausible explanation of the bonding / connectivity based on this data and some chemical intuition and energy penalties. My intuition is that such a "similarity measure" may make a more robust loss function for this problem, and it has been used successfully in at least one case for some organic compounds. [3]
Thanks!
-Peter
[3] S. Habermehl, C. Schlesinger, and D. Prill, “Comparison and evaluation of pair distribution functions, using a similarity measure based on cross-correlation functions,” Journal of Applied Crystallography, vol. 54. International Union of Crystallography, pp. 612–623, 2021.
dx.doi.org/10.1107/S1600576721002569