Hi Jacob,

It seems reasonable for your straight line case that the two would be equivalent. I would guess though that as the function to be fitted against becomes more complex (e.g. a rectangular hyperbola in enzyme kinetics) it would be advantageous to have more points of the independent variable measured a fewer number of times (assuming you measured each one enough to get a reasonable estimate of the experimental error). Another reason I can think of to measure more points less frequently is so you can detect possible deviations in the data from the expected function (there might be a peak in the middle of your straight line for some reason). Looking forward to reading other responses!

Philip

On Thu, Jan 22, 2015 at 5:20 PM, Keller, Jacob <[log in to unmask]> wrote:

Dear Crystallographers,

This is more general than crystallography, but has applications therein, particularly in understanding fine phi-slicing.

The general question is:

Given one needs to collect data to fit parameters for a known function, and given a limited total number of measurements, is it generally better to measure a small group of points multiple times or to distribute each individual measurement over the measureable extent of the function? I have a strong intuition that it is the latter, but all errors being equal, it would seem prima facie that both are equivalent. For example, a line (y = mx + b) can be fit from two points. One could either measure the line at two points A and B five times each for a total of 10 independent measurements, or measure ten points evenly-spaced from A to B. Are these equivalent in terms of fitting and information content or not? Which is better? Again, conjecture and intuition suggest the evenly-spaced experiment is better, but I cannot formulate or prove to myself why, yet.

The application of this to crystallography might be another reason that fine phi-slicing (0.1 degrees * 3600 frames) is better than coarse (1 degree * 3600 frames), even though the number of times one measures reflections is tenfold higher in the second case (assuming no radiation damage). In the first case, one never measures the same phi angle twice, but one does have multiple measurements in a sense, i.e., of different parts of the same reflection.

Yes, 3D profile-fitting may be a big reason fine phi-slicing works, but beyond that, perhaps this sampling choice plays a role as well. Or maybe the profile-fitting works so well precisely because of this diffuse-single type of sampling rather than coarse-multiple sampling?

This general math/science concept must have been discussed somewhere--can anyone point to where?

JPK

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Jacob Pearson Keller, PhD
Looger Lab/HHMI Janelia Research Campus
19700 Helix Dr, Ashburn, VA 20147
email: [log in to unmask]
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