I have a dataset representing the transition between two materials, A and B (graph (a) in the attached image). I wish to estimate the (x-axis) location of the A-B transition and to provide an associated uncertainty in that location.
A professor of mine thinks that the way to do this is to take the derivative of the datapoints, determine the best-fitting Gaussian curve to those derivatives, and use the full width at half maximum (FWHM) parameter to somehow estimate the uncertainty (graph (b) in the attached image)
My questions are: 1) Is this the correct approach? and 2) How exactly should I estimate the uncertainty using the FWHM parameter?
Thanks a lot for your help!
Best wishes,
Robin
Hi, this method is "ok", but first you need to have certain about the normal distribution, some time the comportament is quite different. In your case gaussian curve looks good, but try some Lorenzent function or a combination of gaussian-lorentizian, to fitting better with the upper point. The FWHM as uncertain measurent is good, because is an indirect measure of standart deviation... Pay attention both curves must be the same type.
The method suggested by your professor is acceptable and the “uncertainty” will be +/- one half of the FWHH. There other options: you may trace the tangent to the curve through the inflexion point and locate the intercepts with the “baselines” before and after the event, to define the interval of change (the “uncertainty” will be again +/- one half of this interval). However, none of these intervals are true confidence interval and you will be hard pressed to associate a probability to them...
So far you are considering that there is a transition point, whose position is smudged in the experimental data because of gaussian error or inadequacies in the experimental setup. Frequently, this is NOT the case: the transition process may (intrinsically) extends during an interval, not (only) due to uncertainties in the measurement procedure. In that case you are seeking a “characteristic value” for the transition (not its “true value”, that doesn’t exist). The intervals found are then estimates of the genuine extension of the transition process. The may even be used to obtain a rough estimate the kinetics of the transition...
If you can't be sure the underlying derivative curve is actually Gaussian, you could use standard linear regression techniques, e.g.. fit a local quadratic curve to the raw data and compute its derivative. An analysis of the residuals will give you and indication of your errors, and help you to place bounds on your derivative estimate.See
"Introduction to Probability and Mathematical Statistics " by Bain & Engelhardt
Dear Robin Duncan Wylie,
The above answers seem relevant. Still, I make some remarks, of which some may seem irrelevant to you but others may be helpful.
1. A material is some matter applied for some technical purpose. So I got the impression that the subject of your study is a chemical or physical transformation.
2. Transformations develop in time, so kinetics is essential, given the thermodynamical conditions satisfied.
3. What is plotted on the Y- and X-axis of your graph a?
4. The accuracy of the transition will have values along the X- and the Y-axis.
5. What is the interpretation you would give to the transition point on the X-axis:
a. transformation completed for 50 % or b. maximum of transformation rate.
6. Supposing the transformation studied is thermally activated, would you like to use the transformation point for an activation energy analysis?
Dear Robin Duncan Wylie,
Some remarks/questions:
1. The basic point in the above discussion is of course what about the physics/chemistry of the transformation studied to assume a Gaussian curve depicting the derivative of graph a.
2. The accuracy of independent variable plotted along the X-axis should be known in advance and can be compared to FWHM?
3. The concept of standard deviation belongs to the world of statistics; the above graphs reflect a number of measurements?
4. Your plot suggests a thermally activated transformation, so I would suggest to analyse your data correspondingly.
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