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2

Some brief answers, and references for further reading below. 1) You are right to be concerned about your sample having crystals with a preferred orientation. A single crystal would produce points instead of circles. The points would fall somewhere on the circles. But the image you provided shows that it is possible for the point to be off the film. ...

8

If there are enough data and the prior is not completely unreasonable, the frequentist and the Bayesian approach give essentially the same answer. This is related to the central limit theorem. If data are fairly scarce, the two approaches may differ a lot. In this case the Bayesian approach is far preferable but only if the prior reflects true prior ...

1

If you want to take 4 points, I would still stick with linear approximation. It mostly depends on the noisiness of your data and rate of change of the temperature. You want higher order fit (like Floris described) in case you have low noise and time between observation is big. Your sample data do not seem as the case. The linear fit is analytical and easy. ...

0

I would take two points below and two point above the point where the threshold is crossed and fit a parabola. If the points are equally spaced the equations are quite simple. It is a little bit more work if they are not. You will be solving three equations with three unknowns, where the coefficients of the equation are a function of the $x_i, y_i$ data ...

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