Diffraction Limit Discussion Continuation

Started 10 months ago | Discussions thread
knickerhawk
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Re: This analysis is meaningless
In reply to gollywop, 10 months ago

gollywop wrote:

knickerhawk wrote:

One objection that's been raised is that the DXOMark measurements are too coarse because they're only taken at full f/stop settings. The speculation is that the peak acutance might be occurring somewhere between the measured stops and therefore we can't rely on the DXOMark data. Having looked at enough examples, this struck me as pretty preposterous. Surely, the peaks wouldn't always average out exactly to the same major f-stop setting. Putting that aside, I thought that the data points we do have should be sufficient to infer with some degree a certainty exactly where the peaks occur.

There certainly is not sufficient data here to draw any such inference. You have 4 data points, and the curves you depict are at least cubics, requiring estimating 4 parameters. This problem has zero degrees of freedom and is not amenable to statistical analysis. You can draw no meaningful inferences.

Even if you were to restrict yourself to a quadratic approximation (with 1 degree of freedom), you would end up with completely meaningless (and completely insignificant) statistical results. Here is a regression using a quadratic:

Dependent variable is y[1]
RSquared = 0.731113 RBarSquared = 0.193338
R2uncentered = 0.999931 SER = 1.14872
Num of Observations = 4 Degrees of Freedom = 1
dw = 3.33739 with 0 missing obs.

coef. st. err. t
Const 67.506 5.508 12.257
f-stop 1.775 3.178 0.559
f-stop^2 -0.312 0.411 -0.760

As you can see, the only significant coefficient is the constant term (t = 12.257), while the f-stop terms are completely statistically insignificant with t's, respectively, at 0.559 and -0.760. With their insignificance, so also goes any inference about the maximum.

I'm certainly no math whiz

Understood, and agreed.

Ouch! 

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