Copying comments from a source file is totally the same as writing a blog post, right? Has anyone really considered the full-width N-max of a gaussian before, and then extended that parameter to non gaussian data? Here's how they would start, if they did:
/* Discussion on FWHM for Gaussian lines:
Given a gaussian distribution defined by PDF = 1 / sqrt(2 * pi) * exp(-0.5 * u^2),
then you can define a fractional max location by:
PDF / PDF_MAX = exp(-0.5 * u^2) / exp(-0.5 * 0^2);
PDF / PDF_MAX = exp(-0.5 * u^2)
sqrt(-2.0 * log(R)) = u
So, for a given ratio of the maximum R, the half-width R-max is located at u.
This leads to the full-width R-max location being at
FWRM = 2.0 * sqrt(-2.0 * log(R))
Since we've used u, we can add the sigma dependence back in by noting that u = x / s:
FWRM = 2.0 * sqrt(-2.0 * log(R)) * sigma
The standard value is the FWHM, or R = 0.5:
FWHM = 2.0 * sqrt(-2.0 * log(0.5)) = 2.0 * sqrt(2.0 * log(2.0))
Do this for quarter max, R = 0.25:
FWQM = 2.0 * sqrt(-2.0 * log(0.25)) = 2.0 * sqrt(2.0 * log(4))
= 2.0 * sqrt(2.0 * 2.0 * log(2.0)) = sqrt(2) * FWHM
And for 3/4 max, R = 0.75:
FW3QM = 2.0 * sqrt(-2.0 * log(0.75)) = 2.0 * sqrt(2.0 * log(4/3))
= 2.0 * sqrt(2.0 * (log(4) - log(3)))
= 2.0 * sqrt(2.0 * (2.0 * log(2.0) - log(3)))
=~ 0.64423 * FWHM
If a distribution is "wingier" than a Gaussian, then FWQM will be inflated, such that
FWQM_wingy > FWQM_Gaussian
*/
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| Here's a sleepy bear. |
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| And here's Neil Armstrong making pizzas before flying to the moon and becoming super cool. |
- OCD dog.
- Snorlax is best.
- Comics are run by morons now, so it's good that someone I've liked is getting out of the thrashing.
- Also on comics: really? Facepalm. Followed by washing to get the dirt off.
- Politics (also fluid dynamics): Really? Facepalm.
- How long will it be before people in power realize that dragging people into the street to charge them with minor traffic violations is not the way to crush dissension. Forcing them to come together and present a coherent set of goals with a means to obtain them is far more likely to crush the rebellious feeling.
Mind you, don't take that as a disparagement of the occupy movement, which I fully support. I'm merely commenting that if I were to want to put such a thing down, I'd do it not by police action to arrest people, but rather by forcing them to deal with impossible situations and come up with solutions that are likely to not fully solve the problem, thereby marginalizing the dissidents as ineffectual. That's the true way to defeat dissent.
