Talk:Hyperbolic secant distribution

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Hypersecant distribution is symmetrical around zero in terms of x = (value - mean)/stdev and the sech() function expands into two exponential terms:

f(x) = \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\! = 
\frac{1}{\exp\left(\frac{\pi x}{2}\right)+\exp\left( -\frac{\pi x}{2}\right)}

Both tails of this symmetrical form have the same decay slope on a log plot.

A general asymmetrical form is obtained by scaling the stdev of each of the two exponential terms by a factor a and 1/a, respectively:

f(x) = \frac{\left[\frac{a}{2}+\frac{1}{2a}\right]\sin\left(\frac{\pi}{1+a^2}  \right)}
{\exp\left(\frac{\pi x a}{2}\right)+\exp\left( -\frac{\pi x}{2 a}\right)}

Here, "a" may be regarded as "asymmetry factor" and x=0 is no longer the mean. Either tail of this asymmetrical form has a different decay slope on a log plot. The terms in the numerator are required to preserve the integral to unity and revert to the symmetric form when a=1.

This asymmetric form appears to be a good model for the difference in wind speed between adjacent hours d(V) and for the change in wind direction weighted by wind speed V.d(theta).

The main problem with this general asymmetric form is that the CDF and inverse CDF are not amenable to a simple closed form derivation. Expansion of exp terms as Taylor series or numerical integration seems only viable approaches ... Now there's a challenge.

Windnick (talk) 13:16, 18 March 2010 (UTC)

Inverse cosh[edit]

I find the statement "The hyperbolic secant function is equivalent to the inverse hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution" a little odd. There is a difference between a reciprocal and an inverse function, and so between hyperbolic functions such as sech and inverse hyperbolic functions such as arcosh.--Rumping (talk) 13:52, 1 October 2013 (UTC)

Characteristic function[edit]

Shouldn't the characteristic function be for all t? The entire point of using the CF over the MGF is that it always converges.Chitinid (talk) 12:44, 2 October 2013 (UTC)