Sorry to flag something that may seem obvious, but I have tried these two inverse function with inconsistent result:
A. inverse exp( -exp(-(x-a)/b)) results in a - b log(-log(x))
while…
B. inverse exp( -exp(-(x-u)/g)) results in x + g log(-log(u))
Observe the variable a=u and b=g placements.
Did I miss something here? Thank you for any help given.
InverseFunction applied to expressions with more than one symbol does not work, of course. The function has to be declared as a map and for the symbols regarded as the coordinates of the manifold to be mapped. Position in the sequence of variables and its total lenght have to be given in order to constract the partial inverses as a function. Its a similar problem in the algebra of functions of several variables in Derivative, if switching from the mathematical pictorial display to algorithms.
InverseFunction[Function[{x, u, g}, Exp[-Exp[(x - u)/g]]], 2, 3]
Function[{x, u, g}, x + g Log[1/Log[1/u]]]
Apologies, but I am using Wolfram Alpha (web) and the expression as described did not work. Any basic reference that I could read?
This is easy
Solve[ y == Exp[- Exp[-(a-x)/b]], b ]
Attempts to use InverseFunction on ‘alpha’ did not work for me.
Actually, I was trying to reproduce the inverse of the CDF of the Gumbel distribution linked here wiki Gumbel distribution - see the section of Random variate generation - of which the CDF is
, and the inverse should be
, which is not the same as the result of the solve function in Alpha.
You could try “inverse…with respect to x” and see if that improves matters.