weibull | weibull analysis routines in python - quick and dirty | Analytics library

library(sos); findFn("{lambert W}") finds R implementations of the Lambert W function in packages: emdbook, lamW, LambertW, condmixt, VGAM, pracma, spatstat.core ...

It appears either the PDF is not formulated correctly or the parameters are not constrained appropriately. The denominator quickly approaches 0 as vartheta gets large if x > 1:

Re 3: That's kind of a bug in optimx, but one that's hard to avoid. It uses x as a variable name when calculating a numerical gradient; you also use it as an "additional parameter" to your functions. You can work around that by renaming your argument, e.g. call it xdata in your functions.

Re 1 & 2: There are several techniques to handle boundary problems in optimization. Setting to a big constant value tends not to work: if the optimizer goes out of bounds, it finds the objective function really flat. If the exact boundary is legal, then pushing the parameter to the boundary and adding a penalty sometimes works. If the exact boundary is illegal, you might be able to reflect: e.g. if mu > 0 is a requirement, sometimes replacing mu by abs(mu) in the objective function gets things to work. Sometimes the best solution is to get rid of the boundary by transforming the parameters.

Edited to add some more details:

For this problem, it looks to me as though transformations of the parameters might work. I think alpha and sigma must both be positive. Setting alpha <- exp(theta[1]) and sigma <- exp(theta[3]) will guarantee that. Limits on mu are harder, but I think mu > max(xdata) is needed, so mu <- max(xdata) + exp(theta[2]) should keep it in bounds. Of course, making these changes messes up your gradient formula and starting values.

As to resources: I'm afraid I don't know any. This advice is based on years of painful experience.

fitdistrplus::fitdist() can be used to determine the parameters:

Scale the data first giving xx. Also we have shortened and fixed the code. (The input x is shown in the Note at the end. It is taken from the question but shown in reproducible form.)

This should work

In that question, 127 is sum of y. But in some reply, it saying that approach have some issues and provide an expedient way. There exist two ways, SSweibull and dweibull.

I'm really sorry that adding plotstyle = "ggplot need to be added to every line......But I think this may help you from that weird legend locations.

Use separate as follows. First we define Separate to be like separate except the sep pattern is fixed to what we need and convert is fixed to be TRUE.

Note that [^[:alnum:].]+ refers to using strings of characters that are not alphabetic, numeric or dot as separators. Also note that because the fields end in a delimiter (parenthesis or square bracket) that separate will assume that there is an extra field at the end so use NA to skip it.