scipy, lognormal distribution - parameters

Solution 1:

The distributions in scipy are coded in a generic way wrt two parameter location and scale so that location is the parameter (loc) which shifts the distribution to the left or right, while scale is the parameter which compresses or stretches the distribution.

For the two parameter lognormal distribution, the "mean" and "std dev" correspond to log(scale) and shape (you can let loc=0).

The following illustrates how to fit a lognormal distribution to find the two parameters of interest:

In [56]: import numpy as np

In [57]: from scipy import stats

In [58]: logsample = stats.norm.rvs(loc=10, scale=3, size=1000) # logsample ~ N(mu=10, sigma=3)

In [59]: sample = np.exp(logsample) # sample ~ lognormal(10, 3)

In [60]: shape, loc, scale = stats.lognorm.fit(sample, floc=0) # hold location to 0 while fitting

In [61]: shape, loc, scale
Out[61]: (2.9212650122639419, 0, 21318.029350592606)

In [62]: np.log(scale), shape  # mu, sigma
Out[62]: (9.9673084420467362, 2.9212650122639419)

Solution 2:

I just spent some time working this out and wanted to document it here: If you want to get the probability density (at point x) from the three return values of lognorm.fit (lets call them (shape, loc, scale)), you need to use this formula:

x = 1 / (shape*((x-loc)/scale)*sqrt(2*pi)) * exp(-1/2*(log((x-loc)/scale)/shape)**2) / scale

So as an equation that is (loc is µ, shape is σ and scale is α):

$x = \frac{1}{(x-\mu)\cdot\sqrt{2\pi\sigma^2}} \cdot e^{-\frac{log(\frac{x-\mu}{\alpha})^2}{2\sigma^2}}$

scipy, lognormal distribution - parameters

Solution 1:

Solution 2:

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