How do I get a lognormal distribution in Python with Mu and Sigma?

Question

I have been trying to get the result of a lognormal distribution using Scipy. I already have the Mu and Sigma, so I don\'t need to do any other prep work. If I need to be

Answer 1

Even more late, but in case it's helpful to anyone else: I found that the Excel's

LOGNORM.DIST(x,Ln(mean),standard_dev,TRUE)

provides the same results as python's

from scipy.stats import lognorm
lognorm.cdf(x,sigma,0,mean)

Likewise, Excel's

LOGNORM.DIST(x,Ln(mean),standard_dev,FALSE)

seems equivalent to Python's

from scipy.stats import lognorm
lognorm.pdf(x,sigma,0,mean).

Answer 2

If you read this and just want a function with the behaviour similar to lnorm in R. Well, then relieve yourself from violent anger and use numpy's numpy.random.lognormal.

Answer 3

It sounds like you want to instantiate a "frozen" distribution from known parameters. In your example, you could do something like:

from scipy.stats import lognorm
stddev = 0.859455801705594
mean = 0.418749176686875
dist=lognorm([stddev],loc=mean)

which will give you a lognorm distribution object with the mean and standard deviation you specify. You can then get the pdf or cdf like this:

import numpy as np
import pylab as pl
x=np.linspace(0,6,200)
pl.plot(x,dist.pdf(x))
pl.plot(x,dist.cdf(x))

Is this what you had in mind?

Answer 4

from math import exp
from scipy import stats

def lognorm_cdf(x, mu, sigma):
    shape  = sigma
    loc    = 0
    scale  = exp(mu)
    return stats.lognorm.cdf(x, shape, loc, scale)

x      = 25
mu     = 2.0785
sigma  = 1.744
p      = lognorm_cdf(x, mu, sigma)  #yields the expected 0.74341

Similar to Excel and R, The lognorm_cdf function above parameterizes the CDF for the log-normal distribution using mu and sigma.

Although SciPy uses shape, loc and scale parameters to characterize its probability distributions, for the log-normal distribution I find it slightly easier to think of these parameters at the variable level rather than at the distribution level. Here's what I mean...

A log-normal variable X is related to a normal variable Z as follows:

X = exp(mu + sigma * Z)              #Equation 1

which is the same as:

X = exp(mu) * exp(Z)**sigma          #Equation 2

This can be sneakily re-written as follows:

X = exp(mu) * exp(Z-Z0)**sigma       #Equation 3

where Z0 = 0. This equation is of the form:

f(x) = a * ( (x-x0) ** b )           #Equation 4

If you can visualize equations in your head it should be clear that the scale, shape and location parameters in Equation 4 are: a, b and x0, respectively. This means that in Equation 3 the scale, shape and location parameters are: exp(mu), sigma and zero, respectfully.

If you can't visualize that very clearly, let's rewrite Equation 2 as a function:

f(Z) = exp(mu) * exp(Z)**sigma      #(same as Equation 2)

and then look at the effects of mu and sigma on f(Z). The figure below holds sigma constant and varies mu. You should see that mu vertically scales f(Z). However, it does so in a nonlinear manner; the effect of changing mu from 0 to 1 is smaller than the effect of changing mu from 1 to 2. From Equation 2 we see that exp(mu) is actually the linear scaling factor. Hence SciPy's "scale" is exp(mu).

The next figure holds mu constant and varies sigma. You should see that the shape of f(Z) changes. That is, f(Z) has a constant value when Z=0 and sigma affects how quickly f(Z) curves away from the horizontal axis. Hence SciPy's "shape" is sigma.

Answer 5

I know this is a bit late (almost one year!) but I've been doing some research on the lognorm function in scipy.stats. A lot of folks seem confused about the input parameters, so I hope to help these people out. The example above is almost correct, but I found it strange to set the mean to the location ("loc") parameter - this signals that the cdf or pdf doesn't 'take off' until the value is greater than the mean. Also, the mean and standard deviation arguments should be in the form exp(Ln(mean)) and Ln(StdDev), respectively.

Simply put, the arguments are (x, shape, loc, scale), with the parameter definitions below:

loc - No equivalent, this gets subtracted from your data so that 0 becomes the infimum of the range of the data.

scale - exp μ, where μ is the mean of the log of the variate. (When fitting, typically you'd use the sample mean of the log of the data.)

shape - the standard deviation of the log of the variate.

I went through the same frustration as most people with this function, so I'm sharing my solution. Just be careful because the explanations aren't very clear without a compendium of resources.

For more information, I found these sources helpful:

• http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html#scipy.stats.lognorm
• https://stats.stackexchange.com/questions/33036/fitting-log-normal-distribution-in-r-vs-scipy

And here is an example, taken from @serv-inc 's answer, posted on this page here:

import math
from scipy import stats

# standard deviation of normal distribution
sigma = 0.859455801705594
# mean of normal distribution
mu = 0.418749176686875
# hopefully, total is the value where you need the cdf
total = 37

frozen_lognorm = stats.lognorm(s=sigma, scale=math.exp(mu))
frozen_lognorm.cdf(total) # use whatever function and value you need here

Answer 6

@lucas' answer has the usage down pat. As a code example, you could use

import math
from scipy import stats

# standard deviation of normal distribution
sigma = 0.859455801705594
# mean of normal distribution
mu = 0.418749176686875
# hopefully, total is the value where you need the cdf
total = 37

frozen_lognorm = stats.lognorm(s=sigma, scale=math.exp(mu))
frozen_lognorm.cdf(total) # use whatever function and value you need here