Scipy: lognormal fitting

Question

There have been quite a few posts on handling the lognorm distribution (docs) with Scipy but i still don't get the hang of it.

The lognormal is usually described by the 2 parameters \mu and \sigma which correspond to the Scipy parameters loc=0 and \sigma=shape, \mu=np.log(scale).

At scipy, lognormal distribution - parameters, we can read how to generate a lognorm(\mu,\sigma)sample using the exponential of a random distribution. Now lets try something else:

A)

What is the problem in creating a lognorm directly:

    import scipy as sp
    import matplotlib.pyplot as plt

    # lognorm(mu=10,sigma=3)
    # so shape=3, loc=0, scale=np.exp(10) ?

    x=np.linspace(0.01,20,200)
    sample_dist = sp.stats.lognorm.pdf(x, 3, loc=0, scale=np.exp(10))
    shape, loc, scale = sp.stats.lognorm.fit(sample_dist, floc=0)
    print shape, loc, scale
    print np.log(scale), shape # mu and sigma
    # last line: -7.63285693379 0.140259699945  # not 10 and 3

B)

I use the return values of a fit to create a fitted distribution. But again im doing something wrong apparently:

    samp=sp.stats.lognorm(0.5,loc=0,scale=1).rvs(size=2000) # sample
    param=sp.stats.lognorm.fit(samp) # fit the sample data
    print param # does not coincide  with shape, loc, scale above!
    x=np.linspace(0,4,100)
    pdf_fitted = sp.stats.lognorm.pdf(x, param[0], loc=param[1], scale=param[2]) # fitted distribution
    pdf = sp.stats.lognorm.pdf(x, 0.5, loc=0, scale=1) # original distribution
    plt.plot(x,pdf_fitted,'r-',x,pdf,'g-')
    plt.hist(samp,bins=30,normed=True,alpha=.3)

Tum

Answer 1

I made the same observations: a free fit of all parameters fails most of the time. You can help by providing a better initial guess, fixing the parameter is not necessary.

samp = stats.lognorm(0.5,loc=0,scale=1).rvs(size=2000)

# this is where the fit gets it initial guess from
print stats.lognorm._fitstart(samp)

(1.0, 0.66628696413404565, 0.28031095750445462)

print stats.lognorm.fit(samp)
# note that the fit failed completely as the parameters did not change at all

(1.0, 0.66628696413404565, 0.28031095750445462)

# fit again with a better initial guess for loc
print stats.lognorm.fit(samp, loc=0)

(0.50146296628099118, 0.0011019321419653122, 0.99361128537912125)

You can also make up your own function to calculate the initial guess, e.g.:

def your_func(sample):
    # do some magic here
    return guess

stats.lognorm._fitstart = your_func

Enthrone

Answer 2

I realized my mistakes:

A) The samples i am drawing need to come from the .rvs method. Like so: sample_dist = sp.stats.lognorm.rvs(3, loc=0, scale=np.exp(10), size=2000)

B) The fit has some problems. When we fix the loc parameter the fit succeeds much better. param=sp.stats.lognorm.fit(samp, floc=0)

Tum

Answer 3

This problem has been fixed in newer scipy versions. After upgrading scipy0.9 to scipy0.14 the problem dissapears.

Sorensen

Answer 4

If you are just interested in plotting you can use seaborn to get a lognormal distribution.

import seaborn as sns
import numpy as np
from scipy import stats

mu=0
sigma=1
n=1000

x=np.random.normal(mu,sigma,n)
sns.distplot(x, fit=stats.norm) # normal distribution

loc=0
scale=1

x=np.log(np.random.lognormal(loc,scale,n))
sns.distplot(x, fit=stats.lognorm) # log normal distribution

Pierpont

Answer 5

I answered in here

I leave the code here too just for lazy :D

import scipy
import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np

mu = 10 # Mean of sample !!! Make sure your data is positive for the lognormal example 
sigma = 1.5 # Standard deviation of sample
N = 2000 # Number of samples

norm_dist = scipy.stats.norm(loc=mu, scale=sigma) # Create Random Process
x = norm_dist.rvs(size=N) # Generate samples

# Fit normal
fitting_params = scipy.stats.norm.fit(x)
norm_dist_fitted = scipy.stats.norm(*fitting_params)
t = np.linspace(np.min(x), np.max(x), 100)

# Plot normals
f, ax = plt.subplots(1, sharex='col', figsize=(10, 5))
sns.distplot(x, ax=ax, norm_hist=True, kde=False, label='Data X~N(mu={0:.1f}, sigma={1:.1f})'.format(mu, sigma))
ax.plot(t, norm_dist_fitted.pdf(t), lw=2, color='r',
        label='Fitted Model X~N(mu={0:.1f}, sigma={1:.1f})'.format(norm_dist_fitted.mean(), norm_dist_fitted.std()))
ax.plot(t, norm_dist.pdf(t), lw=2, color='g', ls=':',
        label='Original Model X~N(mu={0:.1f}, sigma={1:.1f})'.format(norm_dist.mean(), norm_dist.std()))
ax.legend(loc='lower right')
plt.show()


# The lognormal model fits to a variable whose log is normal
# We create our variable whose log is normal 'exponenciating' the previous variable

x_exp = np.exp(x)
mu_exp = np.exp(mu)
sigma_exp = np.exp(sigma)

fitting_params_lognormal = scipy.stats.lognorm.fit(x_exp, floc=0, scale=mu_exp)
lognorm_dist_fitted = scipy.stats.lognorm(*fitting_params_lognormal)
t = np.linspace(np.min(x_exp), np.max(x_exp), 100)

# Here is the magic I was looking for a long long time
lognorm_dist = scipy.stats.lognorm(s=sigma, loc=0, scale=np.exp(mu))
# Plot lognormals
f, ax = plt.subplots(1, sharex='col', figsize=(10, 5))
sns.distplot(x_exp, ax=ax, norm_hist=True, kde=False,
             label='Data exp(X)~N(mu={0:.1f}, sigma={1:.1f})\n X~LogNorm(mu={0:.1f}, sigma={1:.1f})'.format(mu, sigma))
ax.plot(t, lognorm_dist_fitted.pdf(t), lw=2, color='r',
        label='Fitted Model X~LogNorm(mu={0:.1f}, sigma={1:.1f})'.format(lognorm_dist_fitted.mean(), lognorm_dist_fitted.std()))
ax.plot(t, lognorm_dist.pdf(t), lw=2, color='g', ls=':',
        label='Original Model X~LogNorm(mu={0:.1f}, sigma={1:.1f})'.format(lognorm_dist.mean(), lognorm_dist.std()))
ax.legend(loc='lower right')
plt.show()

The trick is to understand these two things:

1. If the EXP of a variable is NORMAL with MU and STD -> EXP(X) ~ scipy.stats.lognorm(s=sigma, loc=0, scale=np.exp(mu))
2. If your variable (x) HAS THE FORM of a LOGNORMAL, the model will be scipy.stats.lognorm(s=sigmaX, loc=0, scale=muX) with:
   • muX = np.mean(np.log(x))
   • sigmaX = np.std(np.log(x))

Montague