## Saturday, March 14, 2009

### Warmup: Fitting Distributions

As a warmup exercise, I generate some random samples, fit two distributions and plot the results. I also calculate the Kolmogorvo-Smirnov test using scipy.stats.kstest.

This is a shortened, simplified version of a script that I wrote to see how the dostrinutions in scipy.stats can be used to automatically fit some data and select the best fitting distribution.

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

def plothist(x, distfn, args, loc, scale, right=1):
'''plot histogram and pdf, based on matplotlib doc example'''
plt.figure()
# the histogram of the data
n, bins, patches = plt.hist(x, 25, normed=1, facecolor='green', alpha=0.75)
maxheight = max([p.get_height() for p in patches])
axlim = list(plt.axis())
axlim[-1] = maxheight*1.05

# add more points for density plot
pdfpoints = bins + np.diff(bins)[0]*np.linspace(-0.5,0.5,11)[:,np.newaxis]
pdfpoints = np.sort(pdfpoints.ravel())

# calculate and plot the estimated density function
yt = distfn.pdf(pdfpoints, loc=loc, scale=scale, *args)
yt[yt>maxheight] = maxheight
lt = plt.plot(pdfpoints, yt, 'r--', linewidth=1, label='estimated')
# calculate and plot the density function that generated the data
ys = stats.t.pdf(pdfpoints, 10, scale=10,)*right
ls = plt.plot(pdfpoints, ys, 'b-', linewidth=1, label='true')

plt.legend()
plt.xlabel('values')
plt.ylabel('Probability')
plt.title(r'$\mathrm{Fitting\ Distribution\ %s :}\ \mu=%f,\ \sigma=%f$'%(distfn.name,loc,scale))
plt.grid(True)
plt.draw()

n = 500
dgp_arg = 10
dgp_scale = 10
np.random.seed(6543789)
rvs = stats.t.rvs(dgp_arg, scale=dgp_scale, size=n)
sm = rvs.mean()
sstd = np.sqrt(rvs.var())
ssupp = (rvs.min(), rvs.max())

for distr in [stats.norm, stats.t]:
distname = distr.name
# estimate parameters
par_est = distr.fit(rvs,loc=sm, scale=sstd)
print '\nFitting distribution %s' % distname
print 'estimated distribution parameters\n', par_est
arg_est = par_est[:-2]  # get scale parameters if any
loc_est = par_est[-2]
scale_est = par_est[-1]
rvs_normed = (rvs-loc_est)/scale_est
ks_stat, ks_pval = stats.kstest(rvs_normed,distname, arg_est)
print 'ks-stat = %f, ks-pval = %f)' % (ks_stat, ks_pval)
plothist(rvs, distr, arg_est, loc_est, scale_est, right = 1)
plt.savefig('ex_dist1_%s.png'% distname)

#plt.show() # if we want to see it on screen

(Source code)

Output

The script produces the following output for the parameter estimate and the Kolmogorov-Smirnov test

Fitting distribution norm
estimated distribution parameters
[ -0.70287027  11.22248481]
ks-stat = 0.037073, ks-pval = 0.493706)

Fitting distribution t
estimated distribution parameters
[ 7.8518085  -0.69695469  9.71315677]
ks-stat = 0.019562, ks-pval = 0.990926)

The p-values of Kolmogorov-Smirnov test are derived under the assumption that we test against a known distribution and not against a distribution with estimated parameters. But in this example, the numbers look pretty informative, the p-values are large for both distributions, from which I conclude that both distributions fit overall the sample relatively well. The pvalue of the t-distribution is about twice the p-value of the normal distribution, which strongly suggest that the distribution was generated by a t-distribution and not by a normal distribution. Although, if we look only at the pvalue of the normal distribution, we wouldn't reject the hypothesis that the sample was generated by the normal distribution.

However, I want to emphasis that this is an informal interpretation, in which we can be quite confident and which we know to be correct since we generated the data, but that it is not the result of a formal statistical test for it.

script file ex_dist1.py

Sphinx and restructured text work well, and the highlighting with pygments also works without additional intervention, but adding the graphs to blogger requires some manual upload and editing of the formatting, which still takes too much time. I haven't figured out yet how to upload source files, the upload file button seems to be missing from my blog editor.

update: some google searches later, it seems that it is not possible to attach regular text files to blogger, so it needs to be hosted somewhere else