scipy.special.kolmogorov

scipy.special.kolmogorov(y) = <ufunc 'kolmogorov'>

Complementary cumulative distribution (Survival Function) function of Kolmogorov distribution.

Returns the complementary cumulative distribution function of Kolmogorov’s limiting distribution (D_n*\sqrt(n) as n goes to infinity) of a two-sided test for equality between an empirical and a theoretical distribution. It is equal to the (limit as n->infinity of the) probability that sqrt(n) * max absolute deviation > y.

Parameters:

y : float array_like

Absolute deviation between the Empirical CDF (ECDF) and the target CDF, multiplied by sqrt(n).

Returns:

float

The value(s) of kolmogorov(y)

See also

kolmogi

The Inverse Survival Function for the distribution

scipy.stats.kstwobign

Provides the functionality as a continuous distribution

smirnov, smirnovi

Notes

kolmogorov is used by stats.kstest in the application of the Kolmogorov-Smirnov Goodness of Fit test. For historial reasons this function is exposed in scpy.special, but the recommended way to achieve the most accurate CDF/SF/PDF/PPF/ISF computations is to use the stats.kstwobign distrubution.

Examples

Show the probability of a gap at least as big as 0, 0.5 and 1.0.

>>> from scipy.special import kolmogorov
>>> from scipy.stats import kstwobign
>>> kolmogorov([0, 0.5, 1.0])
array([ 1.        ,  0.96394524,  0.26999967])

Compare a sample of size 1000 drawn from a Laplace(0, 1) distribution against the target distribution, a Normal(0, 1) distribution.

>>> from scipy.stats import norm, laplace
>>> n = 1000
>>> np.random.seed(seed=233423)
>>> lap01 = laplace(0, 1)
>>> x = np.sort(lap01.rvs(n))
>>> np.mean(x), np.std(x)
(-0.083073685397609842, 1.3676426568399822)

Construct the Empirical CDF and the K-S statistic Dn.

>>> target = norm(0,1)  # Normal mean 0, stddev 1
>>> cdfs = target.cdf(x)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> gaps = np.column_stack([cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
>>> Dn = np.max(gaps)
>>> Kn = np.sqrt(n) * Dn
>>> print('Dn=%f, sqrt(n)*Dn=%f' % (Dn, Kn))
Dn=0.058286, sqrt(n)*Dn=1.843153
>>> print(chr(10).join(['For a sample of size n drawn from a N(0, 1) distribution:',
...   ' the approximate Kolmogorov probability that sqrt(n)*Dn>=%f is %f' %  (Kn, kolmogorov(Kn)),
...   ' the approximate Kolmogorov probability that sqrt(n)*Dn<=%f is %f' %  (Kn, kstwobign.cdf(Kn))]))
For a sample of size n drawn from a N(0, 1) distribution:
 the approximate Kolmogorov probability that sqrt(n)*Dn>=1.843153 is 0.002240
 the approximate Kolmogorov probability that sqrt(n)*Dn<=1.843153 is 0.997760

Plot the Empirical CDF against the target N(0, 1) CDF.

>>> import matplotlib.pyplot as plt
>>> plt.step(np.concatenate([[-3], x]), ecdfs, where='post', label='Empirical CDF')
>>> x3 = np.linspace(-3, 3, 100)
>>> plt.plot(x3, target.cdf(x3), label='CDF for N(0, 1)')
>>> plt.ylim([0, 1]); plt.grid(True); plt.legend();
>>> # Add vertical lines marking Dn+ and Dn-
>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', linestyle='dashed', lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='r', linestyle='dashed', lw=4)
>>> plt.show()