scipy.special.kolmogorov

scipy.special.kolmogorov(x, out=None) = <wrapped_ufunc 'kolmogorov'>

Complementary cumulative distribution function of the Kolmogorov distribution.

Returns the survival function of Kolmogorov’s limiting distribution, i.e., the distribution of \(\sqrt{n}\, D_n\) as \(n \to \infty\), where \(D_n\) is the Kolmogorov–Smirnov statistic:

\[D_n = \sup_x \left| F_n(x) - F(x) \right|\]

with \(F_n\) the empirical cumulative distribution function (ECDF) and \(F\) the theoretical CDF. Specifically, this function computes:

\[\mathbb{P}(\sqrt{n}\, D_n > x)\]

Parameters:

xarray_like

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

outndarray, optional

Optional output array for the function results.

Returns:

scalar or ndarray

The probability that the test statistic exceeds x, in the range \([0, 1]\).

See also

kolmogi

Inverse survival function of the Kolmogorov distribution.

scipy.stats.kstwobign

Provides the functionality as a continuous distribution.

smirnov, smirnovi

Functions for the one-sided distribution.

Notes

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

Array API Standard Support

kolmogorov has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. kolmogorov does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Marsaglia, G., Tsang, W. W., & Wang, J. (2003). “Evaluating Kolmogorov’s distribution.” Journal of Statistical Software, 8(18), 1-4.

[2]

“Kolmogorov-Smirnov test”, Wikipedia, https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test

Examples

Try it in your browser!

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

>>> import numpy as np
>>> 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
>>> rng = np.random.default_rng()
>>> n = 1000
>>> lap01 = laplace(0, 1)
>>> x = np.sort(lap01.rvs(n, random_state=rng))
>>> np.mean(x), np.std(x)
(np.float64(-0.05841730131499543), np.float64(1.3968109101997568))

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(f'Dn={Dn:f}, sqrt(n)*Dn={Kn:f}')
Dn=0.043363, sqrt(n)*Dn=1.371265
>>> print(chr(10).join(['For a Laplace sample tested against N(0, 1):',
...   f' the approximate Kolmogorov probability that sqrt(n)*Dn>={Kn:f} '
...   f'is {kolmogorov(Kn):f}',
...   f' the approximate Kolmogorov probability that sqrt(n)*Dn<={Kn:f} '
...   f'is {kstwobign.cdf(Kn):f}']))
For a Laplace sample tested against N(0, 1):
 the approximate Kolmogorov probability that sqrt(n)*Dn>=1.371265 is 0.046533
 the approximate Kolmogorov probability that sqrt(n)*Dn<=1.371265 is 0.953467

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()

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