KStwo Distribution¶
This is the distribution of the maximum absolute differences between an
empirical distribution function, computed from \(n\) samples or observations,
and a comparison (or target) cumulative distribution function, which is
assumed to be continuous.
(The “two” in the name is because this is the two-sided difference.
ksone is the distribution of the positive differences, \(D_n^+\),
hence it concerns one-sided differences.
kstwobign is the limiting
distribution of the normalized maximum absolute differences \(\sqrt{n} D_n\).)
Writing \(D_n = \sup_t \left|F_{empirical,n}(t)-F_{target}(t)\right|\),
kstwo is the distribution of the \(D_n\) values.
kstwo can also be used with the differences between two empirical distribution functions,
for sets of observations with \(m\) and \(n\) samples respectively.
Writing \(D_{m,n} = \sup_t \left|F_{1,m}(t)-F_{2,n}(t)\right|\), where
\(F_{1,m}\) and \(F_{2,n}\) are the two empirical distribution functions, then
\(Pr(D_{m,n} \le x) \approx Pr(D_N \le x)\) under appropriate conditions,
where \(N = \sqrt{\left(\frac{mn}{m+n}\right)}\).
There is one shape parameter \(n\), a positive integer, and the support is \(x\in\left[0,1\right]\).
The implementation follows Simard & L’Ecuyer, which combines exact algorithms of Durbin and Pomeranz with asymptotic estimates of Li-Chien, Pelz and Good to compute the CDF with 5-15 accurate digits.
Examples¶
>>> from scipy.stats import kstwo
Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a sample of size 5
>>> kstwo.sf([0, 0.5, 1.0], 5)
array([1. , 0.112, 0. ])
Compare a sample of size 5 drawn from a source N(0.5, 1) distribution against a target N(0, 1) CDF.
>>> from scipy.stats import norm
>>> n = 5
>>> gendist = norm(0.5, 1) # Normal distribution, mean 0.5, stddev 1
>>> x = np.sort(gendist.rvs(size=n, random_state=np.random.default_rng()))
>>> x
array([-1.59113056, -0.66335147, 0.54791569, 0.78009321, 1.27641365])
>>> target = norm(0, 1)
>>> cdfs = target.cdf(x)
>>> cdfs
array([0.0557901 , 0.25355274, 0.7081251 , 0.78233199, 0.89909533])
# Construct the Empirical CDF and the K-S statistics (Dn+, Dn-, Dn)
>>> ecdfs = np.arange(n+1, dtype=float)/n
>>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n], ecdfs[1:] - cdfs])
>>> np.set_printoptions(precision=3)
>>> cols
array([[-1.591, 0.2 , 0.056, 0.056, 0.144],
[-0.663, 0.4 , 0.254, 0.054, 0.146],
[ 0.548, 0.6 , 0.708, 0.308, -0.108],
[ 0.78 , 0.8 , 0.782, 0.182, 0.018],
[ 1.276, 1. , 0.899, 0.099, 0.101]])
>>> gaps = cols[:, -2:]
>>> Dnpm = np.max(gaps, axis=0)
>>> Dn = np.max(Dnpm)
>>> iminus, iplus = np.argmax(gaps, axis=0)
>>> print('Dn- = %f (at x=%.2f)' % (Dnpm[0], x[iminus]))
Dn- = 0.308125 (at x=0.55)
>>> print('Dn+ = %f (at x=%.2f)' % (Dnpm[1], x[iplus]))
Dn+ = 0.146447 (at x=-0.66)
>>> print('Dn = %f' % (Dn))
Dn = 0.308125
>>> probs = kstwo.sf(Dn, n)
>>> print(chr(10).join(['For a sample of size %d drawn from a N(0, 1) distribution:' % n,
... ' Kolmogorov-Smirnov 2-sided n=%d: Prob(Dn >= %f) = %.4f' % (n, Dn, probs)]))
For a sample of size 5 drawn from a N(0, 1) distribution:
Kolmogorov-Smirnov 2-sided n=5: Prob(Dn >= 0.308125) = 0.6319
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();
>>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r', linestyle='solid', lw=4)
>>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m', linestyle='solid', lw=4)
>>> plt.annotate('Dn-', xy=(x[iminus], (ecdfs[iminus]+ cdfs[iminus])/2),
... xytext=(x[iminus]+1, (ecdfs[iminus]+ cdfs[iminus])/2 - 0.02),
... arrowprops=dict(facecolor='white', edgecolor='r', shrink=0.05), size=15, color='r');
>>> plt.annotate('Dn+', xy=(x[iplus], (ecdfs[iplus+1]+ cdfs[iplus])/2),
... xytext=(x[iplus]-2, (ecdfs[iplus+1]+ cdfs[iplus])/2 - 0.02),
... arrowprops=dict(facecolor='white', edgecolor='m', shrink=0.05), size=15, color='m');
>>> plt.show()
References¶
“Kolmogorov-Smirnov test”, Wikipedia https://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test
Durbin J. “The Probability that the Sample Distribution Function Lies Between Two Parallel Straight Lines.” Ann. Math. Statist., 39 (1968) 39, 398-411.
Pomeranz J. “Exact Cumulative Distribution of the Kolmogorov-Smirnov Statistic for Small Samples (Algorithm 487).” Communications of the ACM, 17(12), (1974) 703-704.
Li-Chien, C. “On the exact distribution of the statistics of A. N. Kolmogorov and their asymptotic expansion.” Acta Matematica Sinica, 6, (1956) 55-81.
Pelz W, Good IJ. “Approximating the Lower Tail-areas of the Kolmogorov-Smirnov One-sample Statistic.” Journal of the Royal Statistical Society, Series B, (1976) 38(2), 152-156.
Simard, R., L’Ecuyer, P. “Computing the Two-Sided Kolmogorov-Smirnov Distribution”, Journal of Statistical Software, Vol 39, (2011) 11.
Implementation: scipy.stats.kstwo