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.. _continuous-kstwo:
KStwo Distribution
==================
This is the distribution of the maximum absolute differences between an
empirical distribution function, computed from :math:`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, :math:`D_n^+`,
hence it concerns one-sided differences.
``kstwobign`` is the limiting
distribution of the *normalized* maximum absolute differences :math:`\sqrt{n} D_n`.)
Writing :math:`D_n = \sup_t \left|F_{empirical,n}(t)-F_{target}(t)\right|`,
``kstwo`` is the distribution of the :math:`D_n` values.
``kstwo`` can also be used with the differences between two empirical distribution functions,
for sets of observations with :math:`m` and :math:`n` samples respectively.
Writing :math:`D_{m,n} = \sup_t \left|F_{1,m}(t)-F_{2,n}(t)\right|`, where
:math:`F_{1,m}` and :math:`F_{2,n}` are the two empirical distribution functions, then
:math:`Pr(D_{m,n} \le x) \approx Pr(D_N \le x)` under appropriate conditions,
where :math:`N = \sqrt{\left(\frac{mn}{m+n}\right)}`.
There is one shape parameter :math:`n`, a positive integer, and the support is :math:`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
>>> np.random.seed(seed=423) # Set the seed for reproducibility
>>> x = np.sort(gendist.rvs(size=n))
>>> 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`
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