# 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`