# scipy.special.smirnov#

scipy.special.smirnov(n, d) = <ufunc 'smirnov'>#

Kolmogorov-Smirnov complementary cumulative distribution function

Returns the exact Kolmogorov-Smirnov complementary cumulative distribution function,(aka the Survival Function) of Dn+ (or Dn-) for a one-sided test of equality between an empirical and a theoretical distribution. It is equal to the probability that the maximum difference between a theoretical distribution and an empirical one based on n samples is greater than d.

Parameters
nint

Number of samples

dfloat array_like

Deviation between the Empirical CDF (ECDF) and the target CDF.

Returns
float

The value(s) of smirnov(n, d), Prob(Dn+ >= d) (Also Prob(Dn- >= d))

`smirnovi`

The Inverse Survival Function for the distribution

`scipy.stats.ksone`

Provides the functionality as a continuous distribution

`kolmogorov`, `kolmogi`

Functions for the two-sided distribution

Notes

`smirnov` 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.ksone distribution.

Examples

```>>> from scipy.special import smirnov
```

Show the probability of a gap at least as big as 0, 0.5 and 1.0 for a sample of size 5

```>>> smirnov(5, [0, 0.5, 1.0])
array([ 1.   ,  0.056,  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
>>> rng = np.random.default_rng()
>>> n = 5
>>> gendist = norm(0.5, 1)       # Normal distribution, mean 0.5, stddev 1
>>> x = np.sort(gendist.rvs(size=n, random_state=rng))
>>> x
array([-1.3922078 , -0.13526532,  0.1371477 ,  0.18981686,  1.81948167])
>>> target = norm(0, 1)
>>> cdfs = target.cdf(x)
>>> cdfs
array([0.08192974, 0.44620105, 0.55454297, 0.57527368, 0.96558101])
# 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([[-1.392,  0.2  ,  0.082,  0.082,  0.118],
[-0.135,  0.4  ,  0.446,  0.246, -0.046],
[ 0.137,  0.6  ,  0.555,  0.155,  0.045],
[ 0.19 ,  0.8  ,  0.575, -0.025,  0.225],
[ 1.819,  1.   ,  0.966,  0.166,  0.034]])
>>> gaps = cols[:, -2:]
>>> Dnpm = np.max(gaps, axis=0)
>>> print('Dn-=%f, Dn+=%f' % (Dnpm, Dnpm))
Dn-=0.246201, Dn+=0.224726
>>> probs = smirnov(n, Dnpm)
>>> print(chr(10).join(['For a sample of size %d drawn from a N(0, 1) distribution:' % n,
...      ' Smirnov n=%d: Prob(Dn- >= %f) = %.4f' % (n, Dnpm, probs),
...      ' Smirnov n=%d: Prob(Dn+ >= %f) = %.4f' % (n, Dnpm, probs)]))
For a sample of size 5 drawn from a N(0, 1) distribution:
Smirnov n=5: Prob(Dn- >= 0.246201) = 0.4713
Smirnov n=5: Prob(Dn+ >= 0.224726) = 0.5243
```

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='m', linestyle='dashed', lw=4)
>>> plt.show()
```