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# scipy.special.smirnov

scipy.special.smirnov(n, d, out=None) = <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:
    n int

      Number of samples

    d float array_like

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

    out ndarray, optional

      Optional output array for the function results

  Returns:
    scalar or ndarray

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

  See also

  `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 historical 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

      >>> import numpy as np
      >>> from scipy.special import smirnov
      >>> from scipy.stats import norm

  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 against N(0, 1), the standard normal distribution with mean 0 and standard deviation 1.

  x is the sample.

      >>> x = np.array([-1.392, -0.135, 0.114, 0.190, 1.82])

      >>> target = norm(0, 1)
      >>> cdfs = target.cdf(x)
      >>> cdfs
      array([0.0819612 , 0.44630594, 0.5453811 , 0.57534543, 0.9656205 ])

  Construct the empirical CDF and the K-S statistics (Dn+, Dn-, Dn).

      >>> n = len(x)
      >>> ecdfs = np.arange(n+1, dtype=float)/n
      >>> cols = np.column_stack([x, ecdfs[1:], cdfs, cdfs - ecdfs[:n],
      ...                        ecdfs[1:] - cdfs])
      >>> with np.printoptions(precision=3):
      ...    print(cols)
      [[-1.392  0.2    0.082  0.082  0.118]
       [-0.135  0.4    0.446  0.246 -0.046]
       [ 0.114  0.6    0.545  0.145  0.055]
       [ 0.19   0.8    0.575 -0.025  0.225]
       [ 1.82   1.     0.966  0.166  0.034]]
      >>> gaps = cols[:, -2:]
      >>> Dnpm = np.max(gaps, axis=0)
      >>> print(f'Dn-={Dnpm[0]:f}, Dn+={Dnpm[1]:f}')
      Dn-=0.246306, Dn+=0.224655
      >>> probs = smirnov(n, Dnpm)
      >>> print(f'For a sample of size {n} drawn from N(0, 1):',
      ...       f' Smirnov n={n}: Prob(Dn- >= {Dnpm[0]:f}) = {probs[0]:.4f}',
      ...       f' Smirnov n={n}: Prob(Dn+ >= {Dnpm[1]:f}) = {probs[1]:.4f}',
      ...       sep='\n')
      For a sample of size 5 drawn from N(0, 1):
       Smirnov n=5: Prob(Dn- >= 0.246306) = 0.4711
       Smirnov n=5: Prob(Dn+ >= 0.224655) = 0.5245

  Plot the empirical CDF and the standard normal CDF.

      >>> import matplotlib.pyplot as plt
      >>> plt.step(np.concatenate(([-2.5], x, [2.5])),
      ...          np.concatenate((ecdfs, [1])),
      ...          where='post', label='Empirical CDF')
      >>> xx = np.linspace(-2.5, 2.5, 100)
      >>> plt.plot(xx, target.cdf(xx), '--', label='CDF for N(0, 1)')

  Add vertical lines marking Dn+ and Dn-.

      >>> iminus, iplus = np.argmax(gaps, axis=0)
      >>> plt.vlines([x[iminus]], ecdfs[iminus], cdfs[iminus], color='r',
      ...            alpha=0.5, lw=4)
      >>> plt.vlines([x[iplus]], cdfs[iplus], ecdfs[iplus+1], color='m',
      ...            alpha=0.5, lw=4)

      >>> plt.grid(True)
      >>> plt.legend(framealpha=1, shadow=True)
      >>> plt.show()