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# scipy.stats.landau

scipy.stats.landau = <scipy.stats._continuous_distns.landau_gen object>

  A Landau continuous random variable.

  As an instance of the `rv_continuous` class, `landau` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

  Methods

  rvs(loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, loc=0, scale=1) | Probability density function.
  logpdf(x, loc=0, scale=1) | Log of the probability density function.
  cdf(x, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, loc=0, scale=1) | Log of the survival function.
  ppf(q, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, loc=0, scale=1) | Non-central moment of the specified order.
  stats(loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(loc=0, scale=1) | (Differential) entropy of the RV.
  fit(data) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  expect(func, args=(), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) | Expected value of a function (of one argument) with respect to the distribution.
  median(loc=0, scale=1) | Median of the distribution.
  mean(loc=0, scale=1) | Mean of the distribution.
  var(loc=0, scale=1) | Variance of the distribution.
  std(loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `landau` ([1], [2]) is:

  \[f(x) = \frac{1}{\pi}\int_0^\infty \exp(-t \log t - xt)\sin(\pi t) dt\]

  for a real number \(x\).

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `landau.pdf(x, loc, scale)` is identically equivalent to `landau.pdf(y) / scale` with `y = (x - loc) / scale`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  Often (e.g. [2]), the Landau distribution is parameterized in terms of a location parameter \(\mu\) and scale parameter \(c\), the latter of which also introduces a location shift. If `mu` and `c` are used to represent these parameters, this corresponds with SciPy’s parameterization with `loc = mu + 2*c / np.pi * np.log(c)` and `scale = c`.

  This distribution uses routines from the Boost Math C++ library for the computation of the `pdf`, `cdf`, `ppf`, `sf` and `isf` methods. [1]

  References

  [1] (1,2)

  Landau, L. (1944). “On the energy loss of fast particles by ionization”. J. Phys. (USSR). 8: 201.

  [2] (1,2)

  “Landau Distribution”, Wikipedia, https://en.wikipedia.org/wiki/Landau_distribution

  [3]

  Chambers, J. M., Mallows, C. L., & Stuck, B. (1976). “A method for simulating stable random variables.” Journal of the American Statistical Association, 71(354), 340-344.

  [4]

  The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

  [5]

  Yoshimura, T. “Numerical Evaluation and High Precision Approximation Formula for Landau Distribution”. DOI:10.36227/techrxiv.171822215.53612870/v2

  Examples

      >>> import numpy as np
      >>> from scipy.stats import landau
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Get the support:

      >>> lb, ub = landau.support()

  Calculate the first four moments:

      >>> mean, var, skew, kurt = landau.stats(moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(landau.ppf(0.01),
      ...                 landau.ppf(0.99), 100)
      >>> ax.plot(x, landau.pdf(x),
      ...        'r-', lw=5, alpha=0.6, label='landau pdf')

  Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

  Freeze the distribution and display the frozen `pdf`:

      >>> rv = landau()
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = landau.ppf([0.001, 0.5, 0.999])
      >>> np.allclose([0.001, 0.5, 0.999], landau.cdf(vals))
      True

  Generate random numbers:

      >>> r = landau.rvs(size=1000)

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()