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# scipy.stats.crystalball

scipy.stats.crystalball = <scipy.stats._continuous_distns.crystalball_gen object>

  Crystalball distribution

  As an instance of the `rv_continuous` class, `crystalball` 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(beta, m, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, beta, m, loc=0, scale=1) | Probability density function.
  logpdf(x, beta, m, loc=0, scale=1) | Log of the probability density function.
  cdf(x, beta, m, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, beta, m, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, beta, m, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, beta, m, loc=0, scale=1) | Log of the survival function.
  ppf(q, beta, m, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, beta, m, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, beta, m, loc=0, scale=1) | Non-central moment of the specified order.
  stats(beta, m, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(beta, m, 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=(beta, m), 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(beta, m, loc=0, scale=1) | Median of the distribution.
  mean(beta, m, loc=0, scale=1) | Mean of the distribution.
  var(beta, m, loc=0, scale=1) | Variance of the distribution.
  std(beta, m, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, beta, m, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `crystalball` is:

  \[\begin{split}f(x, \beta, m) = \begin{cases} N \exp(-x^2 / 2), &\text{for } x > -\beta\\ N A (B - x)^{-m} &\text{for } x \le -\beta \end{cases}\end{split}\]

  where \(A = (m / |\beta|)^m \exp(-\beta^2 / 2)\), \(B = m/|\beta| - |\beta|\) and \(N\) is a normalisation constant.

  `crystalball` takes \(\beta > 0\) and \(m > 1\) as shape parameters. \(\beta\) defines the point where the pdf changes from a power-law to a Gaussian distribution. \(m\) is the power of the power-law tail.

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `crystalball.pdf(x, beta, m, loc, scale)` is identically equivalent to `crystalball.pdf(y, beta, m) / 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.

  Added in version 0.19.0.

  References

  [1]

  “Crystal Ball Function”, https://en.wikipedia.org/wiki/Crystal_Ball_function

  Examples

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

  Get the support:

      >>> beta, m = 2, 3
      >>> lb, ub = crystalball.support(beta, m)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = crystalball.stats(beta, m, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(crystalball.ppf(0.01, beta, m),
      ...                 crystalball.ppf(0.99, beta, m), 100)
      >>> ax.plot(x, crystalball.pdf(x, beta, m),
      ...        'r-', lw=5, alpha=0.6, label='crystalball 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 = crystalball(beta, m)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = crystalball.rvs(beta, m, 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()