Skip to main content
`Ctrl`+`K`

# scipy.stats.zipfian

scipy.stats.zipfian = <scipy.stats._discrete_distns.zipfian_gen object>

  A Zipfian discrete random variable.

  As an instance of the `rv_discrete` class, `zipfian` 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(a, n, loc=0, size=1, random_state=None) | Random variates.
  pmf(k, a, n, loc=0) | Probability mass function.
  logpmf(k, a, n, loc=0) | Log of the probability mass function.
  cdf(k, a, n, loc=0) | Cumulative distribution function.
  logcdf(k, a, n, loc=0) | Log of the cumulative distribution function.
  sf(k, a, n, loc=0) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(k, a, n, loc=0) | Log of the survival function.
  ppf(q, a, n, loc=0) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, a, n, loc=0) | Inverse survival function (inverse of `sf`).
  stats(a, n, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(a, n, loc=0) | (Differential) entropy of the RV.
  expect(func, args=(a, n), loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution.
  median(a, n, loc=0) | Median of the distribution.
  mean(a, n, loc=0) | Mean of the distribution.
  var(a, n, loc=0) | Variance of the distribution.
  std(a, n, loc=0) | Standard deviation of the distribution.
  interval(confidence, a, n, loc=0) | Confidence interval with equal areas around the median.

  See also

  `zipf`

  Notes

  The probability mass function for `zipfian` is:

  \[f(k, a, n) = \frac{1}{H_{n,a} k^a}\]

  for \(k \in \{1, 2, \dots, n-1, n\}\), \(a \ge 0\), \(n \in \{1, 2, 3, \dots\}\).

  `zipfian` takes \(a\) and \(n\) as shape parameters. \(H_{n,a}\) is the \(n\)th generalized harmonic number of order \(a\).

  The SciPy implementation of this distribution requires \(1 \le n \le 2^{53}\). For larger values of \(n\), the `zipfian` methods (pmf, cdf, mean, etc.) will return nan.

  When \(a > 1\), the Zipfian distribution reduces to the Zipf (zeta) distribution as \(n \rightarrow \infty\).

  The probability mass function above is defined in the “standardized” form. To shift distribution use the `loc` parameter. Specifically, `zipfian.pmf(k, a, n, loc)` is identically equivalent to `zipfian.pmf(k - loc, a, n)`.

  References

  [1]

  “Zipf’s Law”, Wikipedia, https://en.wikipedia.org/wiki/Zipf’s_law

  [2]

  Larry Leemis, “Zipf Distribution”, Univariate Distribution Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf

  Examples

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

  Get the support:

      >>> a, n = 1.25, 10
      >>> lb, ub = zipfian.support(a, n)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = zipfian.stats(a, n, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(zipfian.ppf(0.01, a, n),
      ...               zipfian.ppf(0.99, a, n))
      >>> ax.plot(x, zipfian.pmf(x, a, n), 'bo', ms=8, label='zipfian pmf')
      >>> ax.vlines(x, 0, zipfian.pmf(x, a, n), colors='b', lw=5, alpha=0.5)

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

  Freeze the distribution and display the frozen `pmf`:

      >>> rv = zipfian(a, n)
      >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
      ...         label='frozen pmf')
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()

  Check accuracy of `cdf` and `ppf`:

      >>> prob = zipfian.cdf(x, a, n)
      >>> np.allclose(x, zipfian.ppf(prob, a, n))
      True

  Generate random numbers:

      >>> r = zipfian.rvs(a, n, size=1000)

  Confirm that `zipfian` reduces to `zipf` for large n, `a > 1`.

      >>> import numpy as np
      >>> from scipy.stats import zipf, zipfian
      >>> k = np.arange(11)
      >>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
      True