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# scipy.stats.kappa4

scipy.stats.kappa4 = <scipy.stats._continuous_distns.kappa4_gen object>

  Kappa 4 parameter distribution.

  As an instance of the `rv_continuous` class, `kappa4` 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(h, k, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, h, k, loc=0, scale=1) | Probability density function.
  logpdf(x, h, k, loc=0, scale=1) | Log of the probability density function.
  cdf(x, h, k, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, h, k, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, h, k, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, h, k, loc=0, scale=1) | Log of the survival function.
  ppf(q, h, k, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, h, k, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, h, k, loc=0, scale=1) | Non-central moment of the specified order.
  stats(h, k, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(h, k, 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=(h, k), 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(h, k, loc=0, scale=1) | Median of the distribution.
  mean(h, k, loc=0, scale=1) | Mean of the distribution.
  var(h, k, loc=0, scale=1) | Variance of the distribution.
  std(h, k, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, h, k, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for kappa4 is:

  \[f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}\]

  if \(h\) and \(k\) are not equal to 0.

  If \(h\) or \(k\) are zero then the pdf can be simplified:

  \(h = 0\) and \(k \neq 0\):

      kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                            exp(-(1.0 - k*x)**(1.0/k))

  \(h \neq 0\) and \(k = 0\):

      kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

  \(h = 0\) and \(k = 0\):

      kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

  kappa4 takes \(h\) and \(k\) as shape parameters.

  The kappa4 distribution returns other distributions when certain \(h\) and \(k\) values are used.

  h | k=0.0 | k=1.0 | -inf<=k<=inf
  --- | --- | --- | ---
  -1.0 | Logistic logistic(x) |  | Generalized Logistic(1)
  0.0 | Gumbel gumbel_r(x) | Reverse Exponential(2) | Generalized Extreme Value genextreme(x, k)
  1.0 | Exponential expon(x) | Uniform uniform(x) | Generalized Pareto genpareto(x, -k)

  1. There are at least five generalized logistic distributions. Four are described here: https://en.wikipedia.org/wiki/Generalized_logistic_distribution The “fifth” one is the one kappa4 should match which currently isn’t implemented in scipy: https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html

  2. This distribution is currently not in scipy.

  References

  J.C. Finney, “Optimization of a Skewed Logistic Distribution With Respect to the Kolmogorov-Smirnov Test”, A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College, (August, 2004), https://digitalcommons.lsu.edu/gradschool_dissertations/3672

  J.R.M. Hosking, “The four-parameter kappa distribution”. IBM J. Res. Develop. 38 (3), 25 1-258 (1994).

  B. Kumphon, A. Kaew-Man, P. Seenoi, “A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand”, Journal of Water Resource and Protection, vol. 4, 866-869, (2012). DOI:10.4236/jwarp.2012.410101

  C. Winchester, “On Estimation of the Four-Parameter Kappa Distribution”, A Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March 2000). http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `kappa4.pdf(x, h, k, loc, scale)` is identically equivalent to `kappa4.pdf(y, h, k) / 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.

  Examples

      >>> import numpy as np
      >>> from scipy.stats import kappa4
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Get the support:

      >>> h, k = 0.1, 0
      >>> lb, ub = kappa4.support(h, k)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = kappa4.stats(h, k, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(kappa4.ppf(0.01, h, k),
      ...                 kappa4.ppf(0.99, h, k), 100)
      >>> ax.plot(x, kappa4.pdf(x, h, k),
      ...        'r-', lw=5, alpha=0.6, label='kappa4 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 = kappa4(h, k)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = kappa4.ppf([0.001, 0.5, 0.999], h, k)
      >>> np.allclose([0.001, 0.5, 0.999], kappa4.cdf(vals, h, k))
      True

  Generate random numbers:

      >>> r = kappa4.rvs(h, k, 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()