# scipy.stats.kappa3¶

scipy.stats.kappa3 = <scipy.stats._continuous_distns.kappa3_gen object>[source]

Kappa 3 parameter distribution.

As an instance of the rv_continuous class, kappa3 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.

Notes

The probability density function for kappa is:

$\begin{split}f(x, a) = \begin{cases} a [a + x^a]^{-(a + 1)/a}, &\text{for } x > 0\\ 0.0, &\text{for } x \le 0 \end{cases}\end{split}$

kappa3 takes $$a$$ as a shape parameter and $$a > 0$$.

References

P.W. Mielke and E.S. Johnson, “Three-Parameter Kappa Distribution Maximum Likelihood and Likelihood Ratio Tests”, Methods in Weather Research, 701-707, (September, 1973), http://docs.lib.noaa.gov/rescue/mwr/101/mwr-101-09-0701.pdf

B. Kumphon, “Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution”, Open Journal of Statistics, vol 2, 415-419 (2012) http://file.scirp.org/pdf/OJS20120400011_95789012.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, kappa3.pdf(x, a, loc, scale) is identically equivalent to kappa3.pdf(y, a) / scale with y = (x - loc) / scale.

Examples

>>> from scipy.stats import kappa3
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)


Calculate a few first moments:

>>> a = 1
>>> mean, var, skew, kurt = kappa3.stats(a, moments='mvsk')


Display the probability density function (pdf):

>>> x = np.linspace(kappa3.ppf(0.01, a),
...                 kappa3.ppf(0.99, a), 100)
>>> ax.plot(x, kappa3.pdf(x, a),
...        'r-', lw=5, alpha=0.6, label='kappa3 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 = kappa3(a)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')


Check accuracy of cdf and ppf:

>>> vals = kappa3.ppf([0.001, 0.5, 0.999], a)
>>> np.allclose([0.001, 0.5, 0.999], kappa3.cdf(vals, a))
True


Generate random numbers:

>>> r = kappa3.rvs(a, size=1000)


And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()


Methods

 rvs(a, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, a, loc=0, scale=1) Probability density function. logpdf(x, a, loc=0, scale=1) Log of the probability density function. cdf(x, a, loc=0, scale=1) Cumulative distribution function. logcdf(x, a, loc=0, scale=1) Log of the cumulative distribution function. sf(x, a, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). logsf(x, a, loc=0, scale=1) Log of the survival function. ppf(q, a, loc=0, scale=1) Percent point function (inverse of cdf — percentiles). isf(q, a, loc=0, scale=1) Inverse survival function (inverse of sf). moment(n, a, loc=0, scale=1) Non-central moment of order n stats(a, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). entropy(a, loc=0, scale=1) (Differential) entropy of the RV. fit(data, a, loc=0, scale=1) Parameter estimates for generic data. expect(func, args=(a,), 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(a, loc=0, scale=1) Median of the distribution. mean(a, loc=0, scale=1) Mean of the distribution. var(a, loc=0, scale=1) Variance of the distribution. std(a, loc=0, scale=1) Standard deviation of the distribution. interval(alpha, a, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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