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def kappa4(*args, **kwds)
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 <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#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:
.. math::
f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}
if :math:`h` and :math:`k` are not equal to 0.
If :math:`h` or :math:`k` are zero then the pdf can be simplified:
h = 0 and k != 0::
kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
exp(-(1.0 - k*x)**(1.0/k))
h != 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 :math:`h` and :math:`k` as shape parameters.
The kappa4 distribution returns other distributions when certain
:math:`h` and :math:`k` values are used.
+------+-------------+----------------+------------------+
| h | k=0.0 | k=1.0 | -inf<=k<=inf |
+======+=============+================+==================+
| -1.0 | Logistic | | Generalized |
| | | | Logistic(1) |
| | | | |
| | logistic(x) | | |
+------+-------------+----------------+------------------+
| 0.0 | Gumbel | Reverse | Generalized |
| | | Exponential(2) | Extreme Value |
| | | | |
| | gumbel_r(x) | | genextreme(x, k) |
+------+-------------+----------------+------------------+
| 1.0 | Exponential | Uniform | Generalized |
| | | | Pareto |
| | | | |
| | expon(x) | uniform(x) | 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)
Calculate the first four moments:
>>> h, k = 0.1, 0
>>> 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()
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