scipy.stats.kappa4¶
-
scipy.stats.
kappa4
= <scipy.stats._continuous_distns.kappa4_gen object>[source]¶ 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.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 != 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 \(h\) 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)
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
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). https://doi.org/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
andscale
parameters. Specifically,kappa4.pdf(x, h, k, loc, scale)
is identically equivalent tokappa4.pdf(y, h, k) / scale
withy = (x - loc) / scale
.Examples
>>> from scipy.stats import kappa4 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate a few first 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
andppf
:>>> 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, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()
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(n, h, k, loc=0, scale=1)
Non-central moment of order n
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, h, k, loc=0, scale=1)
Parameter estimates for generic data.
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(alpha, h, k, loc=0, scale=1)
Endpoints of the range that contains alpha percent of the distribution