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
kappa3
is:\[f(x, a) = a (a + x^a)^{-(a + 1)/a}\]for \(x > 0\) and \(a > 0\).
kappa3
takesa
as a shape parameter for \(a\).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), DOI:10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2
B. Kumphon, “Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution”, Open Journal of Statistics, vol 2, 415-419 (2012), DOI:10.4236/ojs.2012.24050
The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the
loc
andscale
parameters. Specifically,kappa3.pdf(x, a, loc, scale)
is identically equivalent tokappa3.pdf(y, a) / scale
withy = (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 kappa3 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four 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
andppf
:>>> 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, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> 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(order, a, loc=0, scale=1)
Non-central moment of the specified order.
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)
Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
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(confidence, a, loc=0, scale=1)
Confidence interval with equal areas around the median.