scipy.stats.kappa3 — SciPy v1.15.2 Manual (original) (raw)
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 takes a 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 and scale parameters. Specifically, kappa3.pdf(x, a, loc, scale) is identically equivalent to kappa3.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 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, 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. |