scipy.stats.levy — SciPy v1.15.3 Manual (original) (raw)
scipy.stats.levy = <scipy.stats._continuous_distns.levy_gen object>[source]#
A Levy continuous random variable.
As an instance of the rv_continuous class, levy 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 levy is:
\[f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)\]
for \(x > 0\).
This is the same as the Levy-stable distribution with \(a=1/2\) and\(b=1\).
The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, levy.pdf(x, loc, scale) is identically equivalent to levy.pdf(y) / 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 levy import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
mean, var, skew, kurt = levy.stats(moments='mvsk')
Display the probability density function (pdf):
levyis very heavy-tailed.To show a nice plot, let's cut off the upper 40 percent.
a, b = levy.ppf(0), levy.ppf(0.6) x = np.linspace(a, b, 100) ax.plot(x, levy.pdf(x), ... 'r-', lw=5, alpha=0.6, label='levy 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 = levy() ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of cdf and ppf:
vals = levy.ppf([0.001, 0.5, 0.999]) np.allclose([0.001, 0.5, 0.999], levy.cdf(vals)) True
Generate random numbers:
r = levy.rvs(size=1000)
And compare the histogram:
manual binning to ignore the tail
bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)])) ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2) ax.set_xlim([x[0], x[-1]]) ax.legend(loc='best', frameon=False) plt.show()
Methods
| rvs(loc=0, scale=1, size=1, random_state=None) | Random variates. |
|---|---|
| pdf(x, loc=0, scale=1) | Probability density function. |
| logpdf(x, loc=0, scale=1) | Log of the probability density function. |
| cdf(x, loc=0, scale=1) | Cumulative distribution function. |
| logcdf(x, loc=0, scale=1) | Log of the cumulative distribution function. |
| sf(x, loc=0, scale=1) | Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). |
| logsf(x, loc=0, scale=1) | Log of the survival function. |
| ppf(q, loc=0, scale=1) | Percent point function (inverse of cdf — percentiles). |
| isf(q, loc=0, scale=1) | Inverse survival function (inverse of sf). |
| moment(order, loc=0, scale=1) | Non-central moment of the specified order. |
| stats(loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). |
| entropy(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=(), 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(loc=0, scale=1) | Median of the distribution. |
| mean(loc=0, scale=1) | Mean of the distribution. |
| var(loc=0, scale=1) | Variance of the distribution. |
| std(loc=0, scale=1) | Standard deviation of the distribution. |
| interval(confidence, loc=0, scale=1) | Confidence interval with equal areas around the median. |