scipy.stats.levy_l#
- scipy.stats.levy_l = <scipy.stats._continuous_distns.levy_l_gen object>[source]#
A left-skewed Levy continuous random variable.
As an instance of the
rv_continuous
class,levy_l
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.See also
Notes
The probability density function for
levy_l
is:\[f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}\]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
andscale
parameters. Specifically,levy_l.pdf(x, loc, scale)
is identically equivalent tolevy_l.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_l >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> mean, var, skew, kurt = levy_l.stats(moments='mvsk')
Display the probability density function (
pdf
):>>> # `levy_l` is very heavy-tailed. >>> # To show a nice plot, let's cut off the lower 40 percent. >>> a, b = levy_l.ppf(0.4), levy_l.ppf(1) >>> x = np.linspace(a, b, 100) >>> ax.plot(x, levy_l.pdf(x), ... 'r-', lw=5, alpha=0.6, label='levy_l 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_l() >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of
cdf
andppf
:>>> vals = levy_l.ppf([0.001, 0.5, 0.999]) >>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals)) True
Generate random numbers:
>>> r = levy_l.rvs(size=1000)
And compare the histogram:
>>> # manual binning to ignore the tail >>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20))) >>> 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.