scipy.stats.laplace_asymmetric

scipy.stats.laplace_asymmetric = <scipy.stats._continuous_distns.laplace_asymmetric_gen object>

An asymmetric Laplace continuous random variable.

As an instance of the rv_continuous class, laplace_asymmetric 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.

Methods

rvs(kappa, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, kappa, loc=0, scale=1)	Probability density function.
logpdf(x, kappa, loc=0, scale=1)	Log of the probability density function.
cdf(x, kappa, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, kappa, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, kappa, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, kappa, loc=0, scale=1)	Log of the survival function.
ppf(q, kappa, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, kappa, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(order, kappa, loc=0, scale=1)	Non-central moment of the specified order.
stats(kappa, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(kappa, 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=(kappa,), 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(kappa, loc=0, scale=1)	Median of the distribution.
mean(kappa, loc=0, scale=1)	Mean of the distribution.
var(kappa, loc=0, scale=1)	Variance of the distribution.
std(kappa, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, kappa, loc=0, scale=1)	Confidence interval with equal areas around the median.

See also

laplace

Laplace distribution

Notes

The probability density function for laplace_asymmetric is

\[\begin{split}f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\ &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\\end{split}\]

for \(-\infty < x < \infty\), \(\kappa > 0\).

laplace_asymmetric takes kappa as a shape parameter for \(\kappa\). For \(\kappa = 1\), it is identical to a Laplace distribution.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, laplace_asymmetric.pdf(x, kappa, loc, scale) is identically equivalent to laplace_asymmetric.pdf(y, kappa) / scale with y = (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.

Note that the scale parameter of some references is the reciprocal of SciPy’s scale. For example, \(\lambda = 1/2\) in the parameterization of [1] is equivalent to scale = 2 with laplace_asymmetric.

References

[1]

“Asymmetric Laplace distribution”, Wikipedia https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

[2]

Kozubowski TJ and Podgórski K. A Multivariate and Asymmetric Generalization of Laplace Distribution, Computational Statistics 15, 531–540 (2000). DOI:10.1007/PL00022717

Examples

Try it in your browser!

>>> import numpy as np
>>> from scipy.stats import laplace_asymmetric
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> kappa = 2
>>> mean, var, skew, kurt = laplace_asymmetric.stats(kappa, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(laplace_asymmetric.ppf(0.01, kappa),
...                 laplace_asymmetric.ppf(0.99, kappa), 100)
>>> ax.plot(x, laplace_asymmetric.pdf(x, kappa),
...        'r-', lw=5, alpha=0.6, label='laplace_asymmetric 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 = laplace_asymmetric(kappa)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = laplace_asymmetric.ppf([0.001, 0.5, 0.999], kappa)
>>> np.allclose([0.001, 0.5, 0.999], laplace_asymmetric.cdf(vals, kappa))
True

Generate random numbers:

>>> r = laplace_asymmetric.rvs(kappa, 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()

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