scipy.stats.loglaplace

scipy.stats.loglaplace = <scipy.stats._continuous_distns.loglaplace_gen object>

A log-Laplace continuous random variable.

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

Notes

The probability density function for loglaplace is:

\[\begin{split}f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1} &\text{for } 0 < x < 1\\ \frac{c}{2} x^{-c-1} &\text{for } x \ge 1 \end{cases}\end{split}\]

for \(c > 0\).

loglaplace takes c as a shape parameter for \(c\).

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

Suppose a random variable X follows the Laplace distribution with location a and scale b. Then Y = exp(X) follows the log-Laplace distribution with c = 1 / b and scale = exp(a).

References

T.J. Kozubowski and K. Podgorski, “A log-Laplace growth rate model”, The Mathematical Scientist, vol. 28, pp. 49-60, 2003.

Examples

Try it in your browser!

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

Calculate the first four moments:

>>> c = 3.25
>>> mean, var, skew, kurt = loglaplace.stats(c, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = loglaplace.rvs(c, 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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