scipy.stats.loguniform

scipy.stats.loguniform(*args, **kwds) = <scipy.stats._continuous_distns.reciprocal_gen object>

A loguniform or reciprocal continuous random variable.

As an instance of the rv_continuous class, loguniform 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 this class is:

\[f(x, a, b) = \frac{1}{x \log(b/a)}\]

for \(a \le x \le b\), \(b > a > 0\). This class takes \(a\) and \(b\) as shape parameters. The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, loguniform.pdf(x, a, b, loc, scale) is identically equivalent to loguniform.pdf(y, a, b) / scale with y = (x - loc) / scale.

Examples

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

Calculate a few first moments:

>>> a, b = 0.01, 1
>>> mean, var, skew, kurt = loguniform.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = loguniform.ppf([0.001, 0.5, 0.999], a, b)
>>> np.allclose([0.001, 0.5, 0.999], loguniform.cdf(vals, a, b))
True

Generate random numbers:

>>> r = loguniform.rvs(a, b, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

This doesn’t show the equal probability of 0.01, 0.1 and 1. This is best when the x-axis is log-scaled:

>>> import numpy as np
>>> fig, ax = plt.subplots(1, 1)
>>> ax.hist(np.log10(r))
>>> ax.set_ylabel("Frequency")
>>> ax.set_xlabel("Value of random variable")
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
>>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
>>> ax.set_xticklabels(ticks)  
>>> plt.show()

This random variable will be log-uniform regardless of the base chosen for a and b. Let’s specify with base 2 instead:

>>> rvs = loguniform(2**-2, 2**0).rvs(size=1000)

Values of 1/4, 1/2 and 1 are equally likely with this random variable. Here’s the histogram:

>>> fig, ax = plt.subplots(1, 1)
>>> ax.hist(np.log2(rvs))
>>> ax.set_ylabel("Frequency")
>>> ax.set_xlabel("Value of random variable")
>>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
>>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
>>> ax.set_xticklabels(ticks)  
>>> plt.show()

Methods

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