scipy.stats.truncpareto#

scipy.stats.truncpareto = <scipy.stats._continuous_distns.truncpareto_gen object>[source]#

An upper truncated Pareto continuous random variable.

As an instance of the rv_continuous class, truncpareto 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(b, c, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, b, c, loc=0, scale=1)

Probability density function.

logpdf(x, b, c, loc=0, scale=1)

Log of the probability density function.

cdf(x, b, c, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, b, c, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, b, c, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, b, c, loc=0, scale=1)

Log of the survival function.

ppf(q, b, c, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, b, c, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, b, c, loc=0, scale=1)

Non-central moment of the specified order.

stats(b, c, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(b, 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=(b, 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(b, c, loc=0, scale=1)

Median of the distribution.

mean(b, c, loc=0, scale=1)

Mean of the distribution.

var(b, c, loc=0, scale=1)

Variance of the distribution.

std(b, c, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, b, c, loc=0, scale=1)

Confidence interval with equal areas around the median.

See also

pareto

Pareto distribution

Notes

The probability density function for truncpareto is:

for b > 0, c > 1 and 1 \le x \le c.

truncpareto takes b and c as shape parameters for b and c.

Notice that the upper truncation value c is defined in standardized form so that random values of an unscaled, unshifted variable are within the range [1, c]. If u_r is the upper bound to a scaled and/or shifted variable, then c = (u_r - loc) / scale. In other words, the support of the distribution becomes (scale + loc) <= x <= (c*scale + loc) when scale and/or loc are provided.

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

References

[1]

Burroughs, S. M., and Tebbens S. F. “Upper-truncated power laws in natural systems.” Pure and Applied Geophysics 158.4 (2001): 741-757.

Examples

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

Get the support:

>>> b, c = 2, 5
>>> lb, ub = truncpareto.support(b, c)

Calculate the first four moments:

>>> mean, var, skew, kurt = truncpareto.stats(b, c, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = truncpareto.rvs(b, 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()
../../_images/scipy-stats-truncpareto-1.png