scipy.stats.truncweibull_min

scipy.stats.truncweibull_min = <scipy.stats._continuous_distns.truncweibull_min_gen object>

A doubly truncated Weibull minimum continuous random variable.

As an instance of the rv_continuous class, truncweibull_min 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

weibull_min, truncexpon

Notes

The probability density function for truncweibull_min is:

\[f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)}\]

for \(a < x <= b\), \(0 \le a < b\) and \(c > 0\).

truncweibull_min takes \(a\), \(b\), and \(c\) as shape parameters.

Notice that the truncation values, \(a\) and \(b\), are defined in standardized form:

\[a = (u_l - loc)/scale b = (u_r - loc)/scale\]

where \(u_l\) and \(u_r\) are the specific left and right truncation values, respectively. In other words, the support of the distribution becomes \((a*scale + loc) < x <= (b*scale + loc)\) when \(loc\) and/or \(scale\) 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, truncweibull_min.pdf(x, c, a, b, loc, scale) is identically equivalent to truncweibull_min.pdf(y, c, a, b) / 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

Rinne, H. “The Weibull Distribution: A Handbook”. CRC Press (2009).

Examples

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

Calculate the first four moments:

>>> c, a, b = 2.5, 0.25, 1.75
>>> mean, var, skew, kurt = truncweibull_min.stats(c, a, b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = truncweibull_min.rvs(c, 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()

Methods

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