mars.tensor.random.weibull#

mars.tensor.random.weibull(a, size=None, chunk_size=None, gpu=None, dtype=None)[source]#

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.

\[X = (-ln(U))^{1/a}\]

Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \(\lambda\) is just \(X = \lambda(-ln(U))^{1/a}\).

Parameters

  • a (float or array_like of floats) – Shape of the distribution. Should be greater than zero.

  • size (int or tuple of ints, optional) – Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, mt.array(a).size samples are drawn.

  • chunk_size (int or tuple of int or tuple of ints, optional) – Desired chunk size on each dimension

  • gpu (bool, optional) – Allocate the tensor on GPU if True, False as default

  • dtype (data-type, optional) – Data-type of the returned tensor.

Returns

out – Drawn samples from the parameterized Weibull distribution.

Return type

Tensor or scalar

See also

scipy.stats.weibull_max, scipy.stats.weibull_min, scipy.stats.genextreme, gumbel

Notes

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is

\[p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},\]

where \(a\) is the shape and \(\lambda\) the scale.

The function has its peak (the mode) at \(\lambda(\frac{a-1}{a})^{1/a}\).

When a = 1, the Weibull distribution reduces to the exponential distribution.

References

1

Waloddi Weibull, Royal Technical University, Stockholm, 1939 “A Statistical Theory Of The Strength Of Materials”, Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.

2

Waloddi Weibull, “A Statistical Distribution Function of Wide Applicability”, Journal Of Applied Mechanics ASME Paper 1951.

3

Wikipedia, “Weibull distribution”, http://en.wikipedia.org/wiki/Weibull_distribution

Examples

Draw samples from the distribution:

>>> import mars.tensor as mt

>>> a = 5. # shape
>>> s = mt.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = mt.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * mt.exp(-(x / n)**a)

>>> count, bins, ignored = plt.hist(mt.random.weibull(5.,1000).execute())
>>> x = mt.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x.execute(), (weib(x, 1., 5.)*scale).execute())
>>> plt.show()