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import numpy as np
from ... import opcodes as OperandDef
from ...serialization.serializables import AnyField
from ..utils import gen_random_seeds
from .core import TensorDistribution, TensorRandomOperandMixin, handle_array
class TensorGumbel(TensorDistribution, TensorRandomOperandMixin):
_input_fields_ = ["loc", "scale"]
_op_type_ = OperandDef.RAND_GUMBEL
_fields_ = "loc", "scale", "size"
loc = AnyField("loc")
scale = AnyField("scale")
_func_name = "gumbel"
def __call__(self, loc, scale, chunk_size=None):
return self.new_tensor([loc, scale], None, raw_chunk_size=chunk_size)
[docs]def gumbel(
random_state, loc=0.0, scale=1.0, size=None, chunk_size=None, gpu=None, dtype=None
):
r"""
Draw samples from a Gumbel distribution.
Draw samples from a Gumbel distribution with specified location and
scale. For more information on the Gumbel distribution, see
Notes and References below.
Parameters
----------
loc : float or array_like of floats, optional
The location of the mode of the distribution. Default is 0.
scale : float or array_like of floats, optional
The scale parameter of the distribution. Default is 1.
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 ``loc`` and ``scale`` are both scalars.
Otherwise, ``np.broadcast(loc, scale).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 : Tensor or scalar
Drawn samples from the parameterized Gumbel distribution.
See Also
--------
scipy.stats.gumbel_l
scipy.stats.gumbel_r
scipy.stats.genextreme
weibull
Notes
-----
The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
Value Type I) distribution is one of a class of Generalized Extreme
Value (GEV) distributions used in modeling extreme value problems.
The Gumbel is a special case of the Extreme Value Type I distribution
for maximums from distributions with "exponential-like" tails.
The probability density for the Gumbel distribution is
.. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
\beta}},
where :math:`\mu` is the mode, a location parameter, and
:math:`\beta` is the scale parameter.
The Gumbel (named for German mathematician Emil Julius Gumbel) was used
very early in the hydrology literature, for modeling the occurrence of
flood events. It is also used for modeling maximum wind speed and
rainfall rates. It is a "fat-tailed" distribution - the probability of
an event in the tail of the distribution is larger than if one used a
Gaussian, hence the surprisingly frequent occurrence of 100-year
floods. Floods were initially modeled as a Gaussian process, which
underestimated the frequency of extreme events.
It is one of a class of extreme value distributions, the Generalized
Extreme Value (GEV) distributions, which also includes the Weibull and
Frechet.
The function has a mean of :math:`\mu + 0.57721\beta` and a variance
of :math:`\frac{\pi^2}{6}\beta^2`.
References
----------
.. [1] Gumbel, E. J., "Statistics of Extremes,"
New York: Columbia University Press, 1958.
.. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
Values from Insurance, Finance, Hydrology and Other Fields,"
Basel: Birkhauser Verlag, 2001.
Examples
--------
Draw samples from the distribution:
>>> import mars.tensor as mt
>>> mu, beta = 0, 0.1 # location and scale
>>> s = mt.random.gumbel(mu, beta, 1000).execute()
Display the histogram of the samples, along with
the probability density function:
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> count, bins, ignored = plt.hist(s, 30, normed=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
... * np.exp( -np.exp( -(bins - mu) /beta) ),
... linewidth=2, color='r')
>>> plt.show()
Show how an extreme value distribution can arise from a Gaussian process
and compare to a Gaussian:
>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
... a = mt.random.normal(mu, beta, 1000)
... means.append(a.mean().execute())
... maxima.append(a.max().execute())
>>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
>>> beta = mt.std(maxima) * mt.sqrt(6) / mt.pi
>>> mu = mt.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, ((1/beta)*mt.exp(-(bins - mu)/beta)
... * mt.exp(-mt.exp(-(bins - mu)/beta))).execute(),
... linewidth=2, color='r')
>>> plt.plot(bins, (1/(beta * mt.sqrt(2 * mt.pi))
... * mt.exp(-(bins - mu)**2 / (2 * beta**2))).execute(),
... linewidth=2, color='g')
>>> plt.show()
"""
if dtype is None:
dtype = (
np.random.RandomState()
.gumbel(handle_array(loc), handle_array(scale), size=(0,))
.dtype
)
size = random_state._handle_size(size)
seed = gen_random_seeds(1, random_state.to_numpy())[0]
op = TensorGumbel(seed=seed, size=size, gpu=gpu, dtype=dtype)
return op(loc, scale, chunk_size=chunk_size)