gumbel(self, loc=0.0, scale=1.0, size=None, chunks='auto', **kwargs)
This docstring was copied from numpy.random.mtrand.RandomState.gumbel.
Some inconsistencies with the Dask version may exist.
Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.
New code should use the gumbel
method of a default_rng()
instance instead; please see the :None:ref:`random-quick-start`.
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
$$p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/\beta}},$$where $\mu$ is the mode, a location parameter, and $\beta$ 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 $\mu + 0.57721\beta$ and a variance of $\frac{\pi^2}{6}\beta^2$ .
The location of the mode of the distribution. Default is 0.
The scale parameter of the distribution. Default is 1. Must be non- negative.
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.
Drawn samples from the parameterized Gumbel distribution.
Draw samples from a Gumbel distribution.
Generator.gumbel
which should be used for new code.
Draw samples from the distribution:
This example is valid syntax, but we were not able to check execution>>> mu, beta = 0, 0.1 # location and scale # doctest: +SKIP
... s = np.random.gumbel(mu, beta, 1000) # doctest: +SKIP
Display the histogram of the samples, along with the probability density function:
This example is valid syntax, but we were not able to check execution>>> import matplotlib.pyplot as plt # doctest: +SKIP
... count, bins, ignored = plt.hist(s, 30, density=True) # doctest: +SKIP
... plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) # doctest: +SKIP
... * np.exp( -np.exp( -(bins - mu) /beta) ),
... linewidth=2, color='r')
... plt.show() # doctest: +SKIP
Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:
This example is valid syntax, but we were not able to check execution>>> means = [] # doctest: +SKIPSee :
... maxima = [] # doctest: +SKIP
... for i in range(0,1000) : # doctest: +SKIP
... a = np.random.normal(mu, beta, 1000)
... means.append(a.mean())
... maxima.append(a.max())
... count, bins, ignored = plt.hist(maxima, 30, density=True) # doctest: +SKIP
... beta = np.std(maxima) * np.sqrt(6) / np.pi # doctest: +SKIP
... mu = np.mean(maxima) - 0.57721*beta # doctest: +SKIP
... plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) # doctest: +SKIP
... * np.exp(-np.exp(-(bins - mu)/beta)),
... linewidth=2, color='r')
... plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) # doctest: +SKIP
... * np.exp(-(bins - mu)**2 / (2 * beta**2)),
... linewidth=2, color='g')
... plt.show() # doctest: +SKIP
The following pages refer to to this document either explicitly or contain code examples using this.
dask.array.random.RandomState.weibull
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