nlcpy.random.Generator.gumbel

Generator.gumbel(self, loc=0.0, scale=1.0, size=None)

Draws samples from a Gumbel distribution.

Draws samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes below.

Parameters

locfloat, optional: The location of the mode of the distribution. Default is 0.
scalefloat, optional: The scale parameter of the distribution. Default is 1. Must be non-negative.
sizeint or tuple of ints, optional: Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn.

Returns

outndarray: Drawn samples from the parameterized Gumbel distribution.

See also

Generator.weibull: Draws samples from a Weibull distribution.

Note

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 function has a mean of $\mu + 0.57721\beta$ and a variance of $\frac{\pi^2}{6}\beta^2$ .

Restriction

If loc is neither a scalar nor None : NotImplementedError occurs.
If scale is neither a scalar nor None : NotImplementedError occurs.

Examples

Draw samples from the distribution:

>>> import nlcpy as vp
>>> rng = vp.random.default_rng()
>>> mu, beta = 0, 0.1 # location and scale
>>> s = rng.gumbel(mu, beta, 1000)

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

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s.get(), 30, density=True)
>>> plt.plot(bins, (1/beta)*vp.exp(-(bins - mu)/beta)
...          * vp.exp( -vp.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r') 
... 
>>> plt.show()

../../_images/nlcpy-random-Generator-gumbel-1_00_00.png

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> plt.close()
>>> maxima = vp.empty(1000)
>>> for i in range(0,1000) :
...    a = rng.normal(mu, beta, 1000)
...    maxima[i] = a.max()
>>> count, bins, ignored = plt.hist(maxima.get(), 30, density=True)
>>> beta = vp.std(maxima) * vp.sqrt(6) / vp.pi
>>> mu = vp.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*vp.exp(-(bins - mu)/beta)
...          * vp.exp(-vp.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r') 
... 
>>> plt.plot(bins, 1/(beta * vp.sqrt(2 * vp.pi))
...          * vp.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g') 
... 
>>> plt.show()

../../_images/nlcpy-random-Generator-gumbel-1_01_00.png