# mars.tensor.random.power#

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

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

Parameters
• a (float or array_like of floats) – Parameter 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 power distribution.

Return type

Tensor or scalar

Raises

ValueError – If a < 1.

Notes

The probability density function is

$P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.$

The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

References

1

Christian Kleiber, Samuel Kotz, “Statistical size distributions in economics and actuarial sciences”, Wiley, 2003.

2

Heckert, N. A. and Filliben, James J. “NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions”, National Institute of Standards and Technology Handbook Series, June 2003. http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Examples

Draw samples from the distribution:

>>> import mars.tensor as mt

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


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

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s.execute(), bins=30)
>>> x = mt.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*mt.diff(bins)[0]*y
>>> plt.plot(x.execute(), normed_y.execute())
>>> plt.show()


Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats
>>> rvs = mt.random.power(5, 1000000)
>>> rvsp = mt.random.pareto(5, 1000000)
>>> xx = mt.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx.execute(),5)

>>> plt.figure()
>>> plt.hist(rvs.execute(), bins=50, normed=True)
>>> plt.plot(xx.execute(),powpdf,'r-')
>>> plt.title('np.random.power(5)')

>>> plt.figure()
>>> plt.hist((1./(1.+rvsp)).execute(), bins=50, normed=True)
>>> plt.plot(xx.execute(),powpdf,'r-')
>>> plt.title('inverse of 1 + np.random.pareto(5)')

>>> plt.figure()
>>> plt.hist((1./(1.+rvsp)).execute(), bins=50, normed=True)
>>> plt.plot(xx.execute(),powpdf,'r-')
>>> plt.title('inverse of stats.pareto(5)')