mars.tensor.random.lognormal¶

mars.tensor.random.lognormal(mean=0.0, sigma=1.0, size=None, chunk_size=None, gpu=None, dtype=None)[source]

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

Parameters
• mean (float or array_like of floats, optional) – Mean value of the underlying normal distribution. Default is 0.

• sigma (float or array_like of floats, optional) – Standard deviation of the underlying normal distribution. Should be greater than zero. 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 mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).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 log-normal distribution.

Return type

Tensor or scalar

scipy.stats.lognorm

probability density function, distribution, cumulative density function, etc.

Notes

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:

$p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}$

where $$\mu$$ is the mean and $$\sigma$$ is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

References

1

Limpert, E., Stahel, W. A., and Abbt, M., “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, Vol. 51, No. 5, May, 2001. http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

2

Reiss, R.D. and Thomas, M., “Statistical Analysis of Extreme Values,” Basel: Birkhauser Verlag, 2001, pp. 31-32.

Examples

Draw samples from the distribution:

>>> import mars.tensor as mt

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = mt.random.lognormal(mu, sigma, 1000)


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

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s.execute(), 100, normed=True, align='mid')

>>> x = mt.linspace(min(bins), max(bins), 10000)
>>> pdf = (mt.exp(-(mt.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * mt.sqrt(2 * mt.pi)))

>>> plt.plot(x.execute(), pdf.execute(), linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()


Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + mt.random.random(100)
...    b.append(mt.product(a).execute())

>>> b = mt.array(b) / mt.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b.execute(), 100, normed=True, align='mid')
>>> sigma = mt.std(mt.log(b))
>>> mu = mt.mean(mt.log(b))

>>> x = mt.linspace(min(bins), max(bins), 10000)
>>> pdf = (mt.exp(-(mt.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * mt.sqrt(2 * mt.pi)))

>>> plt.plot(x.execute(), pdf.execute(), color='r', linewidth=2)
>>> plt.show()