mars.tensor.random.
lognormal
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.
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.
(m, n, k)
m * n * k
None
mean
sigma
np.broadcast(mean, sigma).size
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.
out – Drawn samples from the parameterized log-normal distribution.
Tensor or scalar
See also
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:
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
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
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))
>>> plt.plot(x.execute(), pdf.execute(), color='r', linewidth=2) >>> plt.show()