flopscope.numpy.random.lognormal

fnp.random.lognormal(mean=0.0, sigma=1.0, size=None)[flopscope source]

Draw samples from a log-normal distribution.

Adapted from NumPy docs np.random.lognormal

Arearandom

Typecustom

NumPy Refnp.random.lognormal

Cost

16×per-operation

Flopscope Context

Sampling; cost = numel(output).

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.

Note.

New code should use the lognormal method of a Generator instance instead; please see the random-quick-start.

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. Must be non-negative. 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, flops.broadcast(mean, sigma).size samples are drawn.

Returns

out:ndarray or scalar: Drawn samples from the parameterized log-normal distribution.

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

footnote

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.
https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

footnote

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:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = flops.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, 100, density=True, align='mid')

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

>>> plt.plot(x, pdf, 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. + flops.random.standard_normal(100)
... b.append(flops.prod(a))

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

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

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