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# derived from copyright 1999-2021 Alibaba Group Holding Ltd.
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# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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# http://www.apache.org/licenses/LICENSE-2.0
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import numpy as np
from ... import opcodes as OperandDef
from ...serialization.serializables import AnyField
from ..utils import gen_random_seeds
from .core import TensorDistribution, TensorRandomOperandMixin, handle_array
class TensorLognormal(TensorDistribution, TensorRandomOperandMixin):
_input_fields_ = ["mean", "sigma"]
_op_type_ = OperandDef.RAND_LOGNORMAL
_fields_ = "mean", "sigma", "size"
mean = AnyField("mean")
sigma = AnyField("sigma")
_func_name = "lognormal"
def __call__(self, mean, sigma, chunk_size=None):
return self.new_tensor([mean, sigma], None, raw_chunk_size=chunk_size)
[docs]def lognormal(
random_state, mean=0.0, sigma=1.0, size=None, chunk_size=None, gpu=None, dtype=None
):
r"""
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 : Tensor or scalar
Drawn samples from the parameterized log-normal distribution.
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:
.. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}
where :math:`\mu` is the mean and :math:`\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()
"""
if dtype is None:
dtype = (
np.random.RandomState()
.lognormal(handle_array(mean), handle_array(sigma), size=(0,))
.dtype
)
size = random_state._handle_size(size)
seed = gen_random_seeds(1, random_state.to_numpy())[0]
op = TensorLognormal(seed=seed, size=size, gpu=gpu, dtype=dtype)
return op(mean, sigma, chunk_size=chunk_size)