# Copyright 2020 Huawei Technologies Co., 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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# ============================================================================
"""Cauchy Distribution"""
import numpy as np
from mindspore.ops import operations as P
from mindspore.ops import composite as C
from mindspore import _checkparam as Validator
from mindspore.common import dtype as mstype
from .distribution import Distribution
from ._utils.utils import check_greater_zero, check_distribution_name, raise_not_defined
from ._utils.custom_ops import exp_generic, log_generic, log1p_generic
[docs]class Cauchy(Distribution):
r"""
Cauchy distribution.
A Cauchy distributio is a continuous distribution with the range of all real numbers
and the probability density function:
.. math::
f(x, a, b) = 1 / \pi b(1 - ((x - a)/b)^2),
where :math:`a, b` are loc and scale parameter respectively.
Args:
loc (int, float, list, numpy.ndarray, Tensor): The location of the Cauchy distribution. Default: ``None`` .
scale (int, float, list, numpy.ndarray, Tensor): The scale of the Cauchy distribution. Default: ``None`` .
seed (int): The seed used in sampling. The global seed is used if it is None. Default: ``None`` .
dtype (mindspore.dtype): The type of the event samples. Default: ``mstype.float32`` .
name (str): The name of the distribution. Default: ``'Cauchy'`` .
Note:
`scale` must be greater than zero.
`dist_spec_args` are `loc` and `scale`.
`dtype` must be a float type because Cauchy distributions are continuous.
Cauchy distribution is not supported on GPU backend.
Raises:
ValueError: When scale <= 0.
TypeError: When the input `dtype` is not a subclass of float.
Supported Platforms:
``Ascend``
Examples:
>>> import mindspore
>>> import mindspore.nn as nn
>>> import mindspore.nn.probability.distribution as msd
>>> from mindspore import Tensor
>>> # To initialize a Cauchy distribution of loc 3.0 and scale 4.0.
>>> cauchy1 = msd.Cauchy(3.0, 4.0, dtype=mindspore.float32)
>>> # A Cauchy distribution can be initialized without arguments.
>>> # In this case, 'loc' and `scale` must be passed in through arguments.
>>> cauchy2 = msd.Cauchy(dtype=mindspore.float32)
>>> # Here are some tensors used below for testing
>>> value = Tensor([1.0, 2.0, 3.0], dtype=mindspore.float32)
>>> loc_a = Tensor([2.0], dtype=mindspore.float32)
>>> scale_a = Tensor([2.0, 2.0, 2.0], dtype=mindspore.float32)
>>> loc_b = Tensor([1.0], dtype=mindspore.float32)
>>> scale_b = Tensor([1.0, 1.5, 2.0], dtype=mindspore.float32)
>>> # Private interfaces of probability functions corresponding to public interfaces, including
>>> # `prob`, `log_prob`, `cdf`, `log_cdf`, `survival_function`, and `log_survival`,
>>> # have the same arguments as follows.
>>> # Args:
>>> # value (Tensor): the value to be evaluated.
>>> # loc (Tensor): the location of the distribution. Default: self.loc.
>>> # scale (Tensor): the scale of the distribution. Default: self.scale.
>>> # Examples of `prob`.
>>> # Similar calls can be made to other probability functions
>>> # by replacing 'prob' by the name of the function
>>> ans = cauchy1.prob(value)
>>> print(ans.shape)
(3,)
>>> # Evaluate with respect to distribution b.
>>> ans = cauchy1.prob(value, loc_b, scale_b)
>>> print(ans.shape)
(3,)
>>> # `loc` and `scale` must be passed in during function calls
>>> ans = cauchy2.prob(value, loc_a, scale_a)
>>> print(ans.shape)
(3,)
>>> # Functions `mode` and `entropy` have the same arguments.
>>> # Args:
>>> # loc (Tensor): the location of the distribution. Default: self.loc.
>>> # scale (Tensor): the scale of the distribution. Default: self.scale.
>>> # Example of `mode`.
>>> ans = cauchy1.mode() # return 3.0
>>> print(ans.shape)
()
>>> ans = cauchy1.mode(loc_b, scale_b) # return loc_b
>>> print(ans.shape)
(3,)
>>> # `loc` and `scale` must be passed in during function calls.
>>> ans = cauchy2.mode(loc_a, scale_a)
>>> print(ans.shape)
(3,)
>>> # Interfaces of 'kl_loss' and 'cross_entropy' are the same:
>>> # Args:
>>> # dist (str): the type of the distributions. Only "Cauchy" is supported.
>>> # loc_b (Tensor): the loc of distribution b.
>>> # scale_b (Tensor): the scale distribution b.
>>> # loc (Tensor): the loc of distribution a. Default: self.loc.
>>> # scale (Tensor): the scale distribution a. Default: self.scale.
>>> # Examples of `kl_loss`. `cross_entropy` is similar.
>>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b)
>>> print(ans.shape)
(3,)
>>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
>>> print(ans.shape)
(3,)
>>> # Additional `loc` and `scale` must be passed in.
>>> ans = cauchy2.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
>>> print(ans.shape)
(3,)
>>> # Examples of `sample`.
>>> # Args:
>>> # shape (tuple): the shape of the sample. Default: ()
>>> # loc (Tensor): the location of the distribution. Default: self.loc.
>>> # scale (Tensor): the scale of the distribution. Default: self.scale.
>>> ans = cauchy1.sample()
>>> print(ans.shape)
()
>>> ans = cauchy1.sample((2,3))
>>> print(ans.shape)
(2, 3)
>>> ans = cauchy1.sample((2,3), loc_b, scale_b)
>>> print(ans.shape)
(2, 3, 3)
>>> ans = cauchy2.sample((2,3), loc_a, scale_a)
>>> print(ans.shape)
(2, 3, 3)
"""
def __init__(self,
loc=None,
scale=None,
seed=None,
dtype=mstype.float32,
name="Cauchy"):
"""
Constructor of Cauchy.
"""
param = dict(locals())
param['param_dict'] = {'loc': loc, 'scale': scale}
valid_dtype = mstype.float_type
Validator.check_type_name(
"dtype", dtype, valid_dtype, type(self).__name__)
super(Cauchy, self).__init__(seed, dtype, name, param)
self._loc = self._add_parameter(loc, 'loc')
self._scale = self._add_parameter(scale, 'scale')
if self._scale is not None:
check_greater_zero(self._scale, "scale")
# ops needed for the class
self.atan = P.Atan()
self.cast = P.Cast()
self.const = P.ScalarToTensor()
self.dtypeop = P.DType()
self.exp = exp_generic
self.less = P.Less()
self.log = log_generic
self.log1p = log1p_generic
self.squeeze = P.Squeeze(0)
self.shape = P.Shape()
self.sq = P.Square()
self.sqrt = P.Sqrt()
self.tan = P.Tan()
self.uniform = C.uniform
self.entropy_const = np.log(4 * np.pi)
def extend_repr(self):
"""Display instance object as string."""
if self.is_scalar_batch:
str_info = 'location = {}, scale = {}'.format(
self._loc, self._scale)
else:
str_info = 'batch_shape = {}'.format(self._broadcast_shape)
return str_info
@property
def loc(self):
"""
Return the location of the distribution after casting to dtype.
Output:
Tensor, the loc parameter of the distribution.
"""
return self._loc
@property
def scale(self):
"""
Return the scale of the distribution after casting to dtype.
Output:
Tensor, the scale parameter of the distribution.
"""
return self._scale
def _get_dist_type(self):
return "Cauchy"
def _get_dist_args(self, loc=None, scale=None):
if scale is not None:
self.checktensor(scale, 'scale')
else:
scale = self.scale
if loc is not None:
self.checktensor(loc, 'loc')
else:
loc = self.loc
return loc, scale
def _mode(self, loc=None, scale=None):
"""
The mode of the distribution.
"""
loc, scale = self._check_param_type(loc, scale)
return loc
def _mean(self, *args, **kwargs):
return raise_not_defined('mean', 'Cauchy', *args, **kwargs)
def _sd(self, *args, **kwargs):
return raise_not_defined('standard deviation', 'Cauchy', *args, **kwargs)
def _var(self, *args, **kwargs):
return raise_not_defined('variance', 'Cauchy', *args, **kwargs)
def _entropy(self, loc=None, scale=None):
r"""
Evaluate entropy.
.. math::
H(X) = \log(4 * \Pi * scale)
"""
loc, scale = self._check_param_type(loc, scale)
return self.log(scale) + self.entropy_const
def _log_prob(self, value, loc=None, scale=None):
r"""
Evaluate log probability.
Args:
value (Tensor): The value to be evaluated.
loc (Tensor): The location of the distribution. Default: self.loc.
scale (Tensor): The scale of the distribution. Default: self.scale.
.. math::
L(x) = \log(\frac{1}{\pi * scale} * \frac{scale^{2}}{(x - loc)^{2} + scale^{2}})
"""
value = self._check_value(value, 'value')
value = self.cast(value, self.dtype)
loc, scale = self._check_param_type(loc, scale)
z = (value - loc) / scale
log_unnormalized_prob = (-1) * self.log1p(self.sq(z))
log_normalization = self.log(np.pi * scale)
return log_unnormalized_prob - log_normalization
def _cdf(self, value, loc=None, scale=None):
r"""
Evaluate the cumulative distribution function on the given value.
Args:
value (Tensor): The value to be evaluated.
loc (Tensor): The location of the distribution. Default: self.loc.
scale (Tensor): The scale the distribution. Default: self.scale.
.. math::
cdf(x) = \frac{\arctan{(x - loc) / scale}}{\pi} + 0.5
"""
value = self._check_value(value, 'value')
value = self.cast(value, self.dtype)
loc, scale = self._check_param_type(loc, scale)
z = (value - loc) / scale
return self.atan(z) / np.pi + 0.5
def _log_cdf(self, value, loc=None, scale=None):
r"""
Evaluate the log cumulative distribution function on the given value.
Args:
value (Tensor): The value to be evaluated.
loc (Tensor): The location of the distribution. Default: self.loc.
scale (Tensor): The scale the distribution. Default: self.scale.
.. math::
log_cdf(x) = \log(\frac{\arctan(\frac{x-loc}{scale})}{\pi} + 0.5)
= \log {\arctan(\frac{x-loc}{scale}) + 0.5pi}{pi}
= \log1p \frac{2 * arctan(\frac{x-loc}{scale})}{pi} - \log2
"""
value = self._check_value(value, 'value')
value = self.cast(value, self.dtype)
loc, scale = self._check_param_type(loc, scale)
z = (value - loc) / scale
return self.log1p(2. * self.atan(z) / np.pi) - self.log(self.const(2., mstype.float32))
def _quantile(self, p, loc=None, scale=None):
loc, scale = self._check_param_type(loc, scale)
return loc + scale * self.tan(np.pi * (p - 0.5))
def _kl_loss(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
r"""
Evaluate Cauchy-Cauchy kl divergence, i.e. KL(a||b).
Args:
dist (str): The type of the distributions. Should be "Cauchy" in this case.
loc_b (Tensor): The loc of distribution b.
scale_b (Tensor): The scale of distribution b.
loc (Tensor): The loc of distribution a. Default: self.loc.
scale (Tensor): The scale of distribution a. Default: self.scale.
.. math::
KL(a||b) = \log(\frac{(scale_a + scale_b)^{2} + (loc_a - loc_b)^{2}}
{4 * scale_a * scale_b})
"""
check_distribution_name(dist, 'Cauchy')
loc_a, scale_a = self._check_param_type(loc_a, scale_a)
loc_b = self._check_value(loc_b, 'loc_b')
loc_b = self.cast(loc_b, self.parameter_type)
scale_b = self._check_value(scale_b, 'scale_b')
scale_b = self.cast(scale_b, self.parameter_type)
sum_square = self.sq(scale_a + scale_b)
square_diff = self.sq(loc_a - loc_b)
return self.log(sum_square + square_diff) - \
self.log(self.const(4.0, mstype.float32)) - self.log(scale_a) - self.log(scale_b)
def _cross_entropy(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
r"""
Evaluate cross entropy between Cauchy distributions.
Args:
dist (str): The type of the distributions. Should be "Cauchy" in this case.
loc_b (Tensor): The loc of distribution b.
scale_b (Tensor): The scale of distribution b.
loc (Tensor): The loc of distribution a. Default: self.loc.
scale (Tensor): The scale of distribution a. Default: self.scale.
"""
check_distribution_name(dist, 'Cauchy')
return self._entropy(loc_a, scale_a) + self._kl_loss(dist, loc_b, scale_b, loc_a, scale_a)
def _sample(self, shape=(), loc=None, scale=None):
"""
Sampling.
Args:
shape (tuple): The shape of the sample. Default: ().
loc (Tensor): The location of the samples. Default: self.loc.
scale (Tensor): The scale of the samples. Default: self.scale.
Returns:
Tensor, with the shape being shape + batch_shape.
"""
shape = self.checktuple(shape, 'shape')
loc, scale = self._check_param_type(loc, scale)
batch_shape = self.shape(loc + scale)
origin_shape = shape + batch_shape
if origin_shape == ():
sample_shape = (1,)
else:
sample_shape = origin_shape
l_zero = self.const(0.0, mstype.float32)
h_one = self.const(1.0, mstype.float32)
sample_uniform = self.uniform(sample_shape, l_zero, h_one, self.seed)
sample = self._quantile(sample_uniform, loc, scale)
value = self.cast(sample, self.dtype)
if origin_shape == ():
value = self.squeeze(value)
return value