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"""Normal Distribution"""
import numpy as np
from mindspore import context
from mindspore.ops import operations as P
from mindspore.ops import composite as C
from mindspore._checkparam import Validator
from mindspore.common import dtype as mstype
from .distribution import Distribution
from ._utils.utils import check_greater_zero, check_distribution_name
from ._utils.custom_ops import exp_generic, log_generic
[docs]class Normal(Distribution):
r"""
Normal distribution.
A Normal distributio is a continuous distribution with the range :math:`(-\inf, \inf)`
and the probability density function:
.. math::
f(x, \mu, \sigma) = 1 / \sigma\sqrt{2\pi} \exp(-(x - \mu)^2 / 2\sigma^2).
where :math:`\mu, \sigma` are the mean and
the standard deviation of the normal distribution respectively.
Args:
mean (int, float, list, numpy.ndarray, Tensor): The mean of the Normal distribution. Default: None.
sd (int, float, list, numpy.ndarray, Tensor): The standard deviation of the Normal 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: 'Normal'.
Inputs and Outputs of APIs:
The accessible APIs of the Normal distribution are defined in the base class, including:
- `prob`, `log_prob`, `cdf`, `log_cdf`, `survival_function`, and `log_survival`
- `mean`, `sd`, `mode`, `var`, and `entropy`
- `kl_loss` and `cross_entropy`
- `sample`
For more details of all APIs, including the inputs and outputs of all APIs of the Normal distribution,
please refer to :class:`mindspore.nn.probability.distribution.Distribution`, and examples below.
Supported Platforms:
``Ascend`` ``GPU``
Note:
`sd` must be greater than zero.
`dist_spec_args` are `mean` and `sd`.
`dtype` must be a float type because Normal distributions are continuous.
Raises:
ValueError: When sd <= 0.
TypeError: When the input `dtype` is not a subclass of float.
Examples:
>>> import mindspore
>>> import mindspore.nn as nn
>>> import mindspore.nn.probability.distribution as msd
>>> from mindspore import Tensor
>>> # To initialize a Normal distribution of the mean 3.0 and the standard deviation 4.0.
>>> n1 = msd.Normal(3.0, 4.0, dtype=mindspore.float32)
>>> # A Normal distribution can be initialized without arguments.
>>> # In this case, `mean` and `sd` must be passed in through arguments.
>>> n2 = msd.Normal(dtype=mindspore.float32)
>>> # Here are some tensors used below for testing
>>> value = Tensor([1.0, 2.0, 3.0], dtype=mindspore.float32)
>>> mean_a = Tensor([2.0], dtype=mindspore.float32)
>>> sd_a = Tensor([2.0, 2.0, 2.0], dtype=mindspore.float32)
>>> mean_b = Tensor([1.0], dtype=mindspore.float32)
>>> sd_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.
>>> # mean (Tensor): the mean of the distribution. Default: self._mean_value.
>>> # sd (Tensor): the standard deviation of the distribution. Default: self._sd_value.
>>> # Examples of `prob`.
>>> # Similar calls can be made to other probability functions
>>> # by replacing 'prob' by the name of the function
>>> ans = n1.prob(value)
>>> print(ans.shape)
(3,)
>>> # Evaluate with respect to the distribution b.
>>> ans = n1.prob(value, mean_b, sd_b)
>>> print(ans.shape)
(3,)
>>> # `mean` and `sd` must be passed in during function calls
>>> ans = n2.prob(value, mean_a, sd_a)
>>> print(ans.shape)
(3,)
>>> # Functions `mean`, `sd`, `var`, and `entropy` have the same arguments.
>>> # Args:
>>> # mean (Tensor): the mean of the distribution. Default: self._mean_value.
>>> # sd (Tensor): the standard deviation of the distribution. Default: self._sd_value.
>>> # Example of `mean`. `sd`, `var`, and `entropy` are similar.
>>> ans = n1.mean() # return 0.0
>>> print(ans.shape)
()
>>> ans = n1.mean(mean_b, sd_b) # return mean_b
>>> print(ans.shape)
(3,)
>>> # `mean` and `sd` must be passed in during function calls.
>>> ans = n2.mean(mean_a, sd_a)
>>> print(ans.shape)
(3,)
>>> # Interfaces of 'kl_loss' and 'cross_entropy' are the same:
>>> # Args:
>>> # dist (str): the type of the distributions. Only "Normal" is supported.
>>> # mean_b (Tensor): the mean of distribution b.
>>> # sd_b (Tensor): the standard deviation of distribution b.
>>> # mean_a (Tensor): the mean of distribution a. Default: self._mean_value.
>>> # sd_a (Tensor): the standard deviation of distribution a. Default: self._sd_value.
>>> # Examples of `kl_loss`. `cross_entropy` is similar.
>>> ans = n1.kl_loss('Normal', mean_b, sd_b)
>>> print(ans.shape)
(3,)
>>> ans = n1.kl_loss('Normal', mean_b, sd_b, mean_a, sd_a)
>>> print(ans.shape)
(3,)
>>> # Additional `mean` and `sd` must be passed in.
>>> ans = n2.kl_loss('Normal', mean_b, sd_b, mean_a, sd_a)
>>> print(ans.shape)
(3,)
>>> # Examples of `sample`.
>>> # Args:
>>> # shape (tuple): the shape of the sample. Default: ()
>>> # mean (Tensor): the mean of the distribution. Default: self._mean_value.
>>> # sd (Tensor): the standard deviation of the distribution. Default: self._sd_value.
>>> ans = n1.sample()
>>> print(ans.shape)
()
>>> ans = n1.sample((2,3))
>>> print(ans.shape)
(2, 3)
>>> ans = n1.sample((2,3), mean_b, sd_b)
>>> print(ans.shape)
(2, 3, 3)
>>> ans = n2.sample((2,3), mean_a, sd_a)
>>> print(ans.shape)
(2, 3, 3)
"""
def __init__(self,
mean=None,
sd=None,
seed=None,
dtype=mstype.float32,
name="Normal"):
"""
Constructor of Normal.
"""
param = dict(locals())
param['param_dict'] = {'mean': mean, 'sd': sd}
valid_dtype = mstype.float_type
Validator.check_type_name(
"dtype", dtype, valid_dtype, type(self).__name__)
super(Normal, self).__init__(seed, dtype, name, param)
self._mean_value = self._add_parameter(mean, 'mean')
self._sd_value = self._add_parameter(sd, 'sd')
if self._sd_value is not None:
check_greater_zero(self._sd_value, "Standard deviation")
# ops needed for the class
self.exp = exp_generic
self.expm1 = P.Expm1()
# when the graph kernel mode is enable
# use Log directly as akg will handle the corner cases
self.log = P.Log() if context.get_context(
"enable_graph_kernel") else log_generic
self.erf = P.Erf()
self.squeeze = P.Squeeze(0)
self.cast = P.Cast()
self.const = P.ScalarToArray()
self.shape = P.Shape()
self.sq = P.Square()
self.sqrt = P.Sqrt()
[docs] def extend_repr(self):
"""Display instance object as string."""
if self.is_scalar_batch:
s = 'mean = {}, standard deviation = {}'.format(
self._mean_value, self._sd_value)
else:
s = 'batch_shape = {}'.format(self._broadcast_shape)
return s
def _get_dist_type(self):
return "Normal"
def _get_dist_args(self, mean=None, sd=None):
if mean is not None:
self.checktensor(mean, 'mean')
else:
mean = self._mean_value
if sd is not None:
self.checktensor(sd, 'sd')
else:
sd = self._sd_value
return mean, sd
def _mean(self, mean=None, sd=None):
"""
The mean of the distribution.
"""
mean, sd = self._check_param_type(mean, sd)
return mean
def _mode(self, mean=None, sd=None):
"""
The mode of the distribution.
"""
mean, sd = self._check_param_type(mean, sd)
return mean
def _sd(self, mean=None, sd=None):
"""
The standard deviation of the distribution.
"""
mean, sd = self._check_param_type(mean, sd)
return sd
def _entropy(self, mean=None, sd=None):
r"""
Evaluate entropy.
.. math::
H(X) = \log(\sqrt(numpy.e * 2. * numpy.pi * \sq(\sigma)))
"""
mean, sd = self._check_param_type(mean, sd)
return self.log(self.sqrt(self.const(np.e * 2. * np.pi))) + self.log(sd)
def _cross_entropy(self, dist, mean_b, sd_b, mean=None, sd=None):
r"""
Evaluate cross entropy between normal distributions.
Args:
dist (str): Type of the distributions. Should be "Normal" in this case.
mean_b (Tensor): Mean of distribution b.
sd_b (Tensor): Standard deviation distribution b.
mean_a (Tensor): Mean of distribution a. Default: self._mean_value.
sd_a (Tensor): Standard deviation distribution a. Default: self._sd_value.
"""
check_distribution_name(dist, 'Normal')
return self._entropy(mean, sd) + self._kl_loss(dist, mean_b, sd_b, mean, sd)
def _log_prob(self, value, mean=None, sd=None):
r"""
Evaluate log probability.
Args:
value (Tensor): The value to be evaluated.
mean (Tensor): The mean of the distribution. Default: self._mean_value.
sd (Tensor): The standard deviation the distribution. Default: self._sd_value.
.. math::
L(x) = -1* \frac{(x - \mu)^2}{2. * \sigma^2} - \log(\sqrt(2* \pi * \sigma^2))
"""
value = self._check_value(value, 'value')
value = self.cast(value, self.dtype)
mean, sd = self._check_param_type(mean, sd)
unnormalized_log_prob = -1. * \
(self.sq(value - mean)) / (2. * self.sq(sd))
neg_normalization = -1. * \
self.log(self.const(2. * np.pi)) / 2. - self.log(sd)
return unnormalized_log_prob + neg_normalization
def _cdf(self, value, mean=None, sd=None):
r"""
Evaluate the cumulative distribution function on the given value.
Args:
value (Tensor): The value to be evaluated.
mean (Tensor): The mean of the distribution. Default: self._mean_value.
sd (Tensor): The standard deviation the distribution. Default: self._sd_value.
.. math::
cdf(x) = 0.5 * (1+ Erf((x - \mu) / ( \sigma * \sqrt(2))))
"""
value = self._check_value(value, 'value')
value = self.cast(value, self.dtype)
mean, sd = self._check_param_type(mean, sd)
sqrt2 = self.sqrt(self.const(2.0))
adjusted = (value - mean) / (sd * sqrt2)
return 0.5 * (1.0 + self.erf(adjusted))
def _kl_loss(self, dist, mean_b, sd_b, mean=None, sd=None):
r"""
Evaluate Normal-Normal KL divergence, i.e. KL(a||b).
Args:
dist (str): The type of the distributions. Should be "Normal" in this case.
mean_b (Tensor): The mean of distribution b.
sd_b (Tensor): The standard deviation distribution b.
mean_a (Tensor): The mean of distribution a. Default: self._mean_value.
sd_a (Tensor): The standard deviation distribution a. Default: self._sd_value.
.. math::
KL(a||b) = 0.5 * (\frac{MEAN(a)}{STD(b)} - \frac{MEAN(b)}{STD(b)}) ^ 2 +
0.5 * EXPM1(2 * (\log(STD(a)) - \log(STD(b))) - (\log(STD(a)) - \log(STD(b)))
"""
check_distribution_name(dist, 'Normal')
mean_b = self._check_value(mean_b, 'mean_b')
sd_b = self._check_value(sd_b, 'sd_b')
mean_b = self.cast(mean_b, self.parameter_type)
sd_b = self.cast(sd_b, self.parameter_type)
mean_a, sd_a = self._check_param_type(mean, sd)
diff_log_scale = self.log(sd_a) - self.log(sd_b)
squared_diff = self.sq(mean_a / sd_b - mean_b / sd_b)
return 0.5 * squared_diff + 0.5 * self.expm1(2 * diff_log_scale) - diff_log_scale
def _sample(self, shape=(), mean=None, sd=None):
"""
Sampling.
Args:
shape (tuple): The shape of the sample. Default: ().
mean (Tensor): The mean of the samples. Default: self._mean_value.
sd (Tensor): The standard deviation of the samples. Default: self._sd_value.
Returns:
Tensor, with the shape being shape + batch_shape.
"""
shape = self.checktuple(shape, 'shape')
mean, sd = self._check_param_type(mean, sd)
batch_shape = self.shape(mean + sd)
origin_shape = shape + batch_shape
if origin_shape == ():
sample_shape = (1,)
else:
sample_shape = origin_shape
sample_norm = C.normal(sample_shape, mean, sd, self.seed)
value = self.cast(sample_norm, self.dtype)
if origin_shape == ():
value = self.squeeze(value)
return value