/
generalized_normal.py
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/
generalized_normal.py
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# Copyright 2020 The TensorFlow Probability Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ============================================================================
"""The Generalized Normal (Generalized Gaussian) distribution class."""
# Dependency imports
import numpy as np
import tensorflow.compat.v2 as tf
from tensorflow_probability.python import math as tfp_math
from tensorflow_probability.python import random as tfp_random
from tensorflow_probability.python.bijectors import identity as identity_bijector
from tensorflow_probability.python.bijectors import softplus as softplus_bijector
from tensorflow_probability.python.distributions import distribution
from tensorflow_probability.python.distributions import gamma
from tensorflow_probability.python.internal import assert_util
from tensorflow_probability.python.internal import dtype_util
from tensorflow_probability.python.internal import parameter_properties
from tensorflow_probability.python.internal import prefer_static as ps
from tensorflow_probability.python.internal import reparameterization
from tensorflow_probability.python.internal import samplers
from tensorflow_probability.python.internal import tensor_util
__all__ = [
'GeneralizedNormal',
]
class GeneralizedNormal(distribution.AutoCompositeTensorDistribution):
"""The Generalized Normal distribution.
The Generalized Normal (or Generalized Gaussian) generalizes the Normal
distribution with an additional shape parameter. It is parameterized by
location `loc`, scale `scale` and shape `power`.
#### Mathematical details
The probability density function (pdf) is,
```none
pdf(x; loc, scale, power) = 1 / (2 * scale * Gamma(1 + 1 / power)) *
exp(-(|x - loc| / scale) ^ power)
```
where `loc` is the mean, `scale` is the scale, and, `power` is the shape
parameter. If the power is above two, the distribution becomes platykurtic.
A power equal to two results in a Normal distribution. A power smaller than
two produces a leptokurtic (heavy-tailed) distribution. Mean and scale behave
the same way as in the equivalent Normal distribution.
See
https://en.wikipedia.org/w/index.php?title=Generalized_normal_distribution&oldid=954254464
for the definitions used here, including CDF, variance and entropy. See
https://sccn.ucsd.edu/wiki/Generalized_Gaussian_Probability_Density_Function
for the sampling method used here.
#### Examples
```python
import tensorflow_probability as tfp
tfd = tfp.distributions
dist = tfd.GeneralizedNormal(loc=3.0, scale=2.0, power=1.0)
dist2 = tfd.GeneralizedNormal(loc=0, scale=[3.0, 4.0], power=[[2.0], [3.0]])
```
"""
def __init__(self,
loc,
scale,
power,
validate_args=False,
allow_nan_stats=True,
name='GeneralizedNormal'):
"""Construct Generalized Normal distributions.
The Generalized Normal is parametrized with mean `loc`, scale
`scale` and shape parameter `power`. The parameters must be shaped
in a way that supports broadcasting (e.g. `loc + scale` is a valid
operation).
Args:
loc: Floating point tensor; the means of the distribution(s).
scale: Floating point tensor; the scale of the
distribution(s). Must contain only positive values.
power: Floating point tensor; the shape parameter of the distribution(s).
Must contain only positive values. `loc`, `scale` and `power` must have
compatible shapes for broadcasting.
validate_args: Python `bool`, default `False`. When `True` distribution
parameters are checked for validity despite possibly degrading runtime
performance. When `False` invalid inputs may silently render incorrect
outputs.
allow_nan_stats: Python `bool`, default `True`. When `True`,
statistics (e.g., mean, mode, variance) use the value "`NaN`" to
indicate the result is undefined. When `False`, an exception is raised
if one or more of the statistic's batch members are undefined.
name: Python `str` name prefixed to Ops created by this class.
Raises:
TypeError: if `loc`, `scale`, and `power` have different `dtype`.
"""
parameters = dict(locals())
with tf.name_scope(name) as name:
dtype = dtype_util.common_dtype([loc, scale, power],
dtype_hint=tf.float32)
self._loc = tensor_util.convert_nonref_to_tensor(
loc, dtype=dtype, name='loc')
self._scale = tensor_util.convert_nonref_to_tensor(
scale, dtype=dtype, name='scale')
self._power = tensor_util.convert_nonref_to_tensor(
power, dtype=dtype, name='power')
super(GeneralizedNormal, self).__init__(
dtype=dtype,
reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
name=name)
@classmethod
def _parameter_properties(cls, dtype, num_classes=None):
# pylint: disable=g-long-lambda
return dict(
loc=parameter_properties.ParameterProperties(),
scale=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))),
power=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
# pylint: enable=g-long-lambda
@property
def loc(self):
"""Distribution parameter for the mean."""
return self._loc
@property
def scale(self):
"""Distribution parameter for scale."""
return self._scale
@property
def power(self):
"""Distribution parameter for shape."""
return self._power
def _event_shape_tensor(self):
return tf.constant([], dtype=tf.int32)
def _event_shape(self):
return tf.TensorShape([])
def _sample_n(self, n, seed=None, name=None):
n = ps.convert_to_shape_tensor(n, name='num', dtype=tf.int32)
loc = tf.convert_to_tensor(self.loc)
scale = tf.convert_to_tensor(self.scale)
power = tf.convert_to_tensor(self.power)
batch_shape = self._batch_shape_tensor(loc=loc, scale=scale, power=power)
result_shape = ps.concat([[n], batch_shape], axis=0)
ipower = tf.broadcast_to(tf.math.reciprocal(power), batch_shape)
gamma_dist = gamma.Gamma(ipower, 1.)
rademacher_seed, gamma_seed = samplers.split_seed(seed, salt='GenNormal')
gamma_sample = gamma_dist.sample(n, seed=gamma_seed)
binary_sample = tfp_random.rademacher(result_shape, dtype=self.dtype,
seed=rademacher_seed)
sampled = (binary_sample * tf.math.pow(tf.abs(gamma_sample), ipower))
return loc + scale * sampled
def _log_prob(self, x):
loc = tf.convert_to_tensor(self.loc)
scale = tf.convert_to_tensor(self.scale)
power = tf.convert_to_tensor(self.power)
one = tf.constant(1., dtype=self.dtype)
two = tf.constant(2., dtype=self.dtype)
log_normalization = (tf.math.log(two) + tf.math.log(scale) +
tf.math.lgamma(one + tf.math.reciprocal(power)))
log_unnormalized = -tf.pow(tf.abs(x - loc) / scale, power)
return log_unnormalized - log_normalization
def _cdf_zero_mean(self, x):
scale = tf.convert_to_tensor(self.scale)
power = tf.convert_to_tensor(self.power)
zero = tf.constant(0., dtype=self.dtype)
half = tf.constant(0.5, dtype=self.dtype)
one = tf.constant(1., dtype=self.dtype)
# Double tf.where to avoid incorrect gradient at x == 0.
x_is_zero = tf.equal(x, zero)
safe_x = tf.where(x_is_zero, one, x)
half_gamma = half * tf.math.igammac(
tf.math.reciprocal(power),
tf.pow(tf.abs(safe_x) / scale, power))
return tf.where(
x_is_zero,
half,
tf.where(x > zero, one - half_gamma, half_gamma),
)
def _cdf(self, x):
loc = tf.convert_to_tensor(self.loc)
return self._cdf_zero_mean(x - loc)
def _survival_function(self, x):
loc = tf.convert_to_tensor(self.loc)
# sf(x) = cdf(-x) for loc == 0, because distribution is symmetric.
return self._cdf_zero_mean(loc - x)
def _quantile(self, p):
loc = tf.convert_to_tensor(self.loc)
scale = tf.convert_to_tensor(self.scale)
power = tf.convert_to_tensor(self.power)
ipower = tf.math.reciprocal(power)
quantile = tf.where(
p < 0.5,
loc - tf.math.pow(
tfp_math.igammacinv(ipower, 2. * p), ipower) * scale,
loc + tf.math.pow(
tfp_math.igammainv(ipower, 2. * p - 1.), ipower) * scale)
return quantile
def _entropy(self):
scale = tf.convert_to_tensor(self.scale)
power = tf.convert_to_tensor(self.power)
ipower = tf.math.reciprocal(power)
one = tf.constant(1., dtype=self.dtype)
logtwo = tf.constant(np.log(2.), dtype=self.dtype)
entropy = ipower + (logtwo + tf.math.log(scale) +
tf.math.lgamma(one + ipower))
return tf.broadcast_to(entropy,
self._batch_shape_tensor(scale=scale, power=power))
def _mean(self):
loc = tf.convert_to_tensor(self.loc)
return tf.broadcast_to(loc, self._batch_shape_tensor(loc=loc))
def _variance(self):
ipower = tf.math.reciprocal(tf.convert_to_tensor(self.power))
two = tf.constant(2., dtype=self.dtype)
three = tf.constant(3., dtype=self.dtype)
log_var = (two * tf.math.log(self.scale) +
tf.math.lgamma(three * ipower) - tf.math.lgamma(ipower))
var = tf.math.exp(log_var)
return tf.broadcast_to(var, self._batch_shape_tensor())
_mode = _mean
def _default_event_space_bijector(self):
return identity_bijector.Identity(validate_args=self.validate_args)
def _parameter_control_dependencies(self, is_init):
assertions = []
if is_init:
# _batch_shape() will raise error if it can statically prove that `loc`,
# `scale`, and `power` have incompatible shapes.
try:
self._batch_shape()
except ValueError:
raise ValueError(
'Arguments `loc`, `scale` and `power` must have compatible shapes; '
'loc.shape={}, scale.shape={}, power.shape={}.'.format(
self.loc.shape, self.scale.shape, self.power.shape))
# We don't bother checking the shapes in the dynamic case because
# all member functions access the three arguments anyway.
if not self.validate_args:
assert not assertions # Should never happen.
return []
if is_init != tensor_util.is_ref(self.scale):
assertions.append(assert_util.assert_positive(
self.scale, message='Argument `scale` must be positive.'))
if is_init != tensor_util.is_ref(self.power):
assertions.append(assert_util.assert_positive(
self.power, message='Argument `power` must be positive.'))
return assertions