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shifted_gompertz_cdf.py
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shifted_gompertz_cdf.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.
# ============================================================================
"""Shifted Gompertz CDF bijector."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
import numpy as np
import tensorflow.compat.v2 as tf
from tensorflow_probability.python import math as tfp_math
from tensorflow_probability.python.bijectors import bijector
from tensorflow_probability.python.bijectors import softplus as softplus_bijector
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 tensor_util
__all__ = [
'ShiftedGompertzCDF',
]
class ShiftedGompertzCDF(bijector.AutoCompositeTensorBijector):
"""Compute `Y = g(X) = (1 - exp(-rate * X)) * exp(-c * exp(-rate * X))`.
This bijector maps inputs from `[-inf, inf]` to `[0, inf]`. The inverse of the
bijector applied to a uniform random variable `X ~ U(0, 1)` gives back a
random variable with the
[Shifted Gompertz distribution](
https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution):
```none
Y ~ ShiftedGompertzCDF(concentration, rate)
pdf(y; c, r) = r * exp(-r * y - exp(-r * y) / c) * (1 + (1 - exp(-r * y)) / c)
```
Note: Even though this is called `ShiftedGompertzCDF`, when applied to the
`Uniform` distribution, this is not the same as applying a `GompertzCDF` with
a `Shift` bijector (i.e. the Shifted Gompertz distribution is not the same as
a Gompertz distribution with a location parameter).
Note: Because the Shifted Gompertz distribution concentrates its mass close
to zero, for larger rates or larger concentrations, `bijector.forward` will
quickly saturate to 1.
"""
def __init__(self,
concentration,
rate,
validate_args=False,
name='shifted_gompertz_cdf'):
"""Instantiates the `ShiftedGompertzCDF` bijector.
Args:
concentration: Positive Float-like `Tensor` that is the same dtype and is
broadcastable with `concentration`.
This is `c` in
`Y = g(X) = (1 - exp(-rate * X)) * exp(-exp(-rate * X) / c)`.
rate: Positive Float-like `Tensor` that is the same dtype and is
broadcastable with `concentration`.
This is `rate` in
`Y = g(X) = (1 - exp(-rate * X)) * exp(-exp(-rate * X) / c)`.
validate_args: Python `bool` indicating whether arguments should be
checked for correctness.
name: Python `str` name given to ops managed by this object.
"""
parameters = dict(locals())
with tf.name_scope(name) as name:
dtype = dtype_util.common_dtype(
[concentration, rate], dtype_hint=tf.float32)
self._concentration = tensor_util.convert_nonref_to_tensor(
concentration, dtype=dtype, name='concentration')
self._rate = tensor_util.convert_nonref_to_tensor(
rate, dtype=dtype, name='rate')
super(ShiftedGompertzCDF, self).__init__(
validate_args=validate_args,
forward_min_event_ndims=0,
parameters=parameters,
name=name)
@classmethod
def _parameter_properties(cls, dtype):
return dict(
concentration=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))),
rate=parameter_properties.ParameterProperties(
default_constraining_bijector_fn=(
lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
@property
def concentration(self):
"""The `c` in `Y = g(X) = (1 - exp(-r * X)) * exp(-exp(-r * X) / c)`."""
return self._concentration
@property
def rate(self):
"""The `r` in `Y = g(X) = (1 - exp(-r * X)) * exp(-exp(-r * X) / c)`."""
return self._rate
@classmethod
def _is_increasing(cls):
return True
def _forward(self, x):
with tf.control_dependencies(self._maybe_assert_valid_x(x)):
rate = tf.convert_to_tensor(self.rate)
log1mexpx = tfp_math.log1mexp(-rate * x)
return tf.math.exp(
log1mexpx - tf.math.exp(-rate * x) / self.concentration)
def _inverse(self, y):
with tf.control_dependencies(self._maybe_assert_valid_y(y)):
concentration = tf.convert_to_tensor(self.concentration)
reciprocal_concentration = tf.math.reciprocal(concentration)
z = -tfp_math.lambertw(
reciprocal_concentration * tf.math.exp(
reciprocal_concentration + tf.math.log(y))) * concentration
# Due to numerical instability, when y approaches 1, this expression
# can be less than -1. We clip the value to prevent that.
z = tf.clip_by_value(z, -1., np.inf)
return -tf.math.log1p(z) / self.rate
def _forward_log_det_jacobian(self, x):
with tf.control_dependencies(self._maybe_assert_valid_x(x)):
rate = tf.convert_to_tensor(self.rate)
concentration = tf.convert_to_tensor(self.concentration)
z = rate * x
return (-z - tf.math.exp(-z) / concentration + tf.math.log1p(
-tf.math.expm1(-z) / concentration) + tf.math.log(rate))
def _maybe_assert_valid_x(self, x):
if not self.validate_args:
return []
return [assert_util.assert_non_negative(
x, message='Forward transformation input must be non-negative.')]
def _maybe_assert_valid_y(self, y):
if not self.validate_args:
return []
is_positive = assert_util.assert_non_negative(
y, message='Inverse transformation input must be greater than 0.')
less_than_one = assert_util.assert_less_equal(
y,
tf.constant(1., y.dtype),
message='Inverse transformation input must be less than or equal to 1.')
return [is_positive, less_than_one]
def _parameter_control_dependencies(self, is_init):
if not self.validate_args:
return []
assertions = []
if is_init != tensor_util.is_ref(self.rate):
assertions.append(assert_util.assert_positive(
self.rate,
message='Argument `rate` must be positive.'))
if is_init != tensor_util.is_ref(self.concentration):
assertions.append(assert_util.assert_positive(
self.concentration,
message='Argument `concentration` must be positive.'))
return assertions