gev_cdf.py

# 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.
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# ============================================================================
"""GeneralizedExtremeValue bijector."""

import tensorflow.compat.v2 as tf

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__ = [
    'GeneralizedExtremeValueCDF',
]


class GeneralizedExtremeValueCDF(
    bijector.CoordinatewiseBijectorMixin,
    bijector.AutoCompositeTensorBijector):
  """Compute the GeneralizedExtremeValue CDF.

  Compute `Y = g(X) = exp(-t(X))`,

  where `t(x)` is defined to be:
    *`(1 + conc * (x - loc) / scale) ) ** (-1 / conc)` when `conc != 0`;
    *`exp(-(x - loc) / scale)` when `conc = 0`.

  This bijector maps inputs from the domain to `[0, 1]`, where the domain is
    * [loc - scale/conc, inf) when conc > 0;
    * (-inf, loc - scale/conc] when conc < 0;
    * (-inf, inf) when conc = 0;



  The inverse of the bijector applied to a uniform random variable
  `X ~ U(0, 1)` gives back a random variable with the
  [Generalized extreme value distribution](
  https://https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution):

  When `concentration -> +-inf`, the probability mass concentrates near `loc`.

  ```none
  Y ~ GeneralizedExtremeValueCDF(loc, scale, conc)
  pdf(y; loc, scale, conc) = t(y; loc, scale, conc) ** (1 + conc) * exp(
  - t(y; loc, scale, conc) ) / scale
  where t(x) =
    * (1 + conc * (x - loc) / scale) ) ** (-1 / conc) when conc != 0;
    * exp(-(x - loc) / scale) when conc = 0.
  ```
  """

  def __init__(self,
               loc=0.,
               scale=1.,
               concentration=0,
               validate_args=False,
               name='generalizedextremevalue_cdf'):
    """Instantiates the `GeneralizedExtremeValueCDF` bijector.

    Args:
      loc: Float-like `Tensor` that is the same dtype and is broadcastable with
        `scale` and `concentration`.
      scale: Positive Float-like `Tensor` that is the same dtype and is
        broadcastable with `loc` and `concentration`.
      concentration: Nonzero float-like `Tensor` that is the same dtype and is
        broadcastable with `loc` and `scale`.
      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([loc, scale, concentration],
                                      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._concentration = tensor_util.convert_nonref_to_tensor(
          concentration, dtype=dtype, name='concentration')
      super(GeneralizedExtremeValueCDF, self).__init__(
          validate_args=validate_args,
          forward_min_event_ndims=0,
          parameters=parameters,
          name=name)

  @classmethod
  def _parameter_properties(cls, dtype):
    return dict(
        loc=parameter_properties.ParameterProperties(),
        scale=parameter_properties.ParameterProperties(
            default_constraining_bijector_fn=(
                lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))),
        concentration=parameter_properties.ParameterProperties(
            default_constraining_bijector_fn=(
                lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))

  @property
  def loc(self):
    """The location parameter in the Generalized Extreme Value CDF."""
    return self._loc

  @property
  def scale(self):
    """The scale parameter in the Generalized Extreme Value CDF."""
    return self._scale

  @property
  def concentration(self):
    """The concentration parameter in the Generalized Extreme Value CDF."""
    return self._concentration

  @classmethod
  def _is_increasing(cls):
    return True

  def _forward(self, x):
    loc = tf.convert_to_tensor(self.loc)
    scale = tf.convert_to_tensor(self.scale)
    conc = tf.convert_to_tensor(self.concentration)
    with tf.control_dependencies(
        self._maybe_assert_valid_x(
            x, loc=loc, scale=scale, concentration=conc)):
      z = (x - loc) / scale

      equal_zero = tf.equal(conc, 0.)
      # deal with case that gradient is N/A when conc = 0
      safe_conc = tf.where(equal_zero, tf.ones_like(conc), conc)
      t = tf.where(
          equal_zero, tf.math.exp(-z),
          tf.math.exp(-tf.math.log1p(z * safe_conc) / safe_conc))
      return tf.exp(-t)

  def _inverse(self, y):
    with tf.control_dependencies(self._maybe_assert_valid_y(y)):
      t = -tf.math.log(y)

      conc = tf.convert_to_tensor(self.concentration)

      equal_zero = tf.equal(conc, 0.)
      # deal with case that gradient is N/A when conc = 0
      safe_conc = tf.where(equal_zero, tf.ones_like(conc), conc)

      z = tf.where(
          equal_zero, -tf.math.log(t),
          tf.math.expm1(-tf.math.log(t) * safe_conc) / safe_conc)

      return self.loc + self.scale * z

  def _forward_log_det_jacobian(self, x):
    loc = tf.convert_to_tensor(self.loc)
    scale = tf.convert_to_tensor(self.scale)
    conc = tf.convert_to_tensor(self.concentration)
    with tf.control_dependencies(
        self._maybe_assert_valid_x(
            x, loc=loc, scale=scale, concentration=conc)):
      z = (x - loc) / scale

      equal_zero = tf.equal(conc, 0.)
      # deal with case that gradient is N/A when conc = 0
      safe_conc = tf.where(equal_zero, tf.ones_like(conc), conc)

      log_t = tf.where(
          equal_zero, -z,
          -tf.math.log1p(z * safe_conc) / safe_conc)
      return (tf.math.multiply_no_nan(conc + 1., log_t) -
              tf.math.exp(log_t) - tf.math.log(scale))

  def _inverse_log_det_jacobian(self, y):
    with tf.control_dependencies(self._maybe_assert_valid_y(y)):
      t = -tf.math.log(y)

      log_dt = tf.math.xlogy(-self.concentration - 1., t)
      return tf.math.log(self.scale / y) + log_dt

  def _maybe_assert_valid_x(self, x, loc=None, scale=None, concentration=None):
    if not self.validate_args:
      return []
    loc = tf.convert_to_tensor(self.loc) if loc is None else loc
    scale = tf.convert_to_tensor(self.scale) if scale is None else scale
    concentration = (
        tf.convert_to_tensor(self.concentration) if concentration is None else
        concentration)
    # We intentionally compute the boundary with (1.0 / concentration) * scale
    # instead of just scale / concentration.
    # Why?  The sampler returns loc + (foo / concentration) * scale,
    # and at high-ish values of concentration, foo has a decent
    # probability of being numerically exactly -1.  We therefore mimic
    # the pattern of round-off that occurs in the sampler to make sure
    # that samples emitted from this distribution will pass its own
    # validations.  This is sometimes necessary: in TF's float32,
    #   0.69314826 / 37.50019 < (1.0 / 37.50019) * 0.69314826
    boundary = loc - (1.0 / concentration) * scale
    # The support of this bijector depends on the sign of concentration.
    is_in_bounds = tf.where(concentration > 0., x >= boundary, x <= boundary)
    # For concentration 0, the domain is the whole line.
    is_in_bounds = is_in_bounds | tf.math.equal(concentration, 0.)

    return [
        assert_util.assert_equal(
            is_in_bounds,
            True,
            message='Forward transformation input must be inside domain.')
    ]

  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.scale):
      assertions.append(
          assert_util.assert_positive(
              self.scale, message='Argument `scale` must be positive.'))
    return assertions