# Copyright 2018 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. 
# ============================================================================
"""Rational Quadratic kernel.""" 

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import functools

import tensorflow.compat.v2 as tf

from tensorflow_probability.python.internal import assert_util
from tensorflow_probability.python.internal import tensor_util
from tensorflow_probability.python.math.psd_kernels.internal import util
from tensorflow_probability.python.math.psd_kernels.positive_semidefinite_kernel import PositiveSemidefiniteKernel

__all__ = ['RationalQuadratic']


class RationalQuadratic(PositiveSemidefiniteKernel):
  """RationalQuadratic Kernel. 

  This kernel function has the form:

  ```none
  k(x, y) = amplitude**2 * (1.  + ||x - y|| ** 2 / (
    2 * scale_mixture_rate * length_scale**2)) ** -scale_mixture_rate
  ```

  where the double-bars represent vector length (i.e. Euclidean, or L2 Norm). 
  This kernel acts over the space `S = R^(D1 x D2 .. Dd)`.  When
  `scale_mixture_rate` tends towards infinity, this kernel acts like an
  ExponentiatedQuadratic kernel. 

  The name `scale_mixture_rate` comes from the interpretation that that this
  kernel is an Inverse-Gamma weighted mixture of ExponentiatedQuadratic Kernels
  with different length scales. 

  More formally, if `r = ||x - y|||`, then:

  ```none
  integral from 0 to infinity (k_EQ(r | sqrt(t)) p(t | a, a * l ** 2)) dt =
    (1 + r**2 / (2 * a * l ** 2)) ** -a = k(r)
  ```

  where:
    * `a = scale_mixture_rate`
    * `l = length_scale`
    * `p(t | a, b)` is the Inverse-Gamma density. 
      https://en.wikipedia.org/wiki/Inverse-gamma_distribution
    * `k_EQ(r | l) = exp(-r ** 2 / (2 * l ** 2)` is the ExponentiatedQuadratic
      positive semidefinite kernel. 

  #### References

  [1]: Filip Tronarp, Toni Karvonen, and Simo Saarka.  Mixture
       representation of the Matern class with applications in state space
       approximations and Bayesian quadrature.  In _28th IEEE International
       Workshop on Machine Learning for Signal Processing_, 2018. 
       https://users.aalto.fi/~karvont2/pdf/TronarpKarvonenSarkka2018-MLSP.pdf
  """

  def __init__(
      self,
      amplitude=None,
      length_scale=None,
      scale_mixture_rate=None,
      feature_ndims=1,
      validate_args=False,
      name='RationalQuadratic'):
    """Construct a RationalQuadratic kernel instance. 

    Args:
      amplitude: Positive floating point `Tensor` that controls the maximum
        value of the kernel.  Must be broadcastable with `length_scale` and
        `scale_mixture_rate` and inputs to `apply` and `matrix` methods.  A
        value of `None` is treated like 1. 
        Default value: None
      length_scale: Positive floating point `Tensor` that controls how sharp or
        wide the kernel shape is.  This provides a characteristic "unit" of
        length against which `||x - y||` can be compared for scale.  Must be
        broadcastable with `amplitude`, `scale_mixture_rate`  and inputs to
        `apply` and `matrix` methods.  A value of `None` is treated like 1. 
        Default value: None
      scale_mixture_rate: Positive floating point `Tensor` that controls how the
        ExponentiatedQuadratic kernels are mixed.   Must be broadcastable with
        `amplitude`, `length_scale` and inputs to `apply` and `matrix` methods. 
        A value of `None` is treated like 1. 
        Default value: None
      feature_ndims: Python `int` number of rightmost dims to include in the
        squared difference norm in the exponential. 
      validate_args: If `True`, parameters are checked for validity despite
        possibly degrading runtime performance
      name: Python `str` name prefixed to Ops created by this class. 
    """
    parameters = dict(locals())
    with tf.name_scope(name) as name:
      dtype = util.maybe_get_common_dtype(
          [amplitude, scale_mixture_rate, length_scale])

      self._amplitude = tensor_util.convert_nonref_to_tensor(
          amplitude, name='amplitude', dtype=dtype)
      self._scale_mixture_rate = tensor_util.convert_nonref_to_tensor(
          scale_mixture_rate, name='scale_mixture_rate', dtype=dtype)
      self._length_scale = tensor_util.convert_nonref_to_tensor(
          length_scale, name='length_scale', dtype=dtype)

      super(RationalQuadratic, self).__init__(
          feature_ndims,
          dtype=dtype,
          name=name,
          validate_args=validate_args,
          parameters=parameters)

  def _apply_with_distance(
      self, x1, x2, pairwise_square_distance, example_ndims=0):

    pairwise_square_distance /= 2

    if self.length_scale is not None:
      length_scale = tf.convert_to_tensor(self.length_scale)
      length_scale = util.pad_shape_with_ones(
          length_scale, ndims=example_ndims)
      pairwise_square_distance /= length_scale ** 2

    if self.scale_mixture_rate is None:
      power = 1. 
    else:
      scale_mixture_rate = tf.convert_to_tensor(self.scale_mixture_rate)
      power = util.pad_shape_with_ones(
          scale_mixture_rate, ndims=example_ndims)
      pairwise_square_distance /= power

    log_result = -power * tf.math.log1p(pairwise_square_distance)

    if self.amplitude is not None:
      amplitude = tf.convert_to_tensor(self.amplitude)
      amplitude = util.pad_shape_with_ones(amplitude, ndims=example_ndims)
      log_result = log_result + 2 * tf.math.log(amplitude)
    return tf.math.exp(log_result)

  def _apply(self, x1, x2, example_ndims=0):
    pairwise_square_distance = util.sum_rightmost_ndims_preserving_shape(
        tf.math.squared_difference(x1, x2), ndims=self.feature_ndims)
    return self._apply_with_distance(
        x1, x2, pairwise_square_distance, example_ndims=example_ndims)

  def _matrix(self, x1, x2):
    pairwise_square_distance = util.pairwise_square_distance_matrix(
        x1, x2, self.feature_ndims)
    return self._apply_with_distance(
        x1, x2, pairwise_square_distance, example_ndims=2)

  def _tensor(self, x1, x2, x1_example_ndims, x2_example_ndims):
    pairwise_square_distance = util.pairwise_square_distance_tensor(
        x1, x2, self.feature_ndims, x1_example_ndims, x2_example_ndims)
    return self._apply_with_distance(
        x1, x2, pairwise_square_distance,
        example_ndims=(x1_example_ndims + x2_example_ndims))

  @property
  def amplitude(self):
    """Amplitude parameter.""" 
    return self._amplitude

  @property
  def length_scale(self):
    """Length scale parameter.""" 
    return self._length_scale

  @property
  def scale_mixture_rate(self):
    """scale_mixture_rate parameter.""" 
    return self._scale_mixture_rate

  def _batch_shape(self):
    shape_list = [
        x.shape for x in [  # pylint: disable=g-complex-comprehension
            self.amplitude,
            self.scale_mixture_rate,
            self.length_scale
        ] if x is not None
    ]
    if not shape_list:
      return tf. TensorShape([])
    return functools.reduce(tf.broadcast_static_shape, shape_list)

  def _batch_shape_tensor(self):
    shape_list = [
        tf.shape(x)
        for x in [self.amplitude, self.scale_mixture_rate, self.length_scale]
        if x is not None
    ]
    if not shape_list:
      return tf.constant([], dtype=tf.int32)
    return functools.reduce(tf.broadcast_dynamic_shape, shape_list)

  def _parameter_control_dependencies(self, is_init):
    if not self.validate_args:
      return []
    assertions = []
    for arg_name, arg in dict(
        amplitude=self.amplitude,
        length_scale=self.length_scale,
        scale_mixture_rate=self.scale_mixture_rate).items():
      if arg is not None and is_init ! = tensor_util.is_ref(arg):
        assertions.append(assert_util.assert_positive(
            arg,
            message='{} must be positive. '.format(arg_name)))
    return assertions
