# Copyright 2020 Huawei Technologies Co., Ltd
#
# 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,
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# ============================================================================
"""Cauchy Distribution"""
import numpy as np
from mindspore.ops import operations as P
from mindspore.ops import composite as C
from mindspore import _checkparam as Validator
from mindspore.common import dtype as mstype
from .distribution import Distribution
from ._utils.utils import check_greater_zero, check_distribution_name, raise_not_defined
from ._utils.custom_ops import exp_generic, log_generic, log1p_generic


class Cauchy(Distribution):
    r"""
    Cauchy distribution.
    A Cauchy distributio is a continuous distribution with the range of all real numbers
    and the probability density function:

    .. math::
        f(x, a, b) = 1 / \pi b(1 - ((x - a)/b)^2),

    where :math:`a, b` are loc and scale parameter respectively.

    Args:
        loc (int, float, list, numpy.ndarray, Tensor): The location of the Cauchy distribution. Default: ``None`` .
        scale (int, float, list, numpy.ndarray, Tensor): The scale of the Cauchy 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: ``'Cauchy'`` .

    Note:
        `scale` must be greater than zero.
        `dist_spec_args` are `loc` and `scale`.
        `dtype` must be a float type because Cauchy distributions are continuous.
        Cauchy distribution is not supported on GPU backend.

    Raises:
        ValueError: When scale <= 0.
        TypeError: When the input `dtype` is not a subclass of float.

    Supported Platforms:
        ``Ascend``

    Examples:
        >>> import mindspore
        >>> import mindspore.nn as nn
        >>> import mindspore.nn.probability.distribution as msd
        >>> from mindspore import Tensor
        >>> # To initialize a Cauchy distribution of loc 3.0 and scale 4.0.
        >>> cauchy1 = msd.Cauchy(3.0, 4.0, dtype=mindspore.float32)
        >>> # A Cauchy distribution can be initialized without arguments.
        >>> # In this case, 'loc' and `scale` must be passed in through arguments.
        >>> cauchy2 = msd.Cauchy(dtype=mindspore.float32)
        >>> # Here are some tensors used below for testing
        >>> value = Tensor([1.0, 2.0, 3.0], dtype=mindspore.float32)
        >>> loc_a = Tensor([2.0], dtype=mindspore.float32)
        >>> scale_a = Tensor([2.0, 2.0, 2.0], dtype=mindspore.float32)
        >>> loc_b = Tensor([1.0], dtype=mindspore.float32)
        >>> scale_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.
        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
        >>> # Examples of `prob`.
        >>> # Similar calls can be made to other probability functions
        >>> # by replacing 'prob' by the name of the function
        >>> ans = cauchy1.prob(value)
        >>> print(ans.shape)
        (3,)
        >>> # Evaluate with respect to distribution b.
        >>> ans = cauchy1.prob(value, loc_b, scale_b)
        >>> print(ans.shape)
        (3,)
        >>> # `loc` and `scale` must be passed in during function calls
        >>> ans = cauchy2.prob(value, loc_a, scale_a)
        >>> print(ans.shape)
        (3,)
        >>> # Functions `mode` and `entropy` have the same arguments.
        >>> # Args:
        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
        >>> # Example of `mode`.
        >>> ans = cauchy1.mode() # return 3.0
        >>> print(ans.shape)
        ()
        >>> ans = cauchy1.mode(loc_b, scale_b) # return loc_b
        >>> print(ans.shape)
        (3,)
        >>> # `loc` and `scale` must be passed in during function calls.
        >>> ans = cauchy2.mode(loc_a, scale_a)
        >>> print(ans.shape)
        (3,)
        >>> # Interfaces of 'kl_loss' and 'cross_entropy' are the same:
        >>> # Args:
        >>> #     dist (str): the type of the distributions. Only "Cauchy" is supported.
        >>> #     loc_b (Tensor): the loc of distribution b.
        >>> #     scale_b (Tensor): the scale distribution b.
        >>> #     loc (Tensor): the loc of distribution a. Default: self.loc.
        >>> #     scale (Tensor): the scale distribution a. Default: self.scale.
        >>> # Examples of `kl_loss`. `cross_entropy` is similar.
        >>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b)
        >>> print(ans.shape)
        (3,)
        >>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
        >>> print(ans.shape)
        (3,)
        >>> # Additional `loc` and `scale` must be passed in.
        >>> ans = cauchy2.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
        >>> print(ans.shape)
        (3,)
        >>> # Examples of `sample`.
        >>> # Args:
        >>> #     shape (tuple): the shape of the sample. Default: ()
        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
        >>> ans = cauchy1.sample()
        >>> print(ans.shape)
        ()
        >>> ans = cauchy1.sample((2,3))
        >>> print(ans.shape)
        (2, 3)
        >>> ans = cauchy1.sample((2,3), loc_b, scale_b)
        >>> print(ans.shape)
        (2, 3, 3)
        >>> ans = cauchy2.sample((2,3), loc_a, scale_a)
        >>> print(ans.shape)
        (2, 3, 3)
    """

    def __init__(self,
                 loc=None,
                 scale=None,
                 seed=None,
                 dtype=mstype.float32,
                 name="Cauchy"):
        """
        Constructor of Cauchy.
        """
        param = dict(locals())
        param['param_dict'] = {'loc': loc, 'scale': scale}
        valid_dtype = mstype.float_type
        Validator.check_type_name(
            "dtype", dtype, valid_dtype, type(self).__name__)
        super(Cauchy, self).__init__(seed, dtype, name, param)

        self._loc = self._add_parameter(loc, 'loc')
        self._scale = self._add_parameter(scale, 'scale')
        if self._scale is not None:
            check_greater_zero(self._scale, "scale")

        # ops needed for the class
        self.atan = P.Atan()
        self.cast = P.Cast()
        self.const = P.ScalarToTensor()
        self.dtypeop = P.DType()
        self.exp = exp_generic
        self.less = P.Less()
        self.log = log_generic
        self.log1p = log1p_generic
        self.squeeze = P.Squeeze(0)
        self.shape = P.Shape()
        self.sq = P.Square()
        self.sqrt = P.Sqrt()
        self.tan = P.Tan()
        self.uniform = C.uniform

        self.entropy_const = np.log(4 * np.pi)

    def extend_repr(self):
        """Display instance object as string."""
        if self.is_scalar_batch:
            str_info = 'location = {}, scale = {}'.format(
                self._loc, self._scale)
        else:
            str_info = 'batch_shape = {}'.format(self._broadcast_shape)
        return str_info

    @property
    def loc(self):
        """
        Return the location of the distribution after casting to dtype.

        Output:
            Tensor, the loc parameter of the distribution.
        """
        return self._loc

    @property
    def scale(self):
        """
        Return the scale of the distribution after casting to dtype.

        Output:
            Tensor, the scale parameter of the distribution.
        """
        return self._scale

    def _get_dist_type(self):
        return "Cauchy"

    def _get_dist_args(self, loc=None, scale=None):
        if scale is not None:
            self.checktensor(scale, 'scale')
        else:
            scale = self.scale
        if loc is not None:
            self.checktensor(loc, 'loc')
        else:
            loc = self.loc
        return loc, scale

    def _mode(self, loc=None, scale=None):
        """
        The mode of the distribution.
        """
        loc, scale = self._check_param_type(loc, scale)
        return loc

    def _mean(self, *args, **kwargs):
        return raise_not_defined('mean', 'Cauchy', *args, **kwargs)

    def _sd(self, *args, **kwargs):
        return raise_not_defined('standard deviation', 'Cauchy', *args, **kwargs)

    def _var(self, *args, **kwargs):
        return raise_not_defined('variance', 'Cauchy', *args, **kwargs)

    def _entropy(self, loc=None, scale=None):
        r"""
        Evaluate entropy.

        .. math::
            H(X) = \log(4 * \Pi * scale)
        """
        loc, scale = self._check_param_type(loc, scale)
        return self.log(scale) + self.entropy_const

    def _log_prob(self, value, loc=None, scale=None):
        r"""
        Evaluate log probability.

        Args:
            value (Tensor): The value to be evaluated.
            loc (Tensor): The location of the distribution. Default: self.loc.
            scale (Tensor): The scale of the distribution. Default: self.scale.

        .. math::
            L(x) = \log(\frac{1}{\pi * scale} * \frac{scale^{2}}{(x - loc)^{2} + scale^{2}})
        """
        value = self._check_value(value, 'value')
        value = self.cast(value, self.dtype)
        loc, scale = self._check_param_type(loc, scale)
        z = (value - loc) / scale
        log_unnormalized_prob = (-1) * self.log1p(self.sq(z))
        log_normalization = self.log(np.pi * scale)
        return log_unnormalized_prob - log_normalization

    def _cdf(self, value, loc=None, scale=None):
        r"""
        Evaluate the cumulative distribution function on the given value.

        Args:
            value (Tensor): The value to be evaluated.
            loc (Tensor): The location of the distribution. Default: self.loc.
            scale (Tensor): The scale the distribution. Default: self.scale.

        .. math::
            cdf(x) = \frac{\arctan{(x - loc) / scale}}{\pi} + 0.5
        """
        value = self._check_value(value, 'value')
        value = self.cast(value, self.dtype)
        loc, scale = self._check_param_type(loc, scale)
        z = (value - loc) / scale
        return self.atan(z) / np.pi + 0.5

    def _log_cdf(self, value, loc=None, scale=None):
        r"""
        Evaluate the log cumulative distribution function on the given value.

        Args:
            value (Tensor): The value to be evaluated.
            loc (Tensor): The location of the distribution. Default: self.loc.
            scale (Tensor): The scale the distribution. Default: self.scale.

        .. math::
            log_cdf(x) = \log(\frac{\arctan(\frac{x-loc}{scale})}{\pi} + 0.5)
                       = \log {\arctan(\frac{x-loc}{scale}) + 0.5pi}{pi}
                       = \log1p \frac{2 * arctan(\frac{x-loc}{scale})}{pi} - \log2
        """
        value = self._check_value(value, 'value')
        value = self.cast(value, self.dtype)
        loc, scale = self._check_param_type(loc, scale)
        z = (value - loc) / scale
        return self.log1p(2. * self.atan(z) / np.pi) - self.log(self.const(2., mstype.float32))

    def _quantile(self, p, loc=None, scale=None):
        loc, scale = self._check_param_type(loc, scale)
        return loc + scale * self.tan(np.pi * (p - 0.5))

    def _kl_loss(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
        r"""
        Evaluate Cauchy-Cauchy kl divergence, i.e. KL(a||b).

        Args:
            dist (str): The type of the distributions. Should be "Cauchy" in this case.
            loc_b (Tensor): The loc of distribution b.
            scale_b (Tensor): The scale of distribution b.
            loc (Tensor): The loc of distribution a. Default: self.loc.
            scale (Tensor): The scale of distribution a. Default: self.scale.

        .. math::
            KL(a||b) = \log(\frac{(scale_a + scale_b)^{2} + (loc_a - loc_b)^{2}}
                                 {4 * scale_a * scale_b})
        """
        check_distribution_name(dist, 'Cauchy')
        loc_a, scale_a = self._check_param_type(loc_a, scale_a)
        loc_b = self._check_value(loc_b, 'loc_b')
        loc_b = self.cast(loc_b, self.parameter_type)
        scale_b = self._check_value(scale_b, 'scale_b')
        scale_b = self.cast(scale_b, self.parameter_type)
        sum_square = self.sq(scale_a + scale_b)
        square_diff = self.sq(loc_a - loc_b)
        return self.log(sum_square + square_diff) - \
            self.log(self.const(4.0, mstype.float32)) - self.log(scale_a) - self.log(scale_b)

    def _cross_entropy(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
        r"""
        Evaluate cross entropy between Cauchy distributions.

        Args:
            dist (str): The type of the distributions. Should be "Cauchy" in this case.
            loc_b (Tensor): The loc of distribution b.
            scale_b (Tensor): The scale of distribution b.
            loc (Tensor): The loc of distribution a. Default: self.loc.
            scale (Tensor): The scale of distribution a. Default: self.scale.
        """
        check_distribution_name(dist, 'Cauchy')
        return self._entropy(loc_a, scale_a) + self._kl_loss(dist, loc_b, scale_b, loc_a, scale_a)

    def _sample(self, shape=(), loc=None, scale=None):
        """
        Sampling.

        Args:
            shape (tuple): The shape of the sample. Default: ().
            loc (Tensor): The location of the samples. Default: self.loc.
            scale (Tensor): The scale of the samples. Default: self.scale.

        Returns:
            Tensor, with the shape being shape + batch_shape.
        """
        shape = self.checktuple(shape, 'shape')
        loc, scale = self._check_param_type(loc, scale)
        batch_shape = self.shape(loc + scale)
        origin_shape = shape + batch_shape
        if origin_shape == ():
            sample_shape = (1,)
        else:
            sample_shape = origin_shape
        l_zero = self.const(0.0, mstype.float32)
        h_one = self.const(1.0, mstype.float32)
        sample_uniform = self.uniform(sample_shape, l_zero, h_one, self.seed)
        sample = self._quantile(sample_uniform, loc, scale)
        value = self.cast(sample, self.dtype)
        if origin_shape == ():
            value = self.squeeze(value)
        return value
