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1# Copyright 2020 Huawei Technologies Co., Ltd
2#
3# Licensed under the Apache License, Version 2.0 (the "License");
4# you may not use this file except in compliance with the License.
5# You may obtain a copy of the License at
6#
7# http://www.apache.org/licenses/LICENSE-2.0
8#
9# Unless required by applicable law or agreed to in writing, software
10# distributed under the License is distributed on an "AS IS" BASIS,
11# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12# See the License for the specific language governing permissions and
13# limitations under the License.
14# ============================================================================
15"""Cauchy Distribution"""
16import numpy as np
17from mindspore.ops import operations as P
18from mindspore.ops import composite as C
19from mindspore._checkparam import Validator
20from mindspore.common import dtype as mstype
21from .distribution import Distribution
22from ._utils.utils import check_greater_zero, check_distribution_name, raise_not_defined
23from ._utils.custom_ops import exp_generic, log_generic, log1p_generic
24
25
26class Cauchy(Distribution):
27    """
28    Cauchy distribution.
29
30    Args:
31        loc (int, float, list, numpy.ndarray, Tensor): The location of the Cauchy distribution.
32        scale (int, float, list, numpy.ndarray, Tensor): The scale of the Cauchy distribution.
33        seed (int): The seed used in sampling. The global seed is used if it is None. Default: None.
34        dtype (mindspore.dtype): The type of the event samples. Default: mstype.float32.
35        name (str): The name of the distribution. Default: 'Cauchy'.
36
37    Supported Platforms:
38        ``Ascend``
39
40    Note:
41        `scale` must be greater than zero.
42        `dist_spec_args` are `loc` and `scale`.
43        `dtype` must be a float type because Cauchy distributions are continuous.
44        Cauchy distribution is not supported on GPU backend.
45
46    Examples:
47        >>> import mindspore
48        >>> import mindspore.nn as nn
49        >>> import mindspore.nn.probability.distribution as msd
50        >>> from mindspore import Tensor
51        >>> # To initialize a Cauchy distribution of loc 3.0 and scale 4.0.
52        >>> cauchy1 = msd.Cauchy(3.0, 4.0, dtype=mindspore.float32)
53        >>> # A Cauchy distribution can be initialized without arguments.
54        >>> # In this case, 'loc' and `scale` must be passed in through arguments.
55        >>> cauchy2 = msd.Cauchy(dtype=mindspore.float32)
56        >>> # Here are some tensors used below for testing
57        >>> value = Tensor([1.0, 2.0, 3.0], dtype=mindspore.float32)
58        >>> loc_a = Tensor([2.0], dtype=mindspore.float32)
59        >>> scale_a = Tensor([2.0, 2.0, 2.0], dtype=mindspore.float32)
60        >>> loc_b = Tensor([1.0], dtype=mindspore.float32)
61        >>> scale_b = Tensor([1.0, 1.5, 2.0], dtype=mindspore.float32)
62        >>> # Private interfaces of probability functions corresponding to public interfaces, including
63        >>> # `prob`, `log_prob`, `cdf`, `log_cdf`, `survival_function`, and `log_survival`,
64        >>> # have the same arguments as follows.
65        >>> # Args:
66        >>> #     value (Tensor): the value to be evaluated.
67        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
68        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
69        >>> # Examples of `prob`.
70        >>> # Similar calls can be made to other probability functions
71        >>> # by replacing 'prob' by the name of the function
72        >>> ans = cauchy1.prob(value)
73        >>> print(ans.shape)
74        (3,)
75        >>> # Evaluate with respect to distribution b.
76        >>> ans = cauchy1.prob(value, loc_b, scale_b)
77        >>> print(ans.shape)
78        (3,)
79        >>> # `loc` and `scale` must be passed in during function calls
80        >>> ans = cauchy2.prob(value, loc_a, scale_a)
81        >>> print(ans.shape)
82        (3,)
83        >>> # Functions `mode` and `entropy` have the same arguments.
84        >>> # Args:
85        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
86        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
87        >>> # Example of `mode`.
88        >>> ans = cauchy1.mode() # return 3.0
89        >>> print(ans.shape)
90        ()
91        >>> ans = cauchy1.mode(loc_b, scale_b) # return loc_b
92        >>> print(ans.shape)
93        (3,)
94        >>> # `loc` and `scale` must be passed in during function calls.
95        >>> ans = cauchy2.mode(loc_a, scale_a)
96        >>> print(ans.shape)
97        (3,)
98        >>> # Interfaces of 'kl_loss' and 'cross_entropy' are the same:
99        >>> # Args:
100        >>> #     dist (str): the type of the distributions. Only "Cauchy" is supported.
101        >>> #     loc_b (Tensor): the loc of distribution b.
102        >>> #     scale_b (Tensor): the scale distribution b.
103        >>> #     loc (Tensor): the loc of distribution a. Default: self.loc.
104        >>> #     scale (Tensor): the scale distribution a. Default: self.scale.
105        >>> # Examples of `kl_loss`. `cross_entropy` is similar.
106        >>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b)
107        >>> print(ans.shape)
108        (3,)
109        >>> ans = cauchy1.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
110        >>> print(ans.shape)
111        (3,)
112        >>> # Additional `loc` and `scale` must be passed in.
113        >>> ans = cauchy2.kl_loss('Cauchy', loc_b, scale_b, loc_a, scale_a)
114        >>> print(ans.shape)
115        (3,)
116        >>> # Examples of `sample`.
117        >>> # Args:
118        >>> #     shape (tuple): the shape of the sample. Default: ()
119        >>> #     loc (Tensor): the location of the distribution. Default: self.loc.
120        >>> #     scale (Tensor): the scale of the distribution. Default: self.scale.
121        >>> ans = cauchy1.sample()
122        >>> print(ans.shape)
123        ()
124        >>> ans = cauchy1.sample((2,3))
125        >>> print(ans.shape)
126        (2, 3)
127        >>> ans = cauchy1.sample((2,3), loc_b, scale_b)
128        >>> print(ans.shape)
129        (2, 3, 3)
130        >>> ans = cauchy2.sample((2,3), loc_a, scale_a)
131        >>> print(ans.shape)
132        (2, 3, 3)
133    """
134
135    def __init__(self,
136                 loc=None,
137                 scale=None,
138                 seed=None,
139                 dtype=mstype.float32,
140                 name="Cauchy"):
141        """
142        Constructor of Cauchy.
143        """
144        param = dict(locals())
145        param['param_dict'] = {'loc': loc, 'scale': scale}
146        valid_dtype = mstype.float_type
147        Validator.check_type_name("dtype", dtype, valid_dtype, type(self).__name__)
148        super(Cauchy, self).__init__(seed, dtype, name, param)
149
150        self._loc = self._add_parameter(loc, 'loc')
151        self._scale = self._add_parameter(scale, 'scale')
152        if self._scale is not None:
153            check_greater_zero(self._scale, "scale")
154
155        # ops needed for the class
156        self.atan = P.Atan()
157        self.cast = P.Cast()
158        self.const = P.ScalarToArray()
159        self.dtypeop = P.DType()
160        self.exp = exp_generic
161        self.fill = P.Fill()
162        self.less = P.Less()
163        self.log = log_generic
164        self.log1p = log1p_generic
165        self.squeeze = P.Squeeze(0)
166        self.shape = P.Shape()
167        self.sq = P.Square()
168        self.sqrt = P.Sqrt()
169        self.tan = P.Tan()
170        self.uniform = C.uniform
171
172        self.entropy_const = np.log(4 * np.pi)
173
174
175    def extend_repr(self):
176        """Display instance object as string."""
177        if self.is_scalar_batch:
178            str_info = 'location = {}, scale = {}'.format(self._loc, self._scale)
179        else:
180            str_info = 'batch_shape = {}'.format(self._broadcast_shape)
181        return str_info
182
183    @property
184    def loc(self):
185        """
186        Return the location of the distribution after casting to dtype.
187        """
188        return self._loc
189
190    @property
191    def scale(self):
192        """
193        Return the scale of the distribution after casting to dtype.
194        """
195        return self._scale
196
197    def _get_dist_type(self):
198        return "Cauchy"
199
200    def _get_dist_args(self, loc=None, scale=None):
201        if scale is not None:
202            self.checktensor(scale, 'scale')
203        else:
204            scale = self.scale
205        if loc is not None:
206            self.checktensor(loc, 'loc')
207        else:
208            loc = self.loc
209        return loc, scale
210
211    def _mode(self, loc=None, scale=None):
212        """
213        The mode of the distribution.
214        """
215        loc, scale = self._check_param_type(loc, scale)
216        return loc
217
218    def _mean(self, *args, **kwargs):
219        return raise_not_defined('mean', 'Cauchy', *args, **kwargs)
220
221    def _sd(self, *args, **kwargs):
222        return raise_not_defined('standard deviation', 'Cauchy', *args, **kwargs)
223
224    def _var(self, *args, **kwargs):
225        return raise_not_defined('variance', 'Cauchy', *args, **kwargs)
226
227    def _entropy(self, loc=None, scale=None):
228        r"""
229        Evaluate entropy.
230
231        .. math::
232            H(X) = \log(4 * \Pi * scale)
233        """
234        loc, scale = self._check_param_type(loc, scale)
235        return self.log(scale) + self.entropy_const
236
237    def _log_prob(self, value, loc=None, scale=None):
238        r"""
239        Evaluate log probability.
240
241        Args:
242            value (Tensor): The value to be evaluated.
243            loc (Tensor): The location of the distribution. Default: self.loc.
244            scale (Tensor): The scale of the distribution. Default: self.scale.
245
246        .. math::
247            L(x) = \log(\frac{1}{\pi * scale} * \frac{scale^{2}}{(x - loc)^{2} + scale^{2}})
248        """
249        value = self._check_value(value, 'value')
250        value = self.cast(value, self.dtype)
251        loc, scale = self._check_param_type(loc, scale)
252        z = (value - loc) / scale
253        log_unnormalized_prob = (-1) * self.log1p(self.sq(z))
254        log_normalization = self.log(np.pi * scale)
255        return log_unnormalized_prob - log_normalization
256
257    def _cdf(self, value, loc=None, scale=None):
258        r"""
259        Evaluate the cumulative distribution function on the given value.
260
261        Args:
262            value (Tensor): The value to be evaluated.
263            loc (Tensor): The location of the distribution. Default: self.loc.
264            scale (Tensor): The scale the distribution. Default: self.scale.
265
266        .. math::
267            cdf(x) = \frac{\arctan{(x - loc) / scale}}{\pi} + 0.5
268        """
269        value = self._check_value(value, 'value')
270        value = self.cast(value, self.dtype)
271        loc, scale = self._check_param_type(loc, scale)
272        z = (value - loc) / scale
273        return self.atan(z) / np.pi + 0.5
274
275    def _log_cdf(self, value, loc=None, scale=None):
276        r"""
277        Evaluate the log cumulative distribution function on the given value.
278
279        Args:
280            value (Tensor): The value to be evaluated.
281            loc (Tensor): The location of the distribution. Default: self.loc.
282            scale (Tensor): The scale the distribution. Default: self.scale.
283
284        .. math::
285            log_cdf(x) = \log(\frac{\arctan(\frac{x-loc}{scale})}{\pi} + 0.5)
286                       = \log {\arctan(\frac{x-loc}{scale}) + 0.5pi}{pi}
287                       = \log1p \frac{2 * arctan(\frac{x-loc}{scale})}{pi} - \log2
288        """
289        value = self._check_value(value, 'value')
290        value = self.cast(value, self.dtype)
291        loc, scale = self._check_param_type(loc, scale)
292        z = (value - loc) / scale
293        return self.log1p(2. * self.atan(z) / np.pi) - self.log(self.const(2.))
294
295    def _quantile(self, p, loc=None, scale=None):
296        loc, scale = self._check_param_type(loc, scale)
297        return loc + scale * self.tan(np.pi * (p - 0.5))
298
299    def _kl_loss(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
300        r"""
301        Evaluate Cauchy-Cauchy kl divergence, i.e. KL(a||b).
302
303        Args:
304            dist (str): The type of the distributions. Should be "Cauchy" in this case.
305            loc_b (Tensor): The loc of distribution b.
306            scale_b (Tensor): The scale of distribution b.
307            loc (Tensor): The loc of distribution a. Default: self.loc.
308            scale (Tensor): The scale of distribution a. Default: self.scale.
309
310        .. math::
311            KL(a||b) = \log(\frac{(scale_a + scale_b)^{2} + (loc_a - loc_b)^{2}}
312                                 {4 * scale_a * scale_b})
313        """
314        check_distribution_name(dist, 'Cauchy')
315        loc_a, scale_a = self._check_param_type(loc_a, scale_a)
316        loc_b = self._check_value(loc_b, 'loc_b')
317        loc_b = self.cast(loc_b, self.parameter_type)
318        scale_b = self._check_value(scale_b, 'scale_b')
319        scale_b = self.cast(scale_b, self.parameter_type)
320        sum_square = self.sq(scale_a + scale_b)
321        square_diff = self.sq(loc_a - loc_b)
322        return self.log(sum_square + square_diff) - \
323                self.log(self.const(4.0)) - self.log(scale_a) - self.log(scale_b)
324
325    def _cross_entropy(self, dist, loc_b, scale_b, loc_a=None, scale_a=None):
326        r"""
327        Evaluate cross entropy between Cauchy distributions.
328
329        Args:
330            dist (str): The type of the distributions. Should be "Cauchy" in this case.
331            loc_b (Tensor): The loc of distribution b.
332            scale_b (Tensor): The scale of distribution b.
333            loc (Tensor): The loc of distribution a. Default: self.loc.
334            scale (Tensor): The scale of distribution a. Default: self.scale.
335        """
336        check_distribution_name(dist, 'Cauchy')
337        return self._entropy(loc_a, scale_a) + self._kl_loss(dist, loc_b, scale_b, loc_a, scale_a)
338
339    def _sample(self, shape=(), loc=None, scale=None):
340        """
341        Sampling.
342
343        Args:
344            shape (tuple): The shape of the sample. Default: ().
345            loc (Tensor): The location of the samples. Default: self.loc.
346            scale (Tensor): The scale of the samples. Default: self.scale.
347
348        Returns:
349            Tensor, with the shape being shape + batch_shape.
350        """
351        shape = self.checktuple(shape, 'shape')
352        loc, scale = self._check_param_type(loc, scale)
353        batch_shape = self.shape(loc + scale)
354        origin_shape = shape + batch_shape
355        if origin_shape == ():
356            sample_shape = (1,)
357        else:
358            sample_shape = origin_shape
359        l_zero = self.const(0.0)
360        h_one = self.const(1.0)
361        sample_uniform = self.uniform(sample_shape, l_zero, h_one, self.seed)
362        sample = self._quantile(sample_uniform, loc, scale)
363        value = self.cast(sample, self.dtype)
364        if origin_shape == ():
365            value = self.squeeze(value)
366        return value
367