Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.8/site-packages/tensorflow/python/ops/distributions/special_math.py: 19%
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1# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
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# ==============================================================================
16# Functions "ndtr" and "ndtri" are derived from calculations made in:
17# https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
18# In the following email exchange, the author gives his consent to redistribute
19# derived works under an Apache 2.0 license.
20#
21# From: Stephen Moshier <steve@moshier.net>
22# Date: Sat, Jun 9, 2018 at 2:36 PM
23# Subject: Re: Licensing cephes under Apache (BSD-like) license.
24# To: rif <rif@google.com>
25#
26#
27#
28# Hello Rif,
29#
30# Yes, Google may distribute Cephes files under the Apache 2 license.
31#
32# If clarification is needed, I do not favor BSD over other free licenses.
33# I would agree that Apache 2 seems to cover the concern you mentioned
34# about sublicensees.
35#
36# Best wishes for good luck with your projects!
37# Steve Moshier
38#
39#
40#
41# On Thu, 31 May 2018, rif wrote:
42#
43# > Hello Steve.
44# > My name is Rif. I work on machine learning software at Google.
45# >
46# > Your cephes software continues to be incredibly useful and widely used. I
47# > was wondering whether it would be permissible for us to use the Cephes code
48# > under the Apache 2.0 license, which is extremely similar in permissions to
49# > the BSD license (Wikipedia comparisons). This would be quite helpful to us
50# > in terms of avoiding multiple licenses on software.
51# >
52# > I'm sorry to bother you with this (I can imagine you're sick of hearing
53# > about this by now), but I want to be absolutely clear we're on the level and
54# > not misusing your important software. In former conversation with Eugene
55# > Brevdo (ebrevdo@google.com), you wrote "If your licensing is similar to BSD,
56# > the formal way that has been handled is simply to add a statement to the
57# > effect that you are incorporating the Cephes software by permission of the
58# > author." I wanted to confirm that (a) we could use the Apache license, (b)
59# > that we don't need to (and probably you don't want to) keep getting
60# > contacted about individual uses, because your intent is generally to allow
61# > this software to be reused under "BSD-like" license, and (c) you're OK
62# > letting incorporators decide whether a license is sufficiently BSD-like?
63# >
64# > Best,
65# >
66# > rif
67# >
68# >
69# >
71"""Special Math Ops."""
73import numpy as np
75from tensorflow.python.framework import constant_op
76from tensorflow.python.framework import ops
77from tensorflow.python.ops import array_ops
78from tensorflow.python.ops import math_ops
80__all__ = [
81 "erfinv",
82 "ndtr",
83 "ndtri",
84 "log_ndtr",
85 "log_cdf_laplace",
86]
89# log_ndtr uses different functions over the ranges
90# (-infty, lower](lower, upper](upper, infty)
91# Lower bound values were chosen by examining where the support of ndtr
92# appears to be zero, relative to scipy's (which is always 64bit). They were
93# then made more conservative just to be safe. (Conservative means use the
94# expansion more than we probably need to.) See `NdtrTest` in
95# special_math_test.py.
96LOGNDTR_FLOAT64_LOWER = np.array(-20, np.float64)
97LOGNDTR_FLOAT32_LOWER = np.array(-10, np.float32)
99# Upper bound values were chosen by examining for which values of 'x'
100# Log[cdf(x)] is 0, after which point we need to use the approximation
101# Log[cdf(x)] = Log[1 - cdf(-x)] approx -cdf(-x). We chose a value slightly
102# conservative, meaning we use the approximation earlier than needed.
103LOGNDTR_FLOAT64_UPPER = np.array(8, np.float64)
104LOGNDTR_FLOAT32_UPPER = np.array(5, np.float32)
107def ndtr(x, name="ndtr"):
108 """Normal distribution function.
110 Returns the area under the Gaussian probability density function, integrated
111 from minus infinity to x:
113 ```
114 1 / x
115 ndtr(x) = ---------- | exp(-0.5 t**2) dt
116 sqrt(2 pi) /-inf
118 = 0.5 (1 + erf(x / sqrt(2)))
119 = 0.5 erfc(x / sqrt(2))
120 ```
122 Args:
123 x: `Tensor` of type `float32`, `float64`.
124 name: Python string. A name for the operation (default="ndtr").
126 Returns:
127 ndtr: `Tensor` with `dtype=x.dtype`.
129 Raises:
130 TypeError: if `x` is not floating-type.
131 """
133 with ops.name_scope(name, values=[x]):
134 x = ops.convert_to_tensor(x, name="x")
135 if x.dtype.as_numpy_dtype not in [np.float32, np.float64]:
136 raise TypeError(
137 "x.dtype=%s is not handled, see docstring for supported types."
138 % x.dtype)
139 return _ndtr(x)
142def _ndtr(x):
143 """Implements ndtr core logic."""
144 half_sqrt_2 = constant_op.constant(
145 0.5 * np.sqrt(2.), dtype=x.dtype, name="half_sqrt_2")
146 w = x * half_sqrt_2
147 z = math_ops.abs(w)
148 y = array_ops.where_v2(
149 math_ops.less(z, half_sqrt_2), 1. + math_ops.erf(w),
150 array_ops.where_v2(
151 math_ops.greater(w, 0.), 2. - math_ops.erfc(z), math_ops.erfc(z)))
152 return 0.5 * y
155def ndtri(p, name="ndtri"):
156 """The inverse of the CDF of the Normal distribution function.
158 Returns x such that the area under the pdf from minus infinity to x is equal
159 to p.
161 A piece-wise rational approximation is done for the function.
162 This is a port of the implementation in netlib.
164 Args:
165 p: `Tensor` of type `float32`, `float64`.
166 name: Python string. A name for the operation (default="ndtri").
168 Returns:
169 x: `Tensor` with `dtype=p.dtype`.
171 Raises:
172 TypeError: if `p` is not floating-type.
173 """
175 with ops.name_scope(name, values=[p]):
176 p = ops.convert_to_tensor(p, name="p")
177 if p.dtype.as_numpy_dtype not in [np.float32, np.float64]:
178 raise TypeError(
179 "p.dtype=%s is not handled, see docstring for supported types."
180 % p.dtype)
181 return _ndtri(p)
184def _ndtri(p):
185 """Implements ndtri core logic."""
187 # Constants used in piece-wise rational approximations. Taken from the cephes
188 # library:
189 # https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
191 p0 = [
192 -1.23916583867381258016E0, 1.39312609387279679503E1,
193 -5.66762857469070293439E1, 9.80010754185999661536E1,
194 -5.99633501014107895267E1
195 ]
196 q0 = [
197 -1.18331621121330003142E0, 1.59056225126211695515E1,
198 -8.20372256168333339912E1, 2.00260212380060660359E2,
199 -2.25462687854119370527E2, 8.63602421390890590575E1,
200 4.67627912898881538453E0, 1.95448858338141759834E0, 1.0
201 ]
202 p1 = [
203 -8.57456785154685413611E-4, -3.50424626827848203418E-2,
204 -1.40256079171354495875E-1, 2.18663306850790267539E0,
205 1.46849561928858024014E1, 4.40805073893200834700E1,
206 5.71628192246421288162E1, 3.15251094599893866154E1,
207 4.05544892305962419923E0
208 ]
209 q1 = [
210 -9.33259480895457427372E-4, -3.80806407691578277194E-2,
211 -1.42182922854787788574E-1, 2.50464946208309415979E0,
212 1.50425385692907503408E1, 4.13172038254672030440E1,
213 4.53907635128879210584E1, 1.57799883256466749731E1, 1.0
214 ]
215 p2 = [
216 6.23974539184983293730E-9, 2.65806974686737550832E-6,
217 3.01581553508235416007E-4, 1.23716634817820021358E-2,
218 2.01485389549179081538E-1, 1.33303460815807542389E0,
219 3.93881025292474443415E0, 6.91522889068984211695E0,
220 3.23774891776946035970E0
221 ]
222 q2 = [
223 6.79019408009981274425E-9, 2.89247864745380683936E-6,
224 3.28014464682127739104E-4, 1.34204006088543189037E-2,
225 2.16236993594496635890E-1, 1.37702099489081330271E0,
226 3.67983563856160859403E0, 6.02427039364742014255E0, 1.0
227 ]
229 def _create_polynomial(var, coeffs):
230 """Compute n_th order polynomial via Horner's method."""
231 coeffs = np.array(coeffs, var.dtype.as_numpy_dtype)
232 if not coeffs.size:
233 return array_ops.zeros_like(var)
234 return coeffs[0] + _create_polynomial(var, coeffs[1:]) * var
236 maybe_complement_p = array_ops.where_v2(p > -np.expm1(-2.), 1. - p, p)
237 # Write in an arbitrary value in place of 0 for p since 0 will cause NaNs
238 # later on. The result from the computation when p == 0 is not used so any
239 # number that doesn't result in NaNs is fine.
240 sanitized_mcp = array_ops.where_v2(
241 maybe_complement_p <= 0.,
242 array_ops.fill(array_ops.shape(p), np.array(0.5, p.dtype.as_numpy_dtype)),
243 maybe_complement_p)
245 # Compute x for p > exp(-2): x/sqrt(2pi) = w + w**3 P0(w**2)/Q0(w**2).
246 w = sanitized_mcp - 0.5
247 ww = w ** 2
248 x_for_big_p = w + w * ww * (_create_polynomial(ww, p0)
249 / _create_polynomial(ww, q0))
250 x_for_big_p *= -np.sqrt(2. * np.pi)
252 # Compute x for p <= exp(-2): x = z - log(z)/z - (1/z) P(1/z) / Q(1/z),
253 # where z = sqrt(-2. * log(p)), and P/Q are chosen between two different
254 # arrays based on whether p < exp(-32).
255 z = math_ops.sqrt(-2. * math_ops.log(sanitized_mcp))
256 first_term = z - math_ops.log(z) / z
257 second_term_small_p = (
258 _create_polynomial(1. / z, p2) /
259 _create_polynomial(1. / z, q2) / z)
260 second_term_otherwise = (
261 _create_polynomial(1. / z, p1) /
262 _create_polynomial(1. / z, q1) / z)
263 x_for_small_p = first_term - second_term_small_p
264 x_otherwise = first_term - second_term_otherwise
266 x = array_ops.where_v2(
267 sanitized_mcp > np.exp(-2.), x_for_big_p,
268 array_ops.where_v2(z >= 8.0, x_for_small_p, x_otherwise))
270 x = array_ops.where_v2(p > 1. - np.exp(-2.), x, -x)
271 infinity_scalar = constant_op.constant(np.inf, dtype=p.dtype)
272 infinity = array_ops.fill(array_ops.shape(p), infinity_scalar)
273 x_nan_replaced = array_ops.where_v2(p <= 0.0, -infinity,
274 array_ops.where_v2(p >= 1.0, infinity, x))
275 return x_nan_replaced
278def log_ndtr(x, series_order=3, name="log_ndtr"):
279 """Log Normal distribution function.
281 For details of the Normal distribution function see `ndtr`.
283 This function calculates `(log o ndtr)(x)` by either calling `log(ndtr(x))` or
284 using an asymptotic series. Specifically:
285 - For `x > upper_segment`, use the approximation `-ndtr(-x)` based on
286 `log(1-x) ~= -x, x << 1`.
287 - For `lower_segment < x <= upper_segment`, use the existing `ndtr` technique
288 and take a log.
289 - For `x <= lower_segment`, we use the series approximation of erf to compute
290 the log CDF directly.
292 The `lower_segment` is set based on the precision of the input:
294 ```
295 lower_segment = { -20, x.dtype=float64
296 { -10, x.dtype=float32
297 upper_segment = { 8, x.dtype=float64
298 { 5, x.dtype=float32
299 ```
301 When `x < lower_segment`, the `ndtr` asymptotic series approximation is:
303 ```
304 ndtr(x) = scale * (1 + sum) + R_N
305 scale = exp(-0.5 x**2) / (-x sqrt(2 pi))
306 sum = Sum{(-1)^n (2n-1)!! / (x**2)^n, n=1:N}
307 R_N = O(exp(-0.5 x**2) (2N+1)!! / |x|^{2N+3})
308 ```
310 where `(2n-1)!! = (2n-1) (2n-3) (2n-5) ... (3) (1)` is a
311 [double-factorial](https://en.wikipedia.org/wiki/Double_factorial).
314 Args:
315 x: `Tensor` of type `float32`, `float64`.
316 series_order: Positive Python `integer`. Maximum depth to
317 evaluate the asymptotic expansion. This is the `N` above.
318 name: Python string. A name for the operation (default="log_ndtr").
320 Returns:
321 log_ndtr: `Tensor` with `dtype=x.dtype`.
323 Raises:
324 TypeError: if `x.dtype` is not handled.
325 TypeError: if `series_order` is a not Python `integer.`
326 ValueError: if `series_order` is not in `[0, 30]`.
327 """
328 if not isinstance(series_order, int):
329 raise TypeError("series_order must be a Python integer.")
330 if series_order < 0:
331 raise ValueError("series_order must be non-negative.")
332 if series_order > 30:
333 raise ValueError("series_order must be <= 30.")
335 with ops.name_scope(name, values=[x]):
336 x = ops.convert_to_tensor(x, name="x")
338 if x.dtype.as_numpy_dtype == np.float64:
339 lower_segment = LOGNDTR_FLOAT64_LOWER
340 upper_segment = LOGNDTR_FLOAT64_UPPER
341 elif x.dtype.as_numpy_dtype == np.float32:
342 lower_segment = LOGNDTR_FLOAT32_LOWER
343 upper_segment = LOGNDTR_FLOAT32_UPPER
344 else:
345 raise TypeError("x.dtype=%s is not supported." % x.dtype)
347 # The basic idea here was ported from:
348 # https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
349 # We copy the main idea, with a few changes
350 # * For x >> 1, and X ~ Normal(0, 1),
351 # Log[P[X < x]] = Log[1 - P[X < -x]] approx -P[X < -x],
352 # which extends the range of validity of this function.
353 # * We use one fixed series_order for all of 'x', rather than adaptive.
354 # * Our docstring properly reflects that this is an asymptotic series, not a
355 # Taylor series. We also provided a correct bound on the remainder.
356 # * We need to use the max/min in the _log_ndtr_lower arg to avoid nan when
357 # x=0. This happens even though the branch is unchosen because when x=0
358 # the gradient of a select involves the calculation 1*dy+0*(-inf)=nan
359 # regardless of whether dy is finite. Note that the minimum is a NOP if
360 # the branch is chosen.
361 return array_ops.where_v2(
362 math_ops.greater(x, upper_segment),
363 -_ndtr(-x), # log(1-x) ~= -x, x << 1 # pylint: disable=invalid-unary-operand-type
364 array_ops.where_v2(
365 math_ops.greater(x, lower_segment),
366 math_ops.log(_ndtr(math_ops.maximum(x, lower_segment))),
367 _log_ndtr_lower(math_ops.minimum(x, lower_segment), series_order)))
370def _log_ndtr_lower(x, series_order):
371 """Asymptotic expansion version of `Log[cdf(x)]`, appropriate for `x<<-1`."""
372 x_2 = math_ops.square(x)
373 # Log of the term multiplying (1 + sum)
374 log_scale = -0.5 * x_2 - math_ops.log(-x) - 0.5 * np.log(2. * np.pi)
375 return log_scale + math_ops.log(_log_ndtr_asymptotic_series(x, series_order))
378def _log_ndtr_asymptotic_series(x, series_order):
379 """Calculates the asymptotic series used in log_ndtr."""
380 dtype = x.dtype.as_numpy_dtype
381 if series_order <= 0:
382 return np.array(1, dtype)
383 x_2 = math_ops.square(x)
384 even_sum = array_ops.zeros_like(x)
385 odd_sum = array_ops.zeros_like(x)
386 x_2n = x_2 # Start with x^{2*1} = x^{2*n} with n = 1.
387 for n in range(1, series_order + 1):
388 y = np.array(_double_factorial(2 * n - 1), dtype) / x_2n
389 if n % 2:
390 odd_sum += y
391 else:
392 even_sum += y
393 x_2n *= x_2
394 return 1. + even_sum - odd_sum
397def erfinv(x, name="erfinv"):
398 """The inverse function for erf, the error function.
400 Args:
401 x: `Tensor` of type `float32`, `float64`.
402 name: Python string. A name for the operation (default="erfinv").
404 Returns:
405 x: `Tensor` with `dtype=x.dtype`.
407 Raises:
408 TypeError: if `x` is not floating-type.
409 """
411 with ops.name_scope(name, values=[x]):
412 x = ops.convert_to_tensor(x, name="x")
413 if x.dtype.as_numpy_dtype not in [np.float32, np.float64]:
414 raise TypeError(
415 "x.dtype=%s is not handled, see docstring for supported types."
416 % x.dtype)
417 return ndtri((x + 1.0) / 2.0) / np.sqrt(2)
420def _double_factorial(n):
421 """The double factorial function for small Python integer `n`."""
422 return np.prod(np.arange(n, 1, -2))
425def log_cdf_laplace(x, name="log_cdf_laplace"):
426 """Log Laplace distribution function.
428 This function calculates `Log[L(x)]`, where `L(x)` is the cumulative
429 distribution function of the Laplace distribution, i.e.
431 ```L(x) := 0.5 * int_{-infty}^x e^{-|t|} dt```
433 For numerical accuracy, `L(x)` is computed in different ways depending on `x`,
435 ```
436 x <= 0:
437 Log[L(x)] = Log[0.5] + x, which is exact
439 0 < x:
440 Log[L(x)] = Log[1 - 0.5 * e^{-x}], which is exact
441 ```
443 Args:
444 x: `Tensor` of type `float32`, `float64`.
445 name: Python string. A name for the operation (default="log_ndtr").
447 Returns:
448 `Tensor` with `dtype=x.dtype`.
450 Raises:
451 TypeError: if `x.dtype` is not handled.
452 """
454 with ops.name_scope(name, values=[x]):
455 x = ops.convert_to_tensor(x, name="x")
457 # For x < 0, L(x) = 0.5 * exp{x} exactly, so Log[L(x)] = log(0.5) + x.
458 lower_solution = -np.log(2.) + x
460 # safe_exp_neg_x = exp{-x} for x > 0, but is
461 # bounded above by 1, which avoids
462 # log[1 - 1] = -inf for x = log(1/2), AND
463 # exp{-x} --> inf, for x << -1
464 safe_exp_neg_x = math_ops.exp(-math_ops.abs(x))
466 # log1p(z) = log(1 + z) approx z for |z| << 1. This approximation is used
467 # internally by log1p, rather than being done explicitly here.
468 upper_solution = math_ops.log1p(-0.5 * safe_exp_neg_x)
470 return array_ops.where_v2(x < 0., lower_solution, upper_solution)