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1// Copyright John Maddock 2006, 2007.
2// Copyright Paul A. Bristow 2007.
3
4// Use, modification and distribution are subject to the
5// Boost Software License, Version 1.0. (See accompanying file
6// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
7
8#ifndef BOOST_STATS_CAUCHY_HPP
9#define BOOST_STATS_CAUCHY_HPP
10
11#ifdef _MSC_VER
12#pragma warning(push)
13#pragma warning(disable : 4127) // conditional expression is constant
14#endif
15
16#include <boost/math/distributions/fwd.hpp>
17#include <boost/math/constants/constants.hpp>
18#include <boost/math/distributions/complement.hpp>
19#include <boost/math/distributions/detail/common_error_handling.hpp>
20#include <boost/config/no_tr1/cmath.hpp>
21
22#include <utility>
23
24namespace boost{ namespace math
25{
26
27template <class RealType, class Policy>
28class cauchy_distribution;
29
30namespace detail
31{
32
33template <class RealType, class Policy>
34RealType cdf_imp(const cauchy_distribution<RealType, Policy>& dist, const RealType& x, bool complement)
35{
36 //
37 // This calculates the cdf of the Cauchy distribution and/or its complement.
38 //
39 // The usual formula for the Cauchy cdf is:
40 //
41 // cdf = 0.5 + atan(x)/pi
42 //
43 // But that suffers from cancellation error as x -> -INF.
44 //
45 // Recall that for x < 0:
46 //
47 // atan(x) = -pi/2 - atan(1/x)
48 //
49 // Substituting into the above we get:
50 //
51 // CDF = -atan(1/x) ; x < 0
52 //
53 // So the proceedure is to calculate the cdf for -fabs(x)
54 // using the above formula, and then subtract from 1 when required
55 // to get the result.
56 //
57 BOOST_MATH_STD_USING // for ADL of std functions
58 static const char* function = "boost::math::cdf(cauchy<%1%>&, %1%)";
59 RealType result = 0;
60 RealType location = dist.location();
61 RealType scale = dist.scale();
62 if(false == detail::check_location(function, location, &result, Policy()))
63 {
64 return result;
65 }
66 if(false == detail::check_scale(function, scale, &result, Policy()))
67 {
68 return result;
69 }
70 if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
71 { // cdf +infinity is unity.
72 return static_cast<RealType>((complement) ? 0 : 1);
73 }
74 if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
75 { // cdf -infinity is zero.
76 return static_cast<RealType>((complement) ? 1 : 0);
77 }
78 if(false == detail::check_x(function, x, &result, Policy()))
79 { // Catches x == NaN
80 return result;
81 }
82 RealType mx = -fabs((x - location) / scale); // scale is > 0
83 if(mx > -tools::epsilon<RealType>() / 8)
84 { // special case first: x extremely close to location.
85 return 0.5;
86 }
87 result = -atan(1 / mx) / constants::pi<RealType>();
88 return (((x > location) != complement) ? 1 - result : result);
89} // cdf
90
91template <class RealType, class Policy>
92RealType quantile_imp(
93 const cauchy_distribution<RealType, Policy>& dist,
94 const RealType& p,
95 bool complement)
96{
97 // This routine implements the quantile for the Cauchy distribution,
98 // the value p may be the probability, or its complement if complement=true.
99 //
100 // The procedure first performs argument reduction on p to avoid error
101 // when calculating the tangent, then calulates the distance from the
102 // mid-point of the distribution. This is either added or subtracted
103 // from the location parameter depending on whether `complement` is true.
104 //
105 static const char* function = "boost::math::quantile(cauchy<%1%>&, %1%)";
106 BOOST_MATH_STD_USING // for ADL of std functions
107
108 RealType result = 0;
109 RealType location = dist.location();
110 RealType scale = dist.scale();
111 if(false == detail::check_location(function, location, &result, Policy()))
112 {
113 return result;
114 }
115 if(false == detail::check_scale(function, scale, &result, Policy()))
116 {
117 return result;
118 }
119 if(false == detail::check_probability(function, p, &result, Policy()))
120 {
121 return result;
122 }
123 // Special cases:
124 if(p == 1)
125 {
126 return (complement ? -1 : 1) * policies::raise_overflow_error<RealType>(function, 0, Policy());
127 }
128 if(p == 0)
129 {
130 return (complement ? 1 : -1) * policies::raise_overflow_error<RealType>(function, 0, Policy());
131 }
132
133 RealType P = p - floor(p); // argument reduction of p:
134 if(P > 0.5)
135 {
136 P = P - 1;
137 }
138 if(P == 0.5) // special case:
139 {
140 return location;
141 }
142 result = -scale / tan(constants::pi<RealType>() * P);
143 return complement ? RealType(location - result) : RealType(location + result);
144} // quantile
145
146} // namespace detail
147
148template <class RealType = double, class Policy = policies::policy<> >
149class cauchy_distribution
150{
151public:
152 typedef RealType value_type;
153 typedef Policy policy_type;
154
155 cauchy_distribution(RealType l_location = 0, RealType l_scale = 1)
156 : m_a(l_location), m_hg(l_scale)
157 {
158 static const char* function = "boost::math::cauchy_distribution<%1%>::cauchy_distribution";
159 RealType result;
160 detail::check_location(function, l_location, &result, Policy());
161 detail::check_scale(function, l_scale, &result, Policy());
162 } // cauchy_distribution
163
164 RealType location()const
165 {
166 return m_a;
167 }
168 RealType scale()const
169 {
170 return m_hg;
171 }
172
173private:
174 RealType m_a; // The location, this is the median of the distribution.
175 RealType m_hg; // The scale )or shape), this is the half width at half height.
176};
177
178typedef cauchy_distribution<double> cauchy;
179
180template <class RealType, class Policy>
181inline const std::pair<RealType, RealType> range(const cauchy_distribution<RealType, Policy>&)
182{ // Range of permissible values for random variable x.
183 if (std::numeric_limits<RealType>::has_infinity)
184 {
185 return std::pair<RealType, RealType>(-std::numeric_limits<RealType>::infinity(), std::numeric_limits<RealType>::infinity()); // - to + infinity.
186 }
187 else
188 { // Can only use max_value.
189 using boost::math::tools::max_value;
190 return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>()); // - to + max.
191 }
192}
193
194template <class RealType, class Policy>
195inline const std::pair<RealType, RealType> support(const cauchy_distribution<RealType, Policy>& )
196{ // Range of supported values for random variable x.
197 // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
198 if (std::numeric_limits<RealType>::has_infinity)
199 {
200 return std::pair<RealType, RealType>(-std::numeric_limits<RealType>::infinity(), std::numeric_limits<RealType>::infinity()); // - to + infinity.
201 }
202 else
203 { // Can only use max_value.
204 using boost::math::tools::max_value;
205 return std::pair<RealType, RealType>(-tools::max_value<RealType>(), max_value<RealType>()); // - to + max.
206 }
207}
208
209template <class RealType, class Policy>
210inline RealType pdf(const cauchy_distribution<RealType, Policy>& dist, const RealType& x)
211{
212 BOOST_MATH_STD_USING // for ADL of std functions
213
214 static const char* function = "boost::math::pdf(cauchy<%1%>&, %1%)";
215 RealType result = 0;
216 RealType location = dist.location();
217 RealType scale = dist.scale();
218 if(false == detail::check_scale("boost::math::pdf(cauchy<%1%>&, %1%)", scale, &result, Policy()))
219 {
220 return result;
221 }
222 if(false == detail::check_location("boost::math::pdf(cauchy<%1%>&, %1%)", location, &result, Policy()))
223 {
224 return result;
225 }
226 if((boost::math::isinf)(x))
227 {
228 return 0; // pdf + and - infinity is zero.
229 }
230 // These produce MSVC 4127 warnings, so the above used instead.
231 //if(std::numeric_limits<RealType>::has_infinity && abs(x) == std::numeric_limits<RealType>::infinity())
232 //{ // pdf + and - infinity is zero.
233 // return 0;
234 //}
235
236 if(false == detail::check_x(function, x, &result, Policy()))
237 { // Catches x = NaN
238 return result;
239 }
240
241 RealType xs = (x - location) / scale;
242 result = 1 / (constants::pi<RealType>() * scale * (1 + xs * xs));
243 return result;
244} // pdf
245
246template <class RealType, class Policy>
247inline RealType cdf(const cauchy_distribution<RealType, Policy>& dist, const RealType& x)
248{
249 return detail::cdf_imp(dist, x, false);
250} // cdf
251
252template <class RealType, class Policy>
253inline RealType quantile(const cauchy_distribution<RealType, Policy>& dist, const RealType& p)
254{
255 return detail::quantile_imp(dist, p, false);
256} // quantile
257
258template <class RealType, class Policy>
259inline RealType cdf(const complemented2_type<cauchy_distribution<RealType, Policy>, RealType>& c)
260{
261 return detail::cdf_imp(c.dist, c.param, true);
262} // cdf complement
263
264template <class RealType, class Policy>
265inline RealType quantile(const complemented2_type<cauchy_distribution<RealType, Policy>, RealType>& c)
266{
267 return detail::quantile_imp(c.dist, c.param, true);
268} // quantile complement
269
270template <class RealType, class Policy>
271inline RealType mean(const cauchy_distribution<RealType, Policy>&)
272{ // There is no mean:
273 typedef typename Policy::assert_undefined_type assert_type;
274 BOOST_STATIC_ASSERT(assert_type::value == 0);
275
276 return policies::raise_domain_error<RealType>(
277 "boost::math::mean(cauchy<%1%>&)",
278 "The Cauchy distribution does not have a mean: "
279 "the only possible return value is %1%.",
280 std::numeric_limits<RealType>::quiet_NaN(), Policy());
281}
282
283template <class RealType, class Policy>
284inline RealType variance(const cauchy_distribution<RealType, Policy>& /*dist*/)
285{
286 // There is no variance:
287 typedef typename Policy::assert_undefined_type assert_type;
288 BOOST_STATIC_ASSERT(assert_type::value == 0);
289
290 return policies::raise_domain_error<RealType>(
291 "boost::math::variance(cauchy<%1%>&)",
292 "The Cauchy distribution does not have a variance: "
293 "the only possible return value is %1%.",
294 std::numeric_limits<RealType>::quiet_NaN(), Policy());
295}
296
297template <class RealType, class Policy>
298inline RealType mode(const cauchy_distribution<RealType, Policy>& dist)
299{
300 return dist.location();
301}
302
303template <class RealType, class Policy>
304inline RealType median(const cauchy_distribution<RealType, Policy>& dist)
305{
306 return dist.location();
307}
308template <class RealType, class Policy>
309inline RealType skewness(const cauchy_distribution<RealType, Policy>& /*dist*/)
310{
311 // There is no skewness:
312 typedef typename Policy::assert_undefined_type assert_type;
313 BOOST_STATIC_ASSERT(assert_type::value == 0);
314
315 return policies::raise_domain_error<RealType>(
316 "boost::math::skewness(cauchy<%1%>&)",
317 "The Cauchy distribution does not have a skewness: "
318 "the only possible return value is %1%.",
319 std::numeric_limits<RealType>::quiet_NaN(), Policy()); // infinity?
320}
321
322template <class RealType, class Policy>
323inline RealType kurtosis(const cauchy_distribution<RealType, Policy>& /*dist*/)
324{
325 // There is no kurtosis:
326 typedef typename Policy::assert_undefined_type assert_type;
327 BOOST_STATIC_ASSERT(assert_type::value == 0);
328
329 return policies::raise_domain_error<RealType>(
330 "boost::math::kurtosis(cauchy<%1%>&)",
331 "The Cauchy distribution does not have a kurtosis: "
332 "the only possible return value is %1%.",
333 std::numeric_limits<RealType>::quiet_NaN(), Policy());
334}
335
336template <class RealType, class Policy>
337inline RealType kurtosis_excess(const cauchy_distribution<RealType, Policy>& /*dist*/)
338{
339 // There is no kurtosis excess:
340 typedef typename Policy::assert_undefined_type assert_type;
341 BOOST_STATIC_ASSERT(assert_type::value == 0);
342
343 return policies::raise_domain_error<RealType>(
344 "boost::math::kurtosis_excess(cauchy<%1%>&)",
345 "The Cauchy distribution does not have a kurtosis: "
346 "the only possible return value is %1%.",
347 std::numeric_limits<RealType>::quiet_NaN(), Policy());
348}
349
350} // namespace math
351} // namespace boost
352
353#ifdef _MSC_VER
354#pragma warning(pop)
355#endif
356
357// This include must be at the end, *after* the accessors
358// for this distribution have been defined, in order to
359// keep compilers that support two-phase lookup happy.
360#include <boost/math/distributions/detail/derived_accessors.hpp>
361
362#endif // BOOST_STATS_CAUCHY_HPP