# Boost C++ Libraries

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### boost/math/distributions/arcsine.hpp

```// boost/math/distributions/arcsine.hpp

// Copyright Paul A. Bristow 2014.

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.

// http://en.wikipedia.org/wiki/arcsine_distribution

// The arcsine Distribution is a continuous probability distribution.
// http://en.wikipedia.org/wiki/Arcsine_distribution
// http://www.wolframalpha.com/input/?i=ArcSinDistribution

// Standard arcsine distribution is a special case of beta distribution with both a & b = one half,
// and 0 <= x <= 1.

// It is generalized to include any bounded support a <= x <= b from 0 <= x <= 1
// by Wolfram and Wikipedia,
// but using location and scale parameters by
// Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/index.html
// http://www.math.uah.edu/stat/special/Arcsine.html
// The end-point version is simpler and more obvious, so we implement that.
// TODO Perhaps provide location and scale functions?

#ifndef BOOST_MATH_DIST_ARCSINE_HPP
#define BOOST_MATH_DIST_ARCSINE_HPP

#include <cmath>
#include <boost/math/distributions/fwd.hpp>
#include <boost/math/distributions/complement.hpp> // complements.
#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks.
#include <boost/math/constants/constants.hpp>

#include <boost/math/special_functions/fpclassify.hpp> // isnan.

#if defined (BOOST_MSVC)
#  pragma warning(push)
#  pragma warning(disable: 4702) // Unreachable code,
// in domain_error_imp in error_handling.
#endif

#include <utility>
#include <exception>  // For std::domain_error.

namespace boost
{
namespace math
{
namespace arcsine_detail
{
// Common error checking routines for arcsine distribution functions:
// Duplicating for x_min and x_max provides specific error messages.
template <class RealType, class Policy>
inline bool check_x_min(const char* function, const RealType& x, RealType* result, const Policy& pol)
{
if (!(boost::math::isfinite)(x))
{
*result = policies::raise_domain_error<RealType>(
function,
"x_min argument is %1%, but must be finite !", x, pol);
return false;
}
return true;
} // bool check_x_min

template <class RealType, class Policy>
inline bool check_x_max(const char* function, const RealType& x, RealType* result, const Policy& pol)
{
if (!(boost::math::isfinite)(x))
{
*result = policies::raise_domain_error<RealType>(
function,
"x_max argument is %1%, but must be finite !", x, pol);
return false;
}
return true;
} // bool check_x_max

template <class RealType, class Policy>
inline bool check_x_minmax(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)
{ // Check x_min < x_max
if (x_min >= x_max)
{
std::string msg = "x_max argument is %1%, but must be > x_min";
*result = policies::raise_domain_error<RealType>(
function,
msg.c_str(), x_max, pol);
// "x_max argument is %1%, but must be > x_min !", x_max, pol);
// "x_max argument is %1%, but must be > x_min %2!", x_max, x_min, pol); would be better.
// But would require replication of all helpers functions in /policies/error_handling.hpp for two values,
// as well as two value versions of raise_error, raise_domain_error and do_format
return false;
}
return true;
} // bool check_x_minmax

template <class RealType, class Policy>
inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
{
if ((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
{
*result = policies::raise_domain_error<RealType>(
function,
"Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
return false;
}
return true;
} // bool check_prob

template <class RealType, class Policy>
inline bool check_x(const char* function, const RealType& x_min, const RealType& x_max, const RealType& x, RealType* result, const Policy& pol)
{ // Check x finite and x_min < x < x_max.
if (!(boost::math::isfinite)(x))
{
*result = policies::raise_domain_error<RealType>(
function,
"x argument is %1%, but must be finite !", x, pol);
return false;
}
if ((x < x_min) || (x > x_max))
{
// std::cout << x_min << ' ' << x << x_max << std::endl;
*result = policies::raise_domain_error<RealType>(
function,
"x argument is %1%, but must be x_min < x < x_max !", x, pol);
// For example:
// Error in function boost::math::pdf(arcsine_distribution<double> const&, double) : x argument is -1.01, but must be x_min < x < x_max !
// TODO Perhaps show values of x_min and x_max?
return false;
}
return true;
} // bool check_x

template <class RealType, class Policy>
inline bool check_dist(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)
{ // Check both x_min and x_max finite, and x_min  < x_max.
return check_x_min(function, x_min, result, pol)
&& check_x_max(function, x_max, result, pol)
&& check_x_minmax(function, x_min, x_max, result, pol);
} // bool check_dist

template <class RealType, class Policy>
inline bool check_dist_and_x(const char* function, const RealType& x_min, const RealType& x_max, RealType x, RealType* result, const Policy& pol)
{
return check_dist(function, x_min, x_max, result, pol)
&& arcsine_detail::check_x(function, x_min, x_max, x, result, pol);
} // bool check_dist_and_x

template <class RealType, class Policy>
inline bool check_dist_and_prob(const char* function, const RealType& x_min, const RealType& x_max, RealType p, RealType* result, const Policy& pol)
{
return check_dist(function, x_min, x_max, result, pol)
&& check_prob(function, p, result, pol);
} // bool check_dist_and_prob

} // namespace arcsine_detail

template <class RealType = double, class Policy = policies::policy<> >
class arcsine_distribution
{
public:
typedef RealType value_type;
typedef Policy policy_type;

arcsine_distribution(RealType x_min = 0, RealType x_max = 1) : m_x_min(x_min), m_x_max(x_max)
{ // Default beta (alpha = beta = 0.5) is standard arcsine with x_min = 0, x_max = 1.
// Generalized to allow x_min and x_max to be specified.
RealType result;
arcsine_detail::check_dist(
"boost::math::arcsine_distribution<%1%>::arcsine_distribution",
m_x_min,
m_x_max,
&result, Policy());
} // arcsine_distribution constructor.
// Accessor functions:
RealType x_min() const
{
return m_x_min;
}
RealType x_max() const
{
return m_x_max;
}

private:
RealType m_x_min; // Two x min and x max parameters of the arcsine distribution.
RealType m_x_max;
}; // template <class RealType, class Policy> class arcsine_distribution

// Convenient typedef to construct double version.
typedef arcsine_distribution<double> arcsine;

#ifdef __cpp_deduction_guides
template <class RealType>
arcsine_distribution(RealType)->arcsine_distribution<typename boost::math::tools::promote_args<RealType>::type>;
template <class RealType>
arcsine_distribution(RealType, RealType)->arcsine_distribution<typename boost::math::tools::promote_args<RealType>::type>;
#endif

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> range(const arcsine_distribution<RealType, Policy>&  dist)
{ // Range of permissible values for random variable x.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(static_cast<RealType>(dist.x_min()), static_cast<RealType>(dist.x_max()));
}

template <class RealType, class Policy>
inline const std::pair<RealType, RealType> support(const arcsine_distribution<RealType, Policy>&  dist)
{ // Range of supported values for random variable x.
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
return std::pair<RealType, RealType>(static_cast<RealType>(dist.x_min()), static_cast<RealType>(dist.x_max()));
}

template <class RealType, class Policy>
inline RealType mean(const arcsine_distribution<RealType, Policy>& dist)
{ // Mean of arcsine distribution .
RealType result;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

if (false == arcsine_detail::check_dist(
"boost::math::mean(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}
return  (x_min + x_max) / 2;
} // mean

template <class RealType, class Policy>
inline RealType variance(const arcsine_distribution<RealType, Policy>& dist)
{ // Variance of standard arcsine distribution = (1-0)/8 = 0.125.
RealType result;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();
if (false == arcsine_detail::check_dist(
"boost::math::variance(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}
return  (x_max - x_min) * (x_max - x_min) / 8;
} // variance

template <class RealType, class Policy>
inline RealType mode(const arcsine_distribution<RealType, Policy>& /* dist */)
{ //There are always [*two] values for the mode, at ['x_min] and at ['x_max], default 0 and 1,
// so instead we raise the exception domain_error.
return policies::raise_domain_error<RealType>(
"boost::math::mode(arcsine_distribution<%1%>&)",
"The arcsine distribution has two modes at x_min and x_max: "
"so the return value is %1%.",
std::numeric_limits<RealType>::quiet_NaN(), Policy());
} // mode

template <class RealType, class Policy>
inline RealType median(const arcsine_distribution<RealType, Policy>& dist)
{ // Median of arcsine distribution (a + b) / 2 == mean.
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();
RealType result;
if (false == arcsine_detail::check_dist(
"boost::math::median(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}
return  (x_min + x_max) / 2;
}

template <class RealType, class Policy>
inline RealType skewness(const arcsine_distribution<RealType, Policy>& dist)
{
RealType result;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

if (false == arcsine_detail::check_dist(
"boost::math::skewness(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}
return 0;
} // skewness

template <class RealType, class Policy>
inline RealType kurtosis_excess(const arcsine_distribution<RealType, Policy>& dist)
{
RealType result;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

if (false == arcsine_detail::check_dist(
"boost::math::kurtosis_excess(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}
result = -3;
return  result / 2;
} // kurtosis_excess

template <class RealType, class Policy>
inline RealType kurtosis(const arcsine_distribution<RealType, Policy>& dist)
{
RealType result;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

if (false == arcsine_detail::check_dist(
"boost::math::kurtosis(arcsine_distribution<%1%> const&, %1% )",
x_min,
x_max,
&result, Policy())
)
{
return result;
}

return 3 + kurtosis_excess(dist);
} // kurtosis

template <class RealType, class Policy>
inline RealType pdf(const arcsine_distribution<RealType, Policy>& dist, const RealType& xx)
{ // Probability Density/Mass Function arcsine.
BOOST_FPU_EXCEPTION_GUARD
BOOST_MATH_STD_USING // For ADL of std functions.

static const char* function = "boost::math::pdf(arcsine_distribution<%1%> const&, %1%)";

RealType lo = dist.x_min();
RealType hi = dist.x_max();
RealType x = xx;

// Argument checks:
RealType result = 0;
if (false == arcsine_detail::check_dist_and_x(
function,
lo, hi, x,
&result, Policy()))
{
return result;
}
using boost::math::constants::pi;
result = static_cast<RealType>(1) / (pi<RealType>() * sqrt((x - lo) * (hi - x)));
return result;
} // pdf

template <class RealType, class Policy>
inline RealType cdf(const arcsine_distribution<RealType, Policy>& dist, const RealType& x)
{ // Cumulative Distribution Function arcsine.
BOOST_MATH_STD_USING // For ADL of std functions.

static const char* function = "boost::math::cdf(arcsine_distribution<%1%> const&, %1%)";

RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

// Argument checks:
RealType result = 0;
if (false == arcsine_detail::check_dist_and_x(
function,
x_min, x_max, x,
&result, Policy()))
{
return result;
}
// Special cases:
if (x == x_min)
{
return 0;
}
else if (x == x_max)
{
return 1;
}
using boost::math::constants::pi;
result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
return result;
} // arcsine cdf

template <class RealType, class Policy>
inline RealType cdf(const complemented2_type<arcsine_distribution<RealType, Policy>, RealType>& c)
{ // Complemented Cumulative Distribution Function arcsine.
BOOST_MATH_STD_USING // For ADL of std functions.
static const char* function = "boost::math::cdf(arcsine_distribution<%1%> const&, %1%)";

RealType x = c.param;
arcsine_distribution<RealType, Policy> const& dist = c.dist;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();

// Argument checks:
RealType result = 0;
if (false == arcsine_detail::check_dist_and_x(
function,
x_min, x_max, x,
&result, Policy()))
{
return result;
}
if (x == x_min)
{
return 0;
}
else if (x == x_max)
{
return 1;
}
using boost::math::constants::pi;
// Naive version x = 1 - x;
// result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
// is less accurate, so use acos instead of asin for complement.
result = static_cast<RealType>(2) * acos(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
return result;
} // arcsine ccdf

template <class RealType, class Policy>
inline RealType quantile(const arcsine_distribution<RealType, Policy>& dist, const RealType& p)
{
// Quantile or Percent Point arcsine function or
// Inverse Cumulative probability distribution function CDF.
// Return x (0 <= x <= 1),
// for a given probability p (0 <= p <= 1).
// These functions take a probability as an argument
// and return a value such that the probability that a random variable x
// will be less than or equal to that value
// is whatever probability you supplied as an argument.
BOOST_MATH_STD_USING // For ADL of std functions.

using boost::math::constants::half_pi;

static const char* function = "boost::math::quantile(arcsine_distribution<%1%> const&, %1%)";

RealType result = 0; // of argument checks:
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();
if (false == arcsine_detail::check_dist_and_prob(
function,
x_min, x_max, p,
&result, Policy()))
{
return result;
}
// Special cases:
if (p == 0)
{
return 0;
}
if (p == 1)
{
return 1;
}

RealType sin2hpip = sin(half_pi<RealType>() * p);
RealType sin2hpip2 = sin2hpip * sin2hpip;
result = -x_min * sin2hpip2 + x_min + x_max * sin2hpip2;

return result;
} // quantile

template <class RealType, class Policy>
inline RealType quantile(const complemented2_type<arcsine_distribution<RealType, Policy>, RealType>& c)
{
// Complement Quantile or Percent Point arcsine function.
// Return the number of expected x for a given
// complement of the probability q.
BOOST_MATH_STD_USING // For ADL of std functions.

using boost::math::constants::half_pi;
static const char* function = "boost::math::quantile(arcsine_distribution<%1%> const&, %1%)";

// Error checks:
RealType q = c.param;
const arcsine_distribution<RealType, Policy>& dist = c.dist;
RealType result = 0;
RealType x_min = dist.x_min();
RealType x_max = dist.x_max();
if (false == arcsine_detail::check_dist_and_prob(
function,
x_min,
x_max,
q,
&result, Policy()))
{
return result;
}
// Special cases:
if (q == 1)
{
return 0;
}
if (q == 0)
{
return 1;
}
// Naive RealType p = 1 - q; result = sin(half_pi<RealType>() * p); loses accuracy, so use a cos alternative instead.
//result = cos(half_pi<RealType>() * q); // for arcsine(0,1)
//result = result * result;
// For generalized arcsine:
RealType cos2hpip = cos(half_pi<RealType>() * q);
RealType cos2hpip2 = cos2hpip * cos2hpip;
result = -x_min * cos2hpip2 + x_min + x_max * cos2hpip2;

return result;
} // Quantile Complement

} // namespace math
} // namespace boost

// This include must be at the end, *after* the accessors
// for this distribution have been defined, in order to
// keep compilers that support two-phase lookup happy.
#include <boost/math/distributions/detail/derived_accessors.hpp>

#if defined (BOOST_MSVC)
# pragma warning(pop)
#endif

#endif // BOOST_MATH_DIST_ARCSINE_HPP
```