test_beta_dist.cpp

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// test_beta_dist.cpp

// Copyright John Maddock 2006.
// Copyright  Paul A. Bristow 2007.

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt
// or copy at http://www.boost.org/LICENSE_1_0.txt)

// Basic sanity tests for the beta Distribution.

// http://members.aol.com/iandjmsmith/BETAEX.HTM  beta distribution calculator
// Appreas to be a 64-bit calculator showing 17 decimal digit (last is noisy).
// Similar to mathCAD?

// http://www.nuhertz.com/statmat/distributions.html#Beta
// Pretty graphs and explanations for most distributions.

// http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp
// provided 40 decimal digits accuracy incomplete beta aka beta regularized == cdf

// http://www.ausvet.com.au/pprev/content.php?page=PPscript
// mode 0.75    5/95% 0.9    alpha 7.39    beta 3.13
// http://www.epi.ucdavis.edu/diagnostictests/betabuster.html
// Beta Buster also calculates alpha and beta from mode & percentile estimates.
// This is NOT (yet) implemented.

#ifdef _MSC_VER
#  pragma warning(disable: 4127) // conditional expression is constant.
#endif

#include <boost/math/concepts/real_concept.hpp> // for real_concept
using ::boost::math::concepts::real_concept;

#include <boost/math/distributions/beta.hpp> // for beta_distribution
using boost::math::beta_distribution;
using boost::math::beta;

#include <boost/test/included/test_exec_monitor.hpp> // for test_main
#include <boost/test/floating_point_comparison.hpp> // for BOOST_CHECK_CLOSE_FRACTION

#include <iostream>
using std::cout;
using std::endl;
#include <limits>
using std::numeric_limits;

template <class RealType>
void test_spot(
     RealType a,    // alpha a
     RealType b,    // beta b
     RealType x,    // Probability 
     RealType P,    // CDF of beta(a, b)
     RealType Q,    // Complement of CDF
     RealType tol)  // Test tolerance.
{
   boost::math::beta_distribution<RealType> abeta(a, b);
   BOOST_CHECK_CLOSE_FRACTION(cdf(abeta, x), P, tol);
   if((P < 0.99) && (Q < 0.99))
   {  // We can only check this if P is not too close to 1,
      // so that we can guarantee that Q is free of error,
      // (and similarly for Q)
      BOOST_CHECK_CLOSE_FRACTION(
         cdf(complement(abeta, x)), Q, tol);
      if(x != 0)
      {
         BOOST_CHECK_CLOSE_FRACTION(
            quantile(abeta, P), x, tol);
      }
      else
      {
         // Just check quantile is very small:
         if((std::numeric_limits<RealType>::max_exponent <= std::numeric_limits<double>::max_exponent)
           && (boost::is_floating_point<RealType>::value))
         {
            // Limit where this is checked: if exponent range is very large we may
            // run out of iterations in our root finding algorithm.
            BOOST_CHECK(quantile(abeta, P) < boost::math::tools::epsilon<RealType>() * 10);
         }
      } // if k
      if(x != 0)
      {
         BOOST_CHECK_CLOSE_FRACTION(quantile(complement(abeta, Q)), x, tol);
      }
      else
      {  // Just check quantile is very small:
         if((std::numeric_limits<RealType>::max_exponent <= std::numeric_limits<double>::max_exponent) && (boost::is_floating_point<RealType>::value))
         {  // Limit where this is checked: if exponent range is very large we may
            // run out of iterations in our root finding algorithm.
            BOOST_CHECK(quantile(complement(abeta, Q)) < boost::math::tools::epsilon<RealType>() * 10);
         }
      } // if x
      // Estimate alpha & beta from mean and variance:

      BOOST_CHECK_CLOSE_FRACTION(
         beta_distribution<RealType>::find_alpha(mean(abeta), variance(abeta)),
         abeta.alpha(), tol);
      BOOST_CHECK_CLOSE_FRACTION(
         beta_distribution<RealType>::find_beta(mean(abeta), variance(abeta)),
         abeta.beta(), tol);

      // Estimate sample alpha and beta from others:
      BOOST_CHECK_CLOSE_FRACTION(
         beta_distribution<RealType>::find_alpha(abeta.beta(), x, P),
         abeta.alpha(), tol);
      BOOST_CHECK_CLOSE_FRACTION(
         beta_distribution<RealType>::find_beta(abeta.alpha(), x, P),
         abeta.beta(), tol);
   } // if((P < 0.99) && (Q < 0.99)
  
} // template <class RealType> void test_spot

template <class RealType> // Any floating-point type RealType.
void test_spots(RealType)
{
  // Basic sanity checks with 'known good' values.
  // MathCAD test data is to double precision only,
  // so set tolerance to 100 eps expressed as a fraction, or
  // 100 eps of type double expressed as a fraction,
  // whichever is the larger.

  RealType tolerance = (std::max)
      (boost::math::tools::epsilon<RealType>(),
      static_cast<RealType>(std::numeric_limits<double>::epsilon())); // 0 if real_concept.

   cout << "Boost::math::tools::epsilon = " << boost::math::tools::epsilon<RealType>() <<endl;
   cout << "std::numeric_limits::epsilon = " << std::numeric_limits<RealType>::epsilon() <<endl;
   cout << "epsilon = " << tolerance;

   tolerance *= 1000; // Note: NO * 100 because is fraction, NOT %.
   cout  << ", Tolerance = " << tolerance * 100 << "%." << endl;

  // RealType teneps = boost::math::tools::epsilon<RealType>() * 10;

  // Sources of spot test values:

  // MathCAD defines dbeta(x, s1, s2) pdf, s1 == alpha, s2 = beta, x = x in Wolfram
  // pbeta(x, s1, s2) cdf and qbeta(x, s1, s2) inverse of cdf
  // returns pr(X ,= x) when random variable X
  // has the beta distribution with parameters s1)alpha) and s2(beta).
  // s1 > 0 and s2 >0 and 0 < x < 1 (but allows x == 0! and x == 1!)
  // dbeta(0,1,1) = 0
  // dbeta(0.5,1,1) = 1

  using boost::math::beta_distribution;
  using  ::boost::math::cdf;
  using  ::boost::math::pdf;

  // Tests that should throw:
  BOOST_CHECK_THROW(mode(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1))), std::domain_error);
  // mode is undefined, and throws domain_error!

 // BOOST_CHECK_THROW(median(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1))), std::domain_error);
  // median is undefined, and throws domain_error!
  // But now median IS provided via derived accessor as quantile(half).


  BOOST_CHECK_THROW( // For various bad arguments.
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(-1), static_cast<RealType>(1)), // bad alpha < 0.
          static_cast<RealType>(1)), std::domain_error);

  BOOST_CHECK_THROW( 
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(0), static_cast<RealType>(1)), // bad alpha == 0.
          static_cast<RealType>(1)), std::domain_error);

  BOOST_CHECK_THROW(
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(0)), // bad beta == 0.
          static_cast<RealType>(1)), std::domain_error);

  BOOST_CHECK_THROW(
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(-1)), // bad beta < 0.
          static_cast<RealType>(1)), std::domain_error);

  BOOST_CHECK_THROW(
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)), // bad x < 0.
          static_cast<RealType>(-1)), std::domain_error);

  BOOST_CHECK_THROW(
       pdf(
          beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)), // bad x > 1.
          static_cast<RealType>(999)), std::domain_error);

  // Some exact pdf values.

  BOOST_CHECK_EQUAL( // a = b = 1 is uniform distribution.
     pdf(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)),
     static_cast<RealType>(1)),  // x
     static_cast<RealType>(1));
  BOOST_CHECK_EQUAL(
     pdf(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)),
     static_cast<RealType>(0)),  // x
     static_cast<RealType>(1));
  BOOST_CHECK_CLOSE_FRACTION(
     pdf(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)),
     static_cast<RealType>(0.5)),  // x
     static_cast<RealType>(1),
     tolerance);

  BOOST_CHECK_EQUAL(
     beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)).alpha(),
     static_cast<RealType>(1) ); // 

  BOOST_CHECK_EQUAL(
     mean(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1))),
     static_cast<RealType>(0.5) ); // Exact one half.

  BOOST_CHECK_CLOSE_FRACTION(
     pdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.5)),  // x
     static_cast<RealType>(1.5), // Exactly 3/2
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     pdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.5)),  // x
     static_cast<RealType>(1.5), // Exactly 3/2
      tolerance);

  // CDF
  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.1)),  // x
     static_cast<RealType>(0.02800000000000000000000000000000000000000L), // Seems exact.
     // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.1&a=2&b=2&digits=40
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.0001)),  // x
     static_cast<RealType>(2.999800000000000000000000000000000000000e-8L),
     // http://members.aol.com/iandjmsmith/BETAEX.HTM 2.9998000000004
     // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.0001&a=2&b=2&digits=40
      tolerance);


  BOOST_CHECK_CLOSE_FRACTION(
     pdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.0001)),  // x
     static_cast<RealType>(0.0005999400000000004L), // http://members.aol.com/iandjmsmith/BETAEX.HTM
     // Slightly higher tolerance for real concept:
     (std::numeric_limits<RealType>::is_specialized ? 1 : 10) * tolerance);  


  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.9999)),  // x
     static_cast<RealType>(0.999999970002L), // http://members.aol.com/iandjmsmith/BETAEX.HTM
     // Wolfram 0.9999999700020000000000000000000000000000
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(2)),
     static_cast<RealType>(0.9)),  // x
     static_cast<RealType>(0.9961174629530394895796514664963063381217L), 
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.1)),  // x
     static_cast<RealType>(0.2048327646991334516491978475505189480977L), 
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.9)),  // x
     static_cast<RealType>(0.7951672353008665483508021524494810519023L), 
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     quantile(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.7951672353008665483508021524494810519023L)),  // x
     static_cast<RealType>(0.9), 
     // Wolfram
     tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.6)),  // x
     static_cast<RealType>(0.5640942168489749316118742861695149357858L), 
     // Wolfram
      tolerance);