test_beta_dist.cpp

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


  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.6)),  // x
     static_cast<RealType>(0.1778078083562213736802876784474931812329L), 
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     quantile(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.1778078083562213736802876784474931812329L)),  // x
     static_cast<RealType>(0.6), 
     // Wolfram
      tolerance); // gives 

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)),
     static_cast<RealType>(0.1)),  // x 
     static_cast<RealType>(0.1),  // 0.1000000000000000000000000000000000000000
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     quantile(beta_distribution<RealType>(static_cast<RealType>(1), static_cast<RealType>(1)),
     static_cast<RealType>(0.1)),  // x 
     static_cast<RealType>(0.1),  // 0.1000000000000000000000000000000000000000
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     cdf(complement(beta_distribution<RealType>(static_cast<RealType>(0.5), static_cast<RealType>(0.5)),
     static_cast<RealType>(0.1))),  // complement of x
     static_cast<RealType>(0.7951672353008665483508021524494810519023L), 
     // Wolfram
      tolerance);

    BOOST_CHECK_CLOSE_FRACTION(
     quantile(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.0280000000000000000000000000000000000L)),  // x
     static_cast<RealType>(0.1), 
     // Wolfram
      tolerance);


  BOOST_CHECK_CLOSE_FRACTION(
     cdf(complement(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.1))),  // x
     static_cast<RealType>(0.9720000000000000000000000000000000000000L), // Exact.
     // Wolfram
      tolerance);

  BOOST_CHECK_CLOSE_FRACTION(
     pdf(beta_distribution<RealType>(static_cast<RealType>(2), static_cast<RealType>(2)),
     static_cast<RealType>(0.9999)),  // x
     static_cast<RealType>(0.0005999399999999344L), // http://members.aol.com/iandjmsmith/BETAEX.HTM
      tolerance*10); // Note loss of precision calculating 1-p test value.

  //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.

   // These test quantiles and complements, and parameter estimates as well.
  // Spot values using, for example:
  // http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized&ptype=0&z=0.1&a=0.5&b=3&digits=40

  test_spot(
     static_cast<RealType>(1),   // alpha a
     static_cast<RealType>(1),   // beta b
     static_cast<RealType>(0.1), // Probability  p
     static_cast<RealType>(0.1), // Probability of result (CDF of beta), P
     static_cast<RealType>(0.9),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.
  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.1), // Probability  p
     static_cast<RealType>(0.0280000000000000000000000000000000000L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1 - 0.0280000000000000000000000000000000000L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.


  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.5), // Probability  p
     static_cast<RealType>(0.5), // Probability of result (CDF of beta), P
     static_cast<RealType>(0.5),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.9), // Probability  p
     static_cast<RealType>(0.972000000000000), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.972000000000000),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.01), // Probability  p
     static_cast<RealType>(0.0002980000000000000000000000000000000000000L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.0002980000000000000000000000000000000000000L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.001), // Probability  p
     static_cast<RealType>(2.998000000000000000000000000000000000000E-6L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-2.998000000000000000000000000000000000000E-6L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.0001), // Probability  p
     static_cast<RealType>(2.999800000000000000000000000000000000000E-8L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-2.999800000000000000000000000000000000000E-8L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(2),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.99), // Probability  p
     static_cast<RealType>(0.9997020000000000000000000000000000000000L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.9997020000000000000000000000000000000000L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(0.5),   // alpha a
     static_cast<RealType>(2),   // beta b
     static_cast<RealType>(0.5), // Probability  p
     static_cast<RealType>(0.8838834764831844055010554526310612991060L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.8838834764831844055010554526310612991060L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(0.5),   // alpha a
     static_cast<RealType>(3.),   // beta b
     static_cast<RealType>(0.7), // Probability  p
     static_cast<RealType>(0.9903963064097119299191611355232156905687L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.9903963064097119299191611355232156905687L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

  test_spot(
     static_cast<RealType>(0.5),   // alpha a
     static_cast<RealType>(3.),   // beta b
     static_cast<RealType>(0.1), // Probability  p
     static_cast<RealType>(0.5545844446520295253493059553548880128511L), // Probability of result (CDF of beta), P
     static_cast<RealType>(1-0.5545844446520295253493059553548880128511L),  // Complement of CDF Q = 1 - P
     tolerance); // Test tolerance.

} // template <class RealType>void test_spots(RealType)

int test_main(int, char* [])
{
   BOOST_MATH_CONTROL_FP;
   // Check that can generate beta distribution using one convenience methods:
   beta_distribution<> mybeta11(1., 1.); // Using default RealType double.
   // but that
   // boost::math::beta mybeta1(1., 1.); // Using typedef fails.
   // error C2039: 'beta' : is not a member of 'boost::math'
 
   // Basic sanity-check spot values.

   // Some simple checks using double only.
   BOOST_CHECK_EQUAL(mybeta11.alpha(), 1); // 
   BOOST_CHECK_EQUAL(mybeta11.beta(), 1);
   BOOST_CHECK_EQUAL(mean(mybeta11), 0.5); // 1 / (1 + 1) = 1/2 exactly
   BOOST_CHECK_THROW(mode(mybeta11), std::domain_error);
   beta_distribution<> mybeta22(2., 2.); // pdf is dome shape.
   BOOST_CHECK_EQUAL(mode(mybeta22), 0.5); // 2-1 / (2+2-2) = 1/2 exactly.
   beta_distribution<> mybetaH2(0.5, 2.); // 
   beta_distribution<> mybetaH3(0.5, 3.); // 

   // Check a few values using double.
   BOOST_CHECK_EQUAL(pdf(mybeta11, 1), 1); // is uniform unity over 0 to 1,
   BOOST_CHECK_EQUAL(pdf(mybeta11, 0), 1); // including zero and unity.
   // Although these next three have an exact result, internally they're
   // *not* treated as special cases, and may be out by a couple of eps:
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.5), 1.0, 5*std::numeric_limits<double>::epsilon());
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.0001), 1.0, 5*std::numeric_limits<double>::epsilon());
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta11, 0.9999), 1.0, 5*std::numeric_limits<double>::epsilon());
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.1), 0.1, 2 * std::numeric_limits<double>::epsilon());
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.5), 0.5, 2 * std::numeric_limits<double>::epsilon());
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta11, 0.9), 0.9, 2 * std::numeric_limits<double>::epsilon());
   BOOST_CHECK_EQUAL(cdf(mybeta11, 1), 1.); // Exact unity expected.

   double tol = std::numeric_limits<double>::epsilon() * 10;
   BOOST_CHECK_EQUAL(pdf(mybeta22, 1), 0); // is dome shape.
   BOOST_CHECK_EQUAL(pdf(mybeta22, 0), 0);
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.5), 1.5, tol); // top of dome, expect exactly 3/2.
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.0001), 5.9994000000000E-4, tol);
   BOOST_CHECK_CLOSE_FRACTION(pdf(mybeta22, 0.9999), 5.9994000000000E-4, tol*50);

   BOOST_CHECK_EQUAL(cdf(mybeta22, 0.), 0); // cdf is a curved line from 0 to 1.
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.1), 0.028000000000000, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.5), 0.5, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.9), 0.972000000000000, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.0001), 2.999800000000000000000000000000000000000E-8, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.001), 2.998000000000000000000000000000000000000E-6, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.01), 0.0002980000000000000000000000000000000000000, tol);
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.1), 0.02800000000000000000000000000000000000000, tol); // exact
   BOOST_CHECK_CLOSE_FRACTION(cdf(mybeta22, 0.99), 0.9997020000000000000000000000000000000000, tol);

   BOOST_CHECK_EQUAL(cdf(mybeta22, 1), 1.); // Exact unity expected.

   // Complement

   BOOST_CHECK_CLOSE_FRACTION(cdf(complement(mybeta22, 0.9)), 0.028000000000000, tol);

   // quantile.
   BOOST_CHECK_CLOSE_FRACTION(quantile(mybeta22, 0.028), 0.1, tol); 
   BOOST_CHECK_CLOSE_FRACTION(quantile(complement(mybeta22, 1 - 0.028)), 0.1, tol);
   BOOST_CHECK_EQUAL(kurtosis(mybeta11), 3+ kurtosis_excess(mybeta11)); // Check kurtosis_excess = kurtosis - 3;
   BOOST_CHECK_CLOSE_FRACTION(variance(mybeta22), 0.05, tol);
   BOOST_CHECK_CLOSE_FRACTION(mean(mybeta22), 0.5, tol);
   BOOST_CHECK_CLOSE_FRACTION(mode(mybeta22), 0.5, tol);
   BOOST_CHECK_CLOSE_FRACTION(median(mybeta22), 0.5, tol);

   BOOST_CHECK_CLOSE_FRACTION(skewness(mybeta22), 0.0, tol);
   BOOST_CHECK_CLOSE_FRACTION(kurtosis_excess(mybeta22), -144.0 / 168, tol);
   BOOST_CHECK_CLOSE_FRACTION(skewness(beta_distribution<>(3, 5)), 0.30983866769659335081434123198259, tol);

   BOOST_CHECK_CLOSE_FRACTION(beta_distribution<double>::find_alpha(mean(mybeta22), variance(mybeta22)), mybeta22.alpha(), tol); // mean, variance, probability. 
   BOOST_CHECK_CLOSE_FRACTION(beta_distribution<double>::find_beta(mean(mybeta22), variance(mybeta22)), mybeta22.beta(), tol);// mean, variance, probability. 

   BOOST_CHECK_CLOSE_FRACTION(mybeta22.find_alpha(mybeta22.beta(), 0.8, cdf(mybeta22, 0.8)), mybeta22.alpha(), tol);
   BOOST_CHECK_CLOSE_FRACTION(mybeta22.find_beta(mybeta22.alpha(), 0.8, cdf(mybeta22, 0.8)), mybeta22.beta(), tol);


   beta_distribution<real_concept> rcbeta22(2, 2); // Using RealType real_concept.
   cout << "numeric_limits<real_concept>::is_specialized " << numeric_limits<real_concept>::is_specialized << endl;
   cout << "numeric_limits<real_concept>::digits " << numeric_limits<real_concept>::digits << endl;
   cout << "numeric_limits<real_concept>::digits10 " << numeric_limits<real_concept>::digits10 << endl;
   cout << "numeric_limits<real_concept>::epsilon " << numeric_limits<real_concept>::epsilon() << endl;

   // (Parameter value, arbitrarily zero, only communicates the floating point type).
   test_spots(0.0F); // Test float.
   test_spots(0.0); // Test double.
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
   test_spots(0.0L); // Test long double.
#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
   test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
#endif
#endif
   return 0;
} // int test_main(int, char* [])

/*

Output is:

-Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_beta_dist.exe"
Running 1 test case...
numeric_limits<real_concept>::is_specialized 0
numeric_limits<real_concept>::digits 0
numeric_limits<real_concept>::digits10 0
numeric_limits<real_concept>::epsilon 0
Boost::math::tools::epsilon = 1.19209e-007
std::numeric_limits::epsilon = 1.19209e-007
epsilon = 1.19209e-007, Tolerance = 0.0119209%.
Boost::math::tools::epsilon = 2.22045e-016
std::numeric_limits::epsilon = 2.22045e-016
epsilon = 2.22045e-016, Tolerance = 2.22045e-011%.
Boost::math::tools::epsilon = 2.22045e-016
std::numeric_limits::epsilon = 2.22045e-016
epsilon = 2.22045e-016, Tolerance = 2.22045e-011%.
Boost::math::tools::epsilon = 2.22045e-016
std::numeric_limits::epsilon = 0
epsilon = 2.22045e-016, Tolerance = 2.22045e-011%.
*** No errors detected
*/