113 lines · plain
1/*2 * Copyright Nick Thompson, 20193 * Use, modification and distribution are subject to the4 * Boost Software License, Version 1.0. (See accompanying file5 * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)6 */7 8#ifndef BOOST_MATH_STATISTICS_ANDERSON_DARLING_HPP9#define BOOST_MATH_STATISTICS_ANDERSON_DARLING_HPP10 11#include <cmath>12#include <algorithm>13#include <boost/math/statistics/univariate_statistics.hpp>14#include <boost/math/special_functions/erf.hpp>15 16namespace boost { namespace math { namespace statistics {17 18template<class RandomAccessContainer>19auto anderson_darling_normality_statistic(RandomAccessContainer const & v,20 typename RandomAccessContainer::value_type mu = std::numeric_limits<typename RandomAccessContainer::value_type>::quiet_NaN(),21 typename RandomAccessContainer::value_type sd = std::numeric_limits<typename RandomAccessContainer::value_type>::quiet_NaN())22{23 using Real = typename RandomAccessContainer::value_type;24 using std::log;25 using std::sqrt;26 using boost::math::erfc;27 28 if (std::isnan(mu)) {29 mu = boost::math::statistics::mean(v);30 }31 if (std::isnan(sd)) {32 sd = sqrt(boost::math::statistics::sample_variance(v));33 }34 35 typedef boost::math::policies::policy<36 boost::math::policies::promote_float<false>,37 boost::math::policies::promote_double<false> >38 no_promote_policy;39 40 // This is where Knuth's literate programming could really come in handy!41 // I need some LaTeX. The idea is that before any observation, the ecdf is identically zero.42 // So we need to compute:43 // \int_{-\infty}^{v_0} \frac{F(x)F'(x)}{1- F(x)} \, \mathrm{d}x, where F(x) := \frac{1}{2}[1+\erf(\frac{x-\mu}{\sigma \sqrt{2}})]44 // Astonishingly, there is an analytic evaluation to this integral, as you can validate with the following Mathematica command:45 // Integrate[(1/2 (1 + Erf[(x - mu)/Sqrt[2*sigma^2]])*Exp[-(x - mu)^2/(2*sigma^2)]*1/Sqrt[2*\[Pi]*sigma^2])/(1 - 1/2 (1 + Erf[(x - mu)/Sqrt[2*sigma^2]])),46 // {x, -Infinity, x0}, Assumptions -> {x0 \[Element] Reals && mu \[Element] Reals && sigma > 0}]47 // This gives (for s = x-mu/sqrt(2sigma^2))48 // -1/2 + erf(s) + log(2/(1+erf(s)))49 50 51 Real inv_var_scale = 1/(sd*sqrt(Real(2)));52 Real s0 = (v[0] - mu)*inv_var_scale;53 Real erfcs0 = erfc(s0, no_promote_policy());54 // Note that if erfcs0 == 0, then left_tail = inf (numerically), and hence the entire integral is numerically infinite:55 if (erfcs0 <= 0) {56 return std::numeric_limits<Real>::infinity();57 }58 59 // Note that we're going to add erfcs0/2 when we compute the integral over [x_0, x_1], so drop it here:60 Real left_tail = -1 + log(Real(2));61 62 63 // For the right tail, the ecdf is identically 1.64 // Hence we need the integral:65 // \int_{v_{n-1}}^{\infty} \frac{(1-F(x))F'(x)}{F(x)} \, \mathrm{d}x66 // This also has an analytic evaluation! It can be found via the following Mathematica command:67 // Integrate[(E^(-(z^2/2)) *(1 - 1/2 (1 + Erf[z/Sqrt[2]])))/(Sqrt[2 \[Pi]] (1/2 (1 + Erf[z/Sqrt[2]]))),68 // {z, zn, \[Infinity]}, Assumptions -> {zn \[Element] Reals && mu \[Element] Reals}]69 // This gives (for sf = xf-mu/sqrt(2sigma^2))70 // -1/2 + erf(sf)/2 + 2log(2/(1+erf(sf)))71 72 Real sf = (v[v.size()-1] - mu)*inv_var_scale;73 //Real erfcsf = erfc<Real>(sf, no_promote_policy());74 // This is the actual value of the tail integral. However, the -erfcsf/2 cancels from the integral over [v_{n-2}, v_{n-1}]:75 //Real right_tail = -erfcsf/2 + log(Real(2)) - log(2-erfcsf);76 77 // Use erfc(-x) = 2 - erfc(x)78 Real erfcmsf = erfc<Real>(-sf, no_promote_policy());79 // Again if this is precisely zero then the integral is numerically infinite:80 if (erfcmsf == 0) {81 return std::numeric_limits<Real>::infinity();82 }83 Real right_tail = log(2/erfcmsf);84 85 // Now we need each integral:86 // \int_{v_i}^{v_{i+1}} \frac{(i+1/n - F(x))^2F'(x)}{F(x)(1-F(x))} \, \mathrm{d}x87 // Again we get an analytical evaluation via the following Mathematica command:88 // Integrate[((E^(-(z^2/2))/Sqrt[2 \[Pi]])*(k1 - F[z])^2)/(F[z]*(1 - F[z])),89 // {z, z1, z2}, Assumptions -> {z1 \[Element] Reals && z2 \[Element] Reals &&k1 \[Element] Reals}] // FullSimplify90 91 Real integrals = 0;92 int64_t N = v.size();93 for (int64_t i = 0; i < N - 1; ++i) {94 if (v[i] > v[i+1]) {95 throw std::domain_error("Input data must be sorted in increasing order v[0] <= v[1] <= . . . <= v[n-1]");96 }97 98 Real k = (i+1)/Real(N);99 Real s1 = (v[i+1]-mu)*inv_var_scale;100 Real erfcs1 = erfc<Real>(s1, no_promote_policy());101 Real term = k*(k*log(erfcs0*(-2 + erfcs1)/(erfcs1*(-2 + erfcs0))) + 2*log(erfcs1/erfcs0));102 103 integrals += term;104 s0 = s1;105 erfcs0 = erfcs1;106 }107 integrals -= log(erfcs0);108 return v.size()*(left_tail + right_tail + integrals);109}110 111}}}112#endif113