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// (C) Copyright Nick Thompson 2018. // 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) #ifndef BOOST_MATH_TOOLS_UNIVARIATE_STATISTICS_HPP #define BOOST_MATH_TOOLS_UNIVARIATE_STATISTICS_HPP #include #include #include #include #include #include #ifndef BOOST_MATH_STANDALONE #include #ifdef BOOST_MATH_NO_CXX17_IF_CONSTEXPR #error "The header can only be used in C++17 and later." #endif #endif BOOST_MATH_HEADER_DEPRECATED(""); namespace boost::math::tools { template auto mean(ForwardIterator first, ForwardIterator last) { using Real = typename std::iterator_traits::value_type; BOOST_MATH_ASSERT_MSG(first != last, "At least one sample is required to compute the mean."); if constexpr (std::is_integral::value) { double mu = 0; double i = 1; for(auto it = first; it != last; ++it) { mu = mu + (*it - mu)/i; i += 1; } return mu; } else if constexpr (std::is_same_v::iterator_category, std::random_access_iterator_tag>) { size_t elements = std::distance(first, last); Real mu0 = 0; Real mu1 = 0; Real mu2 = 0; Real mu3 = 0; Real i = 1; auto end = last - (elements % 4); for(auto it = first; it != end; it += 4) { Real inv = Real(1)/i; Real tmp0 = (*it - mu0); Real tmp1 = (*(it+1) - mu1); Real tmp2 = (*(it+2) - mu2); Real tmp3 = (*(it+3) - mu3); // please generate a vectorized fma here mu0 += tmp0*inv; mu1 += tmp1*inv; mu2 += tmp2*inv; mu3 += tmp3*inv; i += 1; } Real num1 = Real(elements - (elements %4))/Real(4); Real num2 = num1 + Real(elements % 4); for (auto it = end; it != last; ++it) { mu3 += (*it-mu3)/i; i += 1; } return (num1*(mu0+mu1+mu2) + num2*mu3)/Real(elements); } else { auto it = first; Real mu = *it; Real i = 2; while(++it != last) { mu += (*it - mu)/i; i += 1; } return mu; } } template inline auto mean(Container const & v) { return mean(v.cbegin(), v.cend()); } template auto variance(ForwardIterator first, ForwardIterator last) { using Real = typename std::iterator_traits::value_type; BOOST_MATH_ASSERT_MSG(first != last, "At least one sample is required to compute mean and variance."); // Higham, Accuracy and Stability, equation 1.6a and 1.6b: if constexpr (std::is_integral::value) { double M = *first; double Q = 0; double k = 2; for (auto it = std::next(first); it != last; ++it) { double tmp = *it - M; Q = Q + ((k-1)*tmp*tmp)/k; M = M + tmp/k; k += 1; } return Q/(k-1); } else { Real M = *first; Real Q = 0; Real k = 2; for (auto it = std::next(first); it != last; ++it) { Real tmp = (*it - M)/k; Q += k*(k-1)*tmp*tmp; M += tmp; k += 1; } return Q/(k-1); } } template inline auto variance(Container const & v) { return variance(v.cbegin(), v.cend()); } template auto sample_variance(ForwardIterator first, ForwardIterator last) { size_t n = std::distance(first, last); BOOST_MATH_ASSERT_MSG(n > 1, "At least two samples are required to compute the sample variance."); return n*variance(first, last)/(n-1); } template inline auto sample_variance(Container const & v) { return sample_variance(v.cbegin(), v.cend()); } // Follows equation 1.5 of: // https://prod.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf template auto skewness(ForwardIterator first, ForwardIterator last) { using Real = typename std::iterator_traits::value_type; BOOST_MATH_ASSERT_MSG(first != last, "At least one sample is required to compute skewness."); if constexpr (std::is_integral::value) { double M1 = *first; double M2 = 0; double M3 = 0; double n = 2; for (auto it = std::next(first); it != last; ++it) { double delta21 = *it - M1; double tmp = delta21/n; M3 = M3 + tmp*((n-1)*(n-2)*delta21*tmp - 3*M2); M2 = M2 + tmp*(n-1)*delta21; M1 = M1 + tmp; n += 1; } double var = M2/(n-1); if (var == 0) { // The limit is technically undefined, but the interpretation here is clear: // A constant dataset has no skewness. return static_cast(0); } double skew = M3/(M2*sqrt(var)); return skew; } else { Real M1 = *first; Real M2 = 0; Real M3 = 0; Real n = 2; for (auto it = std::next(first); it != last; ++it) { Real delta21 = *it - M1; Real tmp = delta21/n; M3 += tmp*((n-1)*(n-2)*delta21*tmp - 3*M2); M2 += tmp*(n-1)*delta21; M1 += tmp; n += 1; } Real var = M2/(n-1); if (var == 0) { // The limit is technically undefined, but the interpretation here is clear: // A constant dataset has no skewness. return Real(0); } Real skew = M3/(M2*sqrt(var)); return skew; } } template inline auto skewness(Container const & v) { return skewness(v.cbegin(), v.cend()); } // Follows equation 1.5/1.6 of: // https://prod.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf template auto first_four_moments(ForwardIterator first, ForwardIterator last) { using Real = typename std::iterator_traits::value_type; BOOST_MATH_ASSERT_MSG(first != last, "At least one sample is required to compute the first four moments."); if constexpr (std::is_integral::value) { double M1 = *first; double M2 = 0; double M3 = 0; double M4 = 0; double n = 2; for (auto it = std::next(first); it != last; ++it) { double delta21 = *it - M1; double tmp = delta21/n; M4 = M4 + tmp*(tmp*tmp*delta21*((n-1)*(n*n-3*n+3)) + 6*tmp*M2 - 4*M3); M3 = M3 + tmp*((n-1)*(n-2)*delta21*tmp - 3*M2); M2 = M2 + tmp*(n-1)*delta21; M1 = M1 + tmp; n += 1; } return std::make_tuple(M1, M2/(n-1), M3/(n-1), M4/(n-1)); } else { Real M1 = *first; Real M2 = 0; Real M3 = 0; Real M4 = 0; Real n = 2; for (auto it = std::next(first); it != last; ++it) { Real delta21 = *it - M1; Real tmp = delta21/n; M4 = M4 + tmp*(tmp*tmp*delta21*((n-1)*(n*n-3*n+3)) + 6*tmp*M2 - 4*M3); M3 = M3 + tmp*((n-1)*(n-2)*delta21*tmp - 3*M2); M2 = M2 + tmp*(n-1)*delta21; M1 = M1 + tmp; n += 1; } return std::make_tuple(M1, M2/(n-1), M3/(n-1), M4/(n-1)); } } template inline auto first_four_moments(Container const & v) { return first_four_moments(v.cbegin(), v.cend()); } // Follows equation 1.6 of: // https://prod.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf template auto kurtosis(ForwardIterator first, ForwardIterator last) { auto [M1, M2, M3, M4] = first_four_moments(first, last); if (M2 == 0) { return M2; } return M4/(M2*M2); } template inline auto kurtosis(Container const & v) { return kurtosis(v.cbegin(), v.cend()); } template auto excess_kurtosis(ForwardIterator first, ForwardIterator last) { return kurtosis(first, last) - 3; } template inline auto excess_kurtosis(Container const & v) { return excess_kurtosis(v.cbegin(), v.cend()); } template auto median(RandomAccessIterator first, RandomAccessIterator last) { size_t num_elems = std::distance(first, last); BOOST_MATH_ASSERT_MSG(num_elems > 0, "The median of a zero length vector is undefined."); if (num_elems & 1) { auto middle = first + (num_elems - 1)/2; std::nth_element(first, middle, last); return *middle; } else { auto middle = first + num_elems/2 - 1; std::nth_element(first, middle, last); std::nth_element(middle, middle+1, last); return (*middle + *(middle+1))/2; } } template inline auto median(RandomAccessContainer & v) { return median(v.begin(), v.end()); } template auto gini_coefficient(RandomAccessIterator first, RandomAccessIterator last) { using Real = typename std::iterator_traits::value_type; BOOST_MATH_ASSERT_MSG(first != last && std::next(first) != last, "Computation of the Gini coefficient requires at least two samples."); std::sort(first, last); if constexpr (std::is_integral::value) { double i = 1; double num = 0; double denom = 0; for (auto it = first; it != last; ++it) { num += *it*i; denom += *it; ++i; } // If the l1 norm is zero, all elements are zero, so every element is the same. if (denom == 0) { return static_cast(0); } return ((2*num)/denom - i)/(i-1); } else { Real i = 1; Real num = 0; Real denom = 0; for (auto it = first; it != last; ++it) { num += *it*i; denom += *it; ++i; } // If the l1 norm is zero, all elements are zero, so every element is the same. if (denom == 0) { return Real(0); } return ((2*num)/denom - i)/(i-1); } } template inline auto gini_coefficient(RandomAccessContainer & v) { return gini_coefficient(v.begin(), v.end()); } template inline auto sample_gini_coefficient(RandomAccessIterator first, RandomAccessIterator last) { size_t n = std::distance(first, last); return n*gini_coefficient(first, last)/(n-1); } template inline auto sample_gini_coefficient(RandomAccessContainer & v) { return sample_gini_coefficient(v.begin(), v.end()); } template auto median_absolute_deviation(RandomAccessIterator first, RandomAccessIterator last, typename std::iterator_traits::value_type center=std::numeric_limits::value_type>::quiet_NaN()) { using std::abs; using Real = typename std::iterator_traits::value_type; using std::isnan; if (isnan(center)) { center = boost::math::tools::median(first, last); } size_t num_elems = std::distance(first, last); BOOST_MATH_ASSERT_MSG(num_elems > 0, "The median of a zero-length vector is undefined."); auto comparator = [¢er](Real a, Real b) { return abs(a-center) < abs(b-center);}; if (num_elems & 1) { auto middle = first + (num_elems - 1)/2; std::nth_element(first, middle, last, comparator); return abs(*middle); } else { auto middle = first + num_elems/2 - 1; std::nth_element(first, middle, last, comparator); std::nth_element(middle, middle+1, last, comparator); return (abs(*middle) + abs(*(middle+1)))/abs(static_cast(2)); } } template inline auto median_absolute_deviation(RandomAccessContainer & v, typename RandomAccessContainer::value_type center=std::numeric_limits::quiet_NaN()) { return median_absolute_deviation(v.begin(), v.end(), center); } } #endif