// (C) Copyright Nick Thompson 2018.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_TOOLS_SIGNAL_STATISTICS_HPP
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#define BOOST_MATH_TOOLS_SIGNAL_STATISTICS_HPP
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#include <algorithm>
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#include <iterator>
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#include <boost/assert.hpp>
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#include <boost/math/tools/complex.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/statistics/univariate_statistics.hpp>
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#include <boost/config/header_deprecated.hpp>
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BOOST_HEADER_DEPRECATED("<boost/math/statistics/signal_statistics.hpp>");
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namespace boost::math::tools {
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template<class ForwardIterator>
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auto absolute_gini_coefficient(ForwardIterator first, ForwardIterator last)
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{
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using std::abs;
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using RealOrComplex = typename std::iterator_traits<ForwardIterator>::value_type;
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BOOST_ASSERT_MSG(first != last && std::next(first) != last, "Computation of the Gini coefficient requires at least two samples.");
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std::sort(first, last, [](RealOrComplex a, RealOrComplex b) { return abs(b) > abs(a); });
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decltype(abs(*first)) i = 1;
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decltype(abs(*first)) num = 0;
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decltype(abs(*first)) denom = 0;
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for (auto it = first; it != last; ++it)
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{
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decltype(abs(*first)) tmp = abs(*it);
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num += tmp*i;
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denom += tmp;
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++i;
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}
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// If the l1 norm is zero, all elements are zero, so every element is the same.
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if (denom == 0)
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{
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decltype(abs(*first)) zero = 0;
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return zero;
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}
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return ((2*num)/denom - i)/(i-1);
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}
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template<class RandomAccessContainer>
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inline auto absolute_gini_coefficient(RandomAccessContainer & v)
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{
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return boost::math::tools::absolute_gini_coefficient(v.begin(), v.end());
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}
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template<class ForwardIterator>
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auto sample_absolute_gini_coefficient(ForwardIterator first, ForwardIterator last)
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{
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size_t n = std::distance(first, last);
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return n*boost::math::tools::absolute_gini_coefficient(first, last)/(n-1);
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}
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template<class RandomAccessContainer>
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inline auto sample_absolute_gini_coefficient(RandomAccessContainer & v)
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{
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return boost::math::tools::sample_absolute_gini_coefficient(v.begin(), v.end());
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}
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// The Hoyer sparsity measure is defined in:
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// https://arxiv.org/pdf/0811.4706.pdf
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template<class ForwardIterator>
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auto hoyer_sparsity(const ForwardIterator first, const ForwardIterator last)
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{
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using T = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::abs;
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using std::sqrt;
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BOOST_ASSERT_MSG(first != last && std::next(first) != last, "Computation of the Hoyer sparsity requires at least two samples.");
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if constexpr (std::is_unsigned<T>::value)
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{
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T l1 = 0;
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T l2 = 0;
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size_t n = 0;
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for (auto it = first; it != last; ++it)
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{
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l1 += *it;
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l2 += (*it)*(*it);
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n += 1;
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}
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double rootn = sqrt(n);
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return (rootn - l1/sqrt(l2) )/ (rootn - 1);
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}
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else {
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decltype(abs(*first)) l1 = 0;
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decltype(abs(*first)) l2 = 0;
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// We wouldn't need to count the elements if it was a random access iterator,
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// but our only constraint is that it's a forward iterator.
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size_t n = 0;
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for (auto it = first; it != last; ++it)
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{
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decltype(abs(*first)) tmp = abs(*it);
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l1 += tmp;
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l2 += tmp*tmp;
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n += 1;
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}
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if constexpr (std::is_integral<T>::value)
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{
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double rootn = sqrt(n);
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return (rootn - l1/sqrt(l2) )/ (rootn - 1);
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}
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else
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{
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decltype(abs(*first)) rootn = sqrt(static_cast<decltype(abs(*first))>(n));
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return (rootn - l1/sqrt(l2) )/ (rootn - 1);
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}
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}
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}
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template<class Container>
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inline auto hoyer_sparsity(Container const & v)
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{
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return boost::math::tools::hoyer_sparsity(v.cbegin(), v.cend());
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}
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template<class Container>
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auto oracle_snr(Container const & signal, Container const & noisy_signal)
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{
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using Real = typename Container::value_type;
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BOOST_ASSERT_MSG(signal.size() == noisy_signal.size(),
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"Signal and noisy_signal must be have the same number of elements.");
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if constexpr (std::is_integral<Real>::value)
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{
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double numerator = 0;
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double denominator = 0;
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for (size_t i = 0; i < signal.size(); ++i)
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{
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numerator += signal[i]*signal[i];
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denominator += (noisy_signal[i] - signal[i])*(noisy_signal[i] - signal[i]);
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}
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if (numerator == 0 && denominator == 0)
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{
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return std::numeric_limits<double>::quiet_NaN();
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}
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if (denominator == 0)
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{
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return std::numeric_limits<double>::infinity();
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}
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return numerator/denominator;
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}
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else if constexpr (boost::math::tools::is_complex_type<Real>::value)
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{
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using std::norm;
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typename Real::value_type numerator = 0;
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typename Real::value_type denominator = 0;
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for (size_t i = 0; i < signal.size(); ++i)
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{
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numerator += norm(signal[i]);
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denominator += norm(noisy_signal[i] - signal[i]);
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}
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if (numerator == 0 && denominator == 0)
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{
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return std::numeric_limits<typename Real::value_type>::quiet_NaN();
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}
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if (denominator == 0)
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{
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return std::numeric_limits<typename Real::value_type>::infinity();
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}
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return numerator/denominator;
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}
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else
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{
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Real numerator = 0;
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Real denominator = 0;
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for (size_t i = 0; i < signal.size(); ++i)
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{
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numerator += signal[i]*signal[i];
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denominator += (signal[i] - noisy_signal[i])*(signal[i] - noisy_signal[i]);
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}
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if (numerator == 0 && denominator == 0)
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{
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return std::numeric_limits<Real>::quiet_NaN();
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}
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if (denominator == 0)
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{
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return std::numeric_limits<Real>::infinity();
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}
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return numerator/denominator;
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}
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}
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template<class Container>
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auto mean_invariant_oracle_snr(Container const & signal, Container const & noisy_signal)
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{
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using Real = typename Container::value_type;
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BOOST_ASSERT_MSG(signal.size() == noisy_signal.size(), "Signal and noisy signal must be have the same number of elements.");
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Real mu = boost::math::tools::mean(signal);
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Real numerator = 0;
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Real denominator = 0;
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for (size_t i = 0; i < signal.size(); ++i)
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{
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Real tmp = signal[i] - mu;
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numerator += tmp*tmp;
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denominator += (signal[i] - noisy_signal[i])*(signal[i] - noisy_signal[i]);
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}
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if (numerator == 0 && denominator == 0)
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{
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return std::numeric_limits<Real>::quiet_NaN();
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}
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if (denominator == 0)
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{
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return std::numeric_limits<Real>::infinity();
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}
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return numerator/denominator;
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}
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template<class Container>
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auto mean_invariant_oracle_snr_db(Container const & signal, Container const & noisy_signal)
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{
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using std::log10;
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return 10*log10(boost::math::tools::mean_invariant_oracle_snr(signal, noisy_signal));
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}
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// Follows the definition of SNR given in Mallat, A Wavelet Tour of Signal Processing, equation 11.16.
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template<class Container>
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auto oracle_snr_db(Container const & signal, Container const & noisy_signal)
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{
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using std::log10;
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return 10*log10(boost::math::tools::oracle_snr(signal, noisy_signal));
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}
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// A good reference on the M2M4 estimator:
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// D. R. Pauluzzi and N. C. Beaulieu, "A comparison of SNR estimation techniques for the AWGN channel," IEEE Trans. Communications, Vol. 48, No. 10, pp. 1681-1691, 2000.
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// A nice python implementation:
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// https://github.com/gnuradio/gnuradio/blob/master/gr-digital/examples/snr_estimators.py
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template<class ForwardIterator>
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auto m2m4_snr_estimator(ForwardIterator first, ForwardIterator last, decltype(*first) estimated_signal_kurtosis=1, decltype(*first) estimated_noise_kurtosis=3)
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{
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BOOST_ASSERT_MSG(estimated_signal_kurtosis > 0, "The estimated signal kurtosis must be positive");
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BOOST_ASSERT_MSG(estimated_noise_kurtosis > 0, "The estimated noise kurtosis must be positive.");
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using Real = typename std::iterator_traits<ForwardIterator>::value_type;
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using std::sqrt;
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if constexpr (std::is_floating_point<Real>::value || std::numeric_limits<Real>::max_exponent)
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{
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// If we first eliminate N, we obtain the quadratic equation:
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// (ka+kw-6)S^2 + 2M2(3-kw)S + kw*M2^2 - M4 = 0 =: a*S^2 + bs*N + cs = 0
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// If we first eliminate S, we obtain the quadratic equation:
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// (ka+kw-6)N^2 + 2M2(3-ka)N + ka*M2^2 - M4 = 0 =: a*N^2 + bn*N + cn = 0
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// I believe these equations are totally independent quadratics;
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// if one has a complex solution it is not necessarily the case that the other must also.
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// However, I can't prove that, so there is a chance that this does unnecessary work.
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// Future improvements: There are algorithms which can solve quadratics much more effectively than the naive implementation found here.
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// See: https://stackoverflow.com/questions/48979861/numerically-stable-method-for-solving-quadratic-equations/50065711#50065711
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auto [M1, M2, M3, M4] = boost::math::tools::first_four_moments(first, last);
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if (M4 == 0)
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{
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// The signal is constant. There is no noise:
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return std::numeric_limits<Real>::infinity();
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}
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// Change to notation in Pauluzzi, equation 41:
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auto kw = estimated_noise_kurtosis;
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auto ka = estimated_signal_kurtosis;
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// A common case, since it's the default:
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Real a = (ka+kw-6);
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Real bs = 2*M2*(3-kw);
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Real cs = kw*M2*M2 - M4;
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Real bn = 2*M2*(3-ka);
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Real cn = ka*M2*M2 - M4;
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auto [S0, S1] = boost::math::tools::quadratic_roots(a, bs, cs);
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if (S1 > 0)
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{
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auto N = M2 - S1;
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if (N > 0)
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{
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return S1/N;
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}
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if (S0 > 0)
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{
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N = M2 - S0;
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if (N > 0)
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{
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return S0/N;
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}
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}
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}
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auto [N0, N1] = boost::math::tools::quadratic_roots(a, bn, cn);
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if (N1 > 0)
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{
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auto S = M2 - N1;
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if (S > 0)
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{
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return S/N1;
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}
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if (N0 > 0)
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{
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S = M2 - N0;
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if (S > 0)
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{
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return S/N0;
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}
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}
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}
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// This happens distressingly often. It's a limitation of the method.
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return std::numeric_limits<Real>::quiet_NaN();
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}
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else
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{
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BOOST_ASSERT_MSG(false, "The M2M4 estimator has not been implemented for this type.");
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return std::numeric_limits<Real>::quiet_NaN();
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}
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}
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template<class Container>
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inline auto m2m4_snr_estimator(Container const & noisy_signal, typename Container::value_type estimated_signal_kurtosis=1, typename Container::value_type estimated_noise_kurtosis=3)
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{
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return m2m4_snr_estimator(noisy_signal.cbegin(), noisy_signal.cend(), estimated_signal_kurtosis, estimated_noise_kurtosis);
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}
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template<class ForwardIterator>
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inline auto m2m4_snr_estimator_db(ForwardIterator first, ForwardIterator last, decltype(*first) estimated_signal_kurtosis=1, decltype(*first) estimated_noise_kurtosis=3)
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{
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using std::log10;
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return 10*log10(m2m4_snr_estimator(first, last, estimated_signal_kurtosis, estimated_noise_kurtosis));
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}
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template<class Container>
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inline auto m2m4_snr_estimator_db(Container const & noisy_signal, typename Container::value_type estimated_signal_kurtosis=1, typename Container::value_type estimated_noise_kurtosis=3)
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{
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using std::log10;
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return 10*log10(m2m4_snr_estimator(noisy_signal, estimated_signal_kurtosis, estimated_noise_kurtosis));
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}
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}
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#endif
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