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