/////////////////////////////////////////////////////////////////////////////// | |
// peaks_over_threshold.hpp | |
// | |
// Copyright 2006 Daniel Egloff, Olivier Gygi. Distributed under 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_ACCUMULATORS_STATISTICS_PEAKS_OVER_THRESHOLD_HPP_DE_01_01_2006 | |
#define BOOST_ACCUMULATORS_STATISTICS_PEAKS_OVER_THRESHOLD_HPP_DE_01_01_2006 | |
#include <vector> | |
#include <limits> | |
#include <numeric> | |
#include <functional> | |
#include <boost/config/no_tr1/cmath.hpp> // pow | |
#include <sstream> // stringstream | |
#include <stdexcept> // runtime_error | |
#include <boost/throw_exception.hpp> | |
#include <boost/range.hpp> | |
#include <boost/mpl/if.hpp> | |
#include <boost/mpl/int.hpp> | |
#include <boost/mpl/placeholders.hpp> | |
#include <boost/parameter/keyword.hpp> | |
#include <boost/tuple/tuple.hpp> | |
#include <boost/accumulators/framework/accumulator_base.hpp> | |
#include <boost/accumulators/framework/extractor.hpp> | |
#include <boost/accumulators/numeric/functional.hpp> | |
#include <boost/accumulators/framework/parameters/sample.hpp> | |
#include <boost/accumulators/framework/depends_on.hpp> | |
#include <boost/accumulators/statistics_fwd.hpp> | |
#include <boost/accumulators/statistics/parameters/quantile_probability.hpp> | |
#include <boost/accumulators/statistics/count.hpp> | |
#include <boost/accumulators/statistics/tail.hpp> | |
#ifdef _MSC_VER | |
# pragma warning(push) | |
# pragma warning(disable: 4127) // conditional expression is constant | |
#endif | |
namespace boost { namespace accumulators | |
{ | |
/////////////////////////////////////////////////////////////////////////////// | |
// threshold_probability and threshold named parameters | |
// | |
BOOST_PARAMETER_NESTED_KEYWORD(tag, pot_threshold_value, threshold_value) | |
BOOST_PARAMETER_NESTED_KEYWORD(tag, pot_threshold_probability, threshold_probability) | |
namespace impl | |
{ | |
/////////////////////////////////////////////////////////////////////////////// | |
// peaks_over_threshold_impl | |
// works with an explicit threshold value and does not depend on order statistics | |
/** | |
@brief Peaks over Threshold Method for Quantile and Tail Mean Estimation | |
According to the theorem of Pickands-Balkema-de Haan, the distribution function \f$F_u(x)\f$ of | |
the excesses \f$x\f$ over some sufficiently high threshold \f$u\f$ of a distribution function \f$F(x)\f$ | |
may be approximated by a generalized Pareto distribution | |
\f[ | |
G_{\xi,\beta}(x) = | |
\left\{ | |
\begin{array}{ll} | |
\beta^{-1}\left(1+\frac{\xi x}{\beta}\right)^{-1/\xi-1} & \textrm{if }\xi\neq0\\ | |
\beta^{-1}\exp\left(-\frac{x}{\beta}\right) & \textrm{if }\xi=0, | |
\end{array} | |
\right. | |
\f] | |
with suitable parameters \f$\xi\f$ and \f$\beta\f$ that can be estimated, e.g., with the method of moments, cf. | |
Hosking and Wallis (1987), | |
\f[ | |
\begin{array}{lll} | |
\hat{\xi} & = & \frac{1}{2}\left[1-\frac{(\hat{\mu}-u)^2}{\hat{\sigma}^2}\right]\\ | |
\hat{\beta} & = & \frac{\hat{\mu}-u}{2}\left[\frac{(\hat{\mu}-u)^2}{\hat{\sigma}^2}+1\right], | |
\end{array} | |
\f] | |
\f$\hat{\mu}\f$ and \f$\hat{\sigma}^2\f$ being the empirical mean and variance of the samples over | |
the threshold \f$u\f$. Equivalently, the distribution function | |
\f$F_u(x-u)\f$ of the exceedances \f$x-u\f$ can be approximated by | |
\f$G_{\xi,\beta}(x-u)=G_{\xi,\beta,u}(x)\f$. Since for \f$x\geq u\f$ the distribution function \f$F(x)\f$ | |
can be written as | |
\f[ | |
F(x) = [1 - \P(X \leq u)]F_u(x - u) + \P(X \leq u) | |
\f] | |
and the probability \f$\P(X \leq u)\f$ can be approximated by the empirical distribution function | |
\f$F_n(u)\f$ evaluated at \f$u\f$, an estimator of \f$F(x)\f$ is given by | |
\f[ | |
\widehat{F}(x) = [1 - F_n(u)]G_{\xi,\beta,u}(x) + F_n(u). | |
\f] | |
It can be shown that \f$\widehat{F}(x)\f$ is a generalized | |
Pareto distribution \f$G_{\xi,\bar{\beta},\bar{u}}(x)\f$ with \f$\bar{\beta}=\beta[1-F_n(u)]^{\xi}\f$ | |
and \f$\bar{u}=u-\bar{\beta}\left\{[1-F_n(u)]^{-\xi}-1\right\}/\xi\f$. By inverting \f$\widehat{F}(x)\f$, | |
one obtains an estimator for the \f$\alpha\f$-quantile, | |
\f[ | |
\hat{q}_{\alpha} = \bar{u} + \frac{\bar{\beta}}{\xi}\left[(1-\alpha)^{-\xi}-1\right], | |
\f] | |
and similarly an estimator for the (coherent) tail mean, | |
\f[ | |
\widehat{CTM}_{\alpha} = \hat{q}_{\alpha} - \frac{\bar{\beta}}{\xi-1}(1-\alpha)^{-\xi}, | |
\f] | |
cf. McNeil and Frey (2000). | |
Note that in case extreme values of the left tail are fitted, the distribution is mirrored with respect to the | |
\f$y\f$ axis such that the left tail can be treated as a right tail. The computed fit parameters thus define | |
the Pareto distribution that fits the mirrored left tail. When quantities like a quantile or a tail mean are | |
computed using the fit parameters obtained from the mirrored data, the result is mirrored back, yielding the | |
correct result. | |
For further details, see | |
J. R. M. Hosking and J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, | |
Technometrics, Volume 29, 1987, p. 339-349 | |
A. J. McNeil and R. Frey, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: | |
an Extreme Value Approach, Journal of Empirical Finance, Volume 7, 2000, p. 271-300 | |
@param quantile_probability | |
@param pot_threshold_value | |
*/ | |
template<typename Sample, typename LeftRight> | |
struct peaks_over_threshold_impl | |
: accumulator_base | |
{ | |
typedef typename numeric::functional::average<Sample, std::size_t>::result_type float_type; | |
// for boost::result_of | |
typedef boost::tuple<float_type, float_type, float_type> result_type; | |
// for left tail fitting, mirror the extreme values | |
typedef mpl::int_<is_same<LeftRight, left>::value ? -1 : 1> sign; | |
template<typename Args> | |
peaks_over_threshold_impl(Args const &args) | |
: Nu_(0) | |
, mu_(sign::value * numeric::average(args[sample | Sample()], (std::size_t)1)) | |
, sigma2_(numeric::average(args[sample | Sample()], (std::size_t)1)) | |
, threshold_(sign::value * args[pot_threshold_value]) | |
, fit_parameters_(boost::make_tuple(0., 0., 0.)) | |
, is_dirty_(true) | |
{ | |
} | |
template<typename Args> | |
void operator ()(Args const &args) | |
{ | |
this->is_dirty_ = true; | |
if (sign::value * args[sample] > this->threshold_) | |
{ | |
this->mu_ += args[sample]; | |
this->sigma2_ += args[sample] * args[sample]; | |
++this->Nu_; | |
} | |
} | |
template<typename Args> | |
result_type result(Args const &args) const | |
{ | |
if (this->is_dirty_) | |
{ | |
this->is_dirty_ = false; | |
std::size_t cnt = count(args); | |
this->mu_ = sign::value * numeric::average(this->mu_, this->Nu_); | |
this->sigma2_ = numeric::average(this->sigma2_, this->Nu_); | |
this->sigma2_ -= this->mu_ * this->mu_; | |
float_type threshold_probability = numeric::average(cnt - this->Nu_, cnt); | |
float_type tmp = numeric::average(( this->mu_ - this->threshold_ )*( this->mu_ - this->threshold_ ), this->sigma2_); | |
float_type xi_hat = 0.5 * ( 1. - tmp ); | |
float_type beta_hat = 0.5 * ( this->mu_ - this->threshold_ ) * ( 1. + tmp ); | |
float_type beta_bar = beta_hat * std::pow(1. - threshold_probability, xi_hat); | |
float_type u_bar = this->threshold_ - beta_bar * ( std::pow(1. - threshold_probability, -xi_hat) - 1.)/xi_hat; | |
this->fit_parameters_ = boost::make_tuple(u_bar, beta_bar, xi_hat); | |
} | |
return this->fit_parameters_; | |
} | |
private: | |
std::size_t Nu_; // number of samples larger than threshold | |
mutable float_type mu_; // mean of Nu_ largest samples | |
mutable float_type sigma2_; // variance of Nu_ largest samples | |
float_type threshold_; | |
mutable result_type fit_parameters_; // boost::tuple that stores fit parameters | |
mutable bool is_dirty_; | |
}; | |
/////////////////////////////////////////////////////////////////////////////// | |
// peaks_over_threshold_prob_impl | |
// determines threshold from a given threshold probability using order statistics | |
/** | |
@brief Peaks over Threshold Method for Quantile and Tail Mean Estimation | |
@sa peaks_over_threshold_impl | |
@param quantile_probability | |
@param pot_threshold_probability | |
*/ | |
template<typename Sample, typename LeftRight> | |
struct peaks_over_threshold_prob_impl | |
: accumulator_base | |
{ | |
typedef typename numeric::functional::average<Sample, std::size_t>::result_type float_type; | |
// for boost::result_of | |
typedef boost::tuple<float_type, float_type, float_type> result_type; | |
// for left tail fitting, mirror the extreme values | |
typedef mpl::int_<is_same<LeftRight, left>::value ? -1 : 1> sign; | |
template<typename Args> | |
peaks_over_threshold_prob_impl(Args const &args) | |
: mu_(sign::value * numeric::average(args[sample | Sample()], (std::size_t)1)) | |
, sigma2_(numeric::average(args[sample | Sample()], (std::size_t)1)) | |
, threshold_probability_(args[pot_threshold_probability]) | |
, fit_parameters_(boost::make_tuple(0., 0., 0.)) | |
, is_dirty_(true) | |
{ | |
} | |
void operator ()(dont_care) | |
{ | |
this->is_dirty_ = true; | |
} | |
template<typename Args> | |
result_type result(Args const &args) const | |
{ | |
if (this->is_dirty_) | |
{ | |
this->is_dirty_ = false; | |
std::size_t cnt = count(args); | |
// the n'th cached sample provides an approximate threshold value u | |
std::size_t n = static_cast<std::size_t>( | |
std::ceil( | |
cnt * ( ( is_same<LeftRight, left>::value ) ? this->threshold_probability_ : 1. - this->threshold_probability_ ) | |
) | |
); | |
// If n is in a valid range, return result, otherwise return NaN or throw exception | |
if ( n >= static_cast<std::size_t>(tail(args).size())) | |
{ | |
if (std::numeric_limits<float_type>::has_quiet_NaN) | |
{ | |
return boost::make_tuple( | |
std::numeric_limits<float_type>::quiet_NaN() | |
, std::numeric_limits<float_type>::quiet_NaN() | |
, std::numeric_limits<float_type>::quiet_NaN() | |
); | |
} | |
else | |
{ | |
std::ostringstream msg; | |
msg << "index n = " << n << " is not in valid range [0, " << tail(args).size() << ")"; | |
boost::throw_exception(std::runtime_error(msg.str())); | |
return boost::make_tuple(Sample(0), Sample(0), Sample(0)); | |
} | |
} | |
else | |
{ | |
float_type u = *(tail(args).begin() + n - 1) * sign::value; | |
// compute mean and variance of samples above/under threshold value u | |
for (std::size_t i = 0; i < n; ++i) | |
{ | |
mu_ += *(tail(args).begin() + i); | |
sigma2_ += *(tail(args).begin() + i) * (*(tail(args).begin() + i)); | |
} | |
this->mu_ = sign::value * numeric::average(this->mu_, n); | |
this->sigma2_ = numeric::average(this->sigma2_, n); | |
this->sigma2_ -= this->mu_ * this->mu_; | |
if (is_same<LeftRight, left>::value) | |
this->threshold_probability_ = 1. - this->threshold_probability_; | |
float_type tmp = numeric::average(( this->mu_ - u )*( this->mu_ - u ), this->sigma2_); | |
float_type xi_hat = 0.5 * ( 1. - tmp ); | |
float_type beta_hat = 0.5 * ( this->mu_ - u ) * ( 1. + tmp ); | |
float_type beta_bar = beta_hat * std::pow(1. - threshold_probability_, xi_hat); | |
float_type u_bar = u - beta_bar * ( std::pow(1. - threshold_probability_, -xi_hat) - 1.)/xi_hat; | |
this->fit_parameters_ = boost::make_tuple(u_bar, beta_bar, xi_hat); | |
} | |
} | |
return this->fit_parameters_; | |
} | |
private: | |
mutable float_type mu_; // mean of samples above threshold u | |
mutable float_type sigma2_; // variance of samples above threshold u | |
mutable float_type threshold_probability_; | |
mutable result_type fit_parameters_; // boost::tuple that stores fit parameters | |
mutable bool is_dirty_; | |
}; | |
} // namespace impl | |
/////////////////////////////////////////////////////////////////////////////// | |
// tag::peaks_over_threshold | |
// | |
namespace tag | |
{ | |
template<typename LeftRight> | |
struct peaks_over_threshold | |
: depends_on<count> | |
, pot_threshold_value | |
{ | |
/// INTERNAL ONLY | |
/// | |
typedef accumulators::impl::peaks_over_threshold_impl<mpl::_1, LeftRight> impl; | |
}; | |
template<typename LeftRight> | |
struct peaks_over_threshold_prob | |
: depends_on<count, tail<LeftRight> > | |
, pot_threshold_probability | |
{ | |
/// INTERNAL ONLY | |
/// | |
typedef accumulators::impl::peaks_over_threshold_prob_impl<mpl::_1, LeftRight> impl; | |
}; | |
struct abstract_peaks_over_threshold | |
: depends_on<> | |
{ | |
}; | |
} | |
/////////////////////////////////////////////////////////////////////////////// | |
// extract::peaks_over_threshold | |
// | |
namespace extract | |
{ | |
extractor<tag::abstract_peaks_over_threshold> const peaks_over_threshold = {}; | |
BOOST_ACCUMULATORS_IGNORE_GLOBAL(peaks_over_threshold) | |
} | |
using extract::peaks_over_threshold; | |
// peaks_over_threshold<LeftRight>(with_threshold_value) -> peaks_over_threshold<LeftRight> | |
template<typename LeftRight> | |
struct as_feature<tag::peaks_over_threshold<LeftRight>(with_threshold_value)> | |
{ | |
typedef tag::peaks_over_threshold<LeftRight> type; | |
}; | |
// peaks_over_threshold<LeftRight>(with_threshold_probability) -> peaks_over_threshold_prob<LeftRight> | |
template<typename LeftRight> | |
struct as_feature<tag::peaks_over_threshold<LeftRight>(with_threshold_probability)> | |
{ | |
typedef tag::peaks_over_threshold_prob<LeftRight> type; | |
}; | |
template<typename LeftRight> | |
struct feature_of<tag::peaks_over_threshold<LeftRight> > | |
: feature_of<tag::abstract_peaks_over_threshold> | |
{ | |
}; | |
template<typename LeftRight> | |
struct feature_of<tag::peaks_over_threshold_prob<LeftRight> > | |
: feature_of<tag::abstract_peaks_over_threshold> | |
{ | |
}; | |
// So that peaks_over_threshold can be automatically substituted | |
// with weighted_peaks_over_threshold when the weight parameter is non-void. | |
template<typename LeftRight> | |
struct as_weighted_feature<tag::peaks_over_threshold<LeftRight> > | |
{ | |
typedef tag::weighted_peaks_over_threshold<LeftRight> type; | |
}; | |
template<typename LeftRight> | |
struct feature_of<tag::weighted_peaks_over_threshold<LeftRight> > | |
: feature_of<tag::peaks_over_threshold<LeftRight> > | |
{}; | |
// So that peaks_over_threshold_prob can be automatically substituted | |
// with weighted_peaks_over_threshold_prob when the weight parameter is non-void. | |
template<typename LeftRight> | |
struct as_weighted_feature<tag::peaks_over_threshold_prob<LeftRight> > | |
{ | |
typedef tag::weighted_peaks_over_threshold_prob<LeftRight> type; | |
}; | |
template<typename LeftRight> | |
struct feature_of<tag::weighted_peaks_over_threshold_prob<LeftRight> > | |
: feature_of<tag::peaks_over_threshold_prob<LeftRight> > | |
{}; | |
}} // namespace boost::accumulators | |
#ifdef _MSC_VER | |
# pragma warning(pop) | |
#endif | |
#endif |