| /////////////////////////////////////////////////////////////////////////////// |
| // 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 |