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1 | /////////////////////////////////////////////////////////////////////////////// |
2 | // peaks_over_threshold.hpp | |
3 | // | |
4 | // Copyright 2006 Daniel Egloff, Olivier Gygi. Distributed under the Boost | |
5 | // Software License, Version 1.0. (See accompanying file | |
6 | // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) | |
7 | ||
8 | #ifndef BOOST_ACCUMULATORS_STATISTICS_PEAKS_OVER_THRESHOLD_HPP_DE_01_01_2006 | |
9 | #define BOOST_ACCUMULATORS_STATISTICS_PEAKS_OVER_THRESHOLD_HPP_DE_01_01_2006 | |
10 | ||
11 | #include <vector> | |
12 | #include <limits> | |
13 | #include <numeric> | |
14 | #include <functional> | |
15 | #include <boost/config/no_tr1/cmath.hpp> // pow | |
16 | #include <sstream> // stringstream | |
17 | #include <stdexcept> // runtime_error | |
18 | #include <boost/throw_exception.hpp> | |
19 | #include <boost/range.hpp> | |
20 | #include <boost/mpl/if.hpp> | |
21 | #include <boost/mpl/int.hpp> | |
22 | #include <boost/mpl/placeholders.hpp> | |
23 | #include <boost/parameter/keyword.hpp> | |
24 | #include <boost/tuple/tuple.hpp> | |
25 | #include <boost/accumulators/accumulators_fwd.hpp> | |
26 | #include <boost/accumulators/framework/accumulator_base.hpp> | |
27 | #include <boost/accumulators/framework/extractor.hpp> | |
28 | #include <boost/accumulators/numeric/functional.hpp> | |
29 | #include <boost/accumulators/framework/parameters/sample.hpp> | |
30 | #include <boost/accumulators/framework/depends_on.hpp> | |
31 | #include <boost/accumulators/statistics_fwd.hpp> | |
32 | #include <boost/accumulators/statistics/parameters/quantile_probability.hpp> | |
33 | #include <boost/accumulators/statistics/count.hpp> | |
34 | #include <boost/accumulators/statistics/tail.hpp> | |
35 | ||
36 | #ifdef _MSC_VER | |
37 | # pragma warning(push) | |
38 | # pragma warning(disable: 4127) // conditional expression is constant | |
39 | #endif | |
40 | ||
41 | namespace boost { namespace accumulators | |
42 | { | |
43 | ||
44 | /////////////////////////////////////////////////////////////////////////////// | |
45 | // threshold_probability and threshold named parameters | |
46 | // | |
47 | BOOST_PARAMETER_NESTED_KEYWORD(tag, pot_threshold_value, threshold_value) | |
48 | BOOST_PARAMETER_NESTED_KEYWORD(tag, pot_threshold_probability, threshold_probability) | |
49 | ||
50 | BOOST_ACCUMULATORS_IGNORE_GLOBAL(pot_threshold_value) | |
51 | BOOST_ACCUMULATORS_IGNORE_GLOBAL(pot_threshold_probability) | |
52 | ||
53 | namespace impl | |
54 | { | |
55 | /////////////////////////////////////////////////////////////////////////////// | |
56 | // peaks_over_threshold_impl | |
57 | // works with an explicit threshold value and does not depend on order statistics | |
58 | /** | |
59 | @brief Peaks over Threshold Method for Quantile and Tail Mean Estimation | |
60 | ||
61 | According to the theorem of Pickands-Balkema-de Haan, the distribution function \f$F_u(x)\f$ of | |
62 | the excesses \f$x\f$ over some sufficiently high threshold \f$u\f$ of a distribution function \f$F(x)\f$ | |
63 | may be approximated by a generalized Pareto distribution | |
64 | \f[ | |
65 | G_{\xi,\beta}(x) = | |
66 | \left\{ | |
67 | \begin{array}{ll} | |
68 | \beta^{-1}\left(1+\frac{\xi x}{\beta}\right)^{-1/\xi-1} & \textrm{if }\xi\neq0\\ | |
69 | \beta^{-1}\exp\left(-\frac{x}{\beta}\right) & \textrm{if }\xi=0, | |
70 | \end{array} | |
71 | \right. | |
72 | \f] | |
73 | with suitable parameters \f$\xi\f$ and \f$\beta\f$ that can be estimated, e.g., with the method of moments, cf. | |
74 | Hosking and Wallis (1987), | |
75 | \f[ | |
76 | \begin{array}{lll} | |
77 | \hat{\xi} & = & \frac{1}{2}\left[1-\frac{(\hat{\mu}-u)^2}{\hat{\sigma}^2}\right]\\ | |
78 | \hat{\beta} & = & \frac{\hat{\mu}-u}{2}\left[\frac{(\hat{\mu}-u)^2}{\hat{\sigma}^2}+1\right], | |
79 | \end{array} | |
80 | \f] | |
81 | \f$\hat{\mu}\f$ and \f$\hat{\sigma}^2\f$ being the empirical mean and variance of the samples over | |
82 | the threshold \f$u\f$. Equivalently, the distribution function | |
83 | \f$F_u(x-u)\f$ of the exceedances \f$x-u\f$ can be approximated by | |
84 | \f$G_{\xi,\beta}(x-u)=G_{\xi,\beta,u}(x)\f$. Since for \f$x\geq u\f$ the distribution function \f$F(x)\f$ | |
85 | can be written as | |
86 | \f[ | |
87 | F(x) = [1 - \P(X \leq u)]F_u(x - u) + \P(X \leq u) | |
88 | \f] | |
89 | and the probability \f$\P(X \leq u)\f$ can be approximated by the empirical distribution function | |
90 | \f$F_n(u)\f$ evaluated at \f$u\f$, an estimator of \f$F(x)\f$ is given by | |
91 | \f[ | |
92 | \widehat{F}(x) = [1 - F_n(u)]G_{\xi,\beta,u}(x) + F_n(u). | |
93 | \f] | |
94 | It can be shown that \f$\widehat{F}(x)\f$ is a generalized | |
95 | Pareto distribution \f$G_{\xi,\bar{\beta},\bar{u}}(x)\f$ with \f$\bar{\beta}=\beta[1-F_n(u)]^{\xi}\f$ | |
96 | and \f$\bar{u}=u-\bar{\beta}\left\{[1-F_n(u)]^{-\xi}-1\right\}/\xi\f$. By inverting \f$\widehat{F}(x)\f$, | |
97 | one obtains an estimator for the \f$\alpha\f$-quantile, | |
98 | \f[ | |
99 | \hat{q}_{\alpha} = \bar{u} + \frac{\bar{\beta}}{\xi}\left[(1-\alpha)^{-\xi}-1\right], | |
100 | \f] | |
101 | and similarly an estimator for the (coherent) tail mean, | |
102 | \f[ | |
103 | \widehat{CTM}_{\alpha} = \hat{q}_{\alpha} - \frac{\bar{\beta}}{\xi-1}(1-\alpha)^{-\xi}, | |
104 | \f] | |
105 | cf. McNeil and Frey (2000). | |
106 | ||
107 | Note that in case extreme values of the left tail are fitted, the distribution is mirrored with respect to the | |
108 | \f$y\f$ axis such that the left tail can be treated as a right tail. The computed fit parameters thus define | |
109 | the Pareto distribution that fits the mirrored left tail. When quantities like a quantile or a tail mean are | |
110 | computed using the fit parameters obtained from the mirrored data, the result is mirrored back, yielding the | |
111 | correct result. | |
112 | ||
113 | For further details, see | |
114 | ||
115 | J. R. M. Hosking and J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, | |
116 | Technometrics, Volume 29, 1987, p. 339-349 | |
117 | ||
118 | A. J. McNeil and R. Frey, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: | |
119 | an Extreme Value Approach, Journal of Empirical Finance, Volume 7, 2000, p. 271-300 | |
120 | ||
121 | @param quantile_probability | |
122 | @param pot_threshold_value | |
123 | */ | |
124 | template<typename Sample, typename LeftRight> | |
125 | struct peaks_over_threshold_impl | |
126 | : accumulator_base | |
127 | { | |
128 | typedef typename numeric::functional::fdiv<Sample, std::size_t>::result_type float_type; | |
129 | // for boost::result_of | |
130 | typedef boost::tuple<float_type, float_type, float_type> result_type; | |
131 | // for left tail fitting, mirror the extreme values | |
132 | typedef mpl::int_<is_same<LeftRight, left>::value ? -1 : 1> sign; | |
133 | ||
134 | template<typename Args> | |
135 | peaks_over_threshold_impl(Args const &args) | |
136 | : Nu_(0) | |
137 | , mu_(sign::value * numeric::fdiv(args[sample | Sample()], (std::size_t)1)) | |
138 | , sigma2_(numeric::fdiv(args[sample | Sample()], (std::size_t)1)) | |
139 | , threshold_(sign::value * args[pot_threshold_value]) | |
140 | , fit_parameters_(boost::make_tuple(0., 0., 0.)) | |
141 | , is_dirty_(true) | |
142 | { | |
143 | } | |
144 | ||
145 | template<typename Args> | |
146 | void operator ()(Args const &args) | |
147 | { | |
148 | this->is_dirty_ = true; | |
149 | ||
150 | if (sign::value * args[sample] > this->threshold_) | |
151 | { | |
152 | this->mu_ += args[sample]; | |
153 | this->sigma2_ += args[sample] * args[sample]; | |
154 | ++this->Nu_; | |
155 | } | |
156 | } | |
157 | ||
158 | template<typename Args> | |
159 | result_type result(Args const &args) const | |
160 | { | |
161 | if (this->is_dirty_) | |
162 | { | |
163 | this->is_dirty_ = false; | |
164 | ||
165 | std::size_t cnt = count(args); | |
166 | ||
167 | this->mu_ = sign::value * numeric::fdiv(this->mu_, this->Nu_); | |
168 | this->sigma2_ = numeric::fdiv(this->sigma2_, this->Nu_); | |
169 | this->sigma2_ -= this->mu_ * this->mu_; | |
170 | ||
171 | float_type threshold_probability = numeric::fdiv(cnt - this->Nu_, cnt); | |
172 | ||
173 | float_type tmp = numeric::fdiv(( this->mu_ - this->threshold_ )*( this->mu_ - this->threshold_ ), this->sigma2_); | |
174 | float_type xi_hat = 0.5 * ( 1. - tmp ); | |
175 | float_type beta_hat = 0.5 * ( this->mu_ - this->threshold_ ) * ( 1. + tmp ); | |
176 | float_type beta_bar = beta_hat * std::pow(1. - threshold_probability, xi_hat); | |
177 | float_type u_bar = this->threshold_ - beta_bar * ( std::pow(1. - threshold_probability, -xi_hat) - 1.)/xi_hat; | |
178 | this->fit_parameters_ = boost::make_tuple(u_bar, beta_bar, xi_hat); | |
179 | } | |
180 | ||
181 | return this->fit_parameters_; | |
182 | } | |
183 | ||
184 | private: | |
185 | std::size_t Nu_; // number of samples larger than threshold | |
186 | mutable float_type mu_; // mean of Nu_ largest samples | |
187 | mutable float_type sigma2_; // variance of Nu_ largest samples | |
188 | float_type threshold_; | |
189 | mutable result_type fit_parameters_; // boost::tuple that stores fit parameters | |
190 | mutable bool is_dirty_; | |
191 | }; | |
192 | ||
193 | /////////////////////////////////////////////////////////////////////////////// | |
194 | // peaks_over_threshold_prob_impl | |
195 | // determines threshold from a given threshold probability using order statistics | |
196 | /** | |
197 | @brief Peaks over Threshold Method for Quantile and Tail Mean Estimation | |
198 | ||
199 | @sa peaks_over_threshold_impl | |
200 | ||
201 | @param quantile_probability | |
202 | @param pot_threshold_probability | |
203 | */ | |
204 | template<typename Sample, typename LeftRight> | |
205 | struct peaks_over_threshold_prob_impl | |
206 | : accumulator_base | |
207 | { | |
208 | typedef typename numeric::functional::fdiv<Sample, std::size_t>::result_type float_type; | |
209 | // for boost::result_of | |
210 | typedef boost::tuple<float_type, float_type, float_type> result_type; | |
211 | // for left tail fitting, mirror the extreme values | |
212 | typedef mpl::int_<is_same<LeftRight, left>::value ? -1 : 1> sign; | |
213 | ||
214 | template<typename Args> | |
215 | peaks_over_threshold_prob_impl(Args const &args) | |
216 | : mu_(sign::value * numeric::fdiv(args[sample | Sample()], (std::size_t)1)) | |
217 | , sigma2_(numeric::fdiv(args[sample | Sample()], (std::size_t)1)) | |
218 | , threshold_probability_(args[pot_threshold_probability]) | |
219 | , fit_parameters_(boost::make_tuple(0., 0., 0.)) | |
220 | , is_dirty_(true) | |
221 | { | |
222 | } | |
223 | ||
224 | void operator ()(dont_care) | |
225 | { | |
226 | this->is_dirty_ = true; | |
227 | } | |
228 | ||
229 | template<typename Args> | |
230 | result_type result(Args const &args) const | |
231 | { | |
232 | if (this->is_dirty_) | |
233 | { | |
234 | this->is_dirty_ = false; | |
235 | ||
236 | std::size_t cnt = count(args); | |
237 | ||
238 | // the n'th cached sample provides an approximate threshold value u | |
239 | std::size_t n = static_cast<std::size_t>( | |
240 | std::ceil( | |
241 | cnt * ( ( is_same<LeftRight, left>::value ) ? this->threshold_probability_ : 1. - this->threshold_probability_ ) | |
242 | ) | |
243 | ); | |
244 | ||
245 | // If n is in a valid range, return result, otherwise return NaN or throw exception | |
246 | if ( n >= static_cast<std::size_t>(tail(args).size())) | |
247 | { | |
248 | if (std::numeric_limits<float_type>::has_quiet_NaN) | |
249 | { | |
250 | return boost::make_tuple( | |
251 | std::numeric_limits<float_type>::quiet_NaN() | |
252 | , std::numeric_limits<float_type>::quiet_NaN() | |
253 | , std::numeric_limits<float_type>::quiet_NaN() | |
254 | ); | |
255 | } | |
256 | else | |
257 | { | |
258 | std::ostringstream msg; | |
259 | msg << "index n = " << n << " is not in valid range [0, " << tail(args).size() << ")"; | |
260 | boost::throw_exception(std::runtime_error(msg.str())); | |
261 | return boost::make_tuple(Sample(0), Sample(0), Sample(0)); | |
262 | } | |
263 | } | |
264 | else | |
265 | { | |
266 | float_type u = *(tail(args).begin() + n - 1) * sign::value; | |
267 | ||
268 | // compute mean and variance of samples above/under threshold value u | |
269 | for (std::size_t i = 0; i < n; ++i) | |
270 | { | |
271 | mu_ += *(tail(args).begin() + i); | |
272 | sigma2_ += *(tail(args).begin() + i) * (*(tail(args).begin() + i)); | |
273 | } | |
274 | ||
275 | this->mu_ = sign::value * numeric::fdiv(this->mu_, n); | |
276 | this->sigma2_ = numeric::fdiv(this->sigma2_, n); | |
277 | this->sigma2_ -= this->mu_ * this->mu_; | |
278 | ||
279 | if (is_same<LeftRight, left>::value) | |
280 | this->threshold_probability_ = 1. - this->threshold_probability_; | |
281 | ||
282 | float_type tmp = numeric::fdiv(( this->mu_ - u )*( this->mu_ - u ), this->sigma2_); | |
283 | float_type xi_hat = 0.5 * ( 1. - tmp ); | |
284 | float_type beta_hat = 0.5 * ( this->mu_ - u ) * ( 1. + tmp ); | |
285 | float_type beta_bar = beta_hat * std::pow(1. - threshold_probability_, xi_hat); | |
286 | float_type u_bar = u - beta_bar * ( std::pow(1. - threshold_probability_, -xi_hat) - 1.)/xi_hat; | |
287 | this->fit_parameters_ = boost::make_tuple(u_bar, beta_bar, xi_hat); | |
288 | } | |
289 | } | |
290 | ||
291 | return this->fit_parameters_; | |
292 | } | |
293 | ||
294 | private: | |
295 | mutable float_type mu_; // mean of samples above threshold u | |
296 | mutable float_type sigma2_; // variance of samples above threshold u | |
297 | mutable float_type threshold_probability_; | |
298 | mutable result_type fit_parameters_; // boost::tuple that stores fit parameters | |
299 | mutable bool is_dirty_; | |
300 | }; | |
301 | ||
302 | } // namespace impl | |
303 | ||
304 | /////////////////////////////////////////////////////////////////////////////// | |
305 | // tag::peaks_over_threshold | |
306 | // | |
307 | namespace tag | |
308 | { | |
309 | template<typename LeftRight> | |
310 | struct peaks_over_threshold | |
311 | : depends_on<count> | |
312 | , pot_threshold_value | |
313 | { | |
314 | /// INTERNAL ONLY | |
315 | /// | |
316 | typedef accumulators::impl::peaks_over_threshold_impl<mpl::_1, LeftRight> impl; | |
317 | }; | |
318 | ||
319 | template<typename LeftRight> | |
320 | struct peaks_over_threshold_prob | |
321 | : depends_on<count, tail<LeftRight> > | |
322 | , pot_threshold_probability | |
323 | { | |
324 | /// INTERNAL ONLY | |
325 | /// | |
326 | typedef accumulators::impl::peaks_over_threshold_prob_impl<mpl::_1, LeftRight> impl; | |
327 | }; | |
328 | ||
329 | struct abstract_peaks_over_threshold | |
330 | : depends_on<> | |
331 | { | |
332 | }; | |
333 | } | |
334 | ||
335 | /////////////////////////////////////////////////////////////////////////////// | |
336 | // extract::peaks_over_threshold | |
337 | // | |
338 | namespace extract | |
339 | { | |
340 | extractor<tag::abstract_peaks_over_threshold> const peaks_over_threshold = {}; | |
341 | ||
342 | BOOST_ACCUMULATORS_IGNORE_GLOBAL(peaks_over_threshold) | |
343 | } | |
344 | ||
345 | using extract::peaks_over_threshold; | |
346 | ||
347 | // peaks_over_threshold<LeftRight>(with_threshold_value) -> peaks_over_threshold<LeftRight> | |
348 | template<typename LeftRight> | |
349 | struct as_feature<tag::peaks_over_threshold<LeftRight>(with_threshold_value)> | |
350 | { | |
351 | typedef tag::peaks_over_threshold<LeftRight> type; | |
352 | }; | |
353 | ||
354 | // peaks_over_threshold<LeftRight>(with_threshold_probability) -> peaks_over_threshold_prob<LeftRight> | |
355 | template<typename LeftRight> | |
356 | struct as_feature<tag::peaks_over_threshold<LeftRight>(with_threshold_probability)> | |
357 | { | |
358 | typedef tag::peaks_over_threshold_prob<LeftRight> type; | |
359 | }; | |
360 | ||
361 | template<typename LeftRight> | |
362 | struct feature_of<tag::peaks_over_threshold<LeftRight> > | |
363 | : feature_of<tag::abstract_peaks_over_threshold> | |
364 | { | |
365 | }; | |
366 | ||
367 | template<typename LeftRight> | |
368 | struct feature_of<tag::peaks_over_threshold_prob<LeftRight> > | |
369 | : feature_of<tag::abstract_peaks_over_threshold> | |
370 | { | |
371 | }; | |
372 | ||
373 | // So that peaks_over_threshold can be automatically substituted | |
374 | // with weighted_peaks_over_threshold when the weight parameter is non-void. | |
375 | template<typename LeftRight> | |
376 | struct as_weighted_feature<tag::peaks_over_threshold<LeftRight> > | |
377 | { | |
378 | typedef tag::weighted_peaks_over_threshold<LeftRight> type; | |
379 | }; | |
380 | ||
381 | template<typename LeftRight> | |
382 | struct feature_of<tag::weighted_peaks_over_threshold<LeftRight> > | |
383 | : feature_of<tag::peaks_over_threshold<LeftRight> > | |
384 | {}; | |
385 | ||
386 | // So that peaks_over_threshold_prob can be automatically substituted | |
387 | // with weighted_peaks_over_threshold_prob when the weight parameter is non-void. | |
388 | template<typename LeftRight> | |
389 | struct as_weighted_feature<tag::peaks_over_threshold_prob<LeftRight> > | |
390 | { | |
391 | typedef tag::weighted_peaks_over_threshold_prob<LeftRight> type; | |
392 | }; | |
393 | ||
394 | template<typename LeftRight> | |
395 | struct feature_of<tag::weighted_peaks_over_threshold_prob<LeftRight> > | |
396 | : feature_of<tag::peaks_over_threshold_prob<LeftRight> > | |
397 | {}; | |
398 | ||
399 | }} // namespace boost::accumulators | |
400 | ||
401 | #ifdef _MSC_VER | |
402 | # pragma warning(pop) | |
403 | #endif | |
404 | ||
405 | #endif |