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(t0: 0., t1: 0., t2: 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(t0: 0., t1: 0., t2: 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(e: 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 = {}; |
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 |
406 | |