| /* boost random/normal_distribution.hpp header file |
| * |
| * Copyright Jens Maurer 2000-2001 |
| * Copyright Steven Watanabe 2010-2011 |
| * 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) |
| * |
| * See http://www.boost.org for most recent version including documentation. |
| * |
| * $Id$ |
| * |
| * Revision history |
| * 2001-02-18 moved to individual header files |
| */ |
| |
| #ifndef BOOST_RANDOM_NORMAL_DISTRIBUTION_HPP |
| #define BOOST_RANDOM_NORMAL_DISTRIBUTION_HPP |
| |
| #include <boost/config/no_tr1/cmath.hpp> |
| #include <istream> |
| #include <iosfwd> |
| #include <boost/assert.hpp> |
| #include <boost/limits.hpp> |
| #include <boost/static_assert.hpp> |
| #include <boost/random/detail/config.hpp> |
| #include <boost/random/detail/operators.hpp> |
| #include <boost/random/detail/int_float_pair.hpp> |
| #include <boost/random/uniform_01.hpp> |
| #include <boost/random/exponential_distribution.hpp> |
| |
| namespace boost { |
| namespace random { |
| |
| namespace detail { |
| |
| // tables for the ziggurat algorithm |
| template<class RealType> |
| struct normal_table { |
| static const RealType table_x[129]; |
| static const RealType table_y[129]; |
| }; |
| |
| template<class RealType> |
| const RealType normal_table<RealType>::table_x[129] = { |
| 3.7130862467403632609, 3.4426198558966521214, 3.2230849845786185446, 3.0832288582142137009, |
| 2.9786962526450169606, 2.8943440070186706210, 2.8231253505459664379, 2.7611693723841538514, |
| 2.7061135731187223371, 2.6564064112581924999, 2.6109722484286132035, 2.5690336259216391328, |
| 2.5300096723854666170, 2.4934545220919507609, 2.4590181774083500943, 2.4264206455302115930, |
| 2.3954342780074673425, 2.3658713701139875435, 2.3375752413355307354, 2.3104136836950021558, |
| 2.2842740596736568056, 2.2590595738653295251, 2.2346863955870569803, 2.2110814088747278106, |
| 2.1881804320720206093, 2.1659267937448407377, 2.1442701823562613518, 2.1231657086697899595, |
| 2.1025731351849988838, 2.0824562379877246441, 2.0627822745039633575, 2.0435215366506694976, |
| 2.0246469733729338782, 2.0061338699589668403, 1.9879595741230607243, 1.9701032608497132242, |
| 1.9525457295488889058, 1.9352692282919002011, 1.9182573008597320303, 1.9014946531003176140, |
| 1.8849670357028692380, 1.8686611409895420085, 1.8525645117230870617, 1.8366654602533840447, |
| 1.8209529965910050740, 1.8054167642140487420, 1.7900469825946189862, 1.7748343955807692457, |
| 1.7597702248942318749, 1.7448461281083765085, 1.7300541605582435350, 1.7153867407081165482, |
| 1.7008366185643009437, 1.6863968467734863258, 1.6720607540918522072, 1.6578219209482075462, |
| 1.6436741568569826489, 1.6296114794646783962, 1.6156280950371329644, 1.6017183802152770587, |
| 1.5878768648844007019, 1.5740982160167497219, 1.5603772223598406870, 1.5467087798535034608, |
| 1.5330878776675560787, 1.5195095847593707806, 1.5059690368565502602, 1.4924614237746154081, |
| 1.4789819769830978546, 1.4655259573357946276, 1.4520886428822164926, 1.4386653166774613138, |
| 1.4252512545068615734, 1.4118417124397602509, 1.3984319141236063517, 1.3850170377251486449, |
| 1.3715922024197322698, 1.3581524543224228739, 1.3446927517457130432, 1.3312079496576765017, |
| 1.3176927832013429910, 1.3041418501204215390, 1.2905495919178731508, 1.2769102735516997175, |
| 1.2632179614460282310, 1.2494664995643337480, 1.2356494832544811749, 1.2217602305309625678, |
| 1.2077917504067576028, 1.1937367078237721994, 1.1795873846544607035, 1.1653356361550469083, |
| 1.1509728421389760651, 1.1364898520030755352, 1.1218769225722540661, 1.1071236475235353980, |
| 1.0922188768965537614, 1.0771506248819376573, 1.0619059636836193998, 1.0464709007525802629, |
| 1.0308302360564555907, 1.0149673952392994716, 0.99886423348064351303, 0.98250080350276038481, |
| 0.96585507938813059489, 0.94890262549791195381, 0.93161619660135381056, 0.91396525100880177644, |
| 0.89591535256623852894, 0.87742742909771569142, 0.85845684317805086354, 0.83895221428120745572, |
| 0.81885390668331772331, 0.79809206062627480454, 0.77658398787614838598, 0.75423066443451007146, |
| 0.73091191062188128150, 0.70647961131360803456, 0.68074791864590421664, 0.65347863871504238702, |
| 0.62435859730908822111, 0.59296294244197797913, 0.55869217837551797140, 0.52065603872514491759, |
| 0.47743783725378787681, 0.42654798630330512490, 0.36287143102841830424, 0.27232086470466385065, |
| 0 |
| }; |
| |
| template<class RealType> |
| const RealType normal_table<RealType>::table_y[129] = { |
| 0, 0.0026696290839025035092, 0.0055489952208164705392, 0.0086244844129304709682, |
| 0.011839478657982313715, 0.015167298010672042468, 0.018592102737165812650, 0.022103304616111592615, |
| 0.025693291936149616572, 0.029356317440253829618, 0.033087886146505155566, 0.036884388786968774128, |
| 0.040742868074790604632, 0.044660862200872429800, 0.048636295860284051878, 0.052667401903503169793, |
| 0.056752663481538584188, 0.060890770348566375972, 0.065080585213631873753, 0.069321117394180252601, |
| 0.073611501884754893389, 0.077950982514654714188, 0.082338898242957408243, 0.086774671895542968998, |
| 0.091257800827634710201, 0.09578784912257815216, 0.10036444102954554013, 0.10498725541035453978, |
| 0.10965602101581776100, 0.11437051244988827452, 0.11913054670871858767, 0.12393598020398174246, |
| 0.12878670619710396109, 0.13368265258464764118, 0.13862377998585103702, 0.14361008009193299469, |
| 0.14864157424369696566, 0.15371831220958657066, 0.15884037114093507813, 0.16400785468492774791, |
| 0.16922089223892475176, 0.17447963833240232295, 0.17978427212496211424, 0.18513499701071343216, |
| 0.19053204032091372112, 0.19597565311811041399, 0.20146611007620324118, 0.20700370944187380064, |
| 0.21258877307373610060, 0.21822164655637059599, 0.22390269938713388747, 0.22963232523430270355, |
| 0.23541094226572765600, 0.24123899354775131610, 0.24711694751469673582, 0.25304529850976585934, |
| 0.25902456739871074263, 0.26505530225816194029, 0.27113807914102527343, 0.27727350292189771153, |
| 0.28346220822601251779, 0.28970486044581049771, 0.29600215684985583659, 0.30235482778947976274, |
| 0.30876363800925192282, 0.31522938806815752222, 0.32175291587920862031, 0.32833509837615239609, |
| 0.33497685331697116147, 0.34167914123501368412, 0.34844296754987246935, 0.35526938485154714435, |
| 0.36215949537303321162, 0.36911445366827513952, 0.37613546951445442947, 0.38322381105988364587, |
| 0.39038080824138948916, 0.39760785649804255208, 0.40490642081148835099, 0.41227804010702462062, |
| 0.41972433205403823467, 0.42724699830956239880, 0.43484783025466189638, 0.44252871528024661483, |
| 0.45029164368692696086, 0.45813871627287196483, 0.46607215269457097924, 0.47409430069824960453, |
| 0.48220764633483869062, 0.49041482528932163741, 0.49871863547658432422, 0.50712205108130458951, |
| 0.51562823824987205196, 0.52424057267899279809, 0.53296265938998758838, 0.54179835503172412311, |
| 0.55075179312105527738, 0.55982741271069481791, 0.56902999107472161225, 0.57836468112670231279, |
| 0.58783705444182052571, 0.59745315095181228217, 0.60721953663260488551, 0.61714337082656248870, |
| 0.62723248525781456578, 0.63749547734314487428, 0.64794182111855080873, 0.65858200005865368016, |
| 0.66942766735770616891, 0.68049184100641433355, 0.69178914344603585279, 0.70333609902581741633, |
| 0.71515150742047704368, 0.72725691835450587793, 0.73967724368333814856, 0.75244155918570380145, |
| 0.76558417390923599480, 0.77914608594170316563, 0.79317701178385921053, 0.80773829469612111340, |
| 0.82290721139526200050, 0.83878360531064722379, 0.85550060788506428418, 0.87324304892685358879, |
| 0.89228165080230272301, 0.91304364799203805999, 0.93628268170837107547, 0.96359969315576759960, |
| 1 |
| }; |
| |
| |
| template<class RealType = double> |
| struct unit_normal_distribution |
| { |
| template<class Engine> |
| RealType operator()(Engine& eng) { |
| const double * const table_x = normal_table<double>::table_x; |
| const double * const table_y = normal_table<double>::table_y; |
| for(;;) { |
| std::pair<RealType, int> vals = generate_int_float_pair<RealType, 8>(eng); |
| int i = vals.second; |
| int sign = (i & 1) * 2 - 1; |
| i = i >> 1; |
| RealType x = vals.first * RealType(table_x[i]); |
| if(x < table_x[i + 1]) return x * sign; |
| if(i == 0) return generate_tail(eng) * sign; |
| |
| RealType y01 = uniform_01<RealType>()(eng); |
| RealType y = RealType(table_y[i]) + y01 * RealType(table_y[i + 1] - table_y[i]); |
| |
| // These store the value y - bound, or something proportional to that difference: |
| RealType y_above_ubound, y_above_lbound; |
| |
| // There are three cases to consider: |
| // - convex regions (where x[i] > x[j] >= 1) |
| // - concave regions (where 1 <= x[i] < x[j]) |
| // - region containing the inflection point (where x[i] > 1 > x[j]) |
| // For convex (concave), exp^(-x^2/2) is bounded below (above) by the tangent at |
| // (x[i],y[i]) and is bounded above (below) by the diagonal line from (x[i+1],y[i+1]) to |
| // (x[i],y[i]). |
| // |
| // *If* the inflection point region satisfies slope(x[i+1]) < slope(diagonal), then we |
| // can treat the inflection region as a convex region: this condition is necessary and |
| // sufficient to ensure that the curve lies entirely below the diagonal (that, in turn, |
| // also implies that it will be above the tangent at x[i]). |
| // |
| // For the current table size (128), this is satisfied: slope(x[i+1]) = -0.60653 < |
| // slope(diag) = -0.60649, and so we have only two cases below instead of three. |
| |
| if (table_x[i] >= 1) { // convex (incl. inflection) |
| y_above_ubound = RealType(table_x[i] - table_x[i+1]) * y01 - (RealType(table_x[i]) - x); |
| y_above_lbound = y - (RealType(table_y[i]) + (RealType(table_x[i]) - x) * RealType(table_y[i]) * RealType(table_x[i])); |
| } |
| else { // concave |
| y_above_lbound = RealType(table_x[i] - table_x[i+1]) * y01 - (RealType(table_x[i]) - x); |
| y_above_ubound = y - (RealType(table_y[i]) + (RealType(table_x[i]) - x) * RealType(table_y[i]) * RealType(table_x[i])); |
| } |
| |
| if (y_above_ubound < 0 // if above the upper bound reject immediately |
| && |
| ( |
| y_above_lbound < 0 // If below the lower bound accept immediately |
| || |
| y < f(x) // Otherwise it's between the bounds and we need a full check |
| ) |
| ) { |
| return x * sign; |
| } |
| } |
| } |
| static RealType f(RealType x) { |
| using std::exp; |
| return exp(-(x*x/2)); |
| } |
| // Generate from the tail using rejection sampling from the exponential(x_1) distribution, |
| // shifted by x_1. This looks a little different from the usual rejection sampling because it |
| // transforms the condition by taking the log of both sides, thus avoiding the costly exp() call |
| // on the RHS, then takes advantage of the fact that -log(unif01) is simply generating an |
| // exponential (by inverse cdf sampling) by replacing the log(unif01) on the LHS with a |
| // exponential(1) draw, y. |
| template<class Engine> |
| RealType generate_tail(Engine& eng) { |
| const RealType tail_start = RealType(normal_table<double>::table_x[1]); |
| boost::random::exponential_distribution<RealType> exp_x(tail_start); |
| boost::random::exponential_distribution<RealType> exp_y; |
| for(;;) { |
| RealType x = exp_x(eng); |
| RealType y = exp_y(eng); |
| // If we were doing non-transformed rejection sampling, this condition would be: |
| // if (unif01 < exp(-.5*x*x)) return x + tail_start; |
| if(2*y > x*x) return x + tail_start; |
| } |
| } |
| }; |
| |
| } // namespace detail |
| |
| |
| /** |
| * Instantiations of class template normal_distribution model a |
| * \random_distribution. Such a distribution produces random numbers |
| * @c x distributed with probability density function |
| * \f$\displaystyle p(x) = |
| * \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} |
| * \f$, |
| * where mean and sigma are the parameters of the distribution. |
| * |
| * The implementation uses the "ziggurat" algorithm, as described in |
| * |
| * @blockquote |
| * "The Ziggurat Method for Generating Random Variables", |
| * George Marsaglia and Wai Wan Tsang, Journal of Statistical Software, |
| * Volume 5, Number 8 (2000), 1-7. |
| * @endblockquote |
| */ |
| template<class RealType = double> |
| class normal_distribution |
| { |
| public: |
| typedef RealType input_type; |
| typedef RealType result_type; |
| |
| class param_type { |
| public: |
| typedef normal_distribution distribution_type; |
| |
| /** |
| * Constructs a @c param_type with a given mean and |
| * standard deviation. |
| * |
| * Requires: sigma >= 0 |
| */ |
| explicit param_type(RealType mean_arg = RealType(0.0), |
| RealType sigma_arg = RealType(1.0)) |
| : _mean(mean_arg), |
| _sigma(sigma_arg) |
| {} |
| |
| /** Returns the mean of the distribution. */ |
| RealType mean() const { return _mean; } |
| |
| /** Returns the standand deviation of the distribution. */ |
| RealType sigma() const { return _sigma; } |
| |
| /** Writes a @c param_type to a @c std::ostream. */ |
| BOOST_RANDOM_DETAIL_OSTREAM_OPERATOR(os, param_type, parm) |
| { os << parm._mean << " " << parm._sigma ; return os; } |
| |
| /** Reads a @c param_type from a @c std::istream. */ |
| BOOST_RANDOM_DETAIL_ISTREAM_OPERATOR(is, param_type, parm) |
| { is >> parm._mean >> std::ws >> parm._sigma; return is; } |
| |
| /** Returns true if the two sets of parameters are the same. */ |
| BOOST_RANDOM_DETAIL_EQUALITY_OPERATOR(param_type, lhs, rhs) |
| { return lhs._mean == rhs._mean && lhs._sigma == rhs._sigma; } |
| |
| /** Returns true if the two sets of parameters are the different. */ |
| BOOST_RANDOM_DETAIL_INEQUALITY_OPERATOR(param_type) |
| |
| private: |
| RealType _mean; |
| RealType _sigma; |
| }; |
| |
| /** |
| * Constructs a @c normal_distribution object. @c mean and @c sigma are |
| * the parameters for the distribution. |
| * |
| * Requires: sigma >= 0 |
| */ |
| explicit normal_distribution(const RealType& mean_arg = RealType(0.0), |
| const RealType& sigma_arg = RealType(1.0)) |
| : _mean(mean_arg), _sigma(sigma_arg) |
| { |
| BOOST_ASSERT(_sigma >= RealType(0)); |
| } |
| |
| /** |
| * Constructs a @c normal_distribution object from its parameters. |
| */ |
| explicit normal_distribution(const param_type& parm) |
| : _mean(parm.mean()), _sigma(parm.sigma()) |
| {} |
| |
| /** Returns the mean of the distribution. */ |
| RealType mean() const { return _mean; } |
| /** Returns the standard deviation of the distribution. */ |
| RealType sigma() const { return _sigma; } |
| |
| /** Returns the smallest value that the distribution can produce. */ |
| RealType min BOOST_PREVENT_MACRO_SUBSTITUTION () const |
| { return -std::numeric_limits<RealType>::infinity(); } |
| /** Returns the largest value that the distribution can produce. */ |
| RealType max BOOST_PREVENT_MACRO_SUBSTITUTION () const |
| { return std::numeric_limits<RealType>::infinity(); } |
| |
| /** Returns the parameters of the distribution. */ |
| param_type param() const { return param_type(_mean, _sigma); } |
| /** Sets the parameters of the distribution. */ |
| void param(const param_type& parm) |
| { |
| _mean = parm.mean(); |
| _sigma = parm.sigma(); |
| } |
| |
| /** |
| * Effects: Subsequent uses of the distribution do not depend |
| * on values produced by any engine prior to invoking reset. |
| */ |
| void reset() { } |
| |
| /** Returns a normal variate. */ |
| template<class Engine> |
| result_type operator()(Engine& eng) |
| { |
| detail::unit_normal_distribution<RealType> impl; |
| return impl(eng) * _sigma + _mean; |
| } |
| |
| /** Returns a normal variate with parameters specified by @c param. */ |
| template<class URNG> |
| result_type operator()(URNG& urng, const param_type& parm) |
| { |
| return normal_distribution(parm)(urng); |
| } |
| |
| /** Writes a @c normal_distribution to a @c std::ostream. */ |
| BOOST_RANDOM_DETAIL_OSTREAM_OPERATOR(os, normal_distribution, nd) |
| { |
| os << nd._mean << " " << nd._sigma; |
| return os; |
| } |
| |
| /** Reads a @c normal_distribution from a @c std::istream. */ |
| BOOST_RANDOM_DETAIL_ISTREAM_OPERATOR(is, normal_distribution, nd) |
| { |
| is >> std::ws >> nd._mean >> std::ws >> nd._sigma; |
| return is; |
| } |
| |
| /** |
| * Returns true if the two instances of @c normal_distribution will |
| * return identical sequences of values given equal generators. |
| */ |
| BOOST_RANDOM_DETAIL_EQUALITY_OPERATOR(normal_distribution, lhs, rhs) |
| { |
| return lhs._mean == rhs._mean && lhs._sigma == rhs._sigma; |
| } |
| |
| /** |
| * Returns true if the two instances of @c normal_distribution will |
| * return different sequences of values given equal generators. |
| */ |
| BOOST_RANDOM_DETAIL_INEQUALITY_OPERATOR(normal_distribution) |
| |
| private: |
| RealType _mean, _sigma; |
| |
| }; |
| |
| } // namespace random |
| |
| using random::normal_distribution; |
| |
| } // namespace boost |
| |
| #endif // BOOST_RANDOM_NORMAL_DISTRIBUTION_HPP |