boost/random/normal_distribution.hpp
/* 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