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0001 // @(#)root/mathcore:$Id$ 0002 // Authors: L. Moneta, A. Zsenei 08/2005 0003 0004 0005 // Authors: Andras Zsenei & Lorenzo Moneta 08/2005 0006 0007 0008 /********************************************************************** 0009 * * 0010 * Copyright (c) 2005 , LCG ROOT MathLib Team * 0011 * * 0012 * * 0013 **********************************************************************/ 0014 0015 0016 #ifndef ROOT_Math_QuantFuncMathCore 0017 #define ROOT_Math_QuantFuncMathCore 0018 0019 0020 namespace ROOT { 0021 namespace Math { 0022 0023 0024 0025 /** @defgroup QuantFunc Quantile Functions 0026 * @ingroup StatFunc 0027 * 0028 * Inverse functions of the cumulative distribution functions 0029 * and the inverse of the complement of the cumulative distribution functions 0030 * for various distributions. 0031 * The functions with the extension <em>_quantile</em> calculate the 0032 * inverse of the <em>_cdf</em> function, the 0033 * lower tail integral of the probability density function 0034 * \f$D^{-1}(z)\f$ where 0035 * 0036 * \f[ D(x) = \int_{-\infty}^{x} p(x') dx' \f] 0037 * 0038 * while those with the <em>_quantile_c</em> extension calculate the 0039 * inverse of the <em>_cdf_c</em> functions, the upper tail integral of the probability 0040 * density function \f$D^{-1}(z) \f$ where 0041 * 0042 * \f[ D(x) = \int_{x}^{+\infty} p(x') dx' \f] 0043 * 0044 * These functions are defined in the header file <em>Math/ProbFunc.h</em> or in the global one 0045 * including all statistical functions <em>Math/DistFunc.h</em> 0046 * 0047 * 0048 * <strong>NOTE:</strong> In the old releases (< 5.14) the <em>_quantile</em> functions were called 0049 * <em>_quant_inv</em> and the <em>_quantile_c</em> functions were called 0050 * <em>_prob_inv</em>. 0051 * These names are currently kept for backward compatibility, but 0052 * their usage is deprecated. 0053 * 0054 */ 0055 0056 /** @name Quantile Functions from MathCore 0057 * The implementation is provided in MathCore and for the majority of the function comes from 0058 * <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0059 0060 */ 0061 0062 //@{ 0063 0064 0065 0066 /** 0067 0068 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0069 function of the upper tail of the beta distribution 0070 (#beta_cdf_c). 0071 It is implemented using the function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0072 0073 0074 @ingroup QuantFunc 0075 0076 */ 0077 double beta_quantile(double x, double a, double b); 0078 0079 /** 0080 0081 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0082 function of the lower tail of the beta distribution 0083 (#beta_cdf). 0084 It is implemented using 0085 the function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0086 0087 @ingroup QuantFunc 0088 0089 */ 0090 double beta_quantile_c(double x, double a, double b); 0091 0092 0093 0094 /** 0095 0096 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0097 function of the upper tail of the Cauchy distribution (#cauchy_cdf_c) 0098 which is also called Lorentzian distribution. For 0099 detailed description see 0100 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html"> 0101 Mathworld</A>. 0102 0103 @ingroup QuantFunc 0104 0105 */ 0106 0107 double cauchy_quantile_c(double z, double b); 0108 0109 0110 0111 0112 /** 0113 0114 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0115 function of the lower tail of the Cauchy distribution (#cauchy_cdf) 0116 which is also called Breit-Wigner or Lorentzian distribution. For 0117 detailed description see 0118 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html"> 0119 Mathworld</A>. The implementation used is that of 0120 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC294">GSL</A>. 0121 0122 @ingroup QuantFunc 0123 0124 */ 0125 0126 double cauchy_quantile(double z, double b); 0127 0128 0129 0130 0131 /** 0132 0133 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0134 function of the upper tail of the Breit-Wigner distribution (#breitwigner_cdf_c) 0135 which is similar to the Cauchy distribution. For 0136 detailed description see 0137 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html"> 0138 Mathworld</A>. It is evaluated using the same implementation of 0139 #cauchy_quantile_c. 0140 0141 @ingroup QuantFunc 0142 0143 */ 0144 0145 inline double breitwigner_quantile_c(double z, double gamma) { 0146 return cauchy_quantile_c(z, gamma/2.0); 0147 } 0148 0149 0150 0151 0152 /** 0153 0154 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0155 function of the lower tail of the Breit_Wigner distribution (#breitwigner_cdf) 0156 which is similar to the Cauchy distribution. For 0157 detailed description see 0158 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html"> 0159 Mathworld</A>. It is evaluated using the same implementation of 0160 #cauchy_quantile. 0161 0162 0163 @ingroup QuantFunc 0164 0165 */ 0166 0167 inline double breitwigner_quantile(double z, double gamma) { 0168 return cauchy_quantile(z, gamma/2.0); 0169 } 0170 0171 0172 0173 0174 0175 0176 /** 0177 0178 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0179 function of the upper tail of the \f$\chi^2\f$ distribution 0180 with \f$r\f$ degrees of freedom (#chisquared_cdf_c). For detailed description see 0181 <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html"> 0182 Mathworld</A>. It is implemented using the inverse of the incomplete complement gamma function, using 0183 the function igami from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0184 0185 @ingroup QuantFunc 0186 0187 */ 0188 0189 double chisquared_quantile_c(double z, double r); 0190 0191 0192 0193 /** 0194 0195 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0196 function of the lower tail of the \f$\chi^2\f$ distribution 0197 with \f$r\f$ degrees of freedom (#chisquared_cdf). For detailed description see 0198 <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html"> 0199 Mathworld</A>. 0200 It is implemented using chisquared_quantile_c, therefore is not very precise for small z. 0201 It is recommended to use the MathMore function (ROOT::MathMore::chisquared_quantile )implemented using GSL 0202 0203 @ingroup QuantFunc 0204 0205 */ 0206 0207 double chisquared_quantile(double z, double r); 0208 0209 0210 0211 /** 0212 0213 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0214 function of the upper tail of the exponential distribution 0215 (#exponential_cdf_c). For detailed description see 0216 <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html"> 0217 Mathworld</A>. 0218 0219 @ingroup QuantFunc 0220 0221 */ 0222 0223 double exponential_quantile_c(double z, double lambda); 0224 0225 0226 0227 0228 /** 0229 0230 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0231 function of the lower tail of the exponential distribution 0232 (#exponential_cdf). For detailed description see 0233 <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html"> 0234 Mathworld</A>. 0235 0236 @ingroup QuantFunc 0237 0238 */ 0239 0240 double exponential_quantile(double z, double lambda); 0241 0242 0243 0244 /** 0245 0246 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0247 function of the lower tail of the f distribution 0248 (#fdistribution_cdf). For detailed description see 0249 <A HREF="http://mathworld.wolfram.com/F-Distribution.html"> 0250 Mathworld</A>. 0251 It is implemented using the inverse of the incomplete beta function, 0252 function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0253 0254 @ingroup QuantFunc 0255 0256 */ 0257 double fdistribution_quantile(double z, double n, double m); 0258 0259 /** 0260 0261 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0262 function of the upper tail of the f distribution 0263 (#fdistribution_cdf_c). For detailed description see 0264 <A HREF="http://mathworld.wolfram.com/F-Distribution.html"> 0265 Mathworld</A>. 0266 It is implemented using the inverse of the incomplete beta function, 0267 function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0268 0269 @ingroup QuantFunc 0270 */ 0271 0272 double fdistribution_quantile_c(double z, double n, double m); 0273 0274 0275 /** 0276 0277 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0278 function of the upper tail of the gamma distribution 0279 (#gamma_cdf_c). For detailed description see 0280 <A HREF="http://mathworld.wolfram.com/GammaDistribution.html"> 0281 Mathworld</A>. The implementation used is that of 0282 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC300">GSL</A>. 0283 It is implemented using the function igami taken 0284 from <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0285 0286 @ingroup QuantFunc 0287 0288 */ 0289 0290 double gamma_quantile_c(double z, double alpha, double theta); 0291 0292 0293 0294 0295 /** 0296 0297 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0298 function of the lower tail of the gamma distribution 0299 (#gamma_cdf). For detailed description see 0300 <A HREF="http://mathworld.wolfram.com/GammaDistribution.html"> 0301 Mathworld</A>. 0302 It is implemented using chisquared_quantile_c, therefore is not very precise for small z. 0303 For this special cases it is recommended to use the MathMore function ROOT::MathMore::gamma_quantile 0304 implemented using GSL 0305 0306 0307 @ingroup QuantFunc 0308 0309 */ 0310 0311 double gamma_quantile(double z, double alpha, double theta); 0312 0313 0314 0315 /** 0316 0317 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0318 function of the upper tail of the normal (Gaussian) distribution 0319 (#gaussian_cdf_c). For detailed description see 0320 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html"> 0321 Mathworld</A>. It can also be evaluated using #normal_quantile_c which will 0322 call the same implementation. 0323 0324 @ingroup QuantFunc 0325 0326 */ 0327 0328 double gaussian_quantile_c(double z, double sigma); 0329 0330 0331 0332 0333 /** 0334 0335 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0336 function of the lower tail of the normal (Gaussian) distribution 0337 (#gaussian_cdf). For detailed description see 0338 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html"> 0339 Mathworld</A>. It can also be evaluated using #normal_quantile which will 0340 call the same implementation. 0341 It is implemented using the function ROOT::Math::Cephes::ndtri taken from 0342 <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0343 0344 @ingroup QuantFunc 0345 0346 */ 0347 0348 double gaussian_quantile(double z, double sigma); 0349 0350 0351 0352 0353 /** 0354 0355 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0356 function of the upper tail of the lognormal distribution 0357 (#lognormal_cdf_c). For detailed description see 0358 <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html"> 0359 Mathworld</A>. The implementation used is that of 0360 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC302">GSL</A>. 0361 0362 @ingroup QuantFunc 0363 0364 */ 0365 0366 double lognormal_quantile_c(double x, double m, double s); 0367 0368 0369 0370 0371 /** 0372 0373 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0374 function of the lower tail of the lognormal distribution 0375 (#lognormal_cdf). For detailed description see 0376 <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html"> 0377 Mathworld</A>. The implementation used is that of 0378 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC302">GSL</A>. 0379 0380 @ingroup QuantFunc 0381 0382 */ 0383 0384 double lognormal_quantile(double x, double m, double s); 0385 0386 0387 0388 0389 /** 0390 0391 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0392 function of the upper tail of the normal (Gaussian) distribution 0393 (#normal_cdf_c). For detailed description see 0394 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html"> 0395 Mathworld</A>. It can also be evaluated using #gaussian_quantile_c which will 0396 call the same implementation. 0397 It is implemented using the function ROOT::Math::Cephes::ndtri taken from 0398 <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0399 0400 @ingroup QuantFunc 0401 0402 */ 0403 0404 double normal_quantile_c(double z, double sigma); 0405 /// alternative name for same function 0406 inline double gaussian_quantile_c(double z, double sigma) { 0407 return normal_quantile_c(z,sigma); 0408 } 0409 0410 0411 0412 0413 /** 0414 0415 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0416 function of the lower tail of the normal (Gaussian) distribution 0417 (#normal_cdf). For detailed description see 0418 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html"> 0419 Mathworld</A>. It can also be evaluated using #gaussian_quantile which will 0420 call the same implementation. 0421 It is implemented using the function ROOT::Math::Cephes::ndtri taken from 0422 <A HREF="http://www.netlib.org/cephes">Cephes</A>. 0423 0424 0425 @ingroup QuantFunc 0426 0427 */ 0428 0429 double normal_quantile(double z, double sigma); 0430 /// alternative name for same function 0431 inline double gaussian_quantile(double z, double sigma) { 0432 return normal_quantile(z,sigma); 0433 } 0434 0435 0436 0437 #ifdef LATER // t quantiles are still in MathMore 0438 0439 /** 0440 0441 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0442 function of the upper tail of Student's t-distribution 0443 (#tdistribution_cdf_c). For detailed description see 0444 <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html"> 0445 Mathworld</A>. The implementation used is that of 0446 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC305">GSL</A>. 0447 0448 @ingroup QuantFunc 0449 0450 */ 0451 0452 double tdistribution_quantile_c(double z, double r); 0453 0454 0455 0456 0457 /** 0458 0459 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0460 function of the lower tail of Student's t-distribution 0461 (#tdistribution_cdf). For detailed description see 0462 <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html"> 0463 Mathworld</A>. The implementation used is that of 0464 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC305">GSL</A>. 0465 0466 @ingroup QuantFunc 0467 0468 */ 0469 0470 double tdistribution_quantile(double z, double r); 0471 0472 #endif 0473 0474 0475 /** 0476 0477 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0478 function of the upper tail of the uniform (flat) distribution 0479 (#uniform_cdf_c). For detailed description see 0480 <A HREF="http://mathworld.wolfram.com/UniformDistribution.html"> 0481 Mathworld</A>. 0482 0483 @ingroup QuantFunc 0484 0485 */ 0486 0487 double uniform_quantile_c(double z, double a, double b); 0488 0489 0490 0491 0492 /** 0493 0494 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0495 function of the lower tail of the uniform (flat) distribution 0496 (#uniform_cdf). For detailed description see 0497 <A HREF="http://mathworld.wolfram.com/UniformDistribution.html"> 0498 Mathworld</A>. 0499 0500 @ingroup QuantFunc 0501 0502 */ 0503 0504 double uniform_quantile(double z, double a, double b); 0505 0506 0507 0508 0509 /** 0510 0511 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0512 function of the lower tail of the Landau distribution 0513 (#landau_cdf). 0514 0515 For detailed description see 0516 K.S. Kölbig and B. Schorr, A program package for the Landau distribution, 0517 <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A> 0518 <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>. 0519 The same algorithms as in 0520 <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html"> 0521 CERNLIB</A> (RANLAN) is used. 0522 0523 @param z The argument \f$z\f$ 0524 @param xi The width parameter \f$\xi\f$ 0525 0526 @ingroup QuantFunc 0527 0528 */ 0529 0530 double landau_quantile(double z, double xi = 1); 0531 0532 0533 /** 0534 0535 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution 0536 function of the upper tail of the landau distribution 0537 (#landau_cdf_c). 0538 Implemented using #landau_quantile 0539 0540 @param z The argument \f$z\f$ 0541 @param xi The width parameter \f$\xi\f$ 0542 0543 @ingroup QuantFunc 0544 0545 */ 0546 0547 double landau_quantile_c(double z, double xi = 1); 0548 0549 0550 #ifdef HAVE_OLD_STAT_FUNC 0551 0552 //@} 0553 /** @name Backward compatible functions */ 0554 0555 0556 inline double breitwigner_prob_inv(double x, double gamma) { 0557 return breitwigner_quantile_c(x,gamma); 0558 } 0559 inline double breitwigner_quant_inv(double x, double gamma) { 0560 return breitwigner_quantile(x,gamma); 0561 } 0562 0563 inline double cauchy_prob_inv(double x, double b) { 0564 return cauchy_quantile_c(x,b); 0565 } 0566 inline double cauchy_quant_inv(double x, double b) { 0567 return cauchy_quantile (x,b); 0568 } 0569 0570 inline double exponential_prob_inv(double x, double lambda) { 0571 return exponential_quantile_c(x, lambda ); 0572 } 0573 inline double exponential_quant_inv(double x, double lambda) { 0574 return exponential_quantile (x, lambda ); 0575 } 0576 0577 inline double gaussian_prob_inv(double x, double sigma) { 0578 return gaussian_quantile_c( x, sigma ); 0579 } 0580 inline double gaussian_quant_inv(double x, double sigma) { 0581 return gaussian_quantile ( x, sigma ); 0582 } 0583 0584 inline double lognormal_prob_inv(double x, double m, double s) { 0585 return lognormal_quantile_c( x, m, s ); 0586 } 0587 inline double lognormal_quant_inv(double x, double m, double s) { 0588 return lognormal_quantile ( x, m, s ); 0589 } 0590 0591 inline double normal_prob_inv(double x, double sigma) { 0592 return normal_quantile_c( x, sigma ); 0593 } 0594 inline double normal_quant_inv(double x, double sigma) { 0595 return normal_quantile ( x, sigma ); 0596 } 0597 0598 inline double uniform_prob_inv(double x, double a, double b) { 0599 return uniform_quantile_c( x, a, b ); 0600 } 0601 inline double uniform_quant_inv(double x, double a, double b) { 0602 return uniform_quantile ( x, a, b ); 0603 } 0604 0605 inline double chisquared_prob_inv(double x, double r) { 0606 return chisquared_quantile_c(x, r ); 0607 } 0608 0609 inline double gamma_prob_inv(double x, double alpha, double theta) { 0610 return gamma_quantile_c (x, alpha, theta ); 0611 } 0612 0613 0614 #endif 0615 0616 0617 } // namespace Math 0618 } // namespace ROOT 0619 0620 0621 0622 #endif // ROOT_Math_QuantFuncMathCore
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