Quantile Functions
[Statistical functions]


Quantile Functions from MathCore

The implementation is provided in MathCore and for the majority of the function comes from Cephes.

double ROOT::Math::beta_quantile (double x, double a, double b)
double ROOT::Math::beta_quantile_c (double x, double a, double b)
double ROOT::Math::cauchy_quantile_c (double z, double b)
double ROOT::Math::cauchy_quantile (double z, double b)
double ROOT::Math::breitwigner_quantile_c (double z, double gamma)
double ROOT::Math::breitwigner_quantile (double z, double gamma)
double ROOT::Math::chisquared_quantile_c (double z, double r)
double ROOT::Math::chisquared_quantile (double z, double r)
double ROOT::Math::exponential_quantile_c (double z, double lambda)
double ROOT::Math::exponential_quantile (double z, double lambda)
double ROOT::Math::fdistribution_quantile (double z, double n, double m)
double ROOT::Math::fdistribution_quantile_c (double z, double n, double m)
double ROOT::Math::gamma_quantile_c (double z, double alpha, double theta)
double ROOT::Math::gamma_quantile (double z, double alpha, double theta)
double ROOT::Math::gaussian_quantile_c (double z, double sigma)
 alternative name for same function
double ROOT::Math::gaussian_quantile (double z, double sigma)
 alternative name for same function
double ROOT::Math::lognormal_quantile_c (double x, double m, double s)
double ROOT::Math::lognormal_quantile (double x, double m, double s)
double ROOT::Math::normal_quantile_c (double z, double sigma)
double ROOT::Math::normal_quantile (double z, double sigma)
double ROOT::Math::uniform_quantile_c (double z, double a, double b)
double ROOT::Math::uniform_quantile (double z, double a, double b)
double ROOT::Math::landau_quantile (double z, double xi=1)
double ROOT::Math::landau_quantile_c (double z, double xi=1)

Quantile Functions from MathMore

The implementation used is that of GSL.

double ROOT::Math::tdistribution_quantile_c (double z, double r)
double ROOT::Math::tdistribution_quantile (double z, double r)

Functions

double ROOT::Math::vavilov_accurate_quantile (double z, double kappa, double beta2)
double ROOT::Math::vavilov_accurate_quantile_c (double z, double kappa, double beta2)
double ROOT::Math::vavilov_fast_quantile (double z, double kappa, double beta2)
double ROOT::Math::vavilov_fast_quantile_c (double z, double kappa, double beta2)
double ROOT::MathMore::chisquared_quantile (double z, double r)
double ROOT::MathMore::gamma_quantile (double z, double alpha, double theta)

Detailed Description

Inverse functions of the cumulative distribution functions and the inverse of the complement of the cumulative distribution functions for various distributions. The functions with the extension _quantile calculate the inverse of the _cdf function, the lower tail integral of the probability density function $D^{-1}(z)$ where

\[ D(x) = \int_{-\infty}^{x} p(x') dx' \]

while those with the _quantile_c extension calculate the inverse of the _cdf_c functions, the upper tail integral of the probability density function $D^{-1}(z) $ where

\[ D(x) = \int_{x}^{+\infty} p(x') dx' \]

These functions are defined in the header file Math/ProbFunc.h or in the global one including all statistical dunctions Math/DistFunc.h

NOTE: In the old releases (< 5.14) the _quantile functions were called _quant_inv and the _quantile_c functions were called _prob_inv. These names are currently kept for backward compatibility, but their usage is deprecated.

Function Documentation

double ROOT::Math::beta_quantile ( double  x,
double  a,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the beta distribution (beta_cdf_c). It is implemented using the function incbi from Cephes.

Definition at line 26 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::incbi().

Referenced by TEfficiency::BetaCentralInterval(), TEfficiency::BetaShortestInterval(), TEfficiency::ClopperPearson(), G__G__MathCore_170_0_61(), G__setup_memfuncROOTcLcLMath(), mbeta_quantile(), Beta_interval_length::operator()(), RooMathCoreReg::RooMathCoreReg(), and testBetaFunction().

double ROOT::Math::beta_quantile_c ( double  x,
double  a,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the beta distribution (beta_cdf). It is implemented using the function incbi from Cephes.

Definition at line 16 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::incbi().

Referenced by TEfficiency::BetaShortestInterval(), G__G__MathCore_170_0_62(), G__setup_memfuncROOTcLcLMath(), Beta_interval_length::LowerMax(), mbeta_quantile_c(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::breitwigner_quantile ( double  z,
double  gamma 
) [inline]

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the Breit_Wigner distribution (breitwigner_cdf) which is similar to the Cauchy distribution. For detailed description see Mathworld. It is evaluated using the same implementation of cauchy_quantile.

Definition at line 178 of file QuantFuncMathCore.h.

References ROOT::Math::cauchy_quantile().

Referenced by G__G__MathCore_170_0_66(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::breitwigner_quantile_c ( double  z,
double  gamma 
) [inline]

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the Breit-Wigner distribution (breitwigner_cdf_c) which is similar to the Cauchy distribution. For detailed description see Mathworld. It is evaluated using the same implementation of cauchy_quantile_c.

Definition at line 156 of file QuantFuncMathCore.h.

References ROOT::Math::cauchy_quantile_c().

Referenced by G__G__MathCore_170_0_65(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::cauchy_quantile ( double  z,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the Cauchy distribution (cauchy_cdf) which is also called Breit-Wigner or Lorentzian distribution. For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 46 of file QuantFuncMathCore.cxx.

References RootCsg::infinity, M_PI, and tan().

Referenced by ROOT::Math::breitwigner_quantile(), G__G__MathCore_170_0_64(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::cauchy_quantile_c ( double  z,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the Cauchy distribution (cauchy_cdf_c) which is also called Lorentzian distribution. For detailed description see Mathworld.

Definition at line 33 of file QuantFuncMathCore.cxx.

References RootCsg::infinity, M_PI, and tan().

Referenced by ROOT::Math::breitwigner_quantile_c(), G__G__MathCore_170_0_63(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::MathMore::chisquared_quantile ( double  z,
double  r 
)

Re-implementation in MathMore of the Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the $\chi^2$ distribution with $r$ degrees of freedom (chisquared_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 31 of file QuantFuncMathMore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_68(), G__G__MathMore_100_0_1(), G__setup_memfuncROOTcLcLMath(), G__setup_memfuncROOTcLcLMathMore(), and ROOT::Fit::FitResult::GetConfidenceIntervals().

double ROOT::Math::chisquared_quantile ( double  z,
double  r 
)

Re-implementation in MathMore of the Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the $\chi^2$ distribution with $r$ degrees of freedom (chisquared_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 67 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_68(), G__G__MathMore_100_0_1(), G__setup_memfuncROOTcLcLMath(), G__setup_memfuncROOTcLcLMathMore(), and ROOT::Fit::FitResult::GetConfidenceIntervals().

double ROOT::Math::chisquared_quantile_c ( double  z,
double  r 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the $\chi^2$ distribution with $r$ degrees of freedom (chisquared_cdf_c). For detailed description see Mathworld. It is implemented using the inverse of the incomplete complement gamma function, using the function igami from Cephes.

Definition at line 60 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_67(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::exponential_quantile ( double  z,
double  lambda 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the exponential distribution (exponential_cdf). For detailed description see Mathworld.

Definition at line 82 of file QuantFuncMathCore.cxx.

References ROOT::Math::log1p().

Referenced by G__G__MathCore_170_0_70(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::exponential_quantile_c ( double  z,
double  lambda 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the exponential distribution (exponential_cdf_c). For detailed description see Mathworld.

Definition at line 74 of file QuantFuncMathCore.cxx.

References log().

Referenced by G__G__MathCore_170_0_69(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::fdistribution_quantile ( double  z,
double  n,
double  m 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the f distribution (fdistribution_cdf). For detailed description see Mathworld. It is implemented using the inverse of the incomplete beta function, function incbi from Cephes.

Definition at line 103 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::incbi(), and y.

Referenced by G__G__MathCore_170_0_71(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::fdistribution_quantile_c ( double  z,
double  n,
double  m 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the f distribution (fdistribution_cdf_c). For detailed description see Mathworld. It is implemented using the inverse of the incomplete beta function, function incbi from Cephes.

Definition at line 89 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::incbi(), and y.

Referenced by G__G__MathCore_170_0_72(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::MathMore::gamma_quantile ( double  z,
double  alpha,
double  theta 
)

Re-implementation in MathMore of the Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the gamma distribution (gamma_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 38 of file QuantFuncMathMore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_74(), G__G__MathMore_100_0_2(), G__setup_memfuncROOTcLcLMath(), G__setup_memfuncROOTcLcLMathMore(), and testGammaFunction().

double ROOT::Math::gamma_quantile ( double  z,
double  alpha,
double  theta 
)

Re-implementation in MathMore of the Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the gamma distribution (gamma_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 118 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_74(), G__G__MathMore_100_0_2(), G__setup_memfuncROOTcLcLMath(), G__setup_memfuncROOTcLcLMathMore(), and testGammaFunction().

double ROOT::Math::gamma_quantile_c ( double  z,
double  alpha,
double  theta 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the gamma distribution (gamma_cdf_c). For detailed description see Mathworld. The implementation used is that of GSL. It is implemented using the function igami taken from Cephes.

Definition at line 112 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::igami().

Referenced by G__G__MathCore_170_0_73(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::gaussian_quantile ( double  z,
double  sigma 
) [inline]

alternative name for same function

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the normal (Gaussian) distribution (gaussian_cdf). For detailed description see Mathworld. It can also be evaluated using normal_quantile which will call the same implementation. It is implemented using the function ROOT::Math::Cephes::ndtri taken from Cephes.

Definition at line 442 of file QuantFuncMathCore.h.

References ROOT::Math::normal_quantile().

Referenced by G__G__MathCore_170_0_76(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::gaussian_quantile_c ( double  z,
double  sigma 
) [inline]

alternative name for same function

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the normal (Gaussian) distribution (gaussian_cdf_c). For detailed description see Mathworld. It can also be evaluated using normal_quantile_c which will call the same implementation.

Definition at line 417 of file QuantFuncMathCore.h.

References ROOT::Math::normal_quantile_c().

Referenced by G__G__MathCore_170_0_75(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::landau_quantile ( double  z,
double  xi = 1 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the Landau distribution (landau_cdf).

For detailed description see K.S. Kölbig and B. Schorr, A program package for the Landau distribution, Computer Phys. Comm. 31 (1984) 97-111 [Erratum-ibid. 178 (2008) 972]. The same algorithms as in CERNLIB (RANLAN) is used.

Parameters:
z The argument $z$
xi The width parameter $\xi$

Definition at line 189 of file QuantFuncMathCore.cxx.

References i, RootCsg::infinity, int, log(), and u.

Referenced by G__G__MathCore_170_0_83(), G__setup_memfuncROOTcLcLMath(), TRandom::Landau(), ROOT::Math::landau_quantile_c(), ROOT::Math::VavilovAccurate::Quantile(), and ROOT::Math::VavilovAccurate::Quantile_c().

double ROOT::Math::landau_quantile_c ( double  z,
double  xi = 1 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the landau distribution (landau_cdf_c). Implemented using landau_quantile

Parameters:
z The argument $z$
xi The width parameter $\xi$

Definition at line 396 of file QuantFuncMathCore.cxx.

References ROOT::Math::landau_quantile().

Referenced by G__G__MathCore_170_0_84(), and G__setup_memfuncROOTcLcLMath().

double ROOT::Math::lognormal_quantile ( double  x,
double  m,
double  s 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the lognormal distribution (lognormal_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 151 of file QuantFuncMathCore.cxx.

References exp(), ROOT::Math::Cephes::ndtri(), and y.

Referenced by G__G__MathCore_170_0_78(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::lognormal_quantile_c ( double  x,
double  m,
double  s 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the lognormal distribution (lognormal_cdf_c). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 143 of file QuantFuncMathCore.cxx.

References exp(), ROOT::Math::Cephes::ndtri(), and y.

Referenced by G__G__MathCore_170_0_77(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::normal_quantile ( double  z,
double  sigma 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the normal (Gaussian) distribution (normal_cdf). For detailed description see Mathworld. It can also be evaluated using gaussian_quantile which will call the same implementation. It is implemented using the function ROOT::Math::Cephes::ndtri taken from Cephes.

Definition at line 134 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::ndtri().

Referenced by TEfficiency::AgrestiCoull(), G__G__MathCore_170_0_80(), G__setup_memfuncROOTcLcLMath(), ROOT::Math::gaussian_quantile(), IntervalExamples(), TKDE::LowerConfidenceInterval(), TEfficiency::Normal(), RooMathCoreReg::RooMathCoreReg(), TKDE::UpperConfidenceInterval(), and TEfficiency::Wilson().

double ROOT::Math::normal_quantile_c ( double  z,
double  sigma 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the normal (Gaussian) distribution (normal_cdf_c). For detailed description see Mathworld. It can also be evaluated using gaussian_quantile_c which will call the same implementation. It is implemented using the function ROOT::Math::Cephes::ndtri taken from Cephes.

Definition at line 126 of file QuantFuncMathCore.cxx.

References ROOT::Math::Cephes::ndtri().

Referenced by TGraphAsymmErrors::Divide(), G__G__MathCore_170_0_79(), G__setup_memfuncROOTcLcLMath(), ROOT::Math::gaussian_quantile_c(), IntervalExamples(), RooStats::PValueToSignificance(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::tdistribution_quantile ( double  z,
double  r 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of Student's t-distribution (tdistribution_cdf). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 20 of file QuantFuncMathMore.cxx.

Referenced by G__G__MathMore_99_0_3(), G__setup_memfuncROOTcLcLMath(), RooMathMoreReg::RooMathMoreReg(), and tStudent().

double ROOT::Math::tdistribution_quantile_c ( double  z,
double  r 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of Student's t-distribution (tdistribution_cdf_c). For detailed description see Mathworld. The implementation used is that of GSL.

Definition at line 12 of file QuantFuncMathMore.cxx.

Referenced by G__G__MathMore_99_0_2(), G__setup_memfuncROOTcLcLMath(), and RooMathMoreReg::RooMathMoreReg().

double ROOT::Math::uniform_quantile ( double  z,
double  a,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the lower tail of the uniform (flat) distribution (uniform_cdf). For detailed description see Mathworld.

Definition at line 183 of file QuantFuncMathCore.cxx.

Referenced by G__G__MathCore_170_0_82(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::uniform_quantile_c ( double  z,
double  a,
double  b 
)

Inverse ($D^{-1}(z)$) of the cumulative distribution function of the upper tail of the uniform (flat) distribution (uniform_cdf_c). For detailed description see Mathworld.

Definition at line 175 of file QuantFuncMathCore.cxx.

Referenced by G__G__MathCore_170_0_81(), G__setup_memfuncROOTcLcLMath(), and RooMathCoreReg::RooMathCoreReg().

double ROOT::Math::vavilov_accurate_quantile ( double  z,
double  kappa,
double  beta2 
)

The inverse of the Vavilov cummulative probability density function

Parameters:
z The argument $z$, which must be in the range $0 \le z \le 1$
kappa The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2 The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Definition at line 477 of file VavilovAccurate.cxx.

References ROOT::Math::VavilovAccurate::GetInstance(), and vavilov().

Referenced by G__G__MathMore_99_0_40(), and G__setup_memfuncROOTcLcLMath().

double ROOT::Math::vavilov_accurate_quantile_c ( double  z,
double  kappa,
double  beta2 
)

The inverse of the complementary Vavilov cummulative probability density function

Parameters:
z The argument $z$, which must be in the range $0 \le z \le 1$
kappa The parameter $\kappa$, which must be in the range $\kappa \ge 0.001 $
beta2 The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Definition at line 482 of file VavilovAccurate.cxx.

References ROOT::Math::VavilovAccurate::GetInstance(), and vavilov().

Referenced by G__G__MathMore_99_0_41(), and G__setup_memfuncROOTcLcLMath().

double ROOT::Math::vavilov_fast_quantile ( double  z,
double  kappa,
double  beta2 
)

The inverse of the Vavilov cummulative probability density function

Parameters:
z The argument $z$, which must be in the range $0 \le z \le 1$
kappa The parameter $\kappa$, which must be in the range $0.01 \le \kappa \le 12 $
beta2 The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Definition at line 593 of file VavilovFast.cxx.

References ROOT::Math::VavilovFast::GetInstance(), and vavilov().

Referenced by G__G__MathMore_99_0_45(), and G__setup_memfuncROOTcLcLMath().

double ROOT::Math::vavilov_fast_quantile_c ( double  z,
double  kappa,
double  beta2 
)

The inverse of the complementary Vavilov cummulative probability density function

Parameters:
z The argument $z$, which must be in the range $0 \le z \le 1$
kappa The parameter $\kappa$, which must be in the range $0.01 \le \kappa \le 12 $
beta2 The parameter $\beta^2$, which must be in the range $0 \le \beta^2 \le 1 $

Definition at line 598 of file VavilovFast.cxx.

References ROOT::Math::VavilovFast::GetInstance(), and vavilov().

Referenced by G__G__MathMore_99_0_46(), and G__setup_memfuncROOTcLcLMath().

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