Logo ROOT  
Reference Guide
QuantFuncMathCore.h
Go to the documentation of this file.
1// @(#)root/mathcore:$Id$
2// Authors: L. Moneta, A. Zsenei 08/2005
3
4
5// Authors: Andras Zsenei & Lorenzo Moneta 08/2005
6
7
8/**********************************************************************
9 * *
10 * Copyright (c) 2005 , LCG ROOT MathLib Team *
11 * *
12 * *
13 **********************************************************************/
14
15
16#ifndef ROOT_Math_QuantFuncMathCore
17#define ROOT_Math_QuantFuncMathCore
18
19
20namespace ROOT {
21namespace Math {
22
23
24
25 /** @defgroup QuantFunc Quantile Functions
26 * @ingroup StatFunc
27 *
28 * Inverse functions of the cumulative distribution functions
29 * and the inverse of the complement of the cumulative distribution functions
30 * for various distributions.
31 * The functions with the extension <em>_quantile</em> calculate the
32 * inverse of the <em>_cdf</em> function, the
33 * lower tail integral of the probability density function
34 * \f$D^{-1}(z)\f$ where
35 *
36 * \f[ D(x) = \int_{-\infty}^{x} p(x') dx' \f]
37 *
38 * while those with the <em>_quantile_c</em> extension calculate the
39 * inverse of the <em>_cdf_c</em> functions, the upper tail integral of the probability
40 * density function \f$D^{-1}(z) \f$ where
41 *
42 * \f[ D(x) = \int_{x}^{+\infty} p(x') dx' \f]
43 *
44 * These functions are defined in the header file <em>Math/ProbFunc.h</em> or in the global one
45 * including all statistical functions <em>Math/DistFunc.h</em>
46 *
47 *
48 * <strong>NOTE:</strong> In the old releases (< 5.14) the <em>_quantile</em> functions were called
49 * <em>_quant_inv</em> and the <em>_quantile_c</em> functions were called
50 * <em>_prob_inv</em>.
51 * These names are currently kept for backward compatibility, but
52 * their usage is deprecated.
53 *
54 */
55
56 /** @name Quantile Functions from MathCore
57 * The implementation is provided in MathCore and for the majority of the function comes from
58 * <A HREF="http://www.netlib.org/cephes">Cephes</A>.
59
60 */
61
62 //@{
63
64
65
66 /**
67
68 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
69 function of the upper tail of the beta distribution
70 (#beta_cdf_c).
71 It is implemented using the function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
72
73
74 @ingroup QuantFunc
75
76 */
77 double beta_quantile(double x, double a, double b);
78
79 /**
80
81 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
82 function of the lower tail of the beta distribution
83 (#beta_cdf).
84 It is implemented using
85 the function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
86
87 @ingroup QuantFunc
88
89 */
90 double beta_quantile_c(double x, double a, double b);
91
92
93
94 /**
95
96 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
97 function of the upper tail of the Cauchy distribution (#cauchy_cdf_c)
98 which is also called Lorentzian distribution. For
99 detailed description see
100 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
101 Mathworld</A>.
102
103 @ingroup QuantFunc
104
105 */
106
107 double cauchy_quantile_c(double z, double b);
108
109
110
111
112 /**
113
114 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
115 function of the lower tail of the Cauchy distribution (#cauchy_cdf)
116 which is also called Breit-Wigner or Lorentzian distribution. For
117 detailed description see
118 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
119 Mathworld</A>. The implementation used is that of
120 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC294">GSL</A>.
121
122 @ingroup QuantFunc
123
124 */
125
126 double cauchy_quantile(double z, double b);
127
128
129
130
131 /**
132
133 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
134 function of the upper tail of the Breit-Wigner distribution (#breitwigner_cdf_c)
135 which is similar to the Cauchy distribution. For
136 detailed description see
137 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
138 Mathworld</A>. It is evaluated using the same implementation of
139 #cauchy_quantile_c.
140
141 @ingroup QuantFunc
142
143 */
144
145 inline double breitwigner_quantile_c(double z, double gamma) {
146 return cauchy_quantile_c(z, gamma/2.0);
147 }
148
149
150
151
152 /**
153
154 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
155 function of the lower tail of the Breit_Wigner distribution (#breitwigner_cdf)
156 which is similar to the Cauchy distribution. For
157 detailed description see
158 <A HREF="http://mathworld.wolfram.com/CauchyDistribution.html">
159 Mathworld</A>. It is evaluated using the same implementation of
160 #cauchy_quantile.
161
162
163 @ingroup QuantFunc
164
165 */
166
167 inline double breitwigner_quantile(double z, double gamma) {
168 return cauchy_quantile(z, gamma/2.0);
169 }
170
171
172
173
174
175
176 /**
177
178 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
179 function of the upper tail of the \f$\chi^2\f$ distribution
180 with \f$r\f$ degrees of freedom (#chisquared_cdf_c). For detailed description see
181 <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
182 Mathworld</A>. It is implemented using the inverse of the incomplete complement gamma function, using
183 the function igami from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
184
185 @ingroup QuantFunc
186
187 */
188
189 double chisquared_quantile_c(double z, double r);
190
191
192
193 /**
194
195 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
196 function of the lower tail of the \f$\chi^2\f$ distribution
197 with \f$r\f$ degrees of freedom (#chisquared_cdf). For detailed description see
198 <A HREF="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">
199 Mathworld</A>.
200 It is implemented using chisquared_quantile_c, therefore is not very precise for small z.
201 It is recommended to use the MathMore function (ROOT::MathMore::chisquared_quantile )implemented using GSL
202
203 @ingroup QuantFunc
204
205 */
206
207 double chisquared_quantile(double z, double r);
208
209
210
211 /**
212
213 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
214 function of the upper tail of the exponential distribution
215 (#exponential_cdf_c). For detailed description see
216 <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
217 Mathworld</A>.
218
219 @ingroup QuantFunc
220
221 */
222
223 double exponential_quantile_c(double z, double lambda);
224
225
226
227
228 /**
229
230 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
231 function of the lower tail of the exponential distribution
232 (#exponential_cdf). For detailed description see
233 <A HREF="http://mathworld.wolfram.com/ExponentialDistribution.html">
234 Mathworld</A>.
235
236 @ingroup QuantFunc
237
238 */
239
240 double exponential_quantile(double z, double lambda);
241
242
243
244 /**
245
246 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
247 function of the lower tail of the f distribution
248 (#fdistribution_cdf). For detailed description see
249 <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
250 Mathworld</A>.
251 It is implemented using the inverse of the incomplete beta function,
252 function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
253
254 @ingroup QuantFunc
255
256 */
257 double fdistribution_quantile(double z, double n, double m);
258
259 /**
260
261 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
262 function of the upper tail of the f distribution
263 (#fdistribution_cdf_c). For detailed description see
264 <A HREF="http://mathworld.wolfram.com/F-Distribution.html">
265 Mathworld</A>.
266 It is implemented using the inverse of the incomplete beta function,
267 function incbi from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
268
269 @ingroup QuantFunc
270 */
271
272 double fdistribution_quantile_c(double z, double n, double m);
273
274
275 /**
276
277 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
278 function of the upper tail of the gamma distribution
279 (#gamma_cdf_c). For detailed description see
280 <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
281 Mathworld</A>. The implementation used is that of
282 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC300">GSL</A>.
283 It is implemented using the function igami taken
284 from <A HREF="http://www.netlib.org/cephes">Cephes</A>.
285
286 @ingroup QuantFunc
287
288 */
289
290 double gamma_quantile_c(double z, double alpha, double theta);
291
292
293
294
295 /**
296
297 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
298 function of the lower tail of the gamma distribution
299 (#gamma_cdf). For detailed description see
300 <A HREF="http://mathworld.wolfram.com/GammaDistribution.html">
301 Mathworld</A>.
302 It is implemented using chisquared_quantile_c, therefore is not very precise for small z.
303 For this special cases it is recommended to use the MathMore function ROOT::MathMore::gamma_quantile
304 implemented using GSL
305
306
307 @ingroup QuantFunc
308
309 */
310
311 double gamma_quantile(double z, double alpha, double theta);
312
313
314
315 /**
316
317 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
318 function of the upper tail of the normal (Gaussian) distribution
319 (#gaussian_cdf_c). For detailed description see
320 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
321 Mathworld</A>. It can also be evaluated using #normal_quantile_c which will
322 call the same implementation.
323
324 @ingroup QuantFunc
325
326 */
327
328 double gaussian_quantile_c(double z, double sigma);
329
330
331
332
333 /**
334
335 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
336 function of the lower tail of the normal (Gaussian) distribution
337 (#gaussian_cdf). For detailed description see
338 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
339 Mathworld</A>. It can also be evaluated using #normal_quantile which will
340 call the same implementation.
341 It is implemented using the function ROOT::Math::Cephes::ndtri taken from
342 <A HREF="http://www.netlib.org/cephes">Cephes</A>.
343
344 @ingroup QuantFunc
345
346 */
347
348 double gaussian_quantile(double z, double sigma);
349
350
351
352
353 /**
354
355 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
356 function of the upper tail of the lognormal distribution
357 (#lognormal_cdf_c). For detailed description see
358 <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
359 Mathworld</A>. The implementation used is that of
360 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC302">GSL</A>.
361
362 @ingroup QuantFunc
363
364 */
365
366 double lognormal_quantile_c(double x, double m, double s);
367
368
369
370
371 /**
372
373 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
374 function of the lower tail of the lognormal distribution
375 (#lognormal_cdf). For detailed description see
376 <A HREF="http://mathworld.wolfram.com/LogNormalDistribution.html">
377 Mathworld</A>. The implementation used is that of
378 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC302">GSL</A>.
379
380 @ingroup QuantFunc
381
382 */
383
384 double lognormal_quantile(double x, double m, double s);
385
386
387
388
389 /**
390
391 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
392 function of the upper tail of the normal (Gaussian) distribution
393 (#normal_cdf_c). For detailed description see
394 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
395 Mathworld</A>. It can also be evaluated using #gaussian_quantile_c which will
396 call the same implementation.
397 It is implemented using the function ROOT::Math::Cephes::ndtri taken from
398 <A HREF="http://www.netlib.org/cephes">Cephes</A>.
399
400 @ingroup QuantFunc
401
402 */
403
404 double normal_quantile_c(double z, double sigma);
405 /// alternative name for same function
406 inline double gaussian_quantile_c(double z, double sigma) {
407 return normal_quantile_c(z,sigma);
408 }
409
410
411
412
413 /**
414
415 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
416 function of the lower tail of the normal (Gaussian) distribution
417 (#normal_cdf). For detailed description see
418 <A HREF="http://mathworld.wolfram.com/NormalDistribution.html">
419 Mathworld</A>. It can also be evaluated using #gaussian_quantile which will
420 call the same implementation.
421 It is implemented using the function ROOT::Math::Cephes::ndtri taken from
422 <A HREF="http://www.netlib.org/cephes">Cephes</A>.
423
424
425 @ingroup QuantFunc
426
427 */
428
429 double normal_quantile(double z, double sigma);
430 /// alternative name for same function
431 inline double gaussian_quantile(double z, double sigma) {
432 return normal_quantile(z,sigma);
433 }
434
435
436
437#ifdef LATER // t quantiles are still in MathMore
438
439 /**
440
441 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
442 function of the upper tail of Student's t-distribution
443 (#tdistribution_cdf_c). For detailed description see
444 <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
445 Mathworld</A>. The implementation used is that of
446 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC305">GSL</A>.
447
448 @ingroup QuantFunc
449
450 */
451
452 double tdistribution_quantile_c(double z, double r);
453
454
455
456
457 /**
458
459 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
460 function of the lower tail of Student's t-distribution
461 (#tdistribution_cdf). For detailed description see
462 <A HREF="http://mathworld.wolfram.com/Studentst-Distribution.html">
463 Mathworld</A>. The implementation used is that of
464 <A HREF="http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC305">GSL</A>.
465
466 @ingroup QuantFunc
467
468 */
469
470 double tdistribution_quantile(double z, double r);
471
472#endif
473
474
475 /**
476
477 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
478 function of the upper tail of the uniform (flat) distribution
479 (#uniform_cdf_c). For detailed description see
480 <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
481 Mathworld</A>.
482
483 @ingroup QuantFunc
484
485 */
486
487 double uniform_quantile_c(double z, double a, double b);
488
489
490
491
492 /**
493
494 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
495 function of the lower tail of the uniform (flat) distribution
496 (#uniform_cdf). For detailed description see
497 <A HREF="http://mathworld.wolfram.com/UniformDistribution.html">
498 Mathworld</A>.
499
500 @ingroup QuantFunc
501
502 */
503
504 double uniform_quantile(double z, double a, double b);
505
506
507
508
509 /**
510
511 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
512 function of the lower tail of the Landau distribution
513 (#landau_cdf).
514
515 For detailed description see
516 K.S. K&ouml;lbig and B. Schorr, A program package for the Landau distribution,
517 <A HREF="http://dx.doi.org/10.1016/0010-4655(84)90085-7">Computer Phys. Comm. 31 (1984) 97-111</A>
518 <A HREF="http://dx.doi.org/10.1016/j.cpc.2008.03.002">[Erratum-ibid. 178 (2008) 972]</A>.
519 The same algorithms as in
520 <A HREF="https://cern-tex.web.cern.ch/cern-tex/shortwrupsdir/g110/top.html">
521 CERNLIB</A> (RANLAN) is used.
522
523 @param z The argument \f$z\f$
524 @param xi The width parameter \f$\xi\f$
525
526 @ingroup QuantFunc
527
528 */
529
530 double landau_quantile(double z, double xi = 1);
531
532
533 /**
534
535 Inverse (\f$D^{-1}(z)\f$) of the cumulative distribution
536 function of the upper tail of the landau distribution
537 (#landau_cdf_c).
538 Implemented using #landau_quantile
539
540 @param z The argument \f$z\f$
541 @param xi The width parameter \f$\xi\f$
542
543 @ingroup QuantFunc
544
545 */
546
547 double landau_quantile_c(double z, double xi = 1);
548
549
550#ifdef HAVE_OLD_STAT_FUNC
551
552 //@}
553 /** @name Backward compatible functions */
554
555
556 inline double breitwigner_prob_inv(double x, double gamma) {
558 }
559 inline double breitwigner_quant_inv(double x, double gamma) {
561 }
562
563 inline double cauchy_prob_inv(double x, double b) {
564 return cauchy_quantile_c(x,b);
565 }
566 inline double cauchy_quant_inv(double x, double b) {
567 return cauchy_quantile (x,b);
568 }
569
570 inline double exponential_prob_inv(double x, double lambda) {
571 return exponential_quantile_c(x, lambda );
572 }
573 inline double exponential_quant_inv(double x, double lambda) {
574 return exponential_quantile (x, lambda );
575 }
576
577 inline double gaussian_prob_inv(double x, double sigma) {
578 return gaussian_quantile_c( x, sigma );
579 }
580 inline double gaussian_quant_inv(double x, double sigma) {
581 return gaussian_quantile ( x, sigma );
582 }
583
584 inline double lognormal_prob_inv(double x, double m, double s) {
585 return lognormal_quantile_c( x, m, s );
586 }
587 inline double lognormal_quant_inv(double x, double m, double s) {
588 return lognormal_quantile ( x, m, s );
589 }
590
591 inline double normal_prob_inv(double x, double sigma) {
592 return normal_quantile_c( x, sigma );
593 }
594 inline double normal_quant_inv(double x, double sigma) {
595 return normal_quantile ( x, sigma );
596 }
597
598 inline double uniform_prob_inv(double x, double a, double b) {
599 return uniform_quantile_c( x, a, b );
600 }
601 inline double uniform_quant_inv(double x, double a, double b) {
602 return uniform_quantile ( x, a, b );
603 }
604
605 inline double chisquared_prob_inv(double x, double r) {
606 return chisquared_quantile_c(x, r );
607 }
608
609 inline double gamma_prob_inv(double x, double alpha, double theta) {
610 return gamma_quantile_c (x, alpha, theta );
611 }
612
613
614#endif
615
616
617} // namespace Math
618} // namespace ROOT
619
620
621
622#endif // ROOT_Math_QuantFuncMathCore
ROOT::R::TRInterface & r
Definition: Object.C:4
#define b(i)
Definition: RSha256.hxx:100
double fdistribution_quantile_c(double z, double n, double m)
Inverse ( ) of the cumulative distribution function of the upper tail of the f distribution (fdistrib...
double breitwigner_quantile(double z, double gamma)
Inverse ( ) of the cumulative distribution function of the lower tail of the Breit_Wigner distributio...
double uniform_quantile_c(double z, double a, double b)
Inverse ( ) of the cumulative distribution function of the upper tail of the uniform (flat) distribut...
double landau_quantile(double z, double xi=1)
Inverse ( ) of the cumulative distribution function of the lower tail of the Landau distribution (lan...
double chisquared_quantile_c(double z, double r)
Inverse ( ) of the cumulative distribution function of the upper tail of the distribution with degr...
double normal_quantile(double z, double sigma)
Inverse ( ) of the cumulative distribution function of the lower tail of the normal (Gaussian) distri...
double exponential_quantile_c(double z, double lambda)
Inverse ( ) of the cumulative distribution function of the upper tail of the exponential distribution...
double gamma_quantile_c(double z, double alpha, double theta)
Inverse ( ) of the cumulative distribution function of the upper tail of the gamma distribution (gamm...
double tdistribution_quantile(double z, double r)
Inverse ( ) of the cumulative distribution function of the lower tail of Student's t-distribution (td...
double landau_quantile_c(double z, double xi=1)
Inverse ( ) of the cumulative distribution function of the upper tail of the landau distribution (lan...
double gaussian_quantile_c(double z, double sigma)
Inverse ( ) of the cumulative distribution function of the upper tail of the normal (Gaussian) distri...
double fdistribution_quantile(double z, double n, double m)
Inverse ( ) of the cumulative distribution function of the lower tail of the f distribution (fdistrib...
double cauchy_quantile_c(double z, double b)
Inverse ( ) of the cumulative distribution function of the upper tail of the Cauchy distribution (cau...
double uniform_quantile(double z, double a, double b)
Inverse ( ) of the cumulative distribution function of the lower tail of the uniform (flat) distribut...
double normal_quantile_c(double z, double sigma)
Inverse ( ) of the cumulative distribution function of the upper tail of the normal (Gaussian) distri...
double tdistribution_quantile_c(double z, double r)
Inverse ( ) of the cumulative distribution function of the upper tail of Student's t-distribution (td...
double chisquared_quantile(double z, double r)
Inverse ( ) of the cumulative distribution function of the lower tail of the distribution with degr...
double beta_quantile_c(double x, double a, double b)
Inverse ( ) of the cumulative distribution function of the lower tail of the beta distribution (beta_...
double gamma_quantile(double z, double alpha, double theta)
Inverse ( ) of the cumulative distribution function of the lower tail of the gamma distribution (gamm...
double exponential_quantile(double z, double lambda)
Inverse ( ) of the cumulative distribution function of the lower tail of the exponential distribution...
double lognormal_quantile(double x, double m, double s)
Inverse ( ) of the cumulative distribution function of the lower tail of the lognormal distribution (...
double cauchy_quantile(double z, double b)
Inverse ( ) of the cumulative distribution function of the lower tail of the Cauchy distribution (cau...
double beta_quantile(double x, double a, double b)
Inverse ( ) of the cumulative distribution function of the upper tail of the beta distribution (beta_...
double gaussian_quantile(double z, double sigma)
Inverse ( ) of the cumulative distribution function of the lower tail of the normal (Gaussian) distri...
double lognormal_quantile_c(double x, double m, double s)
Inverse ( ) of the cumulative distribution function of the upper tail of the lognormal distribution (...
double breitwigner_quantile_c(double z, double gamma)
Inverse ( ) of the cumulative distribution function of the upper tail of the Breit-Wigner distributio...
const Double_t sigma
Double_t x[n]
Definition: legend1.C:17
const Int_t n
Definition: legend1.C:16
Namespace for new Math classes and functions.
double gamma(double x)
tbb::task_arena is an alias of tbb::interface7::task_arena, which doesn't allow to forward declare tb...
static constexpr double s
auto * m
Definition: textangle.C:8
auto * a
Definition: textangle.C:12