defects fixed
[u/mrichter/AliRoot.git] / RALICE / AliMath.cxx
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4c039060 1/**************************************************************************
2 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
3 * *
4 * Author: The ALICE Off-line Project. *
5 * Contributors are mentioned in the code where appropriate. *
6 * *
7 * Permission to use, copy, modify and distribute this software and its *
8 * documentation strictly for non-commercial purposes is hereby granted *
9 * without fee, provided that the above copyright notice appears in all *
10 * copies and that both the copyright notice and this permission notice *
11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
15
f531a546 16// $Id$
4c039060 17
959fbac5 18///////////////////////////////////////////////////////////////////////////
19// Class AliMath
20// Various mathematical tools which may be very convenient while
21// performing physics analysis.
22//
23// Example : Probability of a Chi-squared value
24// =========
25//
26// AliMath M;
27// Float_t chi2=20; // The chi-squared value
28// Int_t ndf=12; // The number of degrees of freedom
29// Float_t p=M.Prob(chi2,ndf); // The probability that at least a Chi-squared
30// // value of chi2 will be observed, even for a
31// // correct model
32//
33//--- Author: Nick van Eijndhoven 14-nov-1998 UU-SAP Utrecht
f531a546 34//- Modified: NvE $Date$ UU-SAP Utrecht
959fbac5 35///////////////////////////////////////////////////////////////////////////
36
d88f97cc 37#include "AliMath.h"
c72198f1 38#include "Riostream.h"
d88f97cc 39
40ClassImp(AliMath) // Class implementation to enable ROOT I/O
41
c72198f1 42AliMath::AliMath() : TObject()
d88f97cc 43{
44// Default constructor
45}
46///////////////////////////////////////////////////////////////////////////
47AliMath::~AliMath()
48{
49// Destructor
50}
51///////////////////////////////////////////////////////////////////////////
261c0caf 52AliMath::AliMath(const AliMath& m) : TObject(m)
c72198f1 53{
54// Copy constructor
55}
56///////////////////////////////////////////////////////////////////////////
261c0caf 57Double_t AliMath::Gamma(Double_t z) const
d88f97cc 58{
59// Computation of gamma(z) for all z>0.
60//
61// The algorithm is based on the article by C.Lanczos [1] as denoted in
62// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
63//
64// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
65//
66//--- Nve 14-nov-1998 UU-SAP Utrecht
67
68 if (z<=0.)
69 {
70 cout << "*Gamma(z)* Wrong argument z = " << z << endl;
71 return 0;
72 }
73
29beb80d 74 Double_t v=LnGamma(z);
d88f97cc 75 return exp(v);
76}
77///////////////////////////////////////////////////////////////////////////
261c0caf 78Double_t AliMath::Gamma(Double_t a,Double_t x) const
d88f97cc 79{
80// Computation of the incomplete gamma function P(a,x)
81//
82// The algorithm is based on the formulas and code as denoted in
83// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
84//
85//--- Nve 14-nov-1998 UU-SAP Utrecht
86
87 if (a<=0.)
88 {
89 cout << "*Gamma(a,x)* Invalid argument a = " << a << endl;
90 return 0;
91 }
92
93 if (x<=0.)
94 {
95 if (x<0) cout << "*Gamma(a,x)* Invalid argument x = " << x << endl;
96 return 0;
97 }
98
99 if (x<(a+1.))
100 {
101 return GamSer(a,x);
102 }
103 else
104 {
105 return GamCf(a,x);
106 }
107}
108///////////////////////////////////////////////////////////////////////////
261c0caf 109Double_t AliMath::LnGamma(Double_t z) const
d88f97cc 110{
111// Computation of ln[gamma(z)] for all z>0.
112//
113// The algorithm is based on the article by C.Lanczos [1] as denoted in
114// Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.).
115//
116// [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86.
117//
118// The accuracy of the result is better than 2e-10.
119//
120//--- Nve 14-nov-1998 UU-SAP Utrecht
121
122 if (z<=0.)
123 {
124 cout << "*LnGamma(z)* Wrong argument z = " << z << endl;
125 return 0;
126 }
127
128 // Coefficients for the series expansion
129 Double_t c[7];
130 c[0]= 2.5066282746310005;
131 c[1]= 76.18009172947146;
132 c[2]=-86.50532032941677;
133 c[3]= 24.01409824083091;
134 c[4]= -1.231739572450155;
135 c[5]= 0.1208650973866179e-2;
136 c[6]= -0.5395239384953e-5;
137
138 Double_t x=z;
139 Double_t y=x;
140 Double_t tmp=x+5.5;
141 tmp=(x+0.5)*log(tmp)-tmp;
142 Double_t ser=1.000000000190015;
143 for (Int_t i=1; i<7; i++)
144 {
145 y+=1.;
146 ser+=c[i]/y;
147 }
29beb80d 148 Double_t v=tmp+log(c[0]*ser/x);
d88f97cc 149 return v;
150}
151///////////////////////////////////////////////////////////////////////////
261c0caf 152Double_t AliMath::GamSer(Double_t a,Double_t x) const
d88f97cc 153{
154// Computation of the incomplete gamma function P(a,x)
155// via its series representation.
156//
157// The algorithm is based on the formulas and code as denoted in
158// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
159//
160//--- Nve 14-nov-1998 UU-SAP Utrecht
161
162 Int_t itmax=100; // Maximum number of iterations
29beb80d 163 Double_t eps=3.e-7; // Relative accuracy
d88f97cc 164
165 if (a<=0.)
166 {
167 cout << "*GamSer(a,x)* Invalid argument a = " << a << endl;
168 return 0;
169 }
170
171 if (x<=0.)
172 {
173 if (x<0) cout << "*GamSer(a,x)* Invalid argument x = " << x << endl;
174 return 0;
175 }
176
29beb80d 177 Double_t gln=LnGamma(a);
178 Double_t ap=a;
179 Double_t sum=1./a;
180 Double_t del=sum;
d88f97cc 181 for (Int_t n=1; n<=itmax; n++)
182 {
183 ap+=1.;
184 del=del*x/ap;
185 sum+=del;
186 if (fabs(del)<fabs(sum*eps)) break;
187 if (n==itmax) cout << "*GamSer(a,x)* a too large or itmax too small" << endl;
188 }
29beb80d 189 Double_t v=sum*exp(-x+a*log(x)-gln);
d88f97cc 190 return v;
191}
192///////////////////////////////////////////////////////////////////////////
261c0caf 193Double_t AliMath::GamCf(Double_t a,Double_t x) const
d88f97cc 194{
195// Computation of the incomplete gamma function P(a,x)
196// via its continued fraction representation.
197//
198// The algorithm is based on the formulas and code as denoted in
199// Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.).
200//
201//--- Nve 14-nov-1998 UU-SAP Utrecht
202
203 Int_t itmax=100; // Maximum number of iterations
29beb80d 204 Double_t eps=3.e-7; // Relative accuracy
205 Double_t fpmin=1.e-30; // Smallest Double_t value allowed here
d88f97cc 206
207 if (a<=0.)
208 {
209 cout << "*GamCf(a,x)* Invalid argument a = " << a << endl;
210 return 0;
211 }
212
213 if (x<=0.)
214 {
215 if (x<0) cout << "*GamCf(a,x)* Invalid argument x = " << x << endl;
216 return 0;
217 }
218
29beb80d 219 Double_t gln=LnGamma(a);
220 Double_t b=x+1.-a;
221 Double_t c=1./fpmin;
222 Double_t d=1./b;
223 Double_t h=d;
224 Double_t an,del;
d88f97cc 225 for (Int_t i=1; i<=itmax; i++)
226 {
29beb80d 227 an=double(-i)*(double(i)-a);
d88f97cc 228 b+=2.;
229 d=an*d+b;
230 if (fabs(d)<fpmin) d=fpmin;
231 c=b+an/c;
232 if (fabs(c)<fpmin) c=fpmin;
233 d=1./d;
234 del=d*c;
235 h=h*del;
236 if (fabs(del-1.)<eps) break;
237 if (i==itmax) cout << "*GamCf(a,x)* a too large or itmax too small" << endl;
238 }
29beb80d 239 Double_t v=exp(-x+a*log(x)-gln)*h;
d88f97cc 240 return (1.-v);
241}
242///////////////////////////////////////////////////////////////////////////
261c0caf 243Double_t AliMath::Erf(Double_t x) const
d88f97cc 244{
245// Computation of the error function erf(x).
246//
247//--- NvE 14-nov-1998 UU-SAP Utrecht
248
249 return (1.-Erfc(x));
250}
251///////////////////////////////////////////////////////////////////////////
261c0caf 252Double_t AliMath::Erfc(Double_t x) const
d88f97cc 253{
254// Computation of the complementary error function erfc(x).
255//
256// The algorithm is based on a Chebyshev fit as denoted in
257// Numerical Recipes 2nd ed. on p. 214 (W.H.Press et al.).
258//
259// The fractional error is always less than 1.2e-7.
260//
261//--- Nve 14-nov-1998 UU-SAP Utrecht
262
263 // The parameters of the Chebyshev fit
387a745b 264 const Double_t ka1=-1.26551223, ka2=1.00002368,
265 ka3= 0.37409196, ka4=0.09678418,
266 ka5=-0.18628806, ka6=0.27886807,
267 ka7=-1.13520398, ka8=1.48851587,
268 ka9=-0.82215223, ka10=0.17087277;
d88f97cc 269
29beb80d 270 Double_t v=1.; // The return value
d88f97cc 271
29beb80d 272 Double_t z=fabs(x);
d88f97cc 273
274 if (z <= 0.) return v; // erfc(0)=1
275
29beb80d 276 Double_t t=1./(1.+0.5*z);
d88f97cc 277
278 v=t*exp((-z*z)
387a745b 279 +ka1+t*(ka2+t*(ka3+t*(ka4+t*(ka5+t*(ka6+t*(ka7+t*(ka8+t*(ka9+t*ka10)))))))));
d88f97cc 280
281 if (x < 0.) v=2.-v; // erfc(-x)=2-erfc(x)
282
283 return v;
284}
285///////////////////////////////////////////////////////////////////////////
261c0caf 286Double_t AliMath::Prob(Double_t chi2,Int_t ndf,Int_t mode) const
d88f97cc 287{
288// Computation of the probability for a certain Chi-squared (chi2)
289// and number of degrees of freedom (ndf).
290//
a57b6095 291// A more clear and flexible facility is offered by Chi2Pvalue.
292//
176f88c0 293// According to the value of the parameter "mode" various algorithms
294// can be selected.
295//
296// mode = 0 : Calculations are based on the incomplete gamma function P(a,x),
297// where a=ndf/2 and x=chi2/2.
298//
299// 1 : Same as for mode=0. However, in case ndf=1 an exact expression
300// based on the error function Erf() is used.
301//
302// 2 : Same as for mode=0. However, in case ndf>30 a Gaussian approximation
303// is used instead of the gamma function.
304//
305// When invoked as Prob(chi2,ndf) the default mode=1 is used.
d88f97cc 306//
307// P(a,x) represents the probability that the observed Chi-squared
308// for a correct model should be less than the value chi2.
309//
310// The returned probability corresponds to 1-P(a,x),
311// which denotes the probability that an observed Chi-squared exceeds
312// the value chi2 by chance, even for a correct model.
313//
314//--- NvE 14-nov-1998 UU-SAP Utrecht
315
316 if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0
317
318 if (chi2 <= 0.)
319 {
320 if (chi2 < 0.)
321 {
322 return 0;
323 }
324 else
325 {
326 return 1;
327 }
328 }
176f88c0 329
330 Double_t v=-1.;
331
332 switch (mode)
333 {
334 case 1: // Exact expression for ndf=1 as alternative for the gamma function
335 if (ndf==1) v=1.-Erf(sqrt(chi2)/sqrt(2.));
336 break;
337
338 case 2: // Gaussian approximation for large ndf (i.e. ndf>30) as alternative for the gamma function
339 if (ndf>30)
340 {
341 Double_t q=sqrt(2.*chi2)-sqrt(double(2*ndf-1));
342 if (q>0.) v=0.5*(1.-Erf(q/sqrt(2.)));
343 }
344 break;
345 }
d88f97cc 346
176f88c0 347 if (v<0.)
348 {
349 // Evaluate the incomplete gamma function
350 Double_t a=double(ndf)/2.;
351 Double_t x=chi2/2.;
352 v=1.-Gamma(a,x);
353 }
354
355 return v;
d88f97cc 356}
357///////////////////////////////////////////////////////////////////////////
261c0caf 358Double_t AliMath::BesselI0(Double_t x) const
29beb80d 359{
360// Computation of the modified Bessel function I_0(x) for any real x.
361//
362// The algorithm is based on the article by Abramowitz and Stegun [1]
363// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
364//
365// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
366// Applied Mathematics Series vol. 55 (1964), Washington.
367//
368//--- NvE 12-mar-2000 UU-SAP Utrecht
369
370 // Parameters of the polynomial approximation
387a745b 371 const Double_t kp1=1.0, kp2=3.5156229, kp3=3.0899424,
372 kp4=1.2067492, kp5=0.2659732, kp6=3.60768e-2, kp7=4.5813e-3;
29beb80d 373
387a745b 374 const Double_t kq1= 0.39894228, kq2= 1.328592e-2, kq3= 2.25319e-3,
375 kq4=-1.57565e-3, kq5= 9.16281e-3, kq6=-2.057706e-2,
376 kq7= 2.635537e-2, kq8=-1.647633e-2, kq9= 3.92377e-3;
29beb80d 377
378 Double_t ax=fabs(x);
379
380 Double_t y=0,result=0;
381
382 if (ax < 3.75)
383 {
384 y=pow(x/3.75,2);
387a745b 385 result=kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7)))));
29beb80d 386 }
387 else
388 {
389 y=3.75/ax;
387a745b 390 result=(exp(ax)/sqrt(ax))
391 *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9))))))));
29beb80d 392 }
393
394 return result;
395}
396///////////////////////////////////////////////////////////////////////////
261c0caf 397Double_t AliMath::BesselK0(Double_t x) const
29beb80d 398{
399// Computation of the modified Bessel function K_0(x) for positive real x.
400//
401// The algorithm is based on the article by Abramowitz and Stegun [1]
402// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
403//
404// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
405// Applied Mathematics Series vol. 55 (1964), Washington.
406//
407//--- NvE 12-mar-2000 UU-SAP Utrecht
408
409 // Parameters of the polynomial approximation
387a745b 410 const Double_t kp1=-0.57721566, kp2=0.42278420, kp3=0.23069756,
7a086578 411 kp4= 3.488590e-2, kp5=2.62698e-3, kp6=1.0750e-4, kp7=7.4e-6;
29beb80d 412
387a745b 413 const Double_t kq1= 1.25331414, kq2=-7.832358e-2, kq3= 2.189568e-2,
414 kq4=-1.062446e-2, kq5= 5.87872e-3, kq6=-2.51540e-3, kq7=5.3208e-4;
29beb80d 415
416 if (x <= 0)
417 {
418 cout << " *BesselK0* Invalid argument x = " << x << endl;
419 return 0;
420 }
421
422 Double_t y=0,result=0;
423
424 if (x <= 2)
425 {
426 y=x*x/4.;
387a745b 427 result=(-log(x/2.)*BesselI0(x))
428 +(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
29beb80d 429 }
430 else
431 {
432 y=2./x;
387a745b 433 result=(exp(-x)/sqrt(x))
434 *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7))))));
29beb80d 435 }
436
437 return result;
438}
439///////////////////////////////////////////////////////////////////////////
261c0caf 440Double_t AliMath::BesselI1(Double_t x) const
29beb80d 441{
442// Computation of the modified Bessel function I_1(x) for any real x.
443//
444// The algorithm is based on the article by Abramowitz and Stegun [1]
445// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
446//
447// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
448// Applied Mathematics Series vol. 55 (1964), Washington.
449//
450//--- NvE 12-mar-2000 UU-SAP Utrecht
451
452 // Parameters of the polynomial approximation
387a745b 453 const Double_t kp1=0.5, kp2=0.87890594, kp3=0.51498869,
454 kp4=0.15084934, kp5=2.658733e-2, kp6=3.01532e-3, kp7=3.2411e-4;
29beb80d 455
387a745b 456 const Double_t kq1= 0.39894228, kq2=-3.988024e-2, kq3=-3.62018e-3,
457 kq4= 1.63801e-3, kq5=-1.031555e-2, kq6= 2.282967e-2,
458 kq7=-2.895312e-2, kq8= 1.787654e-2, kq9=-4.20059e-3;
29beb80d 459
460 Double_t ax=fabs(x);
461
462 Double_t y=0,result=0;
463
464 if (ax < 3.75)
465 {
466 y=pow(x/3.75,2);
387a745b 467 result=x*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
29beb80d 468 }
469 else
470 {
471 y=3.75/ax;
387a745b 472 result=(exp(ax)/sqrt(ax))
473 *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9))))))));
29beb80d 474 if (x < 0) result=-result;
475 }
476
477 return result;
478}
479///////////////////////////////////////////////////////////////////////////
261c0caf 480Double_t AliMath::BesselK1(Double_t x) const
29beb80d 481{
482// Computation of the modified Bessel function K_1(x) for positive real x.
483//
484// The algorithm is based on the article by Abramowitz and Stegun [1]
485// as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.).
486//
487// [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions,
488// Applied Mathematics Series vol. 55 (1964), Washington.
489//
490//--- NvE 12-mar-2000 UU-SAP Utrecht
491
492 // Parameters of the polynomial approximation
387a745b 493 const Double_t kp1= 1., kp2= 0.15443144, kp3=-0.67278579,
494 kp4=-0.18156897, kp5=-1.919402e-2, kp6=-1.10404e-3, kp7=-4.686e-5;
29beb80d 495
387a745b 496 const Double_t kq1= 1.25331414, kq2= 0.23498619, kq3=-3.655620e-2,
497 kq4= 1.504268e-2, kq5=-7.80353e-3, kq6= 3.25614e-3, kq7=-6.8245e-4;
29beb80d 498
499 if (x <= 0)
500 {
501 cout << " *BesselK1* Invalid argument x = " << x << endl;
502 return 0;
503 }
504
505 Double_t y=0,result=0;
506
507 if (x <= 2)
508 {
509 y=x*x/4.;
387a745b 510 result=(log(x/2.)*BesselI1(x))
511 +(1./x)*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))));
29beb80d 512 }
513 else
514 {
515 y=2./x;
387a745b 516 result=(exp(-x)/sqrt(x))
517 *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7))))));
29beb80d 518 }
519
520 return result;
521}
522///////////////////////////////////////////////////////////////////////////
261c0caf 523Double_t AliMath::BesselK(Int_t n,Double_t x) const
29beb80d 524{
525// Computation of the Integer Order Modified Bessel function K_n(x)
526// for n=0,1,2,... and positive real x.
527//
528// The algorithm uses the recurrence relation
529//
530// K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x)
531//
532// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
533//
534//--- NvE 12-mar-2000 UU-SAP Utrecht
535
536 if (x <= 0 || n < 0)
537 {
538 cout << " *BesselK* Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl;
539 return 0;
540 }
541
542 if (n==0) return BesselK0(x);
543
544 if (n==1) return BesselK1(x);
545
546 // Perform upward recurrence for all x
547 Double_t tox=2./x;
548 Double_t bkm=BesselK0(x);
549 Double_t bk=BesselK1(x);
550 Double_t bkp=0;
551 for (Int_t j=1; j<n; j++)
552 {
553 bkp=bkm+double(j)*tox*bk;
554 bkm=bk;
555 bk=bkp;
556 }
557
558 return bk;
559}
560///////////////////////////////////////////////////////////////////////////
261c0caf 561Double_t AliMath::BesselI(Int_t n,Double_t x) const
29beb80d 562{
563// Computation of the Integer Order Modified Bessel function I_n(x)
564// for n=0,1,2,... and any real x.
565//
566// The algorithm uses the recurrence relation
567//
568// I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x)
569//
570// as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.).
571//
572//--- NvE 12-mar-2000 UU-SAP Utrecht
573
574 Int_t iacc=40; // Increase to enhance accuracy
575 Double_t bigno=1.e10, bigni=1.e-10;
576
577 if (n < 0)
578 {
579 cout << " *BesselI* Invalid argument (n,x) = (" << n << " , " << x << ")" << endl;
580 return 0;
581 }
582
583 if (n==0) return BesselI0(x);
584
585 if (n==1) return BesselI1(x);
586
587 if (fabs(x) < 1.e-10) return 0;
588
589 Double_t tox=2./fabs(x);
590 Double_t bip=0,bim=0;
591 Double_t bi=1;
592 Double_t result=0;
593 Int_t m=2*((n+int(sqrt(float(iacc*n))))); // Downward recurrence from even m
53ce0aba 594 for (Int_t j=m; j>=1; j--)
29beb80d 595 {
596 bim=bip+double(j)*tox*bi;
597 bip=bi;
598 bi=bim;
599 if (fabs(bi) > bigno) // Renormalise to prevent overflows
600 {
601 result*=bigni;
602 bi*=bigni;
603 bip*=bigni;
604 }
605 if (j==n) result=bip;
606 }
607
608 result*=BesselI0(x)/bi; // Normalise with I0(x)
609 if ((x < 0) && (n%2 == 1)) result=-result;
610
611 return result;
612}
613///////////////////////////////////////////////////////////////////////////
a57b6095 614TF1 AliMath::Chi2Dist(Int_t ndf) const
615{
616// Provide the Chi-squared distribution function corresponding to the
617// specified ndf degrees of freedom.
618//
619// Details can be found in the excellent textbook of Phil Gregory
620// "Bayesian Logical Data Analysis for the Physical Sciences".
621//
622// Note : <chi2>=ndf Var(chi2)=2*ndf
623
624 TF1 chi2dist("chi2dist","1./(TMath::Gamma([0]/2.)*pow(2,[0]/2.))*pow(x,[0]/2.-1.)*exp(-x/2.)");
625 TString title="#chi^{2} distribution for ndf = ";
626 title+=ndf;
627 chi2dist.SetTitle(title.Data());
628 chi2dist.SetParName(0,"ndf");
629 chi2dist.SetParameter(0,ndf);
630
631 return chi2dist;
632}
633///////////////////////////////////////////////////////////////////////////
634TF1 AliMath::StudentDist(Double_t ndf) const
635{
636// Provide the Student's T distribution function corresponding to the
637// specified ndf degrees of freedom.
638//
639// In a frequentist approach, the Student's T distribution is particularly
640// useful in making inferences about the mean of an underlying population
641// based on the data from a random sample.
642//
643// In a Bayesian context it is used to characterise the posterior PDF
644// for a particular state of information.
645//
646// Note : ndf is not restricted to integer values
647//
648// Details can be found in the excellent textbook of Phil Gregory
649// "Bayesian Logical Data Analysis for the Physical Sciences".
650//
651// Note : <T>=0 Var(T)=ndf/(ndf-2)
652
a57b6095 653 TF1 tdist("tdist","(TMath::Gamma(([0]+1.)/2.)/(sqrt(pi*[0])*TMath::Gamma([0]/2.)))*pow(1.+x*x/[0],-([0]+1.)/2.)");
654 TString title="Student's t distribution for ndf = ";
655 title+=ndf;
656 tdist.SetTitle(title.Data());
657 tdist.SetParName(0,"ndf");
658 tdist.SetParameter(0,ndf);
659
660 return tdist;
661}
662///////////////////////////////////////////////////////////////////////////
663TF1 AliMath::FratioDist(Int_t ndf1,Int_t ndf2) const
664{
665// Provide the F (ratio) distribution function corresponding to the
666// specified ndf1 and ndf2 degrees of freedom of the two samples.
667//
668// In a frequentist approach, the F (ratio) distribution is particularly useful
669// in making inferences about the ratio of the variances of two underlying
670// populations based on a measurement of the variance of a random sample taken
671// from each one of the two populations.
672// So the F test provides a means to investigate the degree of equality of
673// two populations.
674//
675// Details can be found in the excellent textbook of Phil Gregory
676// "Bayesian Logical Data Analysis for the Physical Sciences".
677//
678// Note : <F>=ndf2/(ndf2-2) Var(F)=2*ndf2*ndf2*(ndf2+ndf1-2)/(ndf1*(ndf2-1)*(ndf2-1)*(ndf2-4))
679
680 TF1 fdist("fdist",
681 "(TMath::Gamma(([0]+[1])/2.)/(TMath::Gamma([0]/2.)*TMath::Gamma([1]/2.)))*pow([0]/[1],[0]/2.)*pow(x,([0]-2.)/2.)/pow(1.+x*[0]/[1],([0]+[1])/2.)");
682 TString title="F (ratio) distribution for ndf1 = ";
683 title+=ndf1;
684 title+=" ndf2 = ";
685 title+=ndf2;
686 fdist.SetTitle(title.Data());
687 fdist.SetParName(0,"ndf1");
688 fdist.SetParameter(0,ndf1);
689 fdist.SetParName(1,"ndf2");
690 fdist.SetParameter(1,ndf2);
691
692 return fdist;
693}
694///////////////////////////////////////////////////////////////////////////
695TF1 AliMath::BinomialDist(Int_t n,Double_t p) const
696{
697// Provide the Binomial distribution function corresponding to the
698// specified number of trials n and probability p of success.
699//
700// p(k|n,p) = probability to obtain exactly k successes in n trials.
701//
702// Details can be found in the excelent textbook of Phil Gregory
703// "Bayesian Logical Data Analysis for the Physical Sciences".
704//
705// Note : <k>=n*p Var(k)=n*p*(1-p)
706
707 TF1 bindist("bindist","TMath::Binomial(int([0]),int(x))*pow([1],int(x))*pow(1.-[1],int([0])-int(x))");
708 TString title="Binomial distribution for n = ";
709 title+=n;
710 title+=" p = ";
711 title+=p;
712 bindist.SetTitle(title.Data());
713 bindist.SetParName(0,"n");
714 bindist.SetParameter(0,n);
715 bindist.SetParName(1,"p");
716 bindist.SetParameter(1,p);
717
718 return bindist;
719}
720///////////////////////////////////////////////////////////////////////////
721TF1 AliMath::NegBinomialDist(Int_t k,Double_t p) const
722{
723// Provide the Negative Binomial distribution function corresponding to the
724// specified number of successes k and probability p of success.
725//
726// p(n|k,p) = probability to have reached k successes after n trials.
727//
728// Details can be found in the excelent textbook of Phil Gregory
729// "Bayesian Logical Data Analysis for the Physical Sciences".
730//
731// Note : <n>=k*(1-p)/p Var(n)=k*(1-p)/(p*p)
732
733 TF1 negbindist("negbindist","TMath::Binomial(int(x)-1,int([0])-1)*pow([1],int([0]))*pow(1.-[1],int(x)-int([0]))");
734 TString title="Negative Binomial distribution for k = ";
735 title+=k;
736 title+=" p = ";
737 title+=p;
738 negbindist.SetTitle(title.Data());
739 negbindist.SetParName(0,"k");
740 negbindist.SetParameter(0,k);
741 negbindist.SetParName(1,"p");
742 negbindist.SetParameter(1,p);
743
744 return negbindist;
745}
746///////////////////////////////////////////////////////////////////////////
747TF1 AliMath::PoissonDist(Double_t mu) const
748{
749// Provide the Poisson distribution function corresponding to the
750// specified average rate (in time or space) mu of occurrences.
751//
752// p(n|mu) = probability for n occurrences in a certain time or space interval.
753//
754// Details can be found in the excelent textbook of Phil Gregory
755// "Bayesian Logical Data Analysis for the Physical Sciences".
756//
757// Note : <n>=mu Var(n)=mu
758
759 TF1 poissdist("poissdist","exp(-[0])*pow([0],int(x))/TMath::Factorial(int(x))");
760 TString title="Poisson distribution for mu = ";
761 title+=mu;
762 poissdist.SetTitle(title.Data());
763 poissdist.SetParName(0,"mu");
764 poissdist.SetParameter(0,mu);
765
766 return poissdist;
767}
768///////////////////////////////////////////////////////////////////////////
769Double_t AliMath::Chi2Pvalue(Double_t chi2,Int_t ndf,Int_t sides,Int_t sigma,Int_t mode) const
770{
771// Computation of the P-value for a certain specified Chi-squared (chi2) value
772// for a Chi-squared distribution with ndf degrees of freedom.
773//
774// The P-value for a certain Chi-squared value chi2 corresponds to the fraction of
775// repeatedly drawn equivalent samples from a certain population, which is expected
776// to yield a Chi-squared value exceeding (less than) the value chi2 for an
777// upper (lower) tail test in case a certain hypothesis is true.
778//
779// Further details can be found in the excellent textbook of Phil Gregory
780// "Bayesian Logical Data Analysis for the Physical Sciences".
781//
782// Note : <Chi2>=ndf Var(Chi2)=2*ndf
783//
784// With the "sides" parameter a one-sided or two-sided test can be selected
785// using either the upper or lower tail contents.
786// In case of automatic upper/lower selection the decision is made on basis
787// of the location of the input chi2 value w.r.t. <Chi2> of the distribution.
788//
789// sides = 1 : One-sided test using the upper tail contents
790// 2 : Two-sided test using the upper tail contents
791// -1 : One-sided test using the lower tail contents
792// -2 : Two-sided test using the lower tail contents
793// 0 : One-sided test using the auto-selected upper or lower tail contents
794// 3 : Two-sided test using the auto-selected upper or lower tail contents
795//
796// The argument "sigma" allows for the following return values :
797//
798// sigma = 0 : P-value is returned as the above specified fraction
799// 1 : The difference chi2-<Chi2> expressed in units of sigma
800// Note : This difference may be negative.
801//
802// According to the value of the parameter "mode" various algorithms
803// can be selected.
804//
805// mode = 0 : Calculations are based on the incomplete gamma function.
806//
807// 1 : Same as for mode=0. However, in case ndf=1 an exact expression
808// based on the error function Erf() is used.
809//
810// 2 : Same as for mode=0. However, in case ndf>30 a Gaussian approximation
811// is used instead of the gamma function.
812//
813// The default values are sides=0, sigma=0 and mode=1.
814//
815//--- NvE 21-may-2005 Utrecht University
816
817 if (ndf<=0) return 0;
818
819 Double_t mean=ndf;
820
821 if (!sides) // Automatic one-sided test
822 {
823 sides=1;
824 if (chi2<mean) sides=-1;
825 }
826
827 if (sides==3) // Automatic two-sided test
828 {
829 sides=2;
830 if (chi2<mean) sides=-2;
831 }
832
833 Double_t val=0;
834 if (sigma) // P-value in units of sigma
835 {
836 Double_t s=sqrt(double(2*ndf));
837 val=(chi2-mean)/s;
838 }
839 else // P-value from tail contents
840 {
841 if (sides>0) // Upper tail
842 {
843 val=Prob(chi2,ndf,mode);
844 }
845 else // Lower tail
846 {
847 val=1.-Prob(chi2,ndf,mode);
848 }
849 }
850
851 if (abs(sides)==2) val=val*2.;
852
853 return val;
854}
855///////////////////////////////////////////////////////////////////////////
856Double_t AliMath::StudentPvalue(Double_t t,Double_t ndf,Int_t sides,Int_t sigma) const
857{
858// Computation of the P-value for a certain specified Student's t value
859// for a Student's T distribution with ndf degrees of freedom.
860//
861// In a frequentist approach, the Student's T distribution is particularly
862// useful in making inferences about the mean of an underlying population
863// based on the data from a random sample.
864//
865// The P-value for a certain t value corresponds to the fraction of
866// repeatedly drawn equivalent samples from a certain population, which is expected
867// to yield a T value exceeding (less than) the value t for an upper (lower)
868// tail test in case a certain hypothesis is true.
869//
870// Further details can be found in the excellent textbook of Phil Gregory
871// "Bayesian Logical Data Analysis for the Physical Sciences".
872//
873// Note : <T>=0 Var(T)=ndf/(ndf-2)
874//
875// With the "sides" parameter a one-sided or two-sided test can be selected
876// using either the upper or lower tail contents.
877// In case of automatic upper/lower selection the decision is made on basis
878// of the location of the input t value w.r.t. <T> of the distribution.
879//
880// sides = 1 : One-sided test using the upper tail contents
881// 2 : Two-sided test using the upper tail contents
882// -1 : One-sided test using the lower tail contents
883// -2 : Two-sided test using the lower tail contents
884// 0 : One-sided test using the auto-selected upper or lower tail contents
885// 3 : Two-sided test using the auto-selected upper or lower tail contents
886//
887// The argument "sigma" allows for the following return values :
888//
889// sigma = 0 : P-value is returned as the above specified fraction
890// 1 : The difference t-<T> expressed in units of sigma
891// Note : This difference may be negative and sigma
892// is only defined for ndf>2.
893//
894// The default values are sides=0 and sigma=0.
895//
896//--- NvE 21-may-2005 Utrecht University
897
898 if (ndf<=0) return 0;
899
900 Double_t mean=0;
901
902 if (!sides) // Automatic one-sided test
903 {
904 sides=1;
905 if (t<mean) sides=-1;
906 }
907
908 if (sides==3) // Automatic two-sided test
909 {
910 sides=2;
911 if (t<mean) sides=-2;
912 }
913
914 Double_t val=0;
915 if (sigma) // P-value in units of sigma
916 {
917 if (ndf>2) // Sigma is only defined for ndf>2
918 {
919 Double_t s=sqrt(ndf/(ndf-2.));
920 val=t/s;
921 }
922 }
923 else // P-value from tail contents
924 {
925 if (sides>0) // Upper tail
926 {
927 val=1.-TMath::StudentI(t,ndf);
928 }
929 else // Lower tail
930 {
931 val=TMath::StudentI(t,ndf);
932 }
933 }
934
935 if (abs(sides)==2) val=val*2.;
936
937 return val;
938}
939///////////////////////////////////////////////////////////////////////////
940Double_t AliMath::FratioPvalue(Double_t f,Int_t ndf1,Int_t ndf2,Int_t sides,Int_t sigma) const
941{
942// Computation of the P-value for a certain specified F ratio f value
943// for an F (ratio) distribution with ndf1 and ndf2 degrees of freedom
944// for the two samples X,Y respectively to be compared in the ratio X/Y.
945//
946// In a frequentist approach, the F (ratio) distribution is particularly useful
947// in making inferences about the ratio of the variances of two underlying
948// populations based on a measurement of the variance of a random sample taken
949// from each one of the two populations.
950// So the F test provides a means to investigate the degree of equality of
951// two populations.
952//
953// The P-value for a certain f value corresponds to the fraction of
954// repeatedly drawn equivalent samples from a certain population, which is expected
955// to yield an F value exceeding (less than) the value f for an upper (lower)
956// tail test in case a certain hypothesis is true.
957//
958// Further details can be found in the excellent textbook of Phil Gregory
959// "Bayesian Logical Data Analysis for the Physical Sciences".
960//
961// Note : <F>=ndf2/(ndf2-2) Var(F)=2*ndf2*ndf2*(ndf2+ndf1-2)/(ndf1*(ndf2-1)*(ndf2-1)*(ndf2-4))
962//
963// With the "sides" parameter a one-sided or two-sided test can be selected
964// using either the upper or lower tail contents.
965// In case of automatic upper/lower selection the decision is made on basis
966// of the location of the input f value w.r.t. <F> of the distribution.
967//
968// sides = 1 : One-sided test using the upper tail contents
969// 2 : Two-sided test using the upper tail contents
970// -1 : One-sided test using the lower tail contents
971// -2 : Two-sided test using the lower tail contents
972// 0 : One-sided test using the auto-selected upper or lower tail contents
973// 3 : Two-sided test using the auto-selected upper or lower tail contents
974//
975// The argument "sigma" allows for the following return values :
976//
977// sigma = 0 : P-value is returned as the above specified fraction
978// 1 : The difference f-<F> expressed in units of sigma
979// Note : This difference may be negative and sigma
980// is only defined for ndf2>4.
981//
982// The default values are sides=0 and sigma=0.
983//
984//--- NvE 21-may-2005 Utrecht University
985
986 if (ndf1<=0 || ndf2<=0 || f<=0) return 0;
987
988 Double_t mean=double(ndf2/(ndf2-2));
989
990 if (!sides) // Automatic one-sided test
991 {
992 sides=1;
993 if (f<mean) sides=-1;
994 }
995
996 if (sides==3) // Automatic two-sided test
997 {
998 sides=2;
999 if (f<mean) sides=-2;
1000 }
1001
1002 Double_t val=0;
1003 if (sigma) // P-value in units of sigma
1004 {
1005 // Sigma is only defined for ndf2>4
1006 if (ndf2>4)
1007 {
8373d83f 1008 Double_t s=sqrt(double(ndf2*ndf2*(2*ndf2+2*ndf1-4))/double(ndf1*pow(double(ndf2-1),2)*(ndf2-4)));
a57b6095 1009 val=(f-mean)/s;
1010 }
1011 }
1012 else // P-value from tail contents
1013 {
1014 if (sides>0) // Upper tail
1015 {
1016 val=1.-TMath::FDistI(f,ndf1,ndf2);
1017 }
1018 else // Lower tail
1019 {
1020 val=TMath::FDistI(f,ndf1,ndf2);
1021 }
1022 }
1023
1024 if (abs(sides)==2) val=val*2.;
1025
1026 return val;
1027}
1028///////////////////////////////////////////////////////////////////////////
1029Double_t AliMath::BinomialPvalue(Int_t k,Int_t n,Double_t p,Int_t sides,Int_t sigma,Int_t mode) const
1030{
1031// Computation of the P-value for a certain specified number of successes k
1032// for a Binomial distribution with n trials and success probability p.
1033//
1034// The P-value for a certain number of successes k corresponds to the fraction of
1035// repeatedly drawn equivalent samples from a certain population, which is expected
1036// to yield a number of successes exceeding (less than) the value k for an
1037// upper (lower) tail test in case a certain hypothesis is true.
1038//
1039// Further details can be found in the excellent textbook of Phil Gregory
1040// "Bayesian Logical Data Analysis for the Physical Sciences".
1041//
1042// Note : <K>=n*p Var(K)=n*p*(1-p)
1043//
1044// With the "sides" parameter a one-sided or two-sided test can be selected
1045// using either the upper or lower tail contents.
1046// In case of automatic upper/lower selection the decision is made on basis
1047// of the location of the input k value w.r.t. <K> of the distribution.
1048//
1049// sides = 1 : One-sided test using the upper tail contents
1050// 2 : Two-sided test using the upper tail contents
1051// -1 : One-sided test using the lower tail contents
1052// -2 : Two-sided test using the lower tail contents
1053// 0 : One-sided test using the auto-selected upper or lower tail contents
1054// 3 : Two-sided test using the auto-selected upper or lower tail contents
1055//
1056// The argument "sigma" allows for the following return values :
1057//
1058// sigma = 0 : P-value is returned as the above specified fraction
1059// 1 : The difference k-<K> expressed in units of sigma
1060// Note : This difference may be negative.
1061//
1062// mode = 0 : Incomplete Beta function will be used to calculate the tail content.
1063// 1 : Straightforward summation of the Binomial terms is used.
1064//
1065// The Incomplete Beta function in general provides the most accurate values.
1066//
1067// The default values are sides=0, sigma=0 and mode=0.
1068//
1069//--- NvE 24-may-2005 Utrecht University
1070
1071 Double_t mean=double(n)*p;
1072
1073 if (!sides) // Automatic one-sided test
1074 {
1075 sides=1;
1076 if (k<mean) sides=-1;
1077 }
1078
1079 if (sides==3) // Automatic two-sided test
1080 {
1081 sides=2;
1082 if (k<mean) sides=-2;
1083 }
1084
1085 Double_t val=0;
1086
1087 if (sigma) // P-value in units of sigma
1088 {
1089 Double_t s=sqrt(double(n)*p*(1.-p));
1090 val=(double(k)-mean)/s;
1091 }
1092 else // P-value from tail contents
1093 {
1094 if (sides>0)
1095 {
1096 if (!mode) // Use Incomplete Beta function
1097 {
1098 val=TMath::BetaIncomplete(p,k+1,n-k);
1099 }
1100 else // Use straightforward summation
1101 {
1102 for (Int_t i=k+1; i<=n; i++)
1103 {
1104 val+=TMath::Binomial(n,i)*pow(p,i)*pow(1.-p,n-i);
1105 }
1106 }
1107 }
1108 else
1109 {
1110 if (!mode) // Use Incomplete Beta function
1111 {
1112 val=1.-TMath::BetaIncomplete(p,k+1,n-k);
1113 }
1114 else
1115 {
1116 for (Int_t j=0; j<=k; j++)
1117 {
1118 val+=TMath::Binomial(n,j)*pow(p,j)*pow(1.-p,n-j);
1119 }
1120 }
1121 }
1122 }
1123
1124 if (abs(sides)==2) val=val*2.;
1125
1126 return val;
1127}
1128///////////////////////////////////////////////////////////////////////////
1129Double_t AliMath::PoissonPvalue(Int_t k,Double_t mu,Int_t sides,Int_t sigma) const
1130{
1131// Computation of the P-value for a certain specified number of occurrences k
1132// for a Poisson distribution with a specified average rate (in time or space)
1133// mu of occurrences.
1134//
1135// The P-value for a certain number of occurrences k corresponds to the fraction of
1136// repeatedly drawn equivalent samples from a certain population, which is expected
1137// to yield a number of occurrences exceeding (less than) the value k for an
1138// upper (lower) tail test in case a certain hypothesis is true.
1139//
1140// Further details can be found in the excellent textbook of Phil Gregory
1141// "Bayesian Logical Data Analysis for the Physical Sciences".
1142//
1143// Note : <K>=mu Var(K)=mu
1144//
1145// With the "sides" parameter a one-sided or two-sided test can be selected
1146// using either the upper or lower tail contents.
1147// In case of automatic upper/lower selection the decision is made on basis
1148// of the location of the input k value w.r.t. <K> of the distribution.
1149//
1150// sides = 1 : One-sided test using the upper tail contents
1151// 2 : Two-sided test using the upper tail contents
1152// -1 : One-sided test using the lower tail contents
1153// -2 : Two-sided test using the lower tail contents
1154// 0 : One-sided test using the auto-selected upper or lower tail contents
1155// 3 : Two-sided test using the auto-selected upper or lower tail contents
1156//
1157// The argument "sigma" allows for the following return values :
1158//
1159// sigma = 0 : P-value is returned as the above specified fraction
1160// 1 : The difference k-<K> expressed in units of sigma
1161// Note : This difference may be negative.
1162//
1163// The default values are sides=0 and sigma=0.
1164//
1165// Note : The tail contents are given by the incomplete Gamma function P(a,x).
1166// Lower tail contents = 1-P(k,mu)
1167// Upper tail contents = P(k,mu)
1168//
1169//--- NvE 24-may-2005 Utrecht University
1170
1171 Double_t mean=mu;
1172
1173 if (!sides) // Automatic one-sided test
1174 {
1175 sides=1;
1176 if (k<mean) sides=-1;
1177 }
1178
1179 if (sides==3) // Automatic two-sided test
1180 {
1181 sides=2;
1182 if (k<mean) sides=-2;
1183 }
1184
1185 Double_t val=0;
1186
1187 if (sigma) // P-value in units of sigma
1188 {
1189 Double_t s=sqrt(mu);
1190 val=(double(k)-mean)/s;
1191 }
1192 else // P-value from tail contents
1193 {
1194 if (sides>0) // Upper tail
1195 {
1196 val=Gamma(k,mu);
1197 }
1198 else // Lower tail
1199 {
1200 val=1.-Gamma(k,mu);
1201 }
1202 }
1203
1204 if (abs(sides)==2) val=val*2.;
1205
1206 return val;
1207}
1208///////////////////////////////////////////////////////////////////////////
1209Double_t AliMath::NegBinomialPvalue(Int_t n,Int_t k,Double_t p,Int_t sides,Int_t sigma,Int_t mode) const
1210{
1211// Computation of the P-value for a certain specified number of trials n
1212// for a Negative Binomial distribution where exactly k successes are to
1213// be reached which have each a probability p.
1214//
1215// The P-value for a certain number of trials n corresponds to the fraction of
1216// repeatedly drawn equivalent samples from a certain population, which is expected
1217// to yield a number of trials exceeding (less than) the value n for an
1218// upper (lower) tail test in case a certain hypothesis is true.
1219//
1220// Further details can be found in the excellent textbook of Phil Gregory
1221// "Bayesian Logical Data Analysis for the Physical Sciences".
1222//
1223// Note : <N>=k*(1-p)/p Var(N)=k*(1-p)/(p*p)
1224//
1225// With the "sides" parameter a one-sided or two-sided test can be selected
1226// using either the upper or lower tail contents.
1227// In case of automatic upper/lower selection the decision is made on basis
1228// of the location of the input n value w.r.t. <N> of the distribution.
1229//
1230// sides = 1 : One-sided test using the upper tail contents
1231// 2 : Two-sided test using the upper tail contents
1232// -1 : One-sided test using the lower tail contents
1233// -2 : Two-sided test using the lower tail contents
1234// 0 : One-sided test using the auto-selected upper or lower tail contents
1235// 3 : Two-sided test using the auto-selected upper or lower tail contents
1236//
1237// The argument "sigma" allows for the following return values :
1238//
1239// sigma = 0 : P-value is returned as the above specified fraction
1240// 1 : The difference n-<N> expressed in units of sigma
1241// Note : This difference may be negative.
1242//
1243// mode = 0 : Incomplete Beta function will be used to calculate the tail content.
1244// 1 : Straightforward summation of the Negative Binomial terms is used.
1245//
1246// The Incomplete Beta function in general provides the most accurate values.
1247//
1248// The default values are sides=0, sigma=0 and mode=0.
1249//
1250//--- NvE 24-may-2005 Utrecht University
1251
1252 Double_t mean=double(k)*(1.-p)/p;
1253
1254 if (!sides) // Automatic one-sided test
1255 {
1256 sides=1;
1257 if (n<mean) sides=-1;
1258 }
1259
1260 if (sides==3) // Automatic two-sided test
1261 {
1262 sides=2;
1263 if (n<mean) sides=-2;
1264 }
1265
1266 Double_t val=0;
1267
1268 if (sigma) // P-value in units of sigma
1269 {
1270 Double_t s=sqrt(double(k)*(1.-p)/(p*p));
1271 val=(double(n)-mean)/s;
1272 }
1273 else // P-value from tail contents
1274 {
1275 if (sides>0) // Upper tail
1276 {
1277 if (!mode) // Use Incomplete Beta function
1278 {
1279 val=1.-TMath::BetaIncomplete(p,k,n-k);
1280 }
1281 else // Use straightforward summation
1282 {
1283 for (Int_t i=1; i<n; i++)
1284 {
1285 val+=TMath::Binomial(i-1,k-1)*pow(p,k)*pow(1.-p,i-k);
1286 }
1287 val=1.-val;
1288 }
1289 }
1290 else // Lower tail
1291 {
1292 if (!mode) // Use Incomplete Beta function
1293 {
1294 val=TMath::BetaIncomplete(p,k,n-k);
1295 }
1296 else
1297 {
1298 for (Int_t j=1; j<n; j++)
1299 {
1300 val+=TMath::Binomial(j-1,k-1)*pow(p,k)*pow(1.-p,j-k);
1301 }
1302 }
1303 }
1304 }
1305
1306 if (abs(sides)==2) val=val*2.;
1307
1308 return val;
1309}
1310///////////////////////////////////////////////////////////////////////////
cf3100fa 1311Double_t AliMath::Nfac(Int_t n,Int_t mode) const
1312{
1313// Compute n!.
1314// The algorithm can be selected by the "mode" input argument.
1315//
1316// mode : 0 ==> Calculation by means of straightforward multiplication
1317// : 1 ==> Calculation by means of Stirling's approximation
1318// : 2 ==> Calculation by means of n!=Gamma(n+1)
1319//
1320// For large n the calculation modes 1 and 2 will in general be faster.
1321// By default mode=0 is used.
1322// For n<0 the value 0 will be returned.
1323//
1324// Note : Because of Double_t value overflow the maximum value is n=170.
1325//
1326//--- NvE 20-jan-2007 Utrecht University
1327
1328 if (n<0) return 0;
1329 if (n==0) return 1;
1330
1331 Double_t twopi=2.*acos(-1.);
1332 Double_t z=0;
1333 Double_t nfac=1;
1334 Int_t i=n;
1335
1336 switch (mode)
1337 {
1338 case 0: // Straightforward multiplication
1339 while (i>1)
1340 {
1341 nfac*=Double_t(i);
1342 i--;
1343 }
1344 break;
1345
1346 case 1: // Stirling's approximation
1347 z=n;
1348 nfac=sqrt(twopi)*pow(z,z+0.5)*exp(-z)*(1.+1./(12.*z));
1349 break;
1350
1351 case 2: // Use of Gamma(n+1)
1352 z=n+1;
1353 nfac=Gamma(z);
1354 break;
1355
1356 default:
1357 nfac=0;
1358 break;
1359 }
1360
1361 return nfac;
1362}
1363///////////////////////////////////////////////////////////////////////////
1364Double_t AliMath::LnNfac(Int_t n,Int_t mode) const
1365{
1366// Compute ln(n!).
1367// The algorithm can be selected by the "mode" input argument.
1368//
1369// mode : 0 ==> Calculation via evaluation of n! followed by taking ln(n!)
1370// : 1 ==> Calculation via Stirling's approximation ln(n!)=0.5*ln(2*pi)+(n+0.5)*ln(n)-n+1/(12*n)
1371// : 2 ==> Calculation by means of ln(n!)=LnGamma(n+1)
1372//
1373// Note : Because of Double_t value overflow the maximum value is n=170 for mode=0.
1374//
1375// For mode=2 rather accurate results are obtained for both small and large n.
1376// By default mode=2 is used.
1377// For n<1 the value 0 will be returned.
1378//
1379//--- NvE 20-jan-2007 Utrecht University
1380
1381 if (n<=1) return 0;
1382
1383 Double_t twopi=2.*acos(-1.);
1384 Double_t z=0;
1385 Double_t lognfac=0;
1386
1387 switch (mode)
1388 {
1389 case 0: // Straightforward ln(n!)
1390 z=Nfac(n);
1391 lognfac=log(z);
1392 break;
1393
1394 case 1: // Stirling's approximation
1395 z=n;
1396 lognfac=0.5*log(twopi)+(z+0.5)*log(z)-z+1./(12.*z);
1397 break;
1398
1399 case 2: // Use of LnGamma(n+1)
1400 z=n+1;
1401 lognfac=LnGamma(z);
1402 break;
1403
1404 default:
1405 lognfac=0;
1406 break;
1407 }
1408
1409 return lognfac;
1410}
1411///////////////////////////////////////////////////////////////////////////
1412Double_t AliMath::LogNfac(Int_t n,Int_t mode) const
1413{
1414// Compute log_10(n!).
1415// First ln(n!) is evaluated via invokation of LnNfac(n,mode).
1416// Then the algorithm log_10(z)=ln(z)*log_10(e) is used.
1417//
1418//--- NvE 20-jan-2007 Utrecht University
1419
1420 Double_t e=exp(1.);
1421
1422 Double_t val=LnNfac(n,mode);
1423 val*=log10(e);
1424
1425 return val;
1426}
1427///////////////////////////////////////////////////////////////////////////
bf4dd5dd 1428Double_t AliMath::PsiValue(Int_t m,Int_t* n,Double_t* p,Int_t f) const
1429{
1430// Provide the Bayesian Psi value of observations of a counting experiment
1431// w.r.t. a Bernoulli class hypothesis B_m.
1432// The hypothesis B_m represents a counting experiment with m different
1433// possible outcomes and is completely defined by the probabilities
1434// of the various outcomes (and the requirement that the sum of all these
1435// probabilities equals 1).
1436//
1437// The Psi value provides (in dB scale) the amount of support that the
1438// data can maximally give to any Bernoulli class hypothesis different
1439// from the currently specified B_m.
1440//
1441// To be specific : Psi=-10*log[p(D|B_m I)]
1442//
1443// where p(D|B_m I) represents the likelihood of the data D under the condition
1444// that B_m and some prior I are true.
1445//
1446// A Psi value of zero indicates a perfect match between the observations
1447// and the specified hypothesis.
1448// Further mathematical details can be found in astro-ph/0702029.
1449//
1450// m : The number of different possible outcomes of the counting experiment
1451// n : The observed number of different outcomes
1452// p : The probabilities of the different outcomes according to the hypothesis
1453// f : Flag to indicate the use of a frequentist (Stirling) approximation (f=1)
1454// or the exact Bayesian expression (f=0).
1455//
1456// In case no probabilities are given (i.e. pk=0), a uniform distribution
1457// is assumed.
1458//
1459// The default values are pk=0 and f=0.
1460//
1461// In the case of inconsistent input, a Psi value of -1 is returned.
1462//
1463//--- NvE 03-oct-2007 Utrecht University
1464
1465 Double_t psi=-1;
1466
1467 if (m<=0 || !n) return psi;
1468
1469 Int_t ntot=0;
1470 for (Int_t j=0; j<m; j++)
1471 {
1472 if (n[j]>0) ntot+=n[j];
1473 }
1474
1475 psi=0;
1476 Double_t pk=1./float(m); // Prob. of getting outcome k for a uniform distr.
1477 for (Int_t i=0; i<m; i++)
1478 {
1479 if (p) pk=p[i]; // Probabilities from a specific B_m hypothesis
1480 if (n[i]>0 && pk>0)
1481 {
1482 if (!f) // Exact Bayesian expression
1483 {
1484 psi+=double(n[i])*log10(pk)-LogNfac(n[i]);
1485 }
1486 else // Frequentist (Stirling) approximation
1487 {
1488 if (ntot>0) psi+=double(n[i])*log10(n[i]/(ntot*pk));
1489 }
1490 }
1491 }
1492
1493 if (!f) // Exact Bayesian expression
1494 {
1495 psi+=LogNfac(ntot);
1496 psi*=-10.;
1497 }
1498 else // Frequentist (Stirling) approximation
1499 {
1500 psi*=10.;
1501 }
1502
1503 return psi;
1504}
1505///////////////////////////////////////////////////////////////////////////
1506Double_t AliMath::PsiValue(TH1F* his,TH1F* hyp,TF1* pdf,Int_t f) const
1507{
1508// Provide the Bayesian Psi value of observations of a counting experiment
1509// (in histogram format) w.r.t. a Bernoulli class hypothesis B_m.
1510// The hypothesis B_m represents a counting experiment with m different
1511// possible outcomes and is completely defined by the probabilities
1512// of the various outcomes (and the requirement that the sum of all these
1513// probabilities equals 1).
1514// The specification of a hypothesis B_m can be provided either in
1515// histogram format (hyp) or via a probability distribution function (pdf),
1516// as outlined below.
1517// Note : The pdf does not need to be normalised.
1518//
1519// The Psi value provides (in dB scale) the amount of support that the
1520// data can maximally give to any Bernoulli class hypothesis different
1521// from the currently specified B_m.
1522//
1523// To be specific : Psi=-10*log[p(D|B_m I)]
1524//
1525// where p(D|B_m I) represents the likelihood of the data D under the condition
1526// that B_m and some prior I are true.
1527//
1528// A Psi value of zero indicates a perfect match between the observations
1529// and the specified hypothesis.
1530// Further mathematical details can be found in astro-ph/0702029.
1531//
1532// his : The experimental observations in histogram format
1533// hyp : Hypothetical observations according to some hypothesis
1534// pdf : Probability distribution function for the hypothesis
1535// f : Flag to indicate the use of a frequentist (Stirling) approximation (f=1)
1536// or the exact Bayesian expression (f=0).
1537//
1538// In case no hypothesis is specified (i.e. hyp=0 and pdf=0), a uniform
1539// background distribution is assumed.
1540//
1541// Default values are : hyp=0, pdf=0 and f=0.
1542//
1543// In the case of inconsistent input, a Psi value of -1 is returned.
1544//
1545//--- NvE 03-oct-2007 Utrecht University
1546
1547 Double_t psi=-1;
1548
1549 if (!his) return psi;
1550
1551 TAxis* xaxis=his->GetXaxis();
1552 Double_t xmin=xaxis->GetXmin();
1553 Double_t xmax=xaxis->GetXmax();
1554 Double_t range=xmax-xmin;
1555 Int_t nbins=his->GetNbinsX();
1556 Double_t nensig=his->GetEntries();
1557
1558 if (nbins<=0 || nensig<=0 || range<=0) return psi;
1559
1560 Int_t* n=new Int_t[nbins];
1561 Double_t* p=new Double_t[nbins];
1562 Int_t nk;
1563 Double_t pk;
1564
1565 // Uniform hypothesis distribution
1566 if (!hyp && !pdf)
1567 {
1568 for (Int_t i=1; i<=nbins; i++)
1569 {
1570 nk=int(his->GetBinContent(i));
1571 pk=his->GetBinWidth(i)/range;
1572 n[i-1]=0;
1573 p[i-1]=0;
1574 if (nk>0) n[i-1]=nk;
1575 if (pk>0) p[i-1]=pk;
1576 }
1577 psi=PsiValue(nbins,n,p,f);
1578 }
1579
1580 // Hypothesis specified via a pdf
1581 if (pdf && !hyp)
1582 {
1583 pdf->SetRange(xmin,xmax);
1584 Double_t ftot=pdf->Integral(xmin,xmax);
1585 if (ftot>0)
1586 {
1587 Double_t x1,x2;
1588 for (Int_t ipdf=1; ipdf<=nbins; ipdf++)
1589 {
1590 nk=int(his->GetBinContent(ipdf));
1591 x1=his->GetBinLowEdge(ipdf);
1592 x2=x1+his->GetBinWidth(ipdf);
1593 pk=pdf->Integral(x1,x2)/ftot;
1594 n[ipdf-1]=0;
1595 p[ipdf-1]=0;
1596 if (nk>0) n[ipdf-1]=nk;
1597 if (pk>0) p[ipdf-1]=pk;
1598 }
1599 psi=PsiValue(nbins,n,p,f);
1600 }
1601 }
1602
1603 // Hypothesis specified via a histogram
1604 if (hyp && !pdf)
1605 {
1606 TH1F* href=new TH1F(*his);
1607 href->Reset();
1608 Double_t nenhyp=hyp->GetEntries();
1609 Float_t x,y;
1610 for (Int_t ihyp=1; ihyp<=hyp->GetNbinsX(); ihyp++)
1611 {
1612 x=hyp->GetBinCenter(ihyp);
1613 y=hyp->GetBinContent(ihyp);
1614 href->Fill(x,y);
1615 }
1616 for (Int_t j=1; j<=nbins; j++)
1617 {
1618 nk=int(his->GetBinContent(j));
1619 pk=href->GetBinContent(j)/nenhyp;
1620 n[j-1]=0;
1621 p[j-1]=0;
1622 if (nk>0) n[j-1]=nk;
1623 if (pk>0) p[j-1]=pk;
1624 }
1625 psi=PsiValue(nbins,n,p,f);
1626 delete href;
1627 }
1628
1629 delete [] n;
1630 delete [] p;
1631
1632 return psi;
1633}
1634///////////////////////////////////////////////////////////////////////////
99a25cd9 1635Double_t AliMath::Chi2Value(Int_t m,Int_t* n,Double_t* p,Int_t* ndf) const
bf4dd5dd 1636{
1637// Provide the frequentist chi-squared value of observations of a counting
1638// experiment w.r.t. a Bernoulli class hypothesis B_m.
1639// The hypothesis B_m represents a counting experiment with m different
1640// possible outcomes and is completely defined by the probabilities
1641// of the various outcomes (and the requirement that the sum of all these
1642// probabilities equals 1).
1643//
1644// Further mathematical details can be found in astro-ph/0702029.
1645//
99a25cd9 1646// m : The number of different possible outcomes of the counting experiment
1647// n : The observed number of different outcomes
1648// p : The probabilities of the different outcomes according to the hypothesis
1649// ndf : The returned number of degrees of freedom
1650//
1651// Note : Both the arrays "n" and (when provided) "p" should be of dimension "m".
bf4dd5dd 1652//
1653// In case no probabilities are given (i.e. pk=0), a uniform distribution
1654// is assumed.
1655//
99a25cd9 1656// The default values are pk=0 and ndf=0.
bf4dd5dd 1657//
99a25cd9 1658// In the case of inconsistent input, a chi-squared and ndf value of -1 is returned.
bf4dd5dd 1659//
1660//--- NvE 03-oct-2007 Utrecht University
1661
1662 Double_t chi=-1;
99a25cd9 1663 if (ndf) *ndf=-1;
bf4dd5dd 1664
1665 if (m<=0 || !n) return chi;
1666
1667 Int_t ntot=0;
1668 for (Int_t j=0; j<m; j++)
1669 {
1670 if (n[j]>0) ntot+=n[j];
1671 }
1672
1673 chi=0;
1674 Double_t pk=1./float(m); // Prob. of getting outcome k for a uniform distr.
1675 for (Int_t i=0; i<m; i++)
1676 {
1677 if (p) pk=p[i]; // Probabilities from a specific B_m hypothesis
99a25cd9 1678 if (n[i]>0 && pk>0 && ntot>0)
1679 {
1680 chi+=pow(double(n[i])-double(ntot)*pk,2)/(ntot*pk);
1681 if (ndf) (*ndf)=(*ndf)+1;
1682 }
bf4dd5dd 1683 }
1684
1685 return chi;
1686}
1687///////////////////////////////////////////////////////////////////////////
99a25cd9 1688Double_t AliMath::Chi2Value(TH1F* his,TH1F* hyp,TF1* pdf,Int_t* ndf) const
bf4dd5dd 1689{
1690// Provide the frequentist chi-squared value of observations of a counting
1691// experiment (in histogram format) w.r.t. a Bernoulli class hypothesis B_m.
1692// The hypothesis B_m represents a counting experiment with m different
1693// possible outcomes and is completely defined by the probabilities
1694// of the various outcomes (and the requirement that the sum of all these
1695// probabilities equals 1).
1696// The specification of a hypothesis B_m can be provided either in
1697// histogram format (hyp) or via a probability distribution function (pdf),
1698// as outlined below.
1699// Note : The pdf does not need to be normalised.
1700//
1701// Further mathematical details can be found in astro-ph/0702029.
1702//
1703// his : The experimental observations in histogram format
1704// hyp : Hypothetical observations according to some hypothesis
1705// pdf : Probability distribution function for the hypothesis
99a25cd9 1706// ndf : The returned number of degrees of freedom
bf4dd5dd 1707//
1708// In case no hypothesis is specified (i.e. hyp=0 and pdf=0), a uniform
1709// background distribution is assumed.
1710//
99a25cd9 1711// Default values are : hyp=0, pdf=0 and ndf=0.
bf4dd5dd 1712//
99a25cd9 1713// In the case of inconsistent input, a chi-squared and ndf value of -1 is returned.
bf4dd5dd 1714//
1715//--- NvE 03-oct-2007 Utrecht University
1716
1717 Double_t chi=-1;
1718
1719 if (!his) return chi;
1720
1721 TAxis* xaxis=his->GetXaxis();
1722 Double_t xmin=xaxis->GetXmin();
1723 Double_t xmax=xaxis->GetXmax();
1724 Double_t range=xmax-xmin;
1725 Int_t nbins=his->GetNbinsX();
1726 Double_t nensig=his->GetEntries();
1727
1728 if (nbins<=0 || nensig<=0 || range<=0) return chi;
1729
1730 Int_t* n=new Int_t[nbins];
1731 Double_t* p=new Double_t[nbins];
1732 Int_t nk;
1733 Double_t pk;
1734
1735 // Uniform hypothesis distribution
1736 if (!hyp && !pdf)
1737 {
1738 for (Int_t i=1; i<=nbins; i++)
1739 {
1740 nk=int(his->GetBinContent(i));
1741 pk=his->GetBinWidth(i)/range;
1742 n[i-1]=0;
1743 p[i-1]=0;
1744 if (nk>0) n[i-1]=nk;
1745 if (pk>0) p[i-1]=pk;
1746 }
99a25cd9 1747 chi=Chi2Value(nbins,n,p,ndf);
bf4dd5dd 1748 }
1749
1750 // Hypothesis specified via a pdf
1751 if (pdf && !hyp)
1752 {
1753 pdf->SetRange(xmin,xmax);
1754 Double_t ftot=pdf->Integral(xmin,xmax);
1755 if (ftot>0)
1756 {
1757 Double_t x1,x2;
1758 for (Int_t ipdf=1; ipdf<=nbins; ipdf++)
1759 {
1760 nk=int(his->GetBinContent(ipdf));
1761 x1=his->GetBinLowEdge(ipdf);
1762 x2=x1+his->GetBinWidth(ipdf);
1763 pk=pdf->Integral(x1,x2)/ftot;
1764 n[ipdf-1]=0;
1765 p[ipdf-1]=0;
1766 if (nk>0) n[ipdf-1]=nk;
1767 if (pk>0) p[ipdf-1]=pk;
1768 }
99a25cd9 1769 chi=Chi2Value(nbins,n,p,ndf);
bf4dd5dd 1770 }
1771 }
1772
1773 // Hypothesis specified via a histogram
1774 if (hyp && !pdf)
1775 {
1776 TH1F* href=new TH1F(*his);
1777 href->Reset();
1778 Double_t nenhyp=hyp->GetEntries();
1779 Float_t x,y;
1780 for (Int_t ihyp=1; ihyp<=hyp->GetNbinsX(); ihyp++)
1781 {
1782 x=hyp->GetBinCenter(ihyp);
1783 y=hyp->GetBinContent(ihyp);
1784 href->Fill(x,y);
1785 }
1786 for (Int_t j=1; j<=nbins; j++)
1787 {
1788 nk=int(his->GetBinContent(j));
1789 pk=href->GetBinContent(j)/nenhyp;
1790 n[j-1]=0;
1791 p[j-1]=0;
1792 if (nk>0) n[j-1]=nk;
1793 if (pk>0) p[j-1]=pk;
1794 }
99a25cd9 1795 chi=Chi2Value(nbins,n,p,ndf);
bf4dd5dd 1796 delete href;
1797 }
1798
1799 delete [] n;
1800 delete [] p;
1801
1802 return chi;
1803}
1804///////////////////////////////////////////////////////////////////////////