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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 | |
40 | ClassImp(AliMath) // Class implementation to enable ROOT I/O | |
41 | ||
c72198f1 | 42 | AliMath::AliMath() : TObject() |
d88f97cc | 43 | { |
44 | // Default constructor | |
45 | } | |
46 | /////////////////////////////////////////////////////////////////////////// | |
47 | AliMath::~AliMath() | |
48 | { | |
49 | // Destructor | |
50 | } | |
51 | /////////////////////////////////////////////////////////////////////////// | |
261c0caf | 52 | AliMath::AliMath(const AliMath& m) : TObject(m) |
c72198f1 | 53 | { |
54 | // Copy constructor | |
55 | } | |
56 | /////////////////////////////////////////////////////////////////////////// | |
261c0caf | 57 | Double_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 | 78 | Double_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 | 109 | Double_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 | 152 | Double_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 | 193 | Double_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 | 243 | Double_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 | 252 | Double_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 | 286 | Double_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 | 358 | Double_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 | 397 | Double_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 | 440 | Double_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 | 480 | Double_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 | 523 | Double_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 | 561 | Double_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 | |
594 | for (Int_t j=m; j<=1; j--) | |
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 | 614 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
634 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
663 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
695 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
721 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
747 | TF1 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 | /////////////////////////////////////////////////////////////////////////// | |
769 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
856 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
940 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1029 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1129 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1209 | Double_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 | 1311 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1364 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1412 | Double_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 | 1428 | Double_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 | /////////////////////////////////////////////////////////////////////////// | |
1506 | Double_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 | 1635 | Double_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 | 1688 | Double_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 | /////////////////////////////////////////////////////////////////////////// |