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RandBinomial.cc
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1 // $Id:$
2 // -*- C++ -*-
3 //
4 // -----------------------------------------------------------------------
5 // HEP Random
6 // --- RandBinomial ---
7 // class implementation file
8 // -----------------------------------------------------------------------
9 
10 // =======================================================================
11 // John Marraffino - Created: 12th May 1998
12 // M Fischler - put and get to/from streams 12/10/04
13 // M Fischler - put/get to/from streams uses pairs of ulongs when
14 // + storing doubles avoid problems with precision
15 // 4/14/05
16 //
17 // =======================================================================
18 
20 #include "CLHEP/Random/DoubConv.h"
21 #include <algorithm> // for min() and max()
22 #include <cmath> // for exp()
23 
24 namespace CLHEP {
25 
26 std::string RandBinomial::name() const {return "RandBinomial";}
27 HepRandomEngine & RandBinomial::engine() {return *localEngine;}
28 
30 }
31 
32 double RandBinomial::shoot( HepRandomEngine *anEngine, long n,
33  double p ) {
34  return genBinomial( anEngine, n, p );
35 }
36 
37 double RandBinomial::shoot( long n, double p ) {
39  return genBinomial( anEngine, n, p );
40 }
41 
42 double RandBinomial::fire( long n, double p ) {
43  return genBinomial( localEngine.get(), n, p );
44 }
45 
46 void RandBinomial::shootArray( const int size, double* vect,
47  long n, double p )
48 {
49  for( double* v = vect; v != vect+size; ++v )
50  *v = shoot(n,p);
51 }
52 
54  const int size, double* vect,
55  long n, double p )
56 {
57  for( double* v = vect; v != vect+size; ++v )
58  *v = shoot(anEngine,n,p);
59 }
60 
61 void RandBinomial::fireArray( const int size, double* vect)
62 {
63  for( double* v = vect; v != vect+size; ++v )
64  *v = fire(defaultN,defaultP);
65 }
66 
67 void RandBinomial::fireArray( const int size, double* vect,
68  long n, double p )
69 {
70  for( double* v = vect; v != vect+size; ++v )
71  *v = fire(n,p);
72 }
73 
74 /*************************************************************************
75  * *
76  * StirlingCorrection() *
77  * *
78  * Correction term of the Stirling approximation for log(k!) *
79  * (series in 1/k, or table values for small k) *
80  * with long int parameter k *
81  * *
82  *************************************************************************
83  * *
84  * log k! = (k + 1/2)log(k + 1) - (k + 1) + (1/2)log(2Pi) + *
85  * StirlingCorrection(k + 1) *
86  * *
87  * log k! = (k + 1/2)log(k) - k + (1/2)log(2Pi) + *
88  * StirlingCorrection(k) *
89  * *
90  *************************************************************************/
91 
92 static double StirlingCorrection(long int k)
93 {
94  #define C1 8.33333333333333333e-02 // +1/12
95  #define C3 -2.77777777777777778e-03 // -1/360
96  #define C5 7.93650793650793651e-04 // +1/1260
97  #define C7 -5.95238095238095238e-04 // -1/1680
98 
99  static double c[31] = { 0.0,
100  8.106146679532726e-02, 4.134069595540929e-02,
101  2.767792568499834e-02, 2.079067210376509e-02,
102  1.664469118982119e-02, 1.387612882307075e-02,
103  1.189670994589177e-02, 1.041126526197209e-02,
104  9.255462182712733e-03, 8.330563433362871e-03,
105  7.573675487951841e-03, 6.942840107209530e-03,
106  6.408994188004207e-03, 5.951370112758848e-03,
107  5.554733551962801e-03, 5.207655919609640e-03,
108  4.901395948434738e-03, 4.629153749334029e-03,
109  4.385560249232324e-03, 4.166319691996922e-03,
110  3.967954218640860e-03, 3.787618068444430e-03,
111  3.622960224683090e-03, 3.472021382978770e-03,
112  3.333155636728090e-03, 3.204970228055040e-03,
113  3.086278682608780e-03, 2.976063983550410e-03,
114  2.873449362352470e-03, 2.777674929752690e-03,
115  };
116  double r, rr;
117 
118  if (k > 30L) {
119  r = 1.0 / (double) k; rr = r * r;
120  return(r*(C1 + rr*(C3 + rr*(C5 + rr*C7))));
121  }
122  else return(c[k]);
123 }
124 
125 double RandBinomial::genBinomial( HepRandomEngine *anEngine, long n, double p )
126 {
127 /******************************************************************
128  * *
129  * Binomial-Distribution - Acceptance Rejection/Inversion *
130  * *
131  ******************************************************************
132  * *
133  * Acceptance Rejection method combined with Inversion for *
134  * generating Binomial random numbers with parameters *
135  * n (number of trials) and p (probability of success). *
136  * For min(n*p,n*(1-p)) < 10 the Inversion method is applied: *
137  * The random numbers are generated via sequential search, *
138  * starting at the lowest index k=0. The cumulative probabilities *
139  * are avoided by using the technique of chop-down. *
140  * For min(n*p,n*(1-p)) >= 10 Acceptance Rejection is used: *
141  * The algorithm is based on a hat-function which is uniform in *
142  * the centre region and exponential in the tails. *
143  * A triangular immediate acceptance region in the centre speeds *
144  * up the generation of binomial variates. *
145  * If candidate k is near the mode, f(k) is computed recursively *
146  * starting at the mode m. *
147  * The acceptance test by Stirling's formula is modified *
148  * according to W. Hoermann (1992): The generation of binomial *
149  * random variates, to appear in J. Statist. Comput. Simul. *
150  * If p < .5 the algorithm is applied to parameters n, p. *
151  * Otherwise p is replaced by 1-p, and k is replaced by n - k. *
152  * *
153  ******************************************************************
154  * *
155  * FUNCTION: - btpec samples a random number from the binomial *
156  * distribution with parameters n and p and is *
157  * valid for n*min(p,1-p) > 0. *
158  * REFERENCE: - V. Kachitvichyanukul, B.W. Schmeiser (1988): *
159  * Binomial random variate generation, *
160  * Communications of the ACM 31, 216-222. *
161  * SUBPROGRAMS: - StirlingCorrection() *
162  * ... Correction term of the Stirling *
163  * approximation for log(k!) *
164  * (series in 1/k or table values *
165  * for small k) with long int k *
166  * - anEngine ... Pointer to a (0,1)-Uniform *
167  * engine *
168  * *
169  * Implemented by H. Zechner and P. Busswald, September 1992 *
170  ******************************************************************/
171 
172 #define C1_3 0.33333333333333333
173 #define C5_8 0.62500000000000000
174 #define C1_6 0.16666666666666667
175 #define DMAX_KM 20L
176 
177  static long int n_last = -1L, n_prev = -1L;
178  static double par,np,p0,q,p_last = -1.0, p_prev = -1.0;
179  static long b,m,nm;
180  static double pq, rc, ss, xm, xl, xr, ll, lr, c,
181  p1, p2, p3, p4, ch;
182 
183  long bh,i, K, Km, nK;
184  double f, rm, U, V, X, T, E;
185 
186  if (n != n_last || p != p_last) // set-up
187  {
188  n_last = n;
189  p_last = p;
190  par=std::min(p,1.0-p);
191  q=1.0-par;
192  np = n*par;
193 
194 // Check for invalid input values
195 
196  if( np <= 0.0 ) return (-1.0);
197 
198  rm = np + par;
199  m = (long int) rm; // mode, integer
200  if (np<10)
201  {
202  p0=std::exp(n*std::log(q)); // Chop-down
203  bh=(long int)(np+10.0*std::sqrt(np*q));
204  b=std::min(n,bh);
205  }
206  else
207  {
208  rc = (n + 1.0) * (pq = par / q); // recurr. relat.
209  ss = np * q; // variance
210  i = (long int) (2.195*std::sqrt(ss) - 4.6*q); // i = p1 - 0.5
211  xm = m + 0.5;
212  xl = (double) (m - i); // limit left
213  xr = (double) (m + i + 1L); // limit right
214  f = (rm - xl) / (rm - xl*par); ll = f * (1.0 + 0.5*f);
215  f = (xr - rm) / (xr * q); lr = f * (1.0 + 0.5*f);
216  c = 0.134 + 20.5/(15.3 + (double) m); // parallelogram
217  // height
218  p1 = i + 0.5;
219  p2 = p1 * (1.0 + c + c); // probabilities
220  p3 = p2 + c/ll; // of regions 1-4
221  p4 = p3 + c/lr;
222  }
223  }
224  if (np<10) //Inversion Chop-down
225  {
226  double pk;
227 
228  K=0;
229  pk=p0;
230  U=anEngine->flat();
231  while (U>pk)
232  {
233  ++K;
234  if (K>b)
235  {
236  U=anEngine->flat();
237  K=0;
238  pk=p0;
239  }
240  else
241  {
242  U-=pk;
243  pk=(double)(((n-K+1)*par*pk)/(K*q));
244  }
245  }
246  return ((p>0.5) ? (double)(n-K):(double)K);
247  }
248 
249  for (;;)
250  {
251  V = anEngine->flat();
252  if ((U = anEngine->flat() * p4) <= p1) // triangular region
253  {
254  K=(long int) (xm - U + p1*V);
255  return ((p>0.5) ? (double)(n-K):(double)K); // immediate accept
256  }
257  if (U <= p2) // parallelogram
258  {
259  X = xl + (U - p1)/c;
260  if ((V = V*c + 1.0 - std::fabs(xm - X)/p1) >= 1.0) continue;
261  K = (long int) X;
262  }
263  else if (U <= p3) // left tail
264  {
265  if ((X = xl + std::log(V)/ll) < 0.0) continue;
266  K = (long int) X;
267  V *= (U - p2) * ll;
268  }
269  else // right tail
270  {
271  if ((K = (long int) (xr - std::log(V)/lr)) > n) continue;
272  V *= (U - p3) * lr;
273  }
274 
275  // acceptance test : two cases, depending on |K - m|
276  if ((Km = std::labs(K - m)) <= DMAX_KM || Km + Km + 2L >= ss)
277  {
278 
279  // computation of p(K) via recurrence relationship from the mode
280  f = 1.0; // f(m)
281  if (m < K)
282  {
283  for (i = m; i < K; )
284  {
285  if ((f *= (rc / ++i - pq)) < V) break; // multiply f
286  }
287  }
288  else
289  {
290  for (i = K; i < m; )
291  {
292  if ((V *= (rc / ++i - pq)) > f) break; // multiply V
293  }
294  }
295  if (V <= f) break; // acceptance test
296  }
297  else
298  {
299 
300  // lower and upper squeeze tests, based on lower bounds for log p(K)
301  V = std::log(V);
302  T = - Km * Km / (ss + ss);
303  E = (Km / ss) * ((Km * (Km * C1_3 + C5_8) + C1_6) / ss + 0.5);
304  if (V <= T - E) break;
305  if (V <= T + E)
306  {
307  if (n != n_prev || par != p_prev)
308  {
309  n_prev = n;
310  p_prev = par;
311 
312  nm = n - m + 1L;
313  ch = xm * std::log((m + 1.0)/(pq * nm)) +
314  StirlingCorrection(m + 1L) + StirlingCorrection(nm);
315  }
316  nK = n - K + 1L;
317 
318  // computation of log f(K) via Stirling's formula
319  // final acceptance-rejection test
320  if (V <= ch + (n + 1.0)*std::log((double) nm / (double) nK) +
321  (K + 0.5)*std::log(nK * pq / (K + 1.0)) -
322  StirlingCorrection(K + 1L) - StirlingCorrection(nK)) break;
323  }
324  }
325  }
326  return ((p>0.5) ? (double)(n-K):(double)K);
327 }
328 
329 std::ostream & RandBinomial::put ( std::ostream & os ) const {
330  int pr=os.precision(20);
331  std::vector<unsigned long> t(2);
332  os << " " << name() << "\n";
333  os << "Uvec" << "\n";
334  t = DoubConv::dto2longs(defaultP);
335  os << defaultN << " " << defaultP << " " << t[0] << " " << t[1] << "\n";
336  os.precision(pr);
337  return os;
338 }
339 
340 std::istream & RandBinomial::get ( std::istream & is ) {
341  std::string inName;
342  is >> inName;
343  if (inName != name()) {
344  is.clear(std::ios::badbit | is.rdstate());
345  std::cerr << "Mismatch when expecting to read state of a "
346  << name() << " distribution\n"
347  << "Name found was " << inName
348  << "\nistream is left in the badbit state\n";
349  return is;
350  }
351  if (possibleKeywordInput(is, "Uvec", defaultN)) {
352  std::vector<unsigned long> t(2);
353  is >> defaultN >> defaultP;
354  is >> t[0] >> t[1]; defaultP = DoubConv::longs2double(t);
355  return is;
356  }
357  // is >> defaultN encompassed by possibleKeywordInput
358  is >> defaultP;
359  return is;
360 }
361 
362 
363 } // namespace CLHEP