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Reference Guide
TestBinomial.C File Reference
Tutorials » Fit Tutorials

Detailed Description

View in nbviewer Open in SWAN Perform a fit to a set of data with binomial errors like those derived from the division of two histograms.

Three different fits are performed and compared:

The first two methods are biased while the last one is statistical correct. Running the script passing an integer value n larger than 1, n fits are performed and the bias are also shown. To run the script :

to show the bias performing 100 fits for 1000 events per "experiment"

root[0]: .x TestBinomial.C+

to show the bias performing 100 fits for 1000 events per "experiment"

.x TestBinomial.C+(100, 1000)
pict1_TestBinomial.C.png
Processing /mnt/build/workspace/root-makedoc-v614/rootspi/rdoc/src/v6-14-00-patches/tutorials/fit/TestBinomial.C...
32 68
****************************************
Minimizer is Minuit2
Chi2 = 0.350714
NDf = 3
Edm = 1.27001e-06
NCalls = 81
p0 = 0.694132 +/- 0.210029
p1 = 19.3471 +/- 5.85483
p2 = 5.2245 +/- 5.11013
****************************************
Minimizer is Minuit2
Chi2 = 1.64738
NDf = 5
Edm = 1.01171e-06
NCalls = 137
p0 = 0.677329 +/- 0.135567 (limited)
p1 = 15.5028 +/- 4.11118
p2 = 4.10441 +/- 2.77044
****************************************
Minimizer is Minuit2
Chi2 = 0.928456
NDf = 6
Edm = 1.46917e-08
NCalls = 75
p0 = 0.578806 +/- 0.128412 (limited)
p1 = 19.8914 +/- 2.83678
p2 = 3.39323 +/- 1.96156
****************************************
Minimizer is Minuit2
Chi2 = 2.56551
NDf = 7
Edm = 1.19556e-08
NCalls = 109
p0 = 0.57261 +/- 0.16552 (limited)
p1 = 21.4104 +/- 5.10611
p2 = 5.58786 +/- 3.30889
****************************************
Minimizer is Minuit2
Chi2 = 7.37095
NDf = 10
Edm = 1.54804e-06
NCalls = 87
p0 = 0.667689 +/- 0.152841 (limited)
p1 = 24.0146 +/- 2.40617
p2 = 3.63777 +/- 1.74672
****************************************
Minimizer is Minuit2
Chi2 = 0.747776
NDf = 6
Edm = 2.75121e-06
NCalls = 76
p0 = 0.651877 +/- 0.155245 (limited)
p1 = 18.9865 +/- 2.68321
p2 = 2.68989 +/- 1.28756
****************************************
Minimizer is Minuit2
Chi2 = 0.433039
NDf = 4
Edm = 3.07902e-07
NCalls = 104
p0 = 0.500967 +/- 0.254652 (limited)
p1 = 19.4199 +/- 9.30614
p2 = 6.46947 +/- 8.16624
****************************************
Minimizer is Minuit2
Chi2 = 4.49197
NDf = 9
Edm = 4.04441e-06
NCalls = 113
p0 = 0.623236 +/- 0.107197 (limited)
p1 = 20.8129 +/- 1.0507
p2 = 1.22179 +/- 1.09027
****************************************
Minimizer is Minuit2
Chi2 = 2.54966
NDf = 6
Edm = 1.48922e-07
NCalls = 93
p0 = 0.382687 +/- 0.226798 (limited)
p1 = 18.0539 +/- 6.38962
p2 = 3.0464 +/- 5.58751
****************************************
Minimizer is Minuit2
Chi2 = 1.46531
NDf = 7
Edm = 2.36369e-06
NCalls = 90
p0 = 0.575891 +/- 0.187613 (limited)
p1 = 17.6126 +/- 7.68638
p2 = 2.99185 +/- 6.18408
****************************************
Minimizer is Minuit2
Chi2 = 3.31728
NDf = 6
Edm = 8.18927e-07
NCalls = 64
p0 = 0.59853 +/- 0.124912 (limited)
p1 = 17.9048 +/- 3.49062
p2 = 3.52241 +/- 2.09942
****************************************
Minimizer is Minuit2
Chi2 = 0.413866
NDf = 6
Edm = 5.34616e-09
NCalls = 82
p0 = 0.654352 +/- 0.198145 (limited)
p1 = 20.6178 +/- 3.66715
p2 = 4.03019 +/- 2.20627
****************************************
Minimizer is Minuit2
Chi2 = 1.21002
NDf = 4
Edm = 2.90565e-08
NCalls = 82
p0 = 0.651685 +/- 0.159188 (limited)
p1 = 19.3369 +/- 2.66645
p2 = 2.81842 +/- 2.20413
****************************************
Minimizer is Minuit2
Chi2 = 0.752979
NDf = 7
Edm = 5.34806e-06
NCalls = 101
p0 = 0.643852 +/- 0.170146 (limited)
p1 = 21.01 +/- 4.32394
p2 = 5.36976 +/- 3.12567
****************************************
Minimizer is Minuit2
Chi2 = 2.5161
NDf = 8
Edm = 6.68524e-10
NCalls = 61
p0 = 0.628793 +/- 0.103749 (limited)
p1 = 17.4724 +/- 2.12101
p2 = 2.59351 +/- 1.42575
****************************************
Minimizer is Minuit2
Chi2 = 1.27043
NDf = 8
Edm = 6.23845e-06
NCalls = 72
p0 = 0.561127 +/- 0.138134 (limited)
p1 = 18.1601 +/- 3.22498
p2 = 3.50338 +/- 1.84312
****************************************
Minimizer is Minuit2
Chi2 = 2.84972
NDf = 8
Edm = 1.39524e-06
NCalls = 90
p0 = 0.693419 +/- 0.285373 (limited)
p1 = 24.7834 +/- 7.96871
p2 = 5.95505 +/- 3.88805
****************************************
Minimizer is Minuit2
Chi2 = 3.67016
NDf = 5
Edm = 1.80141e-06
NCalls = 82
p0 = 0.502645 +/- 0.176121 (limited)
p1 = 20.1539 +/- 2.79976
p2 = 2.70329 +/- 1.65981
****************************************
Invalid FitResult (status = 3 )
****************************************
Minimizer is Minuit2
Chi2 = 0.936088
NDf = 1
Edm = 0.106873
NCalls = 83
p0 = 0.41816 +/- 0.669502 (limited)
p1 = 18.3049 +/- 3.25491
p2 = 3.06774 +/- 40.715
****************************************
Minimizer is Minuit2
Chi2 = 3.77203
NDf = 8
Edm = 1.56279e-05
NCalls = 84
p0 = 0.642277 +/- 0.195607 (limited)
p1 = 22.9222 +/- 4.97062
p2 = 5.05681 +/- 2.69739
****************************************
Minimizer is Minuit2
Chi2 = 0.47406
NDf = 6
Edm = 1.7448e-06
NCalls = 88
p0 = 0.635811 +/- 0.173792 (limited)
p1 = 21.0478 +/- 3.90109
p2 = 3.91609 +/- 2.85302
****************************************
Minimizer is Minuit2
Chi2 = 2.39556
NDf = 8
Edm = 4.51695e-08
NCalls = 81
p0 = 0.564008 +/- 0.188334 (limited)
p1 = 19.2305 +/- 5.64705
p2 = 4.00431 +/- 3.2294
****************************************
Minimizer is Minuit2
Chi2 = 2.71864
NDf = 3
Edm = 2.02651e-07
NCalls = 84
p0 = 0.430955 +/- 0.103386 (limited)
p1 = 15.3556 +/- 1.58614
p2 = 1.18646 +/- 1.05279
****************************************
Minimizer is Minuit2
Chi2 = 3.91497
NDf = 3
Edm = 9.71484e-06
NCalls = 172
p0 = 0.999991 +/- 0.973989 (limited)
p1 = 26.3679 +/- 4.16357
p2 = 10.2085 +/- 7.97289
****************************************
Minimizer is Minuit2
Chi2 = 3.69474
NDf = 4
Edm = 1.0286e-07
NCalls = 124
p0 = 0.566565 +/- 0.300627 (limited)
p1 = 22.9958 +/- 7.83199
p2 = 6.59909 +/- 5.25595
****************************************
Minimizer is Minuit2
Chi2 = 1.01429
NDf = 10
Edm = 2.60252e-06
NCalls = 94
p0 = 0.610818 +/- 0.175926 (limited)
p1 = 23.3058 +/- 5.77903
p2 = 5.09666 +/- 2.59915
****************************************
Minimizer is Minuit2
Chi2 = 3.63097
NDf = 7
Edm = 2.77767e-07
NCalls = 123
p0 = 0.643028 +/- 0.225348 (limited)
p1 = 21.4306 +/- 4.46994
p2 = 3.95327 +/- 2.04398
****************************************
Minimizer is Minuit2
Chi2 = 1.7669
NDf = 7
Edm = 1.75691e-07
NCalls = 98
p0 = 0.73821 +/- 0.220641 (limited)
p1 = 23.671 +/- 5.18271
p2 = 5.53649 +/- 2.72352
****************************************
Minimizer is Minuit2
Chi2 = 2.70826
NDf = 8
Edm = 5.00596e-07
NCalls = 106
p0 = 0.662591 +/- 0.352545 (limited)
p1 = 24.2783 +/- 12.3384
p2 = 7.60846 +/- 6.66009
****************************************
Minimizer is Minuit2
Chi2 = 9.74855
NDf = 4
Edm = 3.46464e-07
NCalls = 124
p0 = 0.606488 +/- 0.231103 (limited)
p1 = 22.9147 +/- 5.11615
p2 = 4.88595 +/- 2.18611
****************************************
Minimizer is Minuit2
Chi2 = 3.10119
NDf = 5
Edm = 3.71723e-07
NCalls = 234
p0 = 0.999998 +/- 0.501156 (limited)
p1 = 28.5973 +/- 4.11085
p2 = 8.15931 +/- 3.01069
****************************************
Minimizer is Minuit2
Chi2 = 2.78024
NDf = 7
Edm = 1.5102e-06
NCalls = 82
p0 = 0.65308 +/- 0.206386 (limited)
p1 = 23.0571 +/- 4.95138
p2 = 4.86356 +/- 2.47873
****************************************
Minimizer is Minuit2
Chi2 = 3.05436
NDf = 5
Edm = 1.66356e-05
NCalls = 297
p0 = 0.999547 +/- 0.735933 (limited)
p1 = 47.3138 +/- 36.6961
p2 = 43.5419 +/- 61.6394
****************************************
Minimizer is Minuit2
Chi2 = 0.139804
NDf = 2
Edm = 1.06491e-06
NCalls = 91
p0 = 0.660157 +/- 0.258707 (limited)
p1 = 19.2592 +/- 7.6936
p2 = 6.44669 +/- 4.78428
****************************************
Minimizer is Minuit2
Chi2 = 2.55321
NDf = 7
Edm = 5.89829e-06
NCalls = 89
p0 = 0.643594 +/- 0.194962 (limited)
p1 = 21.9893 +/- 5.24123
p2 = 5.90243 +/- 2.90213
****************************************
Minimizer is Minuit2
Chi2 = 2.36128
NDf = 6
Edm = 6.99629e-07
NCalls = 88
p0 = 0.560279 +/- 0.13121 (limited)
p1 = 19.8526 +/- 3.03421
p2 = 2.47661 +/- 2.23047
****************************************
Minimizer is Minuit2
Chi2 = 1.33423
NDf = 7
Edm = 2.89896e-06
NCalls = 102
p0 = 0.564562 +/- 0.203287 (limited)
p1 = 25.8413 +/- 9.79483
p2 = 8.35761 +/- 6.07925
****************************************
Minimizer is Minuit2
Chi2 = 1.74047
NDf = 7
Edm = 7.04231e-07
NCalls = 135
p0 = 0.594764 +/- 0.187954 (limited)
p1 = 25.3877 +/- 8.57244
p2 = 6.52667 +/- 4.25157
****************************************
Minimizer is Minuit2
Chi2 = 0.310648
NDf = 7
Edm = 2.45348e-06
NCalls = 169
p0 = 0.539725 +/- 0.760718 (limited)
p1 = 12.9065 +/- 72.7676
p2 = 22.8828 +/- 100.306
****************************************
Minimizer is Minuit2
Chi2 = 2.52738
NDf = 10
Edm = 1.515e-08
NCalls = 119
p0 = 0.631088 +/- 0.21992 (limited)
p1 = 26.5773 +/- 9.03434
p2 = 8.9826 +/- 4.83446
****************************************
Minimizer is Minuit2
Chi2 = 2.51398
NDf = 6
Edm = 6.94471e-07
NCalls = 84
p0 = 0.484246 +/- 0.226988 (limited)
p1 = 18.2432 +/- 5.68591
p2 = 3.93586 +/- 3.23302
****************************************
Minimizer is Minuit2
Chi2 = 0.946468
NDf = 5
Edm = 5.45857e-06
NCalls = 74
p0 = 0.584615 +/- 0.133896 (limited)
p1 = 19.3728 +/- 3.81107
p2 = 4.20495 +/- 2.53748
****************************************
Minimizer is Minuit2
Chi2 = 5.46724
NDf = 8
Edm = 1.55331e-06
NCalls = 99
p0 = 0.763745 +/- 0.233374 (limited)
p1 = 26.3444 +/- 5.03855
p2 = 6.07972 +/- 2.87905
****************************************
Minimizer is Minuit2
Chi2 = 0.716283
NDf = 7
Edm = 1.89199e-06
NCalls = 85
p0 = 0.422768 +/- 0.179544 (limited)
p1 = 17.3878 +/- 6.43713
p2 = 3.05887 +/- 4.49587
****************************************
Minimizer is Minuit2
Chi2 = 6.40375
NDf = 8
Edm = 5.62321e-07
NCalls = 104
p0 = 0.507615 +/- 0.198193 (limited)
p1 = 22.2924 +/- 5.65071
p2 = 4.8117 +/- 2.67857
****************************************
Minimizer is Minuit2
Chi2 = 3.18171
NDf = 9
Edm = 5.5664e-07
NCalls = 83
p0 = 0.694498 +/- 0.134559 (limited)
p1 = 21.5399 +/- 2.86559
p2 = 3.46906 +/- 1.42287
****************************************
Minimizer is Minuit2
Chi2 = 3.13918
NDf = 6
Edm = 4.11187e-06
NCalls = 101
p0 = 0.725096 +/- 0.145408 (limited)
p1 = 25.2372 +/- 3.82188
p2 = 4.88315 +/- 1.80529
****************************************
Minimizer is Minuit2
Chi2 = 3.15774
NDf = 6
Edm = 2.32136e-06
NCalls = 109
p0 = 0.757722 +/- 0.770168 (limited)
p1 = 26.0106 +/- 13.1967
p2 = 7.01687 +/- 5.25991
****************************************
Minimizer is Minuit2
Chi2 = 1.90008
NDf = 6
Edm = 9.90083e-06
NCalls = 99
p0 = 0.534999 +/- 0.270262 (limited)
p1 = 24.0906 +/- 7.8809
p2 = 5.63935 +/- 3.24361
****************************************
Minimizer is Minuit2
Chi2 = 2.77855
NDf = 8
Edm = 8.13609e-08
NCalls = 85
p0 = 0.702887 +/- 0.199272 (limited)
p1 = 20.9159 +/- 3.68349
p2 = 4.41504 +/- 2.04434
****************************************
Minimizer is Minuit2
Chi2 = 4.37217
NDf = 9
Edm = 2.72083e-07
NCalls = 82
p0 = 0.620959 +/- 0.15843 (limited)
p1 = 21.0013 +/- 3.72087
p2 = 4.56895 +/- 2.37229
****************************************
Minimizer is Minuit2
Chi2 = 1.04047
NDf = 4
Edm = 5.64332e-06
NCalls = 102
p0 = 0.593644 +/- 0.174313 (limited)
p1 = 21.0676 +/- 4.04434
p2 = 3.35047 +/- 2.25854
****************************************
Minimizer is Minuit2
Chi2 = 3.95319
NDf = 10
Edm = 1.32825e-07
NCalls = 96
p0 = 0.574996 +/- 0.152016 (limited)
p1 = 20.2114 +/- 5.1893
p2 = 4.26461 +/- 2.57398
****************************************
Minimizer is Minuit2
Chi2 = 1.2121
NDf = 7
Edm = 3.20149e-06
NCalls = 91
p0 = 0.498388 +/- 0.194573 (limited)
p1 = 16.3792 +/- 10.7582
p2 = 4.04759 +/- 14.7395
****************************************
Minimizer is Minuit2
Chi2 = 1.15843
NDf = 4
Edm = 3.66535e-06
NCalls = 105
p0 = 0.570211 +/- 0.236898 (limited)
p1 = 17.2578 +/- 15.7293
p2 = 9.7494 +/- 16.3952
****************************************
Minimizer is Minuit2
Chi2 = 4.96865
NDf = 8
Edm = 1.88709e-08
NCalls = 80
p0 = 0.666513 +/- 0.140936 (limited)
p1 = 21.2569 +/- 2.26557
p2 = 3.48413 +/- 2.03653
****************************************
Minimizer is Minuit2
Chi2 = 2.88079
NDf = 4
Edm = 8.70733e-06
NCalls = 96
p0 = 0.803697 +/- 0.275939 (limited)
p1 = 22.1124 +/- 5.26388
p2 = 5.42701 +/- 3.06391
****************************************
Minimizer is Minuit2
Chi2 = 3.96392
NDf = 5
Edm = 7.79729e-08
NCalls = 100
p0 = 0.694889 +/- 0.235894 (limited)
p1 = 23.9686 +/- 5.40538
p2 = 5.07476 +/- 3.36335
****************************************
Minimizer is Minuit2
Chi2 = 0.852791
NDf = 5
Edm = 8.88345e-09
NCalls = 84
p0 = 0.310283 +/- 0.164695 (limited)
p1 = 15.0842 +/- 5.86864
p2 = 2.43104 +/- 3.51103
****************************************
Minimizer is Minuit2
Chi2 = 1.32394
NDf = 6
Edm = 6.16879e-07
NCalls = 86
p0 = 0.672106 +/- 0.201669 (limited)
p1 = 23.2133 +/- 5.10201
p2 = 5.04859 +/- 2.50327
****************************************
Minimizer is Minuit2
Chi2 = 2.48447
NDf = 9
Edm = 3.99523e-06
NCalls = 92
p0 = 0.469573 +/- 0.142912 (limited)
p1 = 18.7922 +/- 3.85884
p2 = 2.98044 +/- 2.31127
****************************************
Minimizer is Minuit2
Chi2 = 4.05841
NDf = 7
Edm = 3.35768e-06
NCalls = 93
p0 = 0.667164 +/- 0.148637 (limited)
p1 = 21.9757 +/- 2.55643
p2 = 3.34362 +/- 2.015
****************************************
Minimizer is Minuit2
Chi2 = 3.14114
NDf = 6
Edm = 1.74933e-08
NCalls = 114
p0 = 0.532234 +/- 0.174899 (limited)
p1 = 20.0178 +/- 3.71281
p2 = 3.05935 +/- 2.11129
****************************************
Minimizer is Minuit2
Chi2 = 2.88405
NDf = 5
Edm = 1.25252e-05
NCalls = 77
p0 = 0.698497 +/- 0.183511 (limited)
p1 = 21.6581 +/- 4.18422
p2 = 4.04207 +/- 1.56968
****************************************
Minimizer is Minuit2
Chi2 = 0.624725
NDf = 4
Edm = 6.43892e-06
NCalls = 71
p0 = 0.380579 +/- 0.134096 (limited)
p1 = 13.9323 +/- 2.56334
p2 = 1.96554 +/- 1.64705
****************************************
Minimizer is Minuit2
Chi2 = 1.67529
NDf = 7
Edm = 4.64318e-08
NCalls = 80
p0 = 0.616278 +/- 0.1576 (limited)
p1 = 18.812 +/- 2.91455
p2 = 3.21096 +/- 1.42859
****************************************
Minimizer is Minuit2
Chi2 = 0.374196
NDf = 2
Edm = 8.80185e-06
NCalls = 80
p0 = 0.685904 +/- 0.174442 (limited)
p1 = 19.998 +/- 1.62445
p2 = 1.71179 +/- 1.75684
****************************************
Minimizer is Minuit2
Chi2 = 2.46931
NDf = 7
Edm = 3.32581e-07
NCalls = 93
p0 = 0.640812 +/- 0.135875 (limited)
p1 = 22.3459 +/- 3.96589
p2 = 4.39764 +/- 2.51185
****************************************
Minimizer is Minuit2
Chi2 = 2.49631
NDf = 3
Edm = 6.6234e-06
NCalls = 102
p0 = 0.611704 +/- 0.35871 (limited)
p1 = 20.7057 +/- 8.42105
p2 = 5.08053 +/- 3.9281
****************************************
Minimizer is Minuit2
Chi2 = 0.418946
NDf = 2
Edm = 7.44661e-07
NCalls = 53
p0 = 0.60361 +/- 0.106733 (limited)
p1 = 15.7806 +/- 2
p2 = 0.105266 +/- 2
****************************************
Minimizer is Minuit2
Chi2 = 3.54456
NDf = 5
Edm = 2.33566e-06
NCalls = 85
p0 = 0.527397 +/- 0.217626 (limited)
p1 = 17.9928 +/- 6.38912
p2 = 5.4498 +/- 3.84602
****************************************
Minimizer is Minuit2
Chi2 = 3.23312
NDf = 8
Edm = 9.46056e-07
NCalls = 70
p0 = 0.515942 +/- 0.0980029 (limited)
p1 = 17.2963 +/- 1.44299
p2 = 1.9009 +/- 1.23849
****************************************
Minimizer is Minuit2
Chi2 = 5.28211
NDf = 11
Edm = 1.87776e-06
NCalls = 71
p0 = 0.612568 +/- 0.109907 (limited)
p1 = 19.3198 +/- 2.18618
p2 = 3.37801 +/- 1.31028
****************************************
Minimizer is Minuit2
Chi2 = 2.51155
NDf = 7
Edm = 2.60724e-06
NCalls = 74
p0 = 0.647126 +/- 0.147452 (limited)
p1 = 20.8474 +/- 2.85119
p2 = 3.27715 +/- 1.66359
****************************************
Minimizer is Minuit2
Chi2 = 2.33728
NDf = 6
Edm = 7.37813e-07
NCalls = 109
p0 = 0.624689 +/- 0.141987 (limited)
p1 = 19.3681 +/- 3.57572
p2 = 3.40927 +/- 2.17685
****************************************
Minimizer is Minuit2
Chi2 = 3.75642
NDf = 8
Edm = 5.68096e-06
NCalls = 120
p0 = 0.387014 +/- 0.0702296 (limited)
p1 = 11.5811 +/- 8.22684
p2 = 0.313997 +/- 4.44032
****************************************
Minimizer is Minuit2
Chi2 = 1.68054
NDf = 6
Edm = 5.00607e-06
NCalls = 79
p0 = 0.492763 +/- 0.200749 (limited)
p1 = 18.8758 +/- 5.209
p2 = 3.88829 +/- 2.85902
****************************************
Minimizer is Minuit2
Chi2 = 1.37515
NDf = 4
Edm = 3.40318e-06
NCalls = 106
p0 = 0.759679 +/- 0.234791 (limited)
p1 = 24.2145 +/- 5.44887
p2 = 4.93415 +/- 2.65778
****************************************
Minimizer is Minuit2
Chi2 = 2.35692
NDf = 4
Edm = 1.08773e-07
NCalls = 81
p0 = 0.373315 +/- 0.215845 (limited)
p1 = 12.4572 +/- 7.57085
p2 = 4.00427 +/- 8.97995
****************************************
Minimizer is Minuit2
Chi2 = 1.93137
NDf = 7
Edm = 4.79943e-09
NCalls = 113
p0 = 0.632872 +/- 0.231351 (limited)
p1 = 27.1811 +/- 8.5188
p2 = 6.83092 +/- 3.30519
****************************************
Minimizer is Minuit2
Chi2 = 3.28016
NDf = 6
Edm = 6.04893e-06
NCalls = 99
p0 = 0.696957 +/- 0.193092 (limited)
p1 = 22.9427 +/- 3.44124
p2 = 3.63739 +/- 2.36972
****************************************
Minimizer is Minuit2
Chi2 = 2.56902
NDf = 7
Edm = 5.20615e-07
NCalls = 94
p0 = 0.68832 +/- 0.19603 (limited)
p1 = 24.2988 +/- 4.41739
p2 = 4.93562 +/- 3.1133
****************************************
Minimizer is Minuit2
Chi2 = 3.04459
NDf = 7
Edm = 7.4353e-07
NCalls = 83
p0 = 0.659997 +/- 0.249426 (limited)
p1 = 22.9747 +/- 5.95617
p2 = 5.36022 +/- 2.66473
****************************************
Minimizer is Minuit2
Chi2 = 6.27397
NDf = 8
Edm = 5.29961e-07
NCalls = 81
p0 = 0.662433 +/- 0.15714 (limited)
p1 = 21.5909 +/- 3.29623
p2 = 4.09179 +/- 1.74745
****************************************
Minimizer is Minuit2
Chi2 = 1.94311
NDf = 9
Edm = 1.89133e-07
NCalls = 106
p0 = 0.575157 +/- 0.298322 (limited)
p1 = 25.6868 +/- 11.7079
p2 = 7.54114 +/- 5.74853
****************************************
Minimizer is Minuit2
Chi2 = 2.75509
NDf = 9
Edm = 9.98946e-08
NCalls = 69
p0 = 0.593401 +/- 0.101469 (limited)
p1 = 18.8032 +/- 2.57519
p2 = 2.53183 +/- 1.7654
****************************************
Minimizer is Minuit2
Chi2 = 2.54952
NDf = 6
Edm = 4.57669e-08
NCalls = 96
p0 = 0.660242 +/- 0.259374 (limited)
p1 = 25.0771 +/- 5.98352
p2 = 5.39543 +/- 2.98249
****************************************
Minimizer is Minuit2
Chi2 = 4.62283
NDf = 10
Edm = 1.36401e-08
NCalls = 89
p0 = 0.592713 +/- 0.183874 (limited)
p1 = 20.9349 +/- 5.53213
p2 = 6.39618 +/- 4.21336
****************************************
Minimizer is Minuit2
Chi2 = 5.3127
NDf = 6
Edm = 2.53153e-06
NCalls = 119
p0 = 0.534666 +/- 0.0847463 (limited)
p1 = 12.0856 +/- 13.9766
p2 = 0.685012 +/- 8.82276
****************************************
Minimizer is Minuit2
Chi2 = 0.968364
NDf = 7
Edm = 1.15106e-06
NCalls = 74
p0 = 0.669971 +/- 0.134042 (limited)
p1 = 19.9248 +/- 3.66311
p2 = 4.7186 +/- 3.31403
****************************************
Minimizer is Minuit2
Chi2 = 2.88345
NDf = 10
Edm = 2.24225e-06
NCalls = 102
p0 = 0.629842 +/- 0.174064 (limited)
p1 = 21.9766 +/- 6.0423
p2 = 6.83408 +/- 3.9968
****************************************
Minimizer is Minuit2
Chi2 = 1.80215
NDf = 6
Edm = 2.68316e-07
NCalls = 87
p0 = 0.495591 +/- 0.211145 (limited)
p1 = 19.645 +/- 5.2052
p2 = 3.59211 +/- 2.51889
****************************************
Minimizer is Minuit2
Chi2 = 3.79465
NDf = 6
Edm = 4.35871e-08
NCalls = 69
p0 = 0.611171 +/- 0.128277 (limited)
p1 = 17.467 +/- 2.93532
p2 = 3.02848 +/- 1.64432
****************************************
Minimizer is Minuit2
Chi2 = 2.748
NDf = 7
Edm = 5.78153e-09
NCalls = 140
p0 = 1 +/- 0.733966 (limited)
p1 = 28.7334 +/- 2.38732
p2 = 6.33261 +/- 1.85434
****************************************
Minimizer is Minuit2
Chi2 = 4.29305
NDf = 9
Edm = 3.06585e-06
NCalls = 82
p0 = 0.687721 +/- 0.16847 (limited)
p1 = 22.2779 +/- 3.60769
p2 = 4.13806 +/- 1.75728
****************************************
Minimizer is Minuit2
Chi2 = 1.74986
NDf = 2
Edm = 2.69957e-06
NCalls = 256
p0 = 0.999951 +/- 0.988093 (limited)
p1 = 33.6961 +/- 9.16268
p2 = 13.7955 +/- 8.24703
****************************************
Minimizer is Minuit2
Chi2 = 4.72905
NDf = 5
Edm = 2.89445e-07
NCalls = 67
p0 = 0.417412 +/- 0.107227 (limited)
p1 = 14.5796 +/- 2.95636
p2 = 2.22066 +/- 2.12966
****************************************
Minimizer is Minuit2
Chi2 = 2.45643
NDf = 6
Edm = 2.63752e-06
NCalls = 99
p0 = 0.554001 +/- 0.226655 (limited)
p1 = 23.9121 +/- 6.69531
p2 = 6.04267 +/- 3.44117
****************************************
Minimizer is Minuit2
Chi2 = 2.91637
NDf = 4
Edm = 1.377e-05
NCalls = 84
p0 = 0.769946 +/- 0.185015 (limited)
p1 = 20.5779 +/- 2.52006
p2 = 3.50084 +/- 1.35396
****************************************
Minimizer is Minuit2
Chi2 = 1.21412
NDf = 6
Edm = 1.16053e-08
NCalls = 141
p0 = 0.632519 +/- 0.106789 (limited)
p1 = 17.4834 +/- 1.66297
p2 = 1.44082 +/- 1.10945
#include "TBinomialEfficiencyFitter.h"
#include "TVirtualFitter.h"
#include "TH1.h"
#include "TRandom3.h"
#include "TF1.h"
#include "TFitResult.h"
#include "TStyle.h"
#include "TCanvas.h"
#include "TLegend.h"
#include "TPaveStats.h"
#include "Math/IntegratorOptions.h"
#include <cassert>
#include <iostream>
void TestBinomial(int nloop = 100, int nevts = 100, bool plot = false, bool debug = false, int seed = 111)
{
gStyle->SetMarkerStyle(20);
gStyle->SetLineWidth(2.0);
gStyle->SetOptStat(11);
TObjArray hbiasNorm;
hbiasNorm.Add(new TH1D("h0Norm", "Bias Histogram fit",100,-5,5));
hbiasNorm.Add(new TH1D("h1Norm","Bias Binomial fit",100,-5,5));
TObjArray hbiasThreshold;
hbiasThreshold.Add(new TH1D("h0Threshold", "Bias Histogram fit",100,-5,5));
hbiasThreshold.Add(new TH1D("h1Threshold","Bias Binomial fit",100,-5,5));
TObjArray hbiasWidth;
hbiasWidth.Add(new TH1D("h0Width", "Bias Histogram fit",100,-5,5));
hbiasWidth.Add(new TH1D("h1Width","Bias Binomial fit",100,-5,5));
TH1D* hChisquared = new TH1D("hChisquared",
"#chi^{2} probability (Baker-Cousins)", 200, 0.0, 1.0);
TVirtualFitter::SetDefaultFitter("Minuit2");
ROOT::Math::IntegratorOneDimOptions::SetDefaultIntegrator("Gauss");
// Note: in order to be able to use TH1::FillRandom() to generate
// pseudo-experiments, we use a trick: generate "selected"
// and "non-selected" samples independently. These are
// statistically independent and therefore can be safely
// added to yield the "before selection" sample.
// Define (arbitrarily?) a distribution of input events.
// Here: assume a x^(-2) distribution. Boundaries: [10, 100].
Double_t xmin =10, xmax = 100;
TH1D* hM2D = new TH1D("hM2D", "x^(-2) denominator distribution",
45, xmin, xmax);
TH1D* hM2N = new TH1D("hM2N", "x^(-2) numerator distribution",
45, xmin, xmax);
TH1D* hM2E = new TH1D("hM2E", "x^(-2) efficiency",
45, xmin, xmax);
TF1* fM2D = new TF1("fM2D", "(1-[0]/(1+exp(([1]-x)/[2])))/(x*x)",
xmin, xmax);
TF1* fM2N = new TF1("fM2N", "[0]/(1+exp(([1]-x)/[2]))/(x*x)",
xmin, xmax);
TF1* fM2Fit = new TF1("fM2Fit", "[0]/(1+exp(([1]-x)/[2]))",
xmin, xmax);
TF1* fM2Fit2 = 0;
TRandom3 rb(seed);
// First try: use a single set of parameters.
// For each try, we need to find the overall normalization
Double_t normalization = 0.80;
Double_t threshold = 25.0;
Double_t width = 5.0;
fM2D->SetParameter(0, normalization);
fM2D->SetParameter(1, threshold);
fM2D->SetParameter(2, width);
fM2N->SetParameter(0, normalization);
fM2N->SetParameter(1, threshold);
fM2N->SetParameter(2, width);
Double_t integralN = fM2N->Integral(xmin, xmax);
Double_t integralD = fM2D->Integral(xmin, xmax);
Double_t fracN = integralN/(integralN+integralD);
Int_t nevtsN = rb.Binomial(nevts, fracN);
Int_t nevtsD = nevts - nevtsN;
std::cout << nevtsN << " " << nevtsD << std::endl;
gStyle->SetOptFit(1111);
// generate many times to see the bias
for (int iloop = 0; iloop < nloop; ++iloop) {
// generate pseudo-experiments
hM2D->Reset();
hM2N->Reset();
hM2D->FillRandom(fM2D->GetName(), nevtsD);
hM2N->FillRandom(fM2N->GetName(), nevtsN);
hM2D->Add(hM2N);
// construct the "efficiency" histogram
hM2N->Sumw2();
hM2E->Divide(hM2N, hM2D, 1, 1, "b");
// Fit twice, using the same fit function.
// In the first (standard) fit, initialize to (arbitrary) values.
// In the second fit, use the results from the first fit (this
// makes it easier for the fit -- but the purpose here is not to
// show how easy or difficult it is to obtain results, but whether
// the CORRECT results are obtained or not!).
fM2Fit->SetParameter(0, 0.5);
fM2Fit->SetParameter(1, 15.0);
fM2Fit->SetParameter(2, 2.0);
fM2Fit->SetParError(0, 0.1);
fM2Fit->SetParError(1, 1.0);
fM2Fit->SetParError(2, 0.2);
TH1 * hf = fM2Fit->GetHistogram();
// std::cout << "Function values " << std::endl;
// for (int i = 1; i <= hf->GetNbinsX(); ++i)
// std::cout << hf->GetBinContent(i) << " ";
// std::cout << std::endl;
TCanvas* cEvt;
if (plot) {
cEvt = new TCanvas(Form("cEnv%d",iloop),
Form("plots for experiment %d", iloop),
1000, 600);
cEvt->Divide(1,2);
cEvt->cd(1);
hM2D->DrawCopy("HIST");
hM2N->SetLineColor(kRed);
hM2N->DrawCopy("HIST SAME");
cEvt->cd(2);
}
for (int fit = 0; fit < 2; ++fit) {
Int_t status = 0;
switch (fit) {
case 0:
{
// TVirtualPad * pad = gPad;
// new TCanvas();
// fM2Fit->Draw();
// gPad = pad;
TString optFit = "RN";
if (debug) optFit += TString("SV");
TFitResultPtr res = hM2E->Fit(fM2Fit, optFit);
if (plot) {
hM2E->DrawCopy("E");
fM2Fit->DrawCopy("SAME");
}
if (debug) res->Print();
status = res;
break;
}
case 1:
{
// if (fM2Fit2) delete fM2Fit2;
// fM2Fit2 = dynamic_cast<TF1*>(fM2Fit->Clone("fM2Fit2"));
fM2Fit2 = fM2Fit; // do not clone/copy the function
if (fM2Fit2->GetParameter(0) >= 1.0)
fM2Fit2->SetParameter(0, 0.95);
fM2Fit2->SetParLimits(0, 0.0, 1.0);
// TVirtualPad * pad = gPad;
// new TCanvas();
// fM2Fit2->Draw();
// gPad = pad;
TBinomialEfficiencyFitter bef(hM2N, hM2D);
TString optFit = "RI";
if (debug) optFit += TString("SV");
TFitResultPtr res = bef.Fit(fM2Fit2,optFit);
status = res;
if (status !=0) {
std::cerr << "Error performing binomial efficiency fit, result = "
<< status << std::endl;
res->Print();
continue;
}
if (plot) {
fM2Fit2->SetLineColor(kRed);
fM2Fit2->DrawCopy("SAME");
}
if (debug) {
res->Print();
}
}
}
if (status != 0) break;
Double_t fnorm = fM2Fit->GetParameter(0);
Double_t enorm = fM2Fit->GetParError(0);
Double_t fthreshold = fM2Fit->GetParameter(1);
Double_t ethreshold = fM2Fit->GetParError(1);
Double_t fwidth = fM2Fit->GetParameter(2);
Double_t ewidth = fM2Fit->GetParError(2);
if (fit == 1) {
fnorm = fM2Fit2->GetParameter(0);
enorm = fM2Fit2->GetParError(0);
fthreshold = fM2Fit2->GetParameter(1);
ethreshold = fM2Fit2->GetParError(1);
fwidth = fM2Fit2->GetParameter(2);
ewidth = fM2Fit2->GetParError(2);
hChisquared->Fill(fM2Fit2->GetProb());
}
TH1D* h = dynamic_cast<TH1D*>(hbiasNorm[fit]);
h->Fill((fnorm-normalization)/enorm);
h = dynamic_cast<TH1D*>(hbiasThreshold[fit]);
h->Fill((fthreshold-threshold)/ethreshold);
h = dynamic_cast<TH1D*>(hbiasWidth[fit]);
h->Fill((fwidth-width)/ewidth);
}
}
TCanvas* c1 = new TCanvas("c1",
"Efficiency fit biases",10,10,1000,800);
c1->Divide(2,2);
TH1D *h0, *h1;
c1->cd(1);
h0 = dynamic_cast<TH1D*>(hbiasNorm[0]);
h0->Draw("HIST");
h1 = dynamic_cast<TH1D*>(hbiasNorm[1]);
h1->SetLineColor(kRed);
h1->Draw("HIST SAMES");
TLegend* l1 = new TLegend(0.1, 0.75, 0.5, 0.9,
"plateau parameter", "ndc");
l1->AddEntry(h0, Form("histogram: mean = %4.2f RMS = \
%4.2f", h0->GetMean(), h0->GetRMS()), "l");
l1->AddEntry(h1, Form("binomial : mean = %4.2f RMS = \
%4.2f", h1->GetMean(), h1->GetRMS()), "l");
l1->Draw();
c1->cd(2);
h0 = dynamic_cast<TH1D*>(hbiasThreshold[0]);
h0->Draw("HIST");
h1 = dynamic_cast<TH1D*>(hbiasThreshold[1]);
h1->SetLineColor(kRed);
h1->Draw("HIST SAMES");
TLegend* l2 = new TLegend(0.1, 0.75, 0.5, 0.9,
"threshold parameter", "ndc");
l2->AddEntry(h0, Form("histogram: mean = %4.2f RMS = \
%4.2f", h0->GetMean(), h0->GetRMS()), "l");
l2->AddEntry(h1, Form("binomial : mean = %4.2f RMS = \
%4.2f", h1->GetMean(), h1->GetRMS()), "l");
l2->Draw();
c1->cd(3);
h0 = dynamic_cast<TH1D*>(hbiasWidth[0]);
h0->Draw("HIST");
h1 = dynamic_cast<TH1D*>(hbiasWidth[1]);
h1->SetLineColor(kRed);
h1->Draw("HIST SAMES");
TLegend* l3 = new TLegend(0.1, 0.75, 0.5, 0.9, "width parameter", "ndc");
l3->AddEntry(h0, Form("histogram: mean = %4.2f RMS = \
%4.2f", h0->GetMean(), h0->GetRMS()), "l");
l3->AddEntry(h1, Form("binomial : mean = %4.2f RMS = \
%4.2f", h1->GetMean(), h1->GetRMS()), "l");
l3->Draw();
c1->cd(4);
hChisquared->Draw("HIST");
}
int main() {
TestBinomial();
}
Author
Rene Brun

Definition in file TestBinomial.C.