Logo ROOT   6.08/07
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-v608/rootspi/rdoc/src/v6-08-00-patches/tutorials/fit/TestBinomial.C...
32 68
FCN=0.350713 FROM MIGRAD STATUS=CONVERGED 90 CALLS 91 TOTAL
EDM=1.93032e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.94233e-01 2.10274e-01 7.09547e-05 1.39640e-03
2 p1 1.93528e+01 5.86747e+00 1.52181e-03 -2.57315e-05
3 p2 5.22913e+00 5.11982e+00 1.74728e-03 1.21887e-06
FCN=1.64737 FROM MIGRAD STATUS=CONVERGED 144 CALLS 145 TOTAL
EDM=8.60104e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.77354e-01 1.35581e-01 1.72178e-04 -5.56953e-05
2 p1 1.55035e+01 4.11280e+00 1.94589e-03 -7.61122e-05
3 p2 4.10293e+00 2.76953e+00 1.51435e-03 -4.12560e-05
FCN=0.928456 FROM MIGRAD STATUS=CONVERGED 74 CALLS 75 TOTAL
EDM=2.93834e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.78806e-01 1.28412e-01 1.43805e-04 5.17833e-04
2 p1 1.98914e+01 2.83678e+00 1.23200e-03 -1.31061e-04
3 p2 3.39323e+00 1.96156e+00 1.00309e-03 8.26552e-05
FCN=2.56551 FROM MIGRAD STATUS=CONVERGED 108 CALLS 109 TOTAL
EDM=2.39112e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.72610e-01 1.65520e-01 2.20305e-04 -4.13254e-04
2 p1 2.14104e+01 5.10611e+00 2.26643e-03 6.02706e-05
3 p2 5.58786e+00 3.30889e+00 1.83116e-03 -1.08175e-05
FCN=7.37094 FROM MIGRAD STATUS=CONVERGED 94 CALLS 95 TOTAL
EDM=6.08449e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.67747e-01 1.52883e-01 3.17294e-04 -1.43975e-03
2 p1 2.40154e+01 2.40749e+00 2.18330e-03 8.89241e-05
3 p2 3.63985e+00 1.74757e+00 2.22075e-03 5.13893e-05
FCN=0.747774 FROM MIGRAD STATUS=CONVERGED 83 CALLS 84 TOTAL
EDM=8.40115e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.51650e-01 1.55205e-01 1.63398e-04 3.33967e-04
2 p1 1.89830e+01 2.68235e+00 9.11426e-04 1.72003e-04
3 p2 2.68824e+00 1.28707e+00 5.19033e-04 -4.57823e-04
FCN=0.433039 FROM MIGRAD STATUS=CONVERGED 103 CALLS 104 TOTAL
EDM=6.15804e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.00967e-01 2.54652e-01 1.44005e-04 7.30929e-04
2 p1 1.94199e+01 9.30614e+00 2.29705e-03 -1.15083e-04
3 p2 6.46947e+00 8.16624e+00 3.34841e-03 -5.63798e-05
FCN=4.49196 FROM MIGRAD STATUS=CONVERGED 120 CALLS 121 TOTAL
EDM=2.9653e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.23223e-01 1.07174e-01 2.29200e-04 -3.40333e-03
2 p1 2.08115e+01 1.04914e+00 9.77275e-04 5.82211e-04
3 p2 1.21980e+00 1.08657e+00 1.10736e-03 -3.63081e-04
FCN=2.54966 FROM MIGRAD STATUS=CONVERGED 85 CALLS 86 TOTAL
EDM=1.35928e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 3.82577e-01 2.26238e-01 1.56070e-04 -3.26540e-03
2 p1 1.80514e+01 6.37241e+00 2.05657e-03 -4.46312e-05
3 p2 3.04455e+00 5.57415e+00 2.33742e-03 4.10621e-05
FCN=1.46531 FROM MIGRAD STATUS=CONVERGED 96 CALLS 97 TOTAL
EDM=4.62964e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.75661e-01 1.88224e-01 1.63007e-04 1.02937e-03
2 p1 1.76061e+01 7.72165e+00 1.27377e-03 -1.16992e-04
3 p2 2.98582e+00 6.21368e+00 1.17326e-03 6.03603e-05
FCN=3.31727 FROM MIGRAD STATUS=CONVERGED 71 CALLS 72 TOTAL
EDM=9.09202e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.98638e-01 1.24930e-01 2.13241e-04 -3.15576e-04
2 p1 1.79069e+01 3.49177e+00 2.36705e-03 -6.26345e-06
3 p2 3.52359e+00 2.10055e+00 1.63762e-03 6.94298e-05
FCN=0.413866 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=1.06923e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.54352e-01 1.98145e-01 1.39722e-04 2.03724e-04
2 p1 2.06178e+01 3.66715e+00 9.69398e-04 -9.78269e-06
3 p2 4.03019e+00 2.20627e+00 8.69236e-04 5.14948e-05
FCN=1.21002 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=5.8113e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.51685e-01 1.59188e-01 1.86984e-04 -1.05410e-03
2 p1 1.93369e+01 2.66645e+00 1.45263e-03 1.42183e-04
3 p2 2.81842e+00 2.20413e+00 1.45143e-03 -6.70333e-05
FCN=0.752974 FROM MIGRAD STATUS=CONVERGED 108 CALLS 109 TOTAL
EDM=1.86726e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.43901e-01 1.70123e-01 1.58143e-04 1.75064e-03
2 p1 2.10059e+01 4.31687e+00 1.59552e-03 -2.56571e-05
3 p2 5.36356e+00 3.11878e+00 1.58356e-03 4.38822e-05
FCN=2.5161 FROM MIGRAD STATUS=CONVERGED 60 CALLS 61 TOTAL
EDM=1.33705e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.28793e-01 1.03749e-01 1.79061e-04 1.84828e-04
2 p1 1.74724e+01 2.12101e+00 1.55607e-03 1.02894e-05
3 p2 2.59351e+00 1.42575e+00 1.13739e-03 -3.36572e-06
FCN=1.27042 FROM MIGRAD STATUS=CONVERGED 79 CALLS 80 TOTAL
EDM=1.81074e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.61022e-01 1.38093e-01 1.47286e-04 -5.93345e-04
2 p1 1.81593e+01 3.22444e+00 1.21624e-03 -2.35187e-05
3 p2 3.50041e+00 1.84157e+00 8.67375e-04 4.80780e-05
FCN=2.84971 FROM MIGRAD STATUS=CONVERGED 97 CALLS 98 TOTAL
EDM=1.33716e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.93347e-01 2.85253e-01 2.67502e-04 1.03238e-04
2 p1 2.47780e+01 7.96155e+00 2.43832e-03 1.71428e-06
3 p2 5.95318e+00 3.88495e+00 1.86332e-03 -1.44762e-05
FCN=3.67016 FROM MIGRAD STATUS=CONVERGED 90 CALLS 91 TOTAL
EDM=1.26449e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.02769e-01 1.76135e-01 2.40173e-04 1.10919e-04
2 p1 2.01573e+01 2.79974e+00 1.77805e-03 3.18873e-06
3 p2 2.70345e+00 1.65987e+00 1.54849e-03 4.07855e-06
FCN=0.727165 FROM MIGRAD STATUS=CONVERGED 214 CALLS 215 TOTAL
EDM=7.90505e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.17006e-01 2.25995e-01 2.02138e-04 -1.12767e-03
2 p1 2.08857e+01 7.73194e+00 2.31202e-03 3.16606e-05
3 p2 4.00540e+00 3.80319e+00 1.39800e-03 3.40276e-04
FCN=3.77202 FROM MIGRAD STATUS=CONVERGED 91 CALLS 92 TOTAL
EDM=2.44971e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.41717e-01 1.95414e-01 2.50144e-04 2.14660e-03
2 p1 2.29045e+01 4.96086e+00 2.31890e-03 -3.17749e-04
3 p2 5.04653e+00 2.69026e+00 1.81972e-03 2.99805e-04
FCN=0.474059 FROM MIGRAD STATUS=CONVERGED 95 CALLS 96 TOTAL
EDM=1.11691e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.35578e-01 1.73676e-01 1.31386e-04 -9.71658e-05
2 p1 2.10442e+01 3.89886e+00 1.36952e-03 1.84375e-06
3 p2 3.91418e+00 2.85186e+00 1.29482e-03 2.00106e-05
FCN=2.39556 FROM MIGRAD STATUS=CONVERGED 80 CALLS 81 TOTAL
EDM=9.0339e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.64008e-01 1.88334e-01 1.72514e-04 -7.15441e-05
2 p1 1.92305e+01 5.64705e+00 1.79999e-03 -1.42080e-04
3 p2 4.00431e+00 3.22940e+00 1.34498e-03 1.70013e-04
FCN=2.71864 FROM MIGRAD STATUS=CONVERGED 83 CALLS 84 TOTAL
EDM=4.05302e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.30955e-01 1.03386e-01 1.80051e-04 -1.44117e-03
2 p1 1.53556e+01 1.58614e+00 8.75098e-04 8.49763e-04
3 p2 1.18646e+00 1.05279e+00 6.13845e-04 -1.35262e-03
FCN=3.91496 FROM MIGRAD STATUS=CONVERGED 179 CALLS 180 TOTAL
EDM=2.10519e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.00000e+00 9.61888e-01 3.88943e-03** at limit **
2 p1 2.63612e+01 4.15188e+00 4.42479e-03 4.14319e-05
3 p2 1.01889e+01 7.93361e+00 8.47921e-03 -4.55241e-05
FCN=3.69474 FROM MIGRAD STATUS=CONVERGED 123 CALLS 124 TOTAL
EDM=2.0572e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.66565e-01 3.00627e-01 3.14561e-04 1.86665e-03
2 p1 2.29958e+01 7.83199e+00 3.22937e-03 -2.00067e-04
3 p2 6.59909e+00 5.25595e+00 3.74414e-03 1.64767e-04
FCN=1.01429 FROM MIGRAD STATUS=CONVERGED 101 CALLS 102 TOTAL
EDM=1.2418e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.10841e-01 1.75938e-01 1.51097e-04 5.50849e-05
2 p1 2.33042e+01 5.77849e+00 1.59367e-03 -6.58900e-06
3 p2 5.09786e+00 2.59975e+00 9.36056e-04 -1.06492e-05
FCN=3.63097 FROM MIGRAD STATUS=CONVERGED 122 CALLS 123 TOTAL
EDM=5.55534e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.43028e-01 2.25348e-01 2.29466e-04 6.98755e-04
2 p1 2.14306e+01 4.46994e+00 1.99198e-03 -2.23271e-04
3 p2 3.95327e+00 2.04398e+00 1.43448e-03 -2.09249e-04
FCN=1.7669 FROM MIGRAD STATUS=CONVERGED 97 CALLS 98 TOTAL
EDM=3.51383e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.38210e-01 2.20641e-01 2.36266e-04 3.73388e-04
2 p1 2.36710e+01 5.18271e+00 1.67009e-03 -2.60124e-04
3 p2 5.53649e+00 2.72352e+00 1.26553e-03 1.83613e-04
FCN=2.70826 FROM MIGRAD STATUS=CONVERGED 105 CALLS 106 TOTAL
EDM=1.00119e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.62591e-01 3.52545e-01 2.28812e-04 -5.09675e-03
2 p1 2.42783e+01 1.23384e+01 3.15122e-03 4.00190e-04
3 p2 7.60846e+00 6.66009e+00 3.14641e-03 -1.58839e-04
FCN=9.74855 FROM MIGRAD STATUS=CONVERGED 123 CALLS 124 TOTAL
EDM=6.92929e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.06488e-01 2.31103e-01 4.24420e-04 2.61952e-03
2 p1 2.29147e+01 5.11615e+00 4.17697e-03 -4.23372e-04
3 p2 4.88595e+00 2.18611e+00 3.03805e-03 3.46079e-04
FCN=3.10119 FROM MIGRAD STATUS=CONVERGED 233 CALLS 234 TOTAL
EDM=7.43446e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 9.99998e-01 5.01156e-01 5.03534e-03 -1.58935e-04
2 p1 2.85973e+01 4.11085e+00 2.40610e-03 1.50847e-05
3 p2 8.15931e+00 3.01069e+00 1.76059e-03 2.23514e-04
FCN=2.78024 FROM MIGRAD STATUS=CONVERGED 89 CALLS 90 TOTAL
EDM=9.34305e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.53280e-01 2.06556e-01 2.44343e-04 -1.46860e-03
2 p1 2.30619e+01 4.95707e+00 2.03795e-03 7.70668e-05
3 p2 4.86639e+00 2.48128e+00 1.44516e-03 8.49395e-05
FCN=3.05434 FROM MIGRAD STATUS=CONVERGED 322 CALLS 323 TOTAL
EDM=5.74795e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.00000e+00 6.04950e-01 9.67460e-03 -1.75816e-05
2 p1 4.73259e+01 3.43724e+01 1.39149e-02 -2.62138e-06
3 p2 4.35898e+01 6.16404e+01 2.49642e-02 5.69765e-06
FCN=0.139803 FROM MIGRAD STATUS=CONVERGED 98 CALLS 99 TOTAL
EDM=3.62058e-10 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.60408e-01 2.58771e-01 1.57887e-04 4.96879e-05
2 p1 1.92633e+01 7.69261e+00 1.63815e-03 -8.14507e-06
3 p2 6.44751e+00 4.78166e+00 1.57121e-03 7.26859e-06
FCN=2.5532 FROM MIGRAD STATUS=CONVERGED 96 CALLS 97 TOTAL
EDM=2.6778e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.43241e-01 1.94783e-01 2.19581e-04 -3.17102e-04
2 p1 2.19863e+01 5.23880e+00 2.48598e-03 1.06595e-05
3 p2 5.90008e+00 2.90169e+00 2.11092e-03 8.43728e-05
FCN=2.36127 FROM MIGRAD STATUS=CONVERGED 94 CALLS 95 TOTAL
EDM=2.57586e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.60240e-01 1.31211e-01 1.99840e-04 2.57555e-04
2 p1 1.98535e+01 3.03560e+00 1.29041e-03 -4.27699e-05
3 p2 2.47677e+00 2.23094e+00 1.04724e-03 3.13897e-05
FCN=1.33422 FROM MIGRAD STATUS=CONVERGED 108 CALLS 109 TOTAL
EDM=1.7244e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.64357e-01 2.03239e-01 1.84510e-04 1.11043e-03
2 p1 2.58365e+01 9.79397e+00 3.02148e-03 3.54323e-05
3 p2 8.35188e+00 6.07588e+00 2.62092e-03 -1.03027e-04
FCN=1.74047 FROM MIGRAD STATUS=CONVERGED 134 CALLS 135 TOTAL
EDM=1.40782e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.94764e-01 1.87954e-01 2.00764e-04 -5.42554e-03
2 p1 2.53877e+01 8.57244e+00 2.92043e-03 1.76641e-04
3 p2 6.52667e+00 4.25157e+00 1.82610e-03 2.14805e-04
FCN=0.310646 FROM MIGRAD STATUS=CONVERGED 176 CALLS 177 TOTAL
EDM=1.37875e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.39114e-01 7.59737e-01 1.23060e-04 -3.77729e-03
2 p1 1.28636e+01 7.28133e+01 6.66999e-03 9.31135e-05
3 p2 2.28520e+01 1.01313e+02 1.53513e-02 2.02768e-05
FCN=2.52738 FROM MIGRAD STATUS=CONVERGED 118 CALLS 119 TOTAL
EDM=3.03001e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.31088e-01 2.19920e-01 2.17581e-04 2.73287e-04
2 p1 2.65773e+01 9.03434e+00 3.05477e-03 2.63236e-05
3 p2 8.98260e+00 4.83446e+00 2.54402e-03 -6.58106e-05
FCN=2.51398 FROM MIGRAD STATUS=CONVERGED 83 CALLS 84 TOTAL
EDM=1.38894e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.84246e-01 2.26988e-01 1.78870e-04 2.06407e-03
2 p1 1.82432e+01 5.68591e+00 1.65778e-03 -6.81112e-04
3 p2 3.93586e+00 3.23302e+00 1.50252e-03 1.00572e-03
FCN=0.946462 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=4.48882e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.84726e-01 1.33982e-01 1.48450e-04 -2.16590e-04
2 p1 1.93760e+01 3.81670e+00 1.65819e-03 -3.00781e-04
3 p2 4.21030e+00 2.54214e+00 1.30996e-03 1.91413e-04
FCN=5.46724 FROM MIGRAD STATUS=CONVERGED 112 CALLS 113 TOTAL
EDM=1.71347e-11 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.63804e-01 2.33481e-01 3.87873e-04 1.26706e-05
2 p1 2.63444e+01 5.04025e+00 2.58512e-03 -4.17063e-07
3 p2 6.08162e+00 2.87962e+00 2.29239e-03 -1.23472e-06
FCN=0.716281 FROM MIGRAD STATUS=CONVERGED 92 CALLS 93 TOTAL
EDM=8.18825e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.22972e-01 1.79872e-01 1.17703e-04 -1.38499e-04
2 p1 1.73919e+01 6.44858e+00 1.44903e-03 4.01991e-05
3 p2 3.06066e+00 4.50074e+00 1.23662e-03 -6.14298e-05
FCN=6.40375 FROM MIGRAD STATUS=CONVERGED 103 CALLS 104 TOTAL
EDM=1.12464e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.07615e-01 1.98193e-01 2.96015e-04 -7.00915e-04
2 p1 2.22924e+01 5.65071e+00 3.07661e-03 3.29928e-05
3 p2 4.81170e+00 2.67857e+00 2.15762e-03 5.59284e-04
FCN=3.18171 FROM MIGRAD STATUS=CONVERGED 82 CALLS 83 TOTAL
EDM=1.11328e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.94498e-01 1.34559e-01 2.26273e-04 -5.61242e-03
2 p1 2.15399e+01 2.86559e+00 1.85951e-03 7.21980e-04
3 p2 3.46906e+00 1.42287e+00 1.10674e-03 -3.01146e-04
FCN=3.13917 FROM MIGRAD STATUS=CONVERGED 108 CALLS 109 TOTAL
EDM=4.19132e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.25215e-01 1.45431e-01 2.40324e-04 5.31723e-04
2 p1 2.52445e+01 3.82218e+00 2.70237e-03 -4.77409e-05
3 p2 4.88457e+00 1.80648e+00 1.60801e-03 1.45474e-04
FCN=3.15774 FROM MIGRAD STATUS=CONVERGED 116 CALLS 117 TOTAL
EDM=1.21668e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.58136e-01 7.71130e-01 3.12677e-04 4.38943e-04
2 p1 2.60196e+01 1.32355e+01 2.65914e-03 -5.02490e-05
3 p2 7.01780e+00 5.27024e+00 1.81276e-03 2.23358e-05
FCN=1.90007 FROM MIGRAD STATUS=CONVERGED 106 CALLS 107 TOTAL
EDM=1.04177e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.34562e-01 2.70160e-01 2.51861e-04 -3.37296e-05
2 p1 2.40698e+01 7.86038e+00 2.35647e-03 5.82395e-06
3 p2 5.63409e+00 3.23398e+00 1.42610e-03 -2.16112e-05
FCN=2.77855 FROM MIGRAD STATUS=CONVERGED 84 CALLS 85 TOTAL
EDM=1.62722e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.02887e-01 1.99272e-01 2.25139e-04 6.52094e-06
2 p1 2.09159e+01 3.68349e+00 1.68488e-03 -1.04401e-04
3 p2 4.41504e+00 2.04434e+00 1.45683e-03 -1.18092e-04
FCN=4.37217 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=5.44166e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.20959e-01 1.58430e-01 2.46245e-04 -3.63775e-03
2 p1 2.10013e+01 3.72087e+00 2.11849e-03 1.80435e-04
3 p2 4.56895e+00 2.37229e+00 1.81610e-03 -2.43167e-04
FCN=1.04047 FROM MIGRAD STATUS=CONVERGED 109 CALLS 110 TOTAL
EDM=3.10631e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.93802e-01 1.74397e-01 1.84493e-04 -4.31211e-04
2 p1 2.10650e+01 4.04569e+00 2.01803e-03 -3.32411e-05
3 p2 3.35322e+00 2.26059e+00 1.39612e-03 1.95172e-05
FCN=3.95319 FROM MIGRAD STATUS=CONVERGED 97 CALLS 98 TOTAL
EDM=2.65638e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.74996e-01 1.52016e-01 2.06449e-04 3.80869e-03
2 p1 2.02114e+01 5.18930e+00 2.24185e-03 -2.36065e-04
3 p2 4.26461e+00 2.57398e+00 1.38481e-03 1.38401e-04
FCN=1.21209 FROM MIGRAD STATUS=CONVERGED 98 CALLS 99 TOTAL
EDM=4.82239e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.98486e-01 1.95023e-01 1.29390e-04 1.96944e-04
2 p1 1.63645e+01 1.08118e+01 3.16867e-03 4.27714e-05
3 p2 4.06853e+00 1.48163e+01 2.94776e-03 2.73098e-05
FCN=1.15842 FROM MIGRAD STATUS=CONVERGED 111 CALLS 112 TOTAL
EDM=1.48367e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.69872e-01 2.36523e-01 1.79573e-04 5.75380e-04
2 p1 1.72315e+01 1.57721e+01 6.53577e-03 -1.81013e-05
3 p2 9.74073e+00 1.63762e+01 6.65925e-03 -1.66924e-05
FCN=4.96865 FROM MIGRAD STATUS=CONVERGED 79 CALLS 80 TOTAL
EDM=3.77418e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.66513e-01 1.40936e-01 2.80063e-04 -3.15888e-04
2 p1 2.12569e+01 2.26557e+00 1.96323e-03 -2.21619e-05
3 p2 3.48413e+00 2.03653e+00 2.23961e-03 1.36431e-04
FCN=2.88078 FROM MIGRAD STATUS=CONVERGED 103 CALLS 104 TOTAL
EDM=5.34223e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 8.04166e-01 2.76210e-01 3.22390e-04 6.14828e-05
2 p1 2.21208e+01 5.26480e+00 2.27506e-03 9.62423e-07
3 p2 5.42326e+00 3.06139e+00 2.47150e-03 2.13826e-05
FCN=3.96392 FROM MIGRAD STATUS=CONVERGED 99 CALLS 100 TOTAL
EDM=1.55945e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.94889e-01 2.35894e-01 2.89133e-04 2.80881e-04
2 p1 2.39686e+01 5.40538e+00 2.80292e-03 1.02470e-04
3 p2 5.07476e+00 3.36335e+00 2.65906e-03 -2.12965e-04
FCN=0.852791 FROM MIGRAD STATUS=CONVERGED 83 CALLS 84 TOTAL
EDM=1.77669e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 3.10283e-01 1.64695e-01 1.17775e-04 3.65786e-04
2 p1 1.50842e+01 5.86864e+00 1.27234e-03 2.10603e-05
3 p2 2.43104e+00 3.51103e+00 9.71774e-04 -2.06556e-05
FCN=1.32394 FROM MIGRAD STATUS=CONVERGED 85 CALLS 86 TOTAL
EDM=1.23376e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.72106e-01 2.01669e-01 1.96479e-04 4.35597e-04
2 p1 2.32133e+01 5.10201e+00 1.83079e-03 -1.43377e-04
3 p2 5.04859e+00 2.50327e+00 1.25065e-03 -4.33769e-04
FCN=2.48446 FROM MIGRAD STATUS=CONVERGED 99 CALLS 100 TOTAL
EDM=4.29151e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.69303e-01 1.42681e-01 1.55172e-04 -7.53373e-04
2 p1 1.87854e+01 3.85024e+00 1.47250e-03 -2.45687e-04
3 p2 2.97752e+00 2.30723e+00 1.12137e-03 7.07910e-04
FCN=4.05841 FROM MIGRAD STATUS=CONVERGED 92 CALLS 93 TOTAL
EDM=6.71536e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.67164e-01 1.48637e-01 2.74080e-04 4.60990e-04
2 p1 2.19757e+01 2.55643e+00 1.65670e-03 -3.70915e-05
3 p2 3.34362e+00 2.01500e+00 1.56827e-03 -1.80129e-03
FCN=3.14114 FROM MIGRAD STATUS=CONVERGED 113 CALLS 114 TOTAL
EDM=3.49865e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.32234e-01 1.74899e-01 2.11354e-04 -1.11188e-03
2 p1 2.00178e+01 3.71281e+00 2.18430e-03 4.71661e-05
3 p2 3.05935e+00 2.11129e+00 1.65378e-03 1.02493e-04
FCN=2.88404 FROM MIGRAD STATUS=CONVERGED 84 CALLS 85 TOTAL
EDM=3.75968e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.98819e-01 1.83568e-01 2.72647e-04 -1.88773e-05
2 p1 2.16704e+01 4.18518e+00 2.19682e-03 7.09904e-06
3 p2 4.04339e+00 1.56970e+00 1.04670e-03 -6.53524e-05
FCN=0.624718 FROM MIGRAD STATUS=CONVERGED 78 CALLS 79 TOTAL
EDM=1.36982e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 3.80530e-01 1.34053e-01 1.16531e-04 8.38894e-04
2 p1 1.39337e+01 2.56189e+00 1.04013e-03 -4.07482e-05
3 p2 1.96271e+00 1.64291e+00 8.63756e-04 -5.27194e-05
FCN=1.67529 FROM MIGRAD STATUS=CONVERGED 79 CALLS 80 TOTAL
EDM=9.28635e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.16278e-01 1.57600e-01 1.79784e-04 6.24853e-04
2 p1 1.88120e+01 2.91455e+00 1.31492e-03 1.11021e-04
3 p2 3.21096e+00 1.42859e+00 8.42361e-04 -8.14446e-05
FCN=0.374188 FROM MIGRAD STATUS=CONVERGED 87 CALLS 88 TOTAL
EDM=3.00354e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.85563e-01 1.74411e-01 1.56955e-04 1.58754e-03
2 p1 1.99935e+01 1.62358e+00 7.06760e-04 -4.09027e-04
3 p2 1.70921e+00 1.75420e+00 9.17095e-04 2.64062e-04
FCN=2.46931 FROM MIGRAD STATUS=CONVERGED 92 CALLS 93 TOTAL
EDM=6.65161e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.40812e-01 1.35875e-01 1.99766e-04 1.80810e-03
2 p1 2.23459e+01 3.96589e+00 2.83732e-03 -3.32629e-04
3 p2 4.39765e+00 2.51185e+00 2.19683e-03 -1.94185e-04
FCN=2.4963 FROM MIGRAD STATUS=CONVERGED 109 CALLS 110 TOTAL
EDM=2.24241e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.11899e-01 3.58939e-01 2.85280e-04 -2.85906e-04
2 p1 2.07149e+01 8.44112e+00 2.00820e-03 3.83913e-06
3 p2 5.08169e+00 3.93752e+00 1.63140e-03 6.53492e-05
FCN=0.418945 FROM MIGRAD STATUS=CONVERGED 96 CALLS 97 TOTAL
EDM=9.94923e-22 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.03516e-01 1.07152e-01 0.00000e+00 -2.85714e-10
2 p1 1.57797e+01 1.34564e+00 0.00000e+00 -5.92079e-14
3 p2 1.05537e-01 1.44222e+00 0.00000e+00 -1.43823e-12
FCN=3.54456 FROM MIGRAD STATUS=CONVERGED 91 CALLS 92 TOTAL
EDM=9.81718e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.27390e-01 2.17639e-01 2.12928e-04 2.71218e-04
2 p1 1.79928e+01 6.38787e+00 2.60947e-03 4.54460e-05
3 p2 5.44602e+00 3.84213e+00 2.62500e-03 1.35683e-05
FCN=3.23312 FROM MIGRAD STATUS=CONVERGED 76 CALLS 77 TOTAL
EDM=3.24442e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.15989e-01 9.80035e-02 1.67085e-04 -7.96694e-05
2 p1 1.72976e+01 1.44366e+00 1.11711e-03 -3.24787e-05
3 p2 1.90157e+00 1.23940e+00 1.11600e-03 -3.08323e-05
FCN=5.28211 FROM MIGRAD STATUS=CONVERGED 82 CALLS 83 TOTAL
EDM=1.07673e-12 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.12660e-01 1.09915e-01 2.14740e-04 6.79712e-06
2 p1 1.93227e+01 2.18690e+00 1.78657e-03 -9.53876e-07
3 p2 3.37873e+00 1.31076e+00 1.31940e-03 5.26038e-07
FCN=2.51155 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=7.33093e-10 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.47259e-01 1.47495e-01 2.07067e-04 1.51872e-05
2 p1 2.08502e+01 2.85196e+00 1.66763e-03 2.88970e-06
3 p2 3.27645e+00 1.66427e+00 1.21094e-03 1.79942e-05
FCN=2.33728 FROM MIGRAD STATUS=CONVERGED 115 CALLS 116 TOTAL
EDM=2.09805e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.24772e-01 1.41998e-01 2.10142e-04 -1.49696e-04
2 p1 1.93677e+01 3.57505e+00 1.71511e-03 -1.33117e-05
3 p2 3.40925e+00 2.17675e+00 1.20721e-03 1.71321e-05
FCN=3.7564 FROM MIGRAD STATUS=CONVERGED 144 CALLS 145 TOTAL
EDM=4.17701e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 3.86958e-01 7.02402e-02 1.51474e-04 2.23816e-03
2 p1 1.14739e+01 7.64431e+00 3.28858e-04 -2.97799e-03
3 p2 2.56372e-01 4.13723e+00 1.77749e-04 5.56351e-03
FCN=1.68053 FROM MIGRAD STATUS=CONVERGED 86 CALLS 87 TOTAL
EDM=2.31478e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.92582e-01 2.00597e-01 1.71449e-04 2.29076e-03
2 p1 1.88705e+01 5.20319e+00 1.48847e-03 -1.42308e-04
3 p2 3.88877e+00 2.85838e+00 1.17677e-03 1.67524e-04
FCN=1.37515 FROM MIGRAD STATUS=CONVERGED 97 CALLS 98 TOTAL
EDM=3.77866e-10 STRATEGY= 1 ERROR MATRIX UNCERTAINTY 11.9 per cent
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.59518e-01 2.35466e-01 -3.75759e-04 2.75005e-05
2 p1 2.42056e+01 5.45274e+00 -8.84945e-03 5.93956e-06
3 p2 4.93011e+00 2.65944e+00 -4.03592e-03 -1.82477e-05
FCN=2.35692 FROM MIGRAD STATUS=CONVERGED 98 CALLS 99 TOTAL
EDM=1.62626e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 3.73381e-01 2.12636e-01 1.68572e-04 6.21254e-05
2 p1 1.24594e+01 7.44855e+00 3.60597e-03 5.34898e-06
3 p2 4.00649e+00 8.85394e+00 4.40977e-03 -1.02483e-05
FCN=1.93137 FROM MIGRAD STATUS=CONVERGED 112 CALLS 113 TOTAL
EDM=9.59886e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.32872e-01 2.31351e-01 2.38578e-04 5.95671e-05
2 p1 2.71811e+01 8.51880e+00 2.74598e-03 -1.84591e-05
3 p2 6.83092e+00 3.30519e+00 1.46711e-03 7.02570e-05
FCN=3.28015 FROM MIGRAD STATUS=CONVERGED 106 CALLS 107 TOTAL
EDM=1.61933e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.96619e-01 1.92722e-01 2.83708e-04 1.73660e-03
2 p1 2.29350e+01 3.43141e+00 1.82942e-03 -2.93107e-04
3 p2 3.63195e+00 2.36905e+00 1.65664e-03 1.76615e-04
FCN=2.56902 FROM MIGRAD STATUS=CONVERGED 93 CALLS 94 TOTAL
EDM=1.04123e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.88320e-01 1.96030e-01 2.22173e-04 1.17829e-03
2 p1 2.42988e+01 4.41739e+00 2.13092e-03 3.20435e-04
3 p2 4.93562e+00 3.11330e+00 2.18027e-03 -2.87738e-04
FCN=3.04459 FROM MIGRAD STATUS=CONVERGED 92 CALLS 93 TOTAL
EDM=1.14973e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.60037e-01 2.49483e-01 2.45805e-04 1.12371e-04
2 p1 2.29749e+01 5.95747e+00 2.03878e-03 -3.36129e-05
3 p2 5.36144e+00 2.66551e+00 1.47403e-03 -1.71130e-06
FCN=6.27397 FROM MIGRAD STATUS=CONVERGED 80 CALLS 81 TOTAL
EDM=1.05992e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.62433e-01 1.57140e-01 2.95232e-04 -1.40452e-05
2 p1 2.15909e+01 3.29623e+00 2.81759e-03 -4.11567e-04
3 p2 4.09179e+00 1.74745e+00 2.04603e-03 7.91418e-04
FCN=1.94311 FROM MIGRAD STATUS=CONVERGED 105 CALLS 106 TOTAL
EDM=3.78266e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.75157e-01 2.98322e-01 1.94101e-04 -2.24793e-04
2 p1 2.56868e+01 1.17079e+01 2.49339e-03 9.38828e-05
3 p2 7.54114e+00 5.74853e+00 2.06897e-03 -1.88599e-05
FCN=2.75509 FROM MIGRAD STATUS=CONVERGED 68 CALLS 69 TOTAL
EDM=1.99789e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.93401e-01 1.01469e-01 1.65269e-04 -1.44010e-03
2 p1 1.88032e+01 2.57519e+00 1.96714e-03 2.76180e-04
3 p2 2.53183e+00 1.76540e+00 1.49141e-03 6.57019e-06
FCN=2.54952 FROM MIGRAD STATUS=CONVERGED 95 CALLS 96 TOTAL
EDM=9.15338e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.60242e-01 2.59374e-01 2.74212e-04 1.38114e-03
2 p1 2.50771e+01 5.98352e+00 2.14821e-03 -1.45220e-04
3 p2 5.39543e+00 2.98249e+00 1.67857e-03 4.75046e-05
FCN=4.62283 FROM MIGRAD STATUS=CONVERGED 88 CALLS 89 TOTAL
EDM=2.72802e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.92713e-01 1.83874e-01 2.19186e-04 -7.91561e-04
2 p1 2.09349e+01 5.53213e+00 2.98812e-03 5.24389e-06
3 p2 6.39618e+00 4.21336e+00 3.65952e-03 2.44120e-05
FCN=5.31269 FROM MIGRAD STATUS=CONVERGED 136 CALLS 137 TOTAL
EDM=4.34765e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.34719e-01 8.47458e-02 2.03833e-04 -2.44987e-03
2 p1 1.19371e+01 1.29747e+01 6.25160e-04 -1.41162e-03
3 p2 5.91146e-01 8.18675e+00 3.94117e-04 2.26839e-03
FCN=0.968363 FROM MIGRAD STATUS=CONVERGED 81 CALLS 82 TOTAL
EDM=2.19197e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.69962e-01 1.34012e-01 1.48248e-04 -1.50876e-04
2 p1 1.99232e+01 3.66116e+00 1.91112e-03 6.59680e-05
3 p2 4.71550e+00 3.31047e+00 2.05940e-03 -3.39385e-05
FCN=2.88345 FROM MIGRAD STATUS=CONVERGED 109 CALLS 110 TOTAL
EDM=9.84387e-09 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.30093e-01 1.74116e-01 1.99116e-04 -3.74560e-04
2 p1 2.19822e+01 6.04386e+00 2.37680e-03 5.33604e-05
3 p2 6.83694e+00 3.99757e+00 2.26168e-03 -3.58315e-05
FCN=1.80215 FROM MIGRAD STATUS=CONVERGED 86 CALLS 87 TOTAL
EDM=5.36632e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.95591e-01 2.11145e-01 1.93155e-04 2.00212e-03
2 p1 1.96450e+01 5.20520e+00 1.60029e-03 -4.11896e-04
3 p2 3.59211e+00 2.51889e+00 1.08710e-03 2.49364e-04
FCN=3.79465 FROM MIGRAD STATUS=CONVERGED 68 CALLS 69 TOTAL
EDM=8.71742e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.11171e-01 1.28277e-01 2.07595e-04 -2.07803e-03
2 p1 1.74670e+01 2.93532e+00 1.90435e-03 1.54593e-04
3 p2 3.02848e+00 1.64432e+00 1.25897e-03 -7.03662e-05
FCN=2.748 FROM MIGRAD STATUS=CONVERGED 139 CALLS 140 TOTAL
EDM=1.15631e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.00000e+00 7.33966e-01 1.41931e-03** at limit **
2 p1 2.87334e+01 2.38732e+00 1.81128e-03 -9.24774e-06
3 p2 6.33261e+00 1.85434e+00 1.40636e-03 6.75371e-05
FCN=4.29305 FROM MIGRAD STATUS=CONVERGED 89 CALLS 90 TOTAL
EDM=1.85267e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.87535e-01 1.68381e-01 2.57346e-04 -4.16419e-04
2 p1 2.22757e+01 3.60622e+00 1.89758e-03 3.86784e-05
3 p2 4.13882e+00 1.75747e+00 1.23453e-03 -1.07511e-04
FCN=1.74986 FROM MIGRAD STATUS=CONVERGED 263 CALLS 264 TOTAL
EDM=4.56587e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.00000e+00 9.90225e-01 7.21539e-03** at limit **
2 p1 3.37023e+01 9.08695e+00 4.04147e-03 1.88618e-04
3 p2 1.38012e+01 8.25066e+00 3.67224e-03 -1.82384e-04
FCN=4.72905 FROM MIGRAD STATUS=CONVERGED 68 CALLS 69 TOTAL
EDM=5.78893e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 4.17412e-01 1.07227e-01 2.05293e-04 -1.37460e-03
2 p1 1.45796e+01 2.95636e+00 2.33443e-03 -3.65024e-04
3 p2 2.22066e+00 2.12966e+00 1.87577e-03 5.88181e-04
FCN=2.45643 FROM MIGRAD STATUS=CONVERGED 106 CALLS 107 TOTAL
EDM=1.55102e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 5.54279e-01 2.26682e-01 2.40946e-04 -2.07327e-03
2 p1 2.39214e+01 6.69768e+00 2.60223e-03 1.41625e-04
3 p2 6.04420e+00 3.44221e+00 2.01296e-03 -1.02611e-04
FCN=2.91636 FROM MIGRAD STATUS=CONVERGED 91 CALLS 92 TOTAL
EDM=1.85881e-07 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 7.69304e-01 1.84778e-01 2.92672e-04 -8.48459e-04
2 p1 2.05742e+01 2.51905e+00 1.48433e-03 3.38512e-04
3 p2 3.50052e+00 1.35413e+00 1.12759e-03 -1.89593e-05
FCN=1.21412 FROM MIGRAD STATUS=CONVERGED 140 CALLS 141 TOTAL
EDM=2.32105e-08 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 6.32519e-01 1.06789e-01 1.49393e-04 2.13857e-05
2 p1 1.74834e+01 1.66297e+00 6.60282e-04 -1.77381e-05
3 p2 1.44082e+00 1.10945e+00 4.61581e-04 -1.72797e-04
#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.