TMinuit: error matrix not pos def (once again)

Hi everyone,
I am trying to use minuit to make a likelihood fit and I’m struggling to find a positive defined error matrix (my implementation was originally based in this example https ://root.cern/doc/v610/Ifit_8C_source.html). I have a sample of 15 histograms with two contributions, signal and background, each one of them described by a convolution of a gaussian and a landau functions (I’m using the definition here https ://root.cern/doc/master/langaus_8C.html). The different parameters of each landau-gaus are described by parametric functions along the histograms. In the case of the background is quite simple since the variation of the parameters is soft and it can be described linearly. However, in the case of the signal, parameters vary exponentially at some point, so its description can’t be linear anymore and has to be exponential/rational. The fit converges and if one looks at residuals it performs well, however minuit is always getting a error matrix not positive defined.
I have read different threads in the forum suggesting to try different things:

1. Originally the parametric functions I was using where handmade, but now I’m using orthogonal functions. I’ve been able to decrease the correlation between parameters, however it is still large.
2. I have tried to scale parameters so they are of the same order of magnitude.
3. I have tried to play with the step, fixing and releasing some of the parameters, different minuit strategies, etc.
So far, I haven’t been able to find a convergence without the error matrix problem. I’m posting below the outcome of minuit (with strategy 1, so it is shorter). I would appreciate any suggestion to solve this issue.
`````` **********
**    1 **SET ERR         0.5
**********
PARAMETER DEFINITIONS:
NO.   NAME         VALUE      STEP SIZE      LIMITS
1 slw A        3.19419e+00  1.00000e-01     no limits
2 slw A       -7.35809e+00  1.00000e-01     no limits
3 slw A        6.09946e+00  1.00000e-01     no limits
4 smpv A      -6.57937e+01  1.00000e-01     no limits
5 smpv B       2.48719e+02  1.00000e-01     no limits
6 smpv C      -2.82921e+02  1.00000e-01     no limits
7 smpv D       1.10803e+02  1.00000e-01     no limits
8 sgw A        3.02901e-01  1.00000e-01     no limits
9 sgw B        1.25424e+00  1.00000e-01     no limits
10 sgw C        4.01470e-01  1.00000e-01     no limits
11 snorm        3.96431e-01  1.00000e-01     no limits
12 blw A       -1.10608e+00  1.00000e-01     no limits
13 blw B        6.99002e-01  1.00000e-01     no limits
14 bmpv A       2.12531e+00  1.00000e-01     no limits
15 bmpv B       1.22481e-01  1.00000e-01     no limits
16 bmpv C      -1.53774e+00  1.00000e-01     no limits
17 bgw          1.53570e-02  1.00000e-01     no limits
18 bnorm        8.48840e-02  1.00000e-01     no limits
**********
**    2 **MIGRAD       5e+05           1
**********
FIRST CALL TO USER FUNCTION AT NEW START POINT, WITH IFLAG=4.
START MIGRAD MINIMIZATION.  STRATEGY  1.  CONVERGENCE WHEN EDM .LT. 1.00e-03
FCN=2554.45 FROM MIGRAD    STATUS=INITIATE      114 CALLS         115 TOTAL
EDM= unknown      STRATEGY= 1      NO ERROR MATRIX
EXT PARAMETER               CURRENT GUESS       STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  slw A        3.19419e+00   1.00000e-01   0.00000e+00  -2.78836e+03
2  slw A       -7.35809e+00   1.00000e-01   0.00000e+00  -5.21544e+02
3  slw A        6.09946e+00   1.00000e-01   0.00000e+00   8.21955e+02
4  smpv A      -6.57937e+01   1.00000e-01   0.00000e+00  -8.74304e+02
5  smpv B       2.48719e+02   1.00000e-01   0.00000e+00  -1.08712e+02
6  smpv C      -2.82921e+02   1.00000e-01   0.00000e+00   3.21091e+02
7  smpv D       1.10803e+02   1.00000e-01   0.00000e+00   5.13439e+02
8  sgw A        3.02901e-01   1.00000e-01   0.00000e+00  -3.53344e+02
9  sgw B        1.25424e+00   1.00000e-01   0.00000e+00  -1.31037e+02
10  sgw C        4.01470e-01   1.00000e-01   0.00000e+00   2.81988e+01
11  snorm        3.96431e-01   1.00000e-01   0.00000e+00   5.95406e+02
12  blw A       -1.10608e+00   1.00000e-01   0.00000e+00  -6.62292e+02
13  blw B        6.99002e-01   1.00000e-01   0.00000e+00  -1.33236e+03
14  bmpv A       2.12531e+00   1.00000e-01   0.00000e+00  -4.09754e+02
15  bmpv B       1.22481e-01   1.00000e-01   0.00000e+00   7.81495e+02
16  bmpv C      -1.53774e+00   1.00000e-01   0.00000e+00  -2.74423e+00
17  bgw          9.01066e-02   1.00000e-01   7.47496e-02   3.30492e+00
18  bnorm        8.48840e-02   1.00000e-01   0.00000e+00   1.32578e+03
ERR DEF= 0.5
MIGRAD MINIMIZATION HAS CONVERGED.
MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
EIGENVALUES OF SECOND-DERIVATIVE MATRIX:
-1.0114e-02 -8.4229e-03 -3.3778e-03  1.5594e-03  1.0084e-02  2.7456e-02  4.0005e-02  1.4564e-01  2.7067e-01  3.7230e-01  5.4190e-01  5.8787e-01  6.9819e-01  1.0007e+00  1.0965e+00  3.5637e+00  4.2473e+00  5.4181e+00
MINUIT WARNING IN HESSE
============== MATRIX FORCED POS-DEF BY ADDING 0.015532 TO DIAGONAL.
FCN=2502.23 FROM HESSE     STATUS=NOT POSDEF    260 CALLS        1918 TOTAL
EDM=0.00109231    STRATEGY= 1      ERR MATRIX NOT POS-DEF
EXT PARAMETER                APPROXIMATE        STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  slw A        2.99269e+00   9.13154e-03   4.16811e-05   1.11232e+00
2  slw A       -6.85769e+00   3.15355e-02   2.94437e-04  -5.08921e-02
3  slw A        5.71477e+00   2.07845e-02   1.12780e-04   4.24168e-01
4  smpv A      -6.56473e+01   2.37630e-02   9.77130e-05   3.78658e-01
5  smpv B       2.48528e+02   8.32935e-02   7.81362e-04   2.11647e-02
6  smpv C      -2.82981e+02   5.61563e-02   2.51471e-04   2.70283e-01
7  smpv D       1.11044e+02   4.12117e-02   1.63193e-04   4.33484e-02
8  sgw A       -3.93001e-01   2.25369e-02   9.51033e-05  -5.66321e-01
9  sgw B        2.96129e+00   6.46402e-02   7.38651e-04  -2.53038e-02
10  sgw C       -9.13939e-01   4.97529e-02   2.48670e-04  -1.98799e-01
11  snorm        3.87073e-01   3.26617e-03   9.50566e-05  -6.66382e-02
12  blw A       -1.68016e+00   4.69172e-02   2.79756e-04   7.09820e-02
13  blw B        9.92256e-01   2.23126e-02   1.36271e-04   2.09217e-01
14  bmpv A       2.17053e+00   4.56398e-02   3.24195e-04   2.02575e-02
15  bmpv B       7.48913e-02   1.04711e-02   1.83362e-04  -6.94174e-01
16  bmpv C       1.08541e+00   3.01501e+00   2.60247e-02  -3.13065e-03
17  bgw          7.54284e-06   3.96259e-02   1.36757e-03   5.63713e-03
18  bnorm        9.14224e-02   2.74266e-03   5.57469e-05  -3.20056e-01
ERR DEF= 0.5
MIGRAD MINIMIZATION HAS CONVERGED.
MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
EIGENVALUES OF SECOND-DERIVATIVE MATRIX:
-3.1067e-03  2.7950e-03  6.4404e-03  1.2508e-02  1.8433e-02  3.0416e-02  5.6599e-02  1.6289e-01  2.7964e-01  3.5901e-01  5.2361e-01  5.8476e-01  7.0127e-01  1.0001e+00  1.0948e+00  3.5480e+00  4.2428e+00  5.3791e+00
MINUIT WARNING IN HESSE
============== MATRIX FORCED POS-DEF BY ADDING 0.008486 TO DIAGONAL.
FCN=2502.07 FROM MIGRAD    STATUS=CONVERGED    2843 CALLS        2844 TOTAL
EDM=0.000259052    STRATEGY= 1      ERR MATRIX NOT POS-DEF
EXT PARAMETER                APPROXIMATE        STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  slw A        2.88379e+00   9.54753e-03   4.14949e-05   4.10736e-01
2  slw A       -6.60386e+00   3.13884e-02   2.96285e-04   5.42212e-02
3  slw A        5.50347e+00   1.86586e-02   1.12455e-04   2.77214e-01
4  smpv A      -6.57042e+01   1.99237e-02   9.79021e-05   5.16851e-01
5  smpv B       2.48874e+02   6.42240e-02   7.82631e-04  -5.35246e-02
6  smpv C      -2.83516e+02   4.51017e-02   2.50837e-04  -2.29009e-02
7  smpv D       1.11356e+02   4.08895e-02   1.63294e-04  -1.00032e-01
8  sgw A       -1.80894e-01   1.94104e-02   9.56872e-05  -1.11843e-01
9  sgw B        2.46642e+00   6.64350e-02   7.31068e-04  -2.79255e-02
10  sgw C       -5.02457e-01   3.84029e-02   2.49724e-04   1.17161e-01
11  snorm        3.87064e-01   3.25217e-03   9.50211e-05   3.28914e-01
12  blw A       -1.66884e+00   4.41173e-02   2.80156e-04   7.88147e-02
13  blw B        9.86811e-01   2.15349e-02   1.36257e-04  -1.32711e-01
14  bmpv A       2.17013e+00   5.20274e-02   3.25475e-04   8.13525e-02
15  bmpv B       7.48940e-02   1.15706e-02   1.84907e-04  -8.13597e-01
16  bmpv C       1.09455e+00   3.38409e+00   2.62114e-02  -1.97682e-04
17  bgw         -2.59995e-06   3.92780e-02   1.36313e-03  -1.12753e-03
18  bnorm        9.12969e-02   2.68852e-03   5.56026e-05   9.40593e-01
ERR DEF= 0.5
``````

Hi,

If the correlation is still large this could be one of the cause of your problem. I see however that the parameter error is around 10^-2 / 10^-3 for all parameters a part from one. You can try to rescale all parameter errors to be around 1, this might increase the numerical accuracy when inverting the matrix.
Also, as a cross check you can start the minimisation from different initial points, to check that it always converges to the same value.

Best regards,

Lorenzo

Hi Lorenzo,
Thank you very much for your answer. What do you refer to ‘scale the errors to 1’? Do you mean to set the step size to 1 for every parameter?
Thank you,
Miguel

Hi,

I was meaning to rescale the parameter by a value such that the error comes almost equal to 1. However, I think Minuit might do such rescaling internally, so I am not sure this will be really useful. I think the test to start the minimization from different points and see if it always converges to the same value makes more sense.

Cheers

Lorenzo

Hi Lorenzo,
I finally managed to get the appropriate error matrix. It seems that there is always a better parametrization of the problem.
I have now a different question with respect to the errors. The MPV of my signal in described by a parametric function, whose parameters let’s say are A, B and C, and I obtain their value from the minimization process described above. This A, B and C parameters have some correlation coefficients, and if I want to calculate the error of MPV = f(A,B,C) I do have to consider these coefficients when doing error propagation.
However, if I ask MINOS to compute more accurately the error of the parameters A,B and C, MINOS is supposed to consider also correlations between parameters. Does this mean that after using MINOS, if I want to propagate the error from A,B and C to my MPV, I can forget about the correlations? Or should I keep using them?
Thank you

Hi,
MINOS will compute the confidence interval on the fitted parameter, but it will not be able to compute the confidence interval on the function f(A,B,C). You can use error propagation, but this is correct only in the approximation when the errors are small. Otherwise a possible solution is to use some MC studies. Given the fitted parameters A,B,C re-generate many data sets, with as ample size equal to the original one, fit each one of them and then look at the spread of the fitted f(A,B,C). This will provide a good indication of the uncertainty of your quantity.

Best,

Lorenzo

Hi Lorenzo,
Thank you very much for your answer,
Cheers

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