Fitting

The fitting routines in HBOOK are based on the Minuit package [12]. Minuit is conceived as a tool to find the minimum value of a     multi-parameter function and analyze the shape of the function around the minimum. The principal application is foreseen for statistical analysis, working on chisquare or log-likelihood functions, to compute the best-fit parameter values and uncertainties, including correlations between the parameters. It is especially suited to handle difficult problems, including those which may require guidance in order to find the correct solution.

In the case of $\chi 2$ minimization, the final fitted parameter values correspond to the minimum of the $\chi 2$ function as defined below:

$\chi 2\hspace{0.17em}=\hspace{0.17em}\sum$_i=1ⁿ( {C(I)-F(X(I),A₁,A₂,...,A_k)E(I)})²

where the following definitions are used:

n

Number of channels (cells) in the 1-dimensional or 2-dimensional histogram (HFITH, HFITHN) or number of points of the distribution (HFITV)

C(I)

Contents of channel (cell) I

E(I)

Error on C(I)

X(I)

Coordinate(s) of centre of channel (cell) I or coordinate vector of point I of the distribution

$A$₁..A_k

Parameters of the parametric function.

F

Parametric function to be fitted to the data points.

Remarks:

• If no errors are specified by the user, they are taken to be the square root of the bin contents. In this case:
  • empty channels are ignored by the fitting procedure;
  • If at least one of C(I)<0, then E(I) is set to 1 for all channels (cells).
• If errors are correctly defined via HBARX or HPAKE, then all channels will be taken into account.

The minimization algorithm requires the calculation of derivatives of the function with respect to the parameters and this is normally done numerically by the fitting routine. If the analytical expression of the derivatives is known, the fit can be speeded up by making use of this information (see option D in the control flag of the various fitting routines).

For a log-likelihood fit, the likelihood is formed by determining the Poisson probability that given a number of entries in a particualar bin, the fit would predict its value. This is then done for each bin, and the sum of the logs is taken as the likelihood.    

• One and two-dimensional distributions
• Fitting one-dimensional histograms with special functions
• Fitting one or multi-demensional arrays
• Results of the fit
• Retrieving the fitted parameters
• The user parametric function