Telling the fitting program, how to calculate the standard deviation sta which should be minimised

The user must provide a program calcsta (under windows it is calcsta.bat), which calculates the standard deviation of some experimental data from model calculated data. This program can be a bash script such as /mcphas/examples/cecu2a/fit/calcsta or just a very short program such as:

# do a mcphase simulation. The measured magnetisation data must
# be in directory ./fit/mcphas.fum column 11 12 13.
# the program mcphas\index{mcphas} will compare measured data with calculated
# data and generate an output such as sta=14.3112, which
# contains the standard deviation
system ("mcphas");

... in general the output of this batch (to stdout) must contain a line such as sta = 4122.32, which contains the calculated standard deviation which will be minimized by the fitting procedure.

Important notes:

  1. Before starting a fit it is necessary to test the program calcsta by typing ./calcsta. The calculation should run and there must appear an output the line sta= ... on the screen. If this works, a fit may be started.
  2. the programs McPhas and McDisp always try to calculate a standard deviation and write such sta = ... statement to stdout. So it is possible to use these programs to create the desired output.
  3. Alternatively (or in addition) any other program can be called in calcsta and calculate the standard deviation sta. Useful programs are chi2, rpvalue.
  4. Several ($N$) occurrencies of sta = $\delta_i^2$ contribute additively to the variance $s^2=1/N\sum_i \delta_i^2$, which is minimized in the process.
  5. if sta= is followed by two numbers $\delta_i^2$ and ${\rm err}^2$, the first is interpreted as the $\delta_i^2=$ deviation$^2$ from an experimental data point (which is to be fitted). The second number ${\rm err}^2$ is interpreted as the squared errorbar of this datapoint. If ${\rm err}^2$ is not given, the program minimizes the variance sta $=s^2=1/N\sum_i \delta_i^2$. If ${\rm err}^2$ is given, the program minimizes $\chi^2=1/N\sum_i \delta_i^2/{\rm err}^2$. The error is used at the end of the fitting to calculate a covariance matrix, which can be used to estimate correlations and errors of the fitting parameters: $F_{ij}=\partial \delta_i / {\rm err}_i\partial {\rm par}_j$, the covariance matrix is then ${\rm cov}=\chi^2(F^TF)^{-1}$, it's diagonal elements correspond to the squared error of the fitting parameters par$_j$ and the off diagonal elements to the correlation among these fitting parameters. Useful to generate the required sta= ... statements is the program chi2.
  6. note, that the program simannfit calls the user written program calcsta with a number as argument, e.g. as calcsta 21.3. This number (i.e. 21.3 in our example) denotes a maximum value of the standard deviation. The user written program may use this number and stop, when in the summation process to obtain standard deviation sta reaches a value larger than this maximum number. Using this option is advisable in complex fitting problems in order to optimise the time of the fitting procedure. Note, that the module mcphas can be used with the option -stamax 21.3, which leads to a stop when the value of the standard deviation 21.3 is reached.

Martin Rotter 2017-01-10