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          This folder examines fits to artificial data using a generalized Fermi function and its derivative as the fitting function. 

1        Generalized Fermi Function.htm doc Finds partial derivatives of the Generalized Fermi function and of its derivative with respect to T.

2        Fitting Dfermi.htm  Fits to the derivative of a Fermi function.  This also tests the next section, standard deviation fitting.  Of interest is the fact that the data can collaborate to get a low standard deviation.

3        Standard Deviation fitting.htm Fits to the standard deviation in the data.  The form used here is s2 = max(a + b fA + c fA2,d).  The form is required to be positive definite only if d is.  Values of fA and (fi-fA)2 are written out by nlfit.

4        Fitting sdfermi.htm Fit to sum of dfermi.  This is a test of the derivatives in FPOLY.FOR.  Data for this fit had a larger error in the flat region than some generated later.

5        FermiStd.htm Code for using a fit to the derivative of the Fermi function to find the Fermi function and its standard deviation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dfermi.dir, OUTPUT FILE NAME

DFFERMI.out,    DATA SET NAME                                  

1,1,   DIMENSION OF X IN F(X), power in ((f-fA)/err)**(2*npow) 

R,.0,1,.0,  'R' read data point,err, 'C' err=SQRT(a+b|f|)+c|f| 

5   NUMBER OF CONSTANTS IN POLY                                

400,2  MIN STEPS, Multi 0 NONE, 1_PPCC+MPH, 2 PPCC+MP, 3 P,PPCC,

.1,  INITIAL FR DECREASE DESIRED                               

1 VARY O FIX   FITTED CONS  ERROR IN CONS   -- input  error                   

    1   998.5864254374815       +-  20.2      1000 0.003 sigma

    1  -8.206805201998250       +- 0.291      -8   0.711 sigma

    1   359.2715742295106       +-  1.15      360  0.635 sigma

    1  0.9602883783432931       +- 0.878E-01  .9   0.987 sigma

    1  0.2392332387917320E-16   +- 0.285E-16  0    0.839 sigma

 ended because abs(chb-chl)/chl<1e-9

 CHIS= 502.3     FOR   500 DATA POINTS

 CHI USED IN CALCULATING ERRORS IS  502.3   

 

Changing the line with the error to

C,1e-4,1,.0,  'R' read data point,err, 'C' err=SQRT(a+b|f|)+c|f|

 

FR,CHI,CHB,CHL 0.973151361      269.834422      270.088621      277.540181    Y

FR,CHI,CHB,CHL 0.947023570      267.347523      266.857615      269.834422    N

FR,CHI,CHB,CHL 0.947023570      266.888589      266.865659      267.347523    N

FR,CHI,CHB,CHL 0.947023570      266.887773      266.887755      266.888589    N

FR,CHI,CHB,CHL 0.947023570      266.887770      266.887769      266.887773    N

FR,CHI,CHB,CHL 0.947023570      266.887770      266.887770      266.887770    N

 ROBMIN finds no lower values

 CHI AT END OF FIT     266.8877696341358000

           1     899.9631801507963000 +-      19.0298132302765900  5.24 sigma

           1      -7.4634323997976110 +-       0.2665327605582825  2.02 sigma

           1     358.6590107052419000 +-       1.0657971020038890  1.26 sigma

           1       0.9838298908991070 +-       0.0909048137467139   .93 sigma

           1 -2.5162568757545220D-005 +-       0.0007934167455634   0

 FINAL CHIT=     266.8877696341358000

 CHI/DEG OF FREE        0.5391672113820926

 RESULTS IN FILE dfermi2.dir

 

Note that the chi-square is lower than that using the accurate error estimates.  Change the error estimate to

c,1e-4,1,.0,  'R' read data point,err, 'C' err=SQRT(a+b|f|)+c|f|

The lower case c causes f to be calculated using the current set of constants.

FIT IS BEING MADE TO DATA IN FILE

 DFFERMI.out

 MULTIPLE PRECISION PPCC and MPSMINV

FR,CHI,CHB,CHL 0.100000000      539.027881     0.100000000+307 0.500000000+307N

FR,CHI,CHB,CHL 0.100000000      467.415233      471.465439      539.027881    N

FR,CHI,CHB,CHL 0.100000000      467.205333      467.205933      467.415233    N

FR,CHI,CHB,CHL 0.100000000      467.204700      467.204731      467.205333    N

FR,CHI,CHB,CHL 0.100000000      467.204690      467.204691      467.204700    N

FR,CHI,CHB,CHL 0.100000000      467.204690      467.204690      467.204690    N

 ROBMIN finds no lower values

 CHI AT END OF FIT     467.2046895810433000

           1     998.2001009817087000 +-      18.9436941072828900

           1      -8.1565196388056300 +-       0.2421612739495582

           1     359.7812964888203000 +-       0.9250732527738026

           1       0.9214996954928953 +-       0.0650145772774638

           1  7.7195005742676250D-005 +-       0.0007546439395723

 FINAL CHIT=     467.2046895810433000

 CHI/DEG OF FREE        0.9438478577394814

 RESULTS IN FILE dfermi.dir

Figure 2 The fit comes out in the file fit.dat.  I use gplot to plot the data and fit.dat.

It is numerically integrated in sum.for.  – see ..\..\Weights\ArtDat\Sum.doc -- This was compiled and linked using the command

Wfl386 sum

Figure 3  Sum of the file dfermi.out

          The errors in figure 2 are clearly not random.  The errors to the right of the step are a sum of the errors introduced going up the step.  

 

  The various doc files describe the code in detail, the derivatives and a few graphs.  Directions for fitting data

          1. First plot your data.

              Both of the trial data sets had a single wild point.  This destroys any genuine least squares routine.  Find these points and remove them.

          2. The error is of the form .  Estimate the error from the data fluctuations to determine the coefficients a, b, and c which will be on line 4 in the direction file following a C that instructs the code to calculate the errors.  The third column in the sample data was determined from the code errest which makes a file temp.dat from the data set containing x,y,estat.  The method for doing this is described in my book on using ROBFIT for spectral analysis.  I copied temp.dat to the name of the input data file.

 

          3.  Make a direction file modelled on 1-5.DIR.  Note that the error estimate is of the form It is best to fix some of the constants on the first pass.