8.2 Fitting 2d data sets

Actually there is no principle difference between fitting a 2D or a 1D data set. The only theory function of 2D data currently available in KUPLOT is a set of N 2D Gaussians as defined in equation 8.5. Each Gaussian is defined by eight parameters, peak height p_1,n, peak position p_2,n and p_3,n in x- and y-direction, half widths p_4,n, p_5,n and asymmetry parameters p_7,n, p_8,n for each direction and a new parameter p_6,n that defines the angle between the main axes of the Gaussian and the coordinate system. One additional parameter describes a flat overall background.

 

$$p_{1} + \sum_{n=1}^{N} p_{1,n} \cdot \exp\left\{\frac {r_{x}^{2}}{\sigma_{n}^{2}}\right\} \cdot\exp\left\{\frac {r_{y}^{2}}{\tau_{n}^{2}} \right\}$$ z(x_i,y_i,p) = (8.5)

$$\mbox {with\ } r_{x} \cos(p_{6,n})(x_{i}-p_{2,n}) + \sin(y_{i}-p_{3,n})$$ =

$$-\sin(p_{6,n})(x_{i}-p_{2,n}) + \cos(y_{i}-p_{3,n})$$ r_y =

$$\sigma_{n} \left\{ \begin{array}{ll} p_{4,n} \cdot p_{7,n} & \mbox {for $x_{i} < r_{x}$ }\\ \frac{p_{4,n}}{p_{7,n}} & \mbox {else} \\ \end{array}\right.$$ =

$$\tau_{n} \left\{ \begin{array}{ll} p_{5,n} \cdot p_{8,n} & \mbox {for $y_{i} < r_{y}$ }\\ \frac{p_{5,n}}{p_{8,n}} & \mbox {else} \\ \end{array}\right.$$ =

This equation certainly looks more complex than the 1D equivalent and we will illustrate the fitting of three 2D Gaussians to a section of the diffuse scattering data used in the 2D examples before. The data and the fit result are shown in Figure 8.2. The observed data are shown in the top view graph. The maxima used to determine the starting values for the fit are marked by black circles. One of the major mistakes when using the fit routine of KUPLOT are wrong starting values. The calculated data are shown in the middle view graph using identical contour line settings as for the data. The bottom view graph finally shows the difference between observed and calculated data. The solid black line is the zero level, red lines mark negative values and blue lines stand for positive contour levels. Note that the contour line interval used for the difference plot is only $\frac{1}{5}$ of the interval used for the observed and calculated data in order to make the differences more visible.

  

Figure 8.2: Fit results for 2D example

The macro file used for this example file is shown below. Again we assume the observed data are already loaded as data set one.

     1  fit 1
     2  #
     3  mfen 3
     4  func gau2,3
     5  par
     6  #
     7  run
     8  #
     9  par  7,1
    10  par 23,1
    11  #
    12  run
    13  save
    14  #
    15  exit

First we enter the fit sublevel (line 1) to fit the theory function to data set one. Before we select three 2D Gaussian theory functions in line 4 we set the window size for determination for maxima used as starting values to 3 data points. This will result in the three maxima marked in Figure 8.2. Line 5 displays the current parameter settings for checking. Next (line 7) the fit is started. As default the parameter p_6,n determining the angle between the principle axes of the Gaussian and the coordinate axes and the asymmetry parameters p_7,n and p_8,n will be kept fixed. After the first fit, the orientation of the Gaussian (parameter p_6,n) for Gaussians one and three (the two strong peaks) are refined as well (lines 9-12). Finally the data sets containing the calculated data and the difference as well as the resulting parameters are saved. When fitting interactively the command 'plot' within the fit sublevel will plot the current fit on the screen.

Finally we want to have a look at the produced text file containing the fit results (file extension .erg) for this example fit. The line numbers are given for reference and as for the macro files not part of the actual output file. The first part contains general information about the fit such as a title (lines 2-3), the used theory function (line 4), the data file used (line 5), the number of data points and parameters (lines 6-7). Note that this is the total number of parameters including those not actually refined. Lines 8-10 show the current URF (see 8.1), the maximum number of iterations, the setting for maxima determinations and the used weighting scheme, in our case $w_{i} = \frac{1}{z_{i}}$. The next block of data in the output file show details of the fitting process itself. Line 14 lists the value for $\sum (z_{i} - z(x_{i},y_{i},p))^{2}$ for the last and the previous iteration step. The difference between both is shown in the next line 15. A big difference normally indicates a 'crashed' fit. The value of URF changes during the refinement and the final value is shown in line 16. If this value is not close to 1.0 at the end of the fit something has gone wrong. The next line shows the resulting R value and the expected R value. The ration of these values is also known as the goodness-of-fit. Finally all correlations between fitted parameters that are greater than 0.8 are listed. We are lucky because our fit contains no highly correlated variables.

     1   General fit parameter settings :
     2     Title            : KUPLOT 2D fit demonstration
     3                        January 1998
     4     Fit function     : Gaussian (2D)
     5     Data file name   : test.fit
     6     # of data pts.   :    561
     7     # of parameters  :     25
     8     Urf              :     .1000
     9     Max. cycle       :     30
    10     MFEN for maxima  :      3
    11     Weighting scheme : w(i) = 1.0/i
    12
    13   Information about the fit :
    14     Sum n-1    :  1038.98                 Sum n   :  1040.29
    15     Difference :  1.31653
    16     Urf final  :  1.00047
    17     R4 value   :  .106378                 R exp   :  .077327
    18
    19   Correlations larger than 0.8 :
    20     ** none **
    21

All remaining information of the output file are results of the parameters used in the theory function. In line 22 the number and type of functions fitted are shown. Next we have the result for the overall background p₁ and its standard deviation (line 24). The 'pinc' in the last column indicates whether the parameter was refined (pinc = 1) or kept fixed to its starting value (pinc = 0). The values for each parameter and the corresponding 'pinc' value can be altered via the command 'par'.

    22   Fitted  3 Gaussian(s) :
    23
    24     p( 1) : backgr. 1 :  69.2818     +-  .998061    pinc : 1.
    25

The rest of the output file contains three identical blocks with the resulting parameters of the three Gaussians fitted to the observed data. We show here just the results for Gaussian 1 (left maximum).

    26   Gaussian :   1
    27
    28     p( 2) : peak      :  407.332     +-  11.3879    pinc : 1.
    29     p( 3) : position x:  .665616     +-  .002372    pinc : 1.
    30     p( 4) : position y:  3.74364     +-  .002798    pinc : 1.
    31     p( 5) : fwhm a    :  .237793     +-  .005545    pinc : 1.
    32     p( 6) : fwhm b    :  .235389     +-  .005938    pinc : 1.
    33     p( 7) : angle a,x : -98.8801     +-  85.1452    pinc : 1.
    34     p( 8) : asym. a   :  1.00000     +-  .000000    pinc : 0.
    35     p( 9) : asym. b   :  1.00000     +-  .000000    pinc : 0.
    36             integral  :  25.8344     +-  1.97653

Note that the asymmetry parameters were not refined, thus kept at the default setting for a symmetric function. The orientation for this Gaussian (line 7) has a very large error which can be understood from the resulting half width parameters (lines 31-32) which show a nearly isotropic result. Thus the definition of an orientation is quite arbitrary. This result is displayed here for demonstration purposes, normally the fit needs to be repeated with a fixed orientation parameter for this Gaussian. The third Gaussian however (right maxima) has a significant orientation parameter of $p_{6,n} = 33(8)^{\circ}$. Finally (line 36) the integral for each Gaussian with its standard deviation is given.