Revision: June 25, 1998
A number of existing software packages were surveyed to determine currently used methods for error analysis. This survey is part of the effort by the Standards and Criteria (S&C) committee of the International XAFS Society (IXS) to develop guidelines for standardized error-reporting procedures and minimum reporting requirements. Such guidelines will ultimately be used by authors of present and future software packages to gradually standardize error reporting methods.
The purpose of this survey it to determine currently used methods for error analysis. The decision to conduct the survey was taken at the meeting of the S&C committee of the IXS in July 1997. The goal is to eventually develop guidelines for standardized error-reporting procedures and minimum reporting requirements that are compatible with accepted practices for error analysis and, as much as possible, currently existing procedures. Such guidelines will ultimately be used by authors of present and future software packages to gradually standardize error reporting methods.
This survey is neither exhaustive nor complete. The majority of the software packages surveyed are listed on the ESRF page of XAFS software. In addition, several packages listed on the IXS and UWXAFS home pages were also included. The following authors/maintainers have provided information about their respective packages:
Note that this does not include quite a few of the packages listed at the above locations, whose authors did not respond. Information on other packages, to be included in future revisions of this document, may be submitted here.
Authors were asked to answer the following questions:
, other), including
normalization pre-factor(s), experimental uncertainties, etc.
at the minimum,
doubling of , increasing by a fixed fraction, etc.
This section summarizes the notation used in this document. Some of the notation used by the authors has been changed to maintain consistency of style across the document. The following set of common symbols is used:
is the number of independent parameters
in the fit
is the number of parameters varied during
the fit
is the number of degrees of freedom in the
fit
is the number of data points in the fit
are the experimental data points (either
k-space or r-space)
are the fit points (either k-space or
r-space)
are the individual errors of the experimental
data points (either k-space or r-space)
is the abscissa of the experimental data
(either e, k, or r-space)
is the fit weight (where applicable)
is the covariance matrix of the fit
refers to the k-th parameter varied in the
fit
is the unnormalized "squared residue"
function
is the statistical chi-square function
is the reduced chi-square function
The packages are listed in alphabetical order. The name of the person providing the information is included.
WWW Site: http://lure.u-psud.fr/ Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/pourlemac Contact: Alain Michalowicz
EXAFS pour le Mac minimizes the statistical chi-square function or
Errors in the experimental data may be specified in one of the following ways:
- estimated from a series of identical measurements of the same sample, either point-by-point or a as single (r.m.s.) number.
- estimated from the r.m.s. amplitude of the r-space FT between 15 and 25 A, as a single number for all data points
- specified by the user, point-by-point, or as a single number for all data points
All fits are performed in k-space. Confidence limits are calculated either from the square roots of the diagonal elements of the covariance matrix
where 
is a statistical confidence level, e.g., as defined on p. 536 (Chapter
14) of the 1986 edition of Numerical Recipes. It is also possible to estimate
confidence limits via brute-force methods, i.e., without assumption of parabolicity
of in the vicinity of the minimum. A correlation matrix is calculated
and displayed, but interpretation is left to the user. The following figures
of merit are provided:
- value of
- value of
at the minimum
- normalized residual
Built-in facilities for statistical significance testing (F-test) are also available.
WWW Site: http://www.dl.ac.uk/SRS/XRS/Computing/Programs/excurv97/ex97.html Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/excurve Contact: Paul Stephenson
EXCURVE minimizes the quantity
The weightings w determine the relative significance of the EXAFS, distance, and angle contributions. The sum of the weightings must equal one. Often only the EXAFS contribution is used. The other two terms are restraints, used in special circumstances where the model includes well characterised groups of atoms. The EXAFS contribution is given by:
where
Details on the definitions of the other terms are available here. It is also possible to use experimental
values as the weights in the refinement, where
can be obtained by using a spline fit to the experiment. Confidence limits
for the fit parameters [p] are estimated from the square
roots of the diagonal elements of the covariance matrix [C]
as
The following figures of merit are provided to the user:
- the value of
- a normalized residual
A value of around 25% for R would normally be considered a reasonable fit, with values of 10% or less being difficult to obtain on unfiltered data.
WWW Site: http://krazy.phys.washington.edu/ Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/feffit Contact: Matt Newville
. Fits to multiple data sets are possible in r-space
(with or without phase corrections), unfiltered, and back-transformed k-space.
R-space fits implicitly include k-weighting and FT windows. Complex algebraic
and transcendental relationships may be defined between the fit parameters
with FORTRAN-like syntax, both within a single data set, and accross multiple
data sets. Errors in the experimental data are estimated from the r.m.s.
amplitude of the r-space FT between 15 and 25 A or are specified by the
user, in both cases as a single number for all data points. This number
is transformed as needed between k and r-space with the formula
where 
is the k-grid spacing. Note that the above formula assumes a specific
normalization of the FTs. Confidence limits for the fit parameters [p] are
estimated from the square roots of the diagonal elements of the covariance
matrix [C] as
and correspond to a increase in from its
"optimal" value by a factor of 
. Pair-correlation
coefficients are calculated as
The program documentation discusses the correlation matrix to some extent. The final values of the statistical chi-square function, the reduced chi-square function, and an R-factor are reported:
GNXAS
WWW Site: http://camcnr.unicam.it/www/gnxas.html Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/gnxas Contact: Adriano Filipponi
In the GNXAS package the function minimized is
where w is a suitable, not necessarily integer, k-weighting of the residuals.
is equivalent to a scaled residual function
of the type defined above, provided that 
mimics the energy dependence of the inverse squared mean noise in the data
. The sum is extended over M experimental points.
In practice is evaluated in different regions
of the spectrum by the residual differences between the data points and
a high degree polynomial approximation (for instance 6 degree every 50 points).
This algorithm requires finely sampled spectra possibly with 
. These values
are fitted to the function 
, where 
is a constant which
contributes to the scaling factor between
and . The minimization of
is achieved with the MINUIT subroutine of the CERN library. The confidence intervals
for the fit parameters in the p-dimensional space are defined by the equation
where C is the critical value of the distribution
with p degrees of freedom (p is the number of free parameters) and the 95%
confidence level. The confidence intervals are roughly ellipsoidal regions
in the p-dimensional parameter space. The shape of the region is usually
given by the covariance matrix which can be printed. Intersection of this
region with specific 2-dimensional parameter sub-spaces can be visualized
using the contour MINUIT command. Details on the interpretation of the correlation
matrices and minimization procedures can be found in A. Filipponi, J. Phys.
Condensed Matter 7, 9343 (1995) and A. Filipponi and A. Di Cicco,
Phys. Rev. B 52, 15135 (1995).
WWW Site: ftp://ixs.csrri.iit.edu/programs/uwlib.mac/macxafs.hqx Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/macxafs Contact: Boyan Boyanov
MacXAFS minimizes either the statistical chi-square function or
Up to two data sets may be fit simultaneously in either k-space or r-space, the latter with or without phase corrections. Simple algebraic relationships may be defined between the fit parameters, both within and accross data sets. When applicable, errors in the experimental data may be specified in one of four ways:
- estimated from the r.m.s. amplitude of the r-space FT between 15 and 25 A, as a single number for all data points
- specified by the user, as a single number for all data points
- estimated point by point from a series of identical measurements of the same sample
- specified point by point by the user
In the former two cases the error estimate is transformed between k-space and r-space as needed with a formula identical to that used by FeffIt. Confidence limits for the fit parameters [p] are estimated from the square roots of the diagonal elements of the covariance matrix [C] as
and correspond to one of the following:
- a fixed fractional increase of the residue function (
- a increase in the residue function from its "optimal" value by a factor of
- an absolute increase in the residue function by 1.0 from its "optimal" value
Pair-correlation coefficients are calculated as
No interpretation guidelines are provided. Brute force pair-correlation contours may be calculated at the user's discretion. An R-factor is calculated as an indicator of overall fit quality:
WWW Site: http://www.esrf.fr/computing/expg/subgroups/theory/xafs/SEDEM/SEDEM.html Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/sedem Contact: Daniel Aberdam
SEDEM minimizes the unnormalized "squared residue" function
in unfiltered or back-transformed k-space . Errors
in the experimental data are estimated with one of two methods:
- from the smoothed back-transformed k-space data filtered above a user-specified cutoff in r-space
- from the unsigned difference between the unfiltered and filtered k-space data, where it is assumed that the filtered data includes all shells which contribute significantly to the signal
Either a k-dependent table or the r.m.s. value may be used in both cases. Systematic errors in the background subtraction are accounted for implicitly when the second method described above is used to determine the experimental errors. Confidence limits for the fit parameters [p] are calculated in two steps. In the first stem, the confidence limits are estimated from the square roots of the diagonal elements of the covariance matrix [C] as
where is a statistical confidence level, e.g., as defined on p. 536 (Chapter
14) of the 1986 edition of Numerical Recipes. The user is advised that these
estimates represent a lower limit of the parameter uncertainties, and should
not be given actual statistical interpretation. In the second step the pair
correlations between the parameter and all other
parameters are calculated, and the largest of the uncertainty domains found
is selected. The uncertainty domain is is defined as the width of the constant- contour corresponding to the chosen confidence
level, measured along the parameter axis corresponding to.
Specific formulae may be found in the SEDEM documentation. Pair-correlation
coefficients are calculated as
A global correlation coefficient is also calculated for each parameter
where
and 
is the curvature matrix. Guidelines for the interpretation of the correlation
coefficients will be provided in future releases. A fit quality factor is
also calculated
where Q is the incomplete gamma function.
WWW Site: http://ourworld.compuserve.com/homepages/t_ressler Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/winxas Contact: Thorsten Ressler
WinXAS minimizes the unnormalized "squared residue" function
. An estimate for the experimental error is calculated
from the noise amplitude of the experimental absorption spectrum, and may
be adjusted by the user. Confidence limits are estimated from the square
roots of the diagonal elements of the fit covariance matrix
Pair-correlation coefficients are calculated as
Guidelines for interpretation are provided in the user manual. Several figures of merit are calculated:
WWW Site: http://www.esrf.fr/computing/expg/subgroups/theory/xafs/xafs_software.html#MarkusWinterer Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/winterer Contact: Markus Winterer
XAFS minimizes the reduced chi-square function .
Statistical error of the data are used as described in Koningsberger and
Prins and an average is used for propagation in the analysis procedure,
e.g.,
for transmission data. The user is given the option of switching to other error estimates. Systematic errors in the background subtraction are checked by derivatives of the background with different k-weights and by the Fourier transforms (FTs) of the derivative of the background. Confidence limits for the fit parameters [p] are estimated from the square roots of the diagonal elements of the covariance matrix [C]
where is a user-specified confidence level. Pair-correlation coefficients
are calculated as
and no interpretation guidelines are provided. A goodness-of-fit estimate G, as described in Chapter 14 of Numerical Recipes is calculated
where Q is the incomplete gamma function. An R-factor is planned for future versions.
WWW Site: http://www.xs4all.nl/~xsi Abstract: http://ixs.csrri.iit.edu/catalog/XAFS_Programs/xdap Contact: Marius Vaarkamp
XDAP minimizes either the unnormalized "squared residue" function
or
The experimental errors can be calculated after background subtraction and normalization or after Fourier filtering. Confidence limits are estimated from the square roots of the diagonal elements of the fit covariance matrix
Pair-correlation coefficients are calculated as
Interpretation is left to the user, although examples of good results are supplied. The covariance matrix is calculated using statistical errors in the data and analytical partial derivatives of the model function to account for correlation between parameters. The following figures of merit are supplied:
- "goodness of fit", as defined by the committee on S&C of the IXS
- two kinds of normalized residual functions, both in k-space and r-space
The significance of additional contributions to the model function may be tested with a standard F-test (a.k.a. Joyner test).
Last modified on June 25, 1998 by Boyan Boyanov
