Least-squares fitting of EPR spectra, single and multiple components.
esfit(SimFnc,expSpc,Sys0,Vary,Exp); esfit(SimFnc,expSpc,Sys0,Vary,Exp,SimOpt); esfit(SimFnc,expSpc,Sys0,Vary,Exp,SimOpt,FitOpt); BestSys = esfit(...) [BestSys,BestSpc] = esfit(...)
See also the user guide on fitting.
esfit
fits EPR spectra simulated with
garlic,
pepper,
chili etc. to experimental spectral data
using least-squares fitting algorithms.
If no output is requested, esfit
start an interactive user interface that allows extensive
control over the fitting process and also allows the export of the fitting results to the workspace. In the UI, the fitting algorithm can be changed,
parameters can be excluded from the fit, and multiple fit set can be stored.
If the user interface is not desired, then esfit
should be called with one or two output arguments:
BestSys
contains the spin system parameters of the best fit.
BestSpc
contains the simulated spectrum that gives the best fit.
The input parameters are the following:
SimFnc
is the name of the simulation function: 'garlic'
, 'pepper'
,
'chili'
, 'saffron'
or 'salt'
.
expSpc
is a 1D array containing the experimental spectral data.
Sys0
is a spin system structure containing all the parameters of the spin system.
For parameters that are to varied during the fitting process, the values in the structure represent the centers of the search ranges.
Sys0
is a list of spin system structures, e.g. {Sys1,Sys2}
for a two-component system. The weights for the different components are included in each component structure, e.g. Sys1.weight = 0.7
and Sys2.weight = 0.3
. The weights are relative (0.7 and 0.3 is the same as 14 and 6) and can be included in the fit. If weight is omitted, it is assumed to be 1. There is no limit on the number of components.
Vary
contains the approximate maximum variations for the spin system parameters allowed in the fitting. If a spin system parameter should be included
in the fitting, give it a non-zero value in this structure. If it should not be included, set its value in this structure to zero, or don't include it at all.
Sys0.lwpp = 5; Vary.lwpp = 2; % the linewidth is searched over a range of about 3 to 7 Vary.lwpp = 0; % the linewidth kept constant at the value given in Sys0For multiple components,
Vary
should contain one structure per component, e.g. {Vary1,Vary2}
for two components. If none of the parameters of a component are varied in the fit, []
can be specified, e.g. {Vary1,[]}
for a two-component fit where only parameters of the first component are varied.
Exp
contains the experimental parameters needed for the simulation. Make sure to specify the field or frequency range you used for the experimental data in Exp.Range
or Exp.mwRange
. If you let the simulation function automatically pick a range, it will likely be different from the experimental one, and esfit
cannot do its work. For example, if x
is the experimental field range, use Exp.Range = [min(x) max(x)]
.
SimOpt
contains simulation options that are forwarded directly to the simulation function. Refer to the documentation of
chili,
garlic,
pepper, and
salt for details.
FitOpt
is a structure containing settings for the optimization algorithms used in esfit
. The possible settings are discussed
further down on this page.
The structure FitOpt
contains fitting options. The most important ones select the fitting algorithm and the function to use for computing the residuals:
Method
FitOpt.Method = 'simplex fcn'; FitOpt.Method = 'genetic int';
The keywords for choosing one of the available algorithm are
'simplex'
: Nelder-Mead simplex algorithm (unconstrained)
'levmar'
: Levenberg/Marquardt (unconstrained)
'montecarlo'
: Monte Carlo random search (constrained)
'genetic'
: Genetic algorithm (constrained)
'grid'
: Systematic grid search (constrained)
'swarm'
: Particle swarm algorithm (constrained)
'simplex'
is used.
The second keyword specifies what the residuals and consequently the root-mean-square deviation (rmsd) should be computed from. Usually the residuals are computed by taking the difference of the experimental and the simulated spectrum. However, the spectra can be transformed before computing the residuals.
'fcn'
means take the experimental and simulated spectral data directly.
'int'
: integrate the two spectra.
'dint'
: use the double integrals of the two spectra.
'diff'
: compute the derivatives of the two spectra.
'fft'
: FFT both spectra before computing the residuals.
'fcn'
. If there are many resolved lines, 'int'
gives better convergence.
Scaling
'maxabs'
: make the maximum absolute of simulated and experimental spectrum equal
'minmax'
: rescales and shifts so that minimum and maximum of the two spectra are identical
'lsq'
: obtains the scaling factor by least-squares fitting, without shifting
'lsq0'
: same as 'lsq'
, but includes a constant baseline correction
'lsq1'
: same as 'lsq'
, but includes a linear baseline correction
'lsq2'
: same as 'lsq'
, but includes a quadratic baseline correction
'none'
: no scaling at all
If the baseline of the experimental spectra has been corrected beforehand, the best choice is 'maxabs'
.
This is also the default value.
If the noise level is high, 'lsq'
is a better choice:
FitOpt.Scaling = 'lsq';To include corrections for a baseline in the expermental spectrum, use
'lsq0
, 'lsq1
, or 'lsq2
.
There are some more options which are recognized by all fitting algorithms:
OutArg
[nArgOut useArg]
that tells esfit
how to use the output arguments of the simulation function. nArgOut
is the number of outputs provided by the function. esfit
will call the function asking for this number of outputs. The second number, useArg
, tells esfit
which of the output arguments actually contains the simulated data.
E.g. OutArg = [3 2]
indicates to call the function asking for three outputs and then to use output number 2 for the fitting.
For the common EasySpin simulation function, OutArg
is chosen automatically. For custom simulation functions that deviate in their output behavior, OutArg
might be useful.
maxTime
esfit
will terminate even if the fitting has not yet converged.
PrintLevel
Each of the available fitting algorithms can be fine-tuned using a set of parameters.
Parameters for Nelder/Mead downhill simplex:
RandomStart
RandomStart
is 0.
TolEdgeLength
TolFun
SimplexPars
[rho chi psi sigma]
, where rho
is the reflection coefficient, chi
is the expansion coefficient, psi
is the contraction coefficient, and sigma
is the shrink coefficient. The default values are [1,2,0.5,0.5]
.
delta
Parameters for Levenberg-Marquardt:
RandomStart
RandomStart
is 0.
TolStep
TolFun
lambda
delta
Parameters for Monte Carlo:
nTrials
TolFun
Parameters for the genetic algorithm:
PopulationSize
maxGenerations
maxGenerations
can be decreased.
Parameters for the grid search:
GridSize
Vary.g = [0 0.001 0]; Vary.lw = 0.2;Then
GridSize
can contain 1 or 2 numbers:
FitOpt.GridSize = 10; % 10 points for each parameter, making 100 grid points total FitOpt.GridSize = [20 3]; % 10 points along g and 3 along lw, giving a total of 60
Parameters for the particle swarm algorithm:
nParticles
TolFun
SwarmParams
[k w c1 c2]
. k
is the velocity limit
and determines how far a particle can move in an iteration (default 0.2). w
is the inertial
coefficient (default 0.5) and describes the propensity of the particle of moving in the same direction as
in the iteration before. c1
is the cognitive coefficient (default 2) and determines to what
degree a particle moves towards the currently optimal point. c2
is the social coeffient (default 1)
and describes how much the particle is following the other particles. Together, k
, c1
and c2
determine in which direction a particle will move in a given iteration.
Internally, esfit
rescales the parameters to be fitted. E.g. if the linewidth should be fitted and is expected to be around 5 mT, plus minus about 1.5 mT, Sys.lw = 3;
and Vary.lw = 1.5
. After rescaling, esfit
treats 5 mT as x = 0
, 3.5 mT as -1
and 6.5 mT as +1
.
For details about the algorithms implemented in esfit
, see e.g.
Here is a very simple example. Let's assume the experimental data are stored in expspec
.
Then the following code performs a least-squares fitting using the simplex algorithm.
Exp.mwFreq = 9.5; Sys0.g = [2.1 2.2] Sys0.lwpp = 0.1; Vary.g = [0.05 0.02]; esfit('pepper',expspc,Sys0,Vary,Exp);
See the example section for a full example. Also, read the user guide about fitting.