iFit: iData objects fitting


  1. Fitting a model to the Data
    1. Getting most of the fit results (4th output argument)
  2. Specifying/configuring the optimization method
    1. Choosing an optimization method
    2. Configuring the optimization method (4th input argument)
    3. Monitoring the performance of the optimization method and the fit process
  3. Model parameter constraints
    1. Fixed parameters
    2. Parameters varying within limits
    3. Limiting parameter change
    4. Other constraints/restraints
    5. Additional arguments to the model
    6. Using constraints within the model
  4. Estimating the model parameter uncertainties and the fit quality
  5. Building model functions
  6. Specifying the optimization criteria
  7. Issues when fitting


Commands we use in this page: iData/fits, iFunc, Models and Optimizers routines

This documentation details the procedure that can be used to find optimal parameter set from a model in order to match (fit) an iData object Signal with axes.

fit: { vary vector p d so that distance(Signal, Model) is minimal }

where distance is the criteria for the fit, which measures how far the model is from the data. There is a number of possible distance definitions, detailed below. You can refer to the mathematical optimization definition.

Fitting a model to the Data

The fits function for iData and iFunc objects provides a simple mean to find the best model parameters which fits a data set.
>> a=load(iData, [ ifitpath 'Data/sv1850.scn' ])
>> p=fits(a);
'Amplitude' 'Centre' 'HalfWidth' 'Background'
0.6786 1.0008 0.0035 0.0002
will fit the data Signal/Monitor to a default 1D Gaussian function 'gauss', starting from automatically guessed initial model parameters. Of course, it is possible to chose explicitly the model function, and the starting parameter set with e.g.
>> p=fits(a, 'gauss');                    % specify model function to use: gauss, and use guessed parameters
>> p=fits(a, gauss, [ 0.5 1 0.01 0 ]); % specify the starting parameters for the model function (3rd input arg)
The two first arguments to fits are the data object (iData class) and the model object (iFunc class).
The result of the fits method is the model parameter set that matches best the data. The number of fit parameters is not limited.

A number of model functions is available from the Models sub-library. The list of all available fit functions can be obtained from the command:iFit: default
        functions (in Models)
>> fits(iData); 			% or fits(iFunc)
Refer to the section below in order to learn how to define new fit models with the ifitmakefunc tool, or directly with the iFunc class. This latter supports many operators which allow to build complex models from simpler ones, and store them into model objects (variables). You can get information about a model by inquiring its name. When a single input parameter is specified, starting parameters are guessed :
>> gauss
>> disp(gauss) % return information about the function
>> plot(gauss) % plot the function with a default parameter set and axes
>> gauss(p, x) % evaluate the function with parameters 'p' and axis 'x'
>> gauss(p) % same as above with guessed axes
>> g=gauss % g is now a variable of class iFunc
>> disp(g)
A fast way to define a new model and fit it to the data is to directly execute, with p the parameter vector, and x,y,... the axes:
>> p=fits(a, 'p(1)*x+p(2)');
>> p=fits(a, @(p,x)p(1)*x+p(2)); % same with a function handle
which will transparently create the model function, and then use it for the fit.

fits(iData)In order to estimate the values of the model for the fit parameters onto the iData object data set axes, use the model as argument to the object:
>> b= a(gauss, p) % evaluate the 'gauss' model onto the 'a' axes, with parameters 'p'
>> plot([ a b ]) % with parameters 'p', and plot both data and fit
In this case, the evaluated model is an iData object, which contains Parameter and Model aliases that hold the model parameters and description.

The fits function work with any data and model dimensionality, and can also build multi-dimensional models from lower dimensionality functions (e.g. making a 2D Gaussian from multiplied orthogonal 1D Gaussians).

Warning: if you do not specify the optimization method to use (see below), an automatic guess is made, based on the number of parameters to fit and the level of noise in the data and the model. As this choice is made among a list of applicable optimizers, two consecutive fits may chosse different optimizers and produce different fit results. In that case, it is advisable to select a specific optimizer to use with syntax: fits(data, model, init_pars, optimizer).

Getting most of the fit results (4th output argument)

The fits method can return additional arguments
>> [parameters,criteria,message,output]= fits(a, model, initial_parameters,...)
The input arguments allow to specify:
  1. a: an iData object, or an array of objects
  2. model: the name of the function or a function handle
  3. initial_parameters: a set of starting parameters for the model. Using an empty set will trigger an automatic estimate (guess) of the starting model parameters.
The returned arguments are similar to those returned by the Matlab fminsearch non-linear optimizer:
  1. parameters: is the best fit parameter set for the model obtained during the optimization
  2. criteria: is the criteria value (least square χ by default - see below on how to change this setting)
  3. message: is the final optimization routine state (converged, failed, ...)
  4. output: is a structure that holds all the information gathered during the fit procedure
The most interesting output argument result is the 4th one output, which provides significantly more information such as:
Last, the starting parameter 'guess' can also be specified as a structure which fields contain numbers, or as a string with members separated with the ';' character, such as in the following example:
>> p.Amplitude=0.5; p.Center=1; p.HalfWidth=0.0035; p.Background=1e-4; 		% optimize named parameters 
>> fits(a, gauss, p) % The result is also returned as a structure.
>> fits(a,'gauss','Amplitude=0.5; Center=1; HalfWidth=0.0035; Background=1e-4') % create the structure above and fit...
All model named parameters must be specified.

Specifying/configuring the optimization method

The iData fits method is a wrapper to any of the Optimizers optimizers, with a specified fit criteria. Each optimizer can be customized (e.g. the maximum number of iterations, stop/convergence conditions, ...) with an options structure.
>> [parameters,criteria,message,output]= fits(a, model, initial_parameters, options)

Choosing an optimization method

The Optimizers sub-library provides 21 different optimization techniques. No doubt that choosing one at first sight is difficult. We provide below a set of preferred optimizers, based on a careful comparison explained in the Optimizers documentation.

The process of choosing a sensible optimizer can be automated by using the fmin optimizer (which is the default choice). The objective function is analyzed and a set of best optimizers is determined, depending on the execution time, the number of parameters, and the type of objective - noisy or continuous:
>> fits(a, model, initial_parameters, 'fmin')
When more than one optimizer is suitable, a random choice is performed, with a weighting by the success ratio and the number of function calls (optimization speed). Successive calls to fmin with the same problem to solve may result in different optimizers and solutions.

Optimizer
Description
Mean Success ratio (%)
fminpso Particle Swarm Optimization (slow but very efficient)
97.9 / 84.1
fminimfil Unconstrained Implicit filtering
93.5 / 53.8
fminralg Shor R-algorithm (avoid when noisy) 88.9 / 16.2
fminhooke Hooke-Jeeves direct search 97.1/ 56.3
fmincmaes Evolution Strategy with Covariance Matrix Adaptation 88.9 / 71.2
fminsimpsa simplex/simulated annealing 97.1 / 84.8
fminsce shuffled complex evolution
95.7 / 85.2
fminpowell Powell with Coggins line search
99.2 / 51.7
fminmarkov
Markov Chain Monte Carlo robust, but slow
Table 1: A selection among the most efficient and fast optimization methods. The two success ratio are given for continuous and noisy functions resp.

The choice of the optimizer is done through a 4th input argument options to fits (see below).

The list of all available optimizer methods can be obtained from the command:
>> fits(iData)

Configuring the optimization method (4th input argument)

In order to use these optimizers, one just has to specify their names as a fits input argument, e.g. options='fminimfil' :
>> [parameters,criteria,message,output]= fits(a, model, initial_parameters, 'fminimfil')
will select the Unconstrained Implicit filtering coupled with BFGS fminimfil optimizer to perform the fit. The default optimizer configuration will be used, as obtained from the optimset function, or from the method itself with 'defaults' as parameter. This latter call (i.e. not with optimset) provides more information:
>> options=fminimfil('defaults')	% get the default optimizer configuration parameters
The optimizer configuration is a structure which members enable to tune the behavior of the optimization process. Each of the fields can be changed, e.g.
>> a=load(iData, [ ifitpath 'Data/sv1850.scn' ])
>> options=fminimfil('defaults')
>> options.TolFun=0.01;
>> p=fits(a, model, [], options); % fit with the customized options, and guessed starting parameters
The options specify the internal optimiser configuration, as returned by optimset:
>> optimset
Display: [ off | iter | notify | final ]
MaxFunEvals: [ positive scalar ]
MaxIter: [ positive scalar ]
TolFun: [ positive scalar ]
TolX: [ positive scalar ]
FunValCheck: [ on | {off} ]
OutputFcn: [ function | {[]} ]
PlotFcns: [ function | {[]} ]
Diagnostics: [ on | {off} ]
In addition to the default optimset structure fields, the iData fits and the Optimizers optimizers use additional members in the structure:
The options can also be entered as a single string such as in :
>> p=fits(a, model, [], 'optimizer=fminimfil; Display=iter; OutputFcn=fminplot; TolFun=0.01;');
which avoids creating a structure.

Monitoring function, to be used in OutputFcn and PlotFcns, are:

Monitoring the performance of the optimization method and the fit process

In order to follow the optimization process, you may define a call to a user function at each optimization iteration. A default plotting facility has been implemented as the fminplot function:
>> a=load(iData, [ ifitpath 'Data/sv1850.scn' ])
>> options=fminimfil('defaults')
>> options.OutputFcn='fminplot';
>> p=fits(a, gauss, [], options)
p =

0.6263 1.0008 -0.0037 0.0002
>> b = a(gauss, p)
>> figure; plot([ a b ])
fits with
          options.OutputFcn=fminplot iData fits(gauss)
            with plot
Left: The options.OutputFcn='fminplot' window on the right, showing the criteria evolution with the optimization iteration and up to the 3 first fit parameters. The red dot indicates the current/final parameter set and criteria. Right: The final fit result on the left.

Also, as explained above, it is possible to obtain, on optimization completion, the whole criteria, and parameter set history, as a function of iterations/criteria evaluations.
>> [p,criteria,message,output]= fits(a, gauss, [], options)
>> output.parsHistory
>> output.criteriaHistory
The monitoring window above can be re-displayed with:
>> fminplot(output);

Model parameter constraints

In many cases, the model parameters are to be constrained. Some optimization specialists call these restraints, that is parameter values constraints. This includes some fixed values, or bounds within which parameters should be restricted. This is given to the fits method by mean of a 5th input argument constraints :
>> [parameters,criteria,message,output]= fits(a, model, initial_parameters, options, constraints)

In short, the constraints is a structure with the following members, which should all have the same length as the model parameter vector:
All these constraints may be used simultaneously.

The constraints input argument can also be entered as a character string, like the input parameters and options :
constraints='min=[0 0 0 0]; max=[1 10 3 0.1]; eval=p(4)=0';
As constraints apply on the iFunc model, it is possible to manipulate all constraints using the object syntax and methods, as explained below.

Fixed parameters

To fix some of the model parameters to their starting value, you just need to define constraints as a vector with 0 for free parameters, and 1 for fixed parameters, e.g. :
>> a=load(iData, [ ifitpath 'Data/sv1850.scn' ])
>> p=fits(a, gauss, [], 'fminimfil', [ 1 0 0 0 ])
** Minimization performed on parameters:
'Amplitude' 'Centre' 'HalfWidth' 'Background'

0.5936 0.9998 0.0018 0.0050

p =

0.5936 1.0008 -0.0037 0.0002

will fix the first model parameter, which is here the Amplitude. This parameter name can be checked by simply entering the name of the model, which returns some information structure :
>> disp(gauss)
ans = iFunc 1D model:
Expression: @(p,x)p(1)*exp(-0.5*((x-p(2))/p(3)).^2)+p(4)
Description: Single 1D Gaussian model
Tag: 'iF420775'
Date: '17-Jul-2012 15:50:11'
Name: 'Gaussian (1D) [gauss]'
Parameters: {'Amplitude' 'Centre' 'HalfWidth' 'Background'}
Guess: [function_handle]
Dimension: 1
ParameterValues: []
UserData: ''

Parameters:
p( 1)= Amplitude
p( 2)= Centre
p( 3)= HalfWidth
p( 4)= Background
A similar behavior is obtained when setting constraints as a structure with a fixed member :
>> constraints.fixed = [ 1 0 0 0 ];
>> p=fits(a, gauss, [], 'fminimfil', constraints);
The constraints.fixed vector should have the same length as the model parameter vector.

Once the iFunc model object has been instantiated, it is possible to fix/free its parameters with the lock/munlock methods, or more explicitly:
>> f=gauss;			% create a Gaussian model
>> f.Amplitude = 'fix'; % fix its Amplitude. 'free' would free it
>> mlock(f,
'Amplitude') % same as above. munlock would free it.

Parameters varying within limits

If one needs to restrict the exploration range of parameters, it is possible to define the lower and upper bounds of the model parameters. This can be done by setting the 5th fits argument to the lower bounds lb, and the 6th to the upper ub, e.g. :
>> a=load(iData, [ ifitpath 'Data/sv1850.scn' ])
>> p=fits(a, 'gauss', [], 'fminimfil', [ 0.5 0.8 0 0 ], [ 1 1.2 1 1 ])
lb ub
A similar behavior is obtained by setting constraints as a structure with members min and max :
>> constraints.min = [ 0.5 0.8 0 0 ];
>> constraints.max = [ 1 1.2 1 1 ];
>> p=fits(a, 'gauss', [], 'fminimfil', constraints);
The constraints min and max vectors should have the same length as the model parameter vector.

Once the iFunc model object has been instantiated, it is possible to bound its parameters with the xlim method, or more explicitly:
>> f=gauss;			% create a Gaussian model
>> f.Amplitude = [ 0 10 ]; % bound Amplitude between 0 and 10
>> xlim(f, 'Amplitude', [0 10]) % same as above

Limiting parameter change

Last, it is possible to restrict the change rate of parameters by assigning the constraints.steps field to a vector. Each non-zero value then specifies the absolute change that the corresponding parameter can vary between two optimizer iterations.

In short, the constraints structure can have the following members, which all should have the same length as the model parameter vector:

Other constraints/restraints

The constraints.eval member can be used to specify any other constraint/restraint by mean of

For instance one could use constraints.eval='p(3)=p(1);'.


All these constraints may be used simultaneously.
The constraints input argument can also be entered as a character string, like the input parameters and options :

constraints='min=[0 0 0 0]; max=[1 10 3 0.1]; eval=p(4)=0';

NaN values in these constraints are ignored (the corresponding parameters are not constrained).

Once the iFunc model object has been instantiated, it is possible to restraint its parameters explicitly with expressions:
>> f=gauss;			% create a Gaussian model
>> f.Amplitude = 'abs(p(1))'; % set Amplitude as its absolute value (Amplitude is p(1), as listed from disp(f))
>> f.Amplitude = 'abs("Amplitude")'; % same as above, double-quoting requests parameter name search in model
>> f.Amplitude = '"Amplitude" - "Background"'; % it is possible to use any parameter in expressions

Additional arguments to the model

When defining a model with n parameters and rank m (that is m axes), any fit process will optimize the n parameters by evaluating the model on the m axes. If you need to use additional arguments to the model, which are fixed during a fit/optimization, you can access them from the 'varargin' variable in the Expression and Constraints. varargin{1} is the first additional argument, varargin{2} the second...

>> f=gauss;
>> disp(f)
f = iFunc 1D model:
Expression: signal = feval(@(p,x)p(1)*exp(-0.5*((x-p(2))/p(3)).^2)+p(4), p, x);
Description: Single 1D Gaussian model
Tag: 'iF457990'
Date: '16-Sep-2013 10:03:45'
Name: 'Gaussian (1D) [gauss]'
Parameters: {'Amplitude' 'Centre' 'HalfWidth' 'Background'}
Guess: @(x,s)[NaN,m1(x,s-min(s(:))),m2(x,s-min(s(:))),NaN]
Constraint: [1x1 struct]
Dimension: 1
ParameterValues: []
UserData: ''

Parameters (4):
p( 1)= Amplitude
p( 2)= Centre
p( 3)= HalfWidth
p( 4)= Background

>> f([1 2 3 4], 1:10) % evaluate with parameters p=[1 2 3 4] and axis x=1:10
ans =
4.9460 5.0000 4.9460 4.8007 4.6065 4.4111 4.2494 4.1353 4.0657 4.0286
>> f.Constraint.eval='disp(varargin)'; % add a constraint to be evaluated, using additional arguments
>> f([1 2 3 4], 1:10, 'eee') % evaluate again, with additional arguments
'eee' <-- here is displayed 'varargin' when evaluating Constraint.eval
ans =
4.9460 5.0000 4.9460 4.8007 4.6065 4.4111 4.2494 4.1353 4.0657 4.0286
>> f.Expression=@(p,x, varargin)p(1)*exp(-0.5*((x-p(2))/p(3)).^2)+p(4)+varargin{:}
>> f([1 2 3 4], 1:10, 1)
[1] <-- here is displayed 'varargin' when evaluating Constraint.eval ans = <-- and we add it the the result as specified in the new Expression
5.9460 6.0000 5.9460 5.8007 5.6065 5.4111 5.2494 5.1353 5.0657 5.0286

Using constraints within the model

It may be easier to store constraints directly within the model used during the fit. The main advantage is that constraints can be directly related to the name of the model parameters.

To set a constraint on a model parameter, define the 'constraint' input argument when calling fits or set the constraint directly on the model parameters with:
>> model.parameter='fix'     % to lock its value during a fit process
>> model.parameter='clear' % or 'free' to unlock value during a fit process >> model.parameter=[min max] % to bound value
>> model.parameter=[nan nan] % to remove bound constraint
>> model.parameter='' % to remove all constraints on 'parameter'
>> model.parameter='expression' % to set the parameter from an expression
>> model.Constraint='' % to remove all constraints
>> model.Constraint = 0; % to unlock/free all Parameters during a fit process
>> model.Constraint = 1; % to lock/fix all Parameters during a fit process
Any parameter name surrounded by double quotes, e.g. "Amplitude", is replaced by the corresponding p(n) value in an expression used for setting a parameter value (cross-constraints). For instance
>> f=gauss;			% create a Gaussian model
>> f.Amplitude = 'fix'; % fix its Amplitude
>> f.Background = [0 1e-4]; % bound its background
>> f.Width = 'p(1)/1000'; % use an expression referring to p(1)/1000 value
>> f.Width = '"Amplitude"/1000'; % same as above with direct naming as p(1)=Amplitude
Alternatively, you can use the mlock, munlock and xlim methods:
>> mlock(f, {'Amplitude','Background'})	% fix these 2 parameters, same as setting parameters to 'fix' 
>> munlock(f, 'Background') % unlock that parameter, same as f.Background='clear'
>> xlim(f, 'Background', [0 1e-3]) % force parameter within range, same as f.Background=[min max]
>> xlim(f, 'Background', []) % remove limits constraint
Last, you can fix/clear/bound parameters based on a regular expression search such as:
>> mlock(f, regexp(f.Parameters, 'Amplitude|Background'})
where we have used the '|' OR operator.

To list parameters which are fixed, free and bounded, use:
>> mlock(f)
>> munlock(f)
>> xlim(f)
which return the number of parameters in each category.

Estimating the model parameter uncertainties and the fit quality

Some theoretical notes  about the goodness of fit are indicated in the Optimizers help.

To get the fit statistics, you need to use the syntax:
>> [parameters,criteria,message,output]= fits(data, model, ...)
i.e. retrieve all output arguments. Then the 'output' 4-th returned argument holds lots of information.

The uncertainty on the fit parameters can be obtained from:
Additionally, the output.parsHessianCorrelation indicates the cross-correlations between parameters (off-diagonal values larger that 0.7 indicate cross-correlated parameters).
The Hessian statistics are explicitly computed when options.Diagnostics='on', or when the time required for computation does not exceed about a minute.
The parameter uncertainties are displayed when options.Display is set to 'final' or 'iter'.
The Hessian method is very sensitive to noise in the objective function. All these Hessian final computations (which may take time) can be skipped when using options.Diagnostics='off'.

We also explain in the Optimizers help how the parameter distributions (their variation around the best optimum value) can be plotted. This is well known from heuristic optimizers and e.g. the Markov Chain Monte-Carlo.

The fit quality can be assessed from the output.corrcoef returned value which goes from 0 (very bad fit) to 1 (perfect fit). A value higher than 0.95 is usually very good.

Building model functions

ifitmakefuncThe ifitmakefunc tool has been designed to automatically create model functions from either a single expression, or a more detailed description. Refer to the Models/Model Builder.
>> h=ifitmakefunc;                              % pops-up a dialog to define the new fit function/model
>> h=iFunc('p(1)*exp( (x-p(2))/p(3) )'); % directly create an iFunc model
>> fits(a, h)
The resulting model has the ability to identify itself ('identify', provide detailed information), compute automatic starting parameters ('guess'), display itself ('plot'), and evaluate its value of course. It can be directly used with fits, either with their name, or their function handle.

It is even possible to directly call the fitting method and create the model function on the fly, which makes the fit much easier for simple functions that can be written as a single expression:
>> fits(a, 'p(1)*exp( (x-p(2))/p(3) )');	% does the same as above, in a single command
To assemble existing functions into new ones, you may use e.g. :
>> h=gauss+lorz; h.Constraint = 'p(8)=0;';
which creates a new function which is the sum of a Gaussian and a Lorentzian. The second redundant Lorentzian Background p(8) parameter is forced to 0 so that it does not correlate with the Gaussian Background p(4). Other function information/parameter names (here not specified) are guessed/defaulted.

It is even possible to convolute and correlate functions and data sets as new function definitions. Refer to the Models page.

In case the model requires additional arguments, just pass them to the fits method (arguments above the 5th)
>> p=fits(a, model, [ 0.5 1 0.01 0 ],'','',additional_arguments);
assumes the model function has syntax
model(p, axes from object, additional_arguments)

Specifying the optimization criteria

The default optimization criteria is the 'least square error':

χ2 = ∑ (Signal-Model)2.

When the Monitor is defined (see iData definition), the Signal is normalized, so that

χ2 = ∑ (Signal/Monitor-Model)2.

When the Error on the Signal is available, the weighted criteria reads

χ2 = ∑ [(Signal/Monitor-Model)/Error]2.

The options.criteria can be defined as the name of a criteria function with syntax criteria(Signal, Error, Model) where all arguments should have the same dimension, and the default criteria is:

options.criteria = 'least_square'; % criteria definition

The Signal will be normalized to the Monitor prior to calling the criteria. The Error is used as weight.
The criteria is then normalized to the number of degrees of freedom by dividing it by

χ2 ← χ2 /(number_of_data_points_in_the_Signal - number_of_parameters -1)

The following pre-defined functions can be used as criteria:

criteria
(Signal,Error,Model)
Expression
comment
least_square
(|Signal-Model|/Error)2
non-robust. Ref.
least_absolute
|Signal-Model|/Error
robust. Ref.
least_median
median(|Signal-Model|/Error)
robust, scalar. Ref.
least_mean
mean(|Signal-Model|/Error) Ref.
least_max
max(|Signal-Model|/Error)
non-robust, scalar. Ref.
least_rfactor
(|Signal-Model|/Error)2/(Signal/Error)2 non-robust. Ref.
max_likelihood
(|Signal-Model|/σ)2 + 1/2 log(2πσ) Ref. Minimize -log(L).
The σ 'error' is refined during the fit.
max_corrcoef
1-corrcoef(Signal, Model)
Ref.

Issues when fitting

If you can not obtain a good fit, you can think about the following things to do:
Good luck !


E. Farhi - iFit/iData fitting - Mar. 22, 2017 $Revision: 1229 $ - back to Main iFit Page ILL,
          Grenoble, France <www.ill.eu>