A titolo di esempio, si immagini di avere i dati sperimentali

FITTING di dati sperimentali
un esempio
A titolo di esempio, si immagini di avere i dati sperimentali
Xdata, Ydata nonché l’errore associato eYdata
Su questi dati si vuole fittare un modello del tipo Y=A+B*X+C*X 2
Di seguito riporto una procedura Matlab che genera i dati (secondo il modello + rumore) e poi ci
fitta il modello dato minimizzando una funzione che calcola il χ 2 del fit. La minimizzazione può
essere fatta sia usando il comando di Matlab FMINSEARCH , che generalmente è sufficiente,
sia usando il molto più sofisticato e potente pacchetto di funzioni di FMINUIT (della libreria di
routines CERNlibrary) così come è stato interfacciato nell’ambiente Matlab da G. Allodi.
Fminsearch restituisce i parametri che danno il miglior fit ed il χ 2 del fit.
In aggiunta Fminuit può restituire l’accuratezza della stima di ciascun parametro. Inoltre consente di
seguire strategia di minimizzazione diverse, e di chiamare la funzione minimizzanda per eseguire
dei plot intermedi.
PROCEDURA MATLAB:
clear all
Xdata=[0:.05:5]';
Ydata=-.2+.1*Xdata+2.5*Xdata.^2;
NOISE=.1; %rumore sperimentale che affligge le misure Ydata
Ydata=Ydata+NOISE*randn(size(Ydata));
eYdata=zeros(size(Ydata))+NOISE;
data=[Xdata,Ydata,eYdata];
figure(1)
clf
errorbar(Xdata,Ydata,eYdata/2,eYdata/2,'.');
pause
startpar=[-.1 .1 1];
%[bestpar,chival]=fminsearch('chi2',startpar,[],data)
[bestpar,ebestpar,chival]=fminuit('chi2',startpar,[],data)
hold on
plot(Xdata,bestpar(1)+bestpar(2)*Xdata+bestpar(3)*Xdata.^2,'-k');
FUNZIONE χ 2 DA MINIMIZZARE:
function chi2=chi2(parametri, data)
%Calcola il Chi2 ridotto (Taylor, 12.2 pag 186)
A=parametri(1);
B=parametri(2);
C=parametri(3);
Xdata=data(:,1);
Ydata=data(:,2);
eYdata=data(:,3);
N=length(Xdata); %Numero di osservazioni
p=length(parametri); %Numero di parametri del fit
model=A + B*Xdata + C*Xdata .^2 ;
chi2=sum ( (model - Ydata ).^2 ./ eYdata.^2 ) / (N-p);

HELP DELLA FUNZIONE FMINSEARCH
(disponibile anche ondine dentro a Matlab)
FMINSEARCH Multidimensional unconstrained nonlinear minimization (Nelder-Mead).
X = FMINSEARCH(FUN,X0) starts at X0 and finds a local minimizer X of the
function FUN. FUN accepts input X and returns a scalar function value
F evaluated at X. X0 can be a scalar, vector or matrix.
X = FMINSEARCH(FUN,X0,OPTIONS) minimizes with the default optimization
parameters replaced by values in the structure OPTIONS, created
with the OPTIMSET function. See OPTIMSET for details. FMINSEARCH uses
these options: Display, TolX, TolFun, MaxFunEvals, and MaxIter.
X = FMINSEARCH(FUN,X0,OPTIONS,P1,P2,...) provides for additional
arguments which are passed to the objective function,
F=feval(FUN,X,P1,P2,...).
Pass an empty matrix for OPTIONS to use the default values.
(Use OPTIONS = [] as a place holder if no options are set.)
[X,FVAL]= FMINSEARCH(...) returns the value of the objective function,
described in FUN, at X.
[X,FVAL,EXITFLAG] = FMINSEARCH(...) returns a string EXITFLAG that
describes the exit condition of FMINSEARCH.
If EXITFLAG is:
1 then FMINSEARCH converged with a solution X.
0 then the maximum number of iterations was reached.
[X,FVAL,EXITFLAG,OUTPUT] = FMINSEARCH(...) returns a structure
OUTPUT with the number of iterations taken in OUTPUT.iterations.
Examples
FUN can be specified using @:
X = fminsearch(@sin,3)
finds a minimum of the SIN function near 3.
In this case, SIN is a function that returns a scalar function value
SIN evaluated at X.
FUN can also be an inline object:
X = fminsearch(inline('norm(x)'),[1;2;3])
returns a minimum near [0;0;0].
FMINSEARCH uses the Nelder-Mead simplex (direct search) method.
See also OPTIMSET, FMINBND, @, INLINE.

HELP DELLA FUNZIONE FMINUIT
(disponibile anche ondine dentro a Matlab)
FMINUIT Multidimensional nonlinear minimization by means of the MINUIT
engine. Usage is similar (but not identical) to that of Matlab Fmins.
For a description of the MINUIT commands, read the MINUIT writeup
or type HELP during the interactive execution of Fminuit.
[BestPars, Errs, Chi2, ErrMatrix] = fminuit(FunName, InitGuess, ...);
minimizes a scalar user function named FunName, using InitGuess as initial
values of the variational parameters. Here FunName is a string and
InitGuess a vector.
Dots ... stand for a variable number of optional extra-arguments.
The user function depends on one or two variables: a vector of variational
parameters (same dimension as InitGuess) and, optionally, on arbitrary
structure of constant data (any data type except a string), respectively.
Functions of more than 2 variables are not supported.
- Note -
If the user function would need to be passed many matrices of constant
data, such matrices should rather be arranged into a cell array or a struct
(Matlab 5 or later) or a list (Scilab 2.5 or later), and passed as a single
argument to the user function.
[BestPars, Errs, Chi2, ErrMatrix] = fminuit(FunName, AuxFunName, InitGuess,
...);
(here AuxFunName is a string) behaves as in the previous case. In addition,
an auxiliary function named AuxFunName is called whenever a 'CALL 5'
Minuit command is issued. Such a function is passed the following
argument list:
(BestPars, FunName, ErrMatrix), or
(BestPars, Data, FunName, ErrMatrix) if the argument Data (constant data,
see below) is defined.
Values of BestPars and ErrMatrix are those currently returned by Minuit.
Return arguments of the auxiliary function are ignored.
The interface to AuxFunName is mainly intended for logging, or to draw a
run-time plot during the minimization process.
OPTIONAL ARGUMENTS (following InitGuess, in any order)
Data constant data: any data type supported by Matlab (Scilab),
except a string. If this argument is supplied to Fminuit, it is passed to
the user function FunName as its second argument. In the case of chi-square
fitting, Data usually consists of a 3-column (or 3-row) matrix of
experimental data: the independent variable, measured values of the
dependent variable, and error bars.
- Note to the case of a scalar Data -
The first occurrence of a scalar argument is interpreted as ConstError
(see below). Therefore, in the case that a scalar Data is required by the
user function to be minimized, ConstError has to be specified as well, and
Data must follow in the argument list.
ConstError a scalar parameter. The value returned by the user function to
Fminuit is internally divided by ConstError^2. In the case of chi-square
fitting, ConstError has the meaning of a uniform error bar assigned to
the experimental points. Default value is ConstError = 1.
'-a' or 'a' option switch. Interactive mode.
If the string constant '-a' or 'a' is supplied, Fminuit prompts for Minuit
commands interactively, until 'EXIT', or 'RETURN' is issued.
This is the default behaviour.

'-b' or 'b' option switch. Batch mode.
If the string constant '-b' or 'b' is supplied, Fminuit runs a sequence of
Minuit commands in batch mode without user control. The command sequence
can be defined by the (...,'-c', MinuitCommands, ...) option, or by
creating a global string variable named 'commands' in the caller workspace
(see below).
The default sequence is 'MINIMIZE; IMPROVE' .
..., '-c', MinuitCommands, .... option switch followed by a string
argument. Flow control.
If the string constant '-c' followed by a string argument MinuitCommands
is passed, Fminuit executes the Minuit commands defined in MinuitCommands
when in batch mode. Defining a global non-empty string variable named
'commands' in the caller workspace controls the program flow in the same
way. The latter feature is kept in Fminuit for backward compatibility,
and might be removed in future releases. Controlling the program flow via
the command line should be preferred.
If both the '-c' option and the global variable commands are present, the
latter is overridden.
Minuit commands are separated by one of the characters: ';', '\', '|',
'/', or '@'; for instance,
fminuit(... ,'-c', 'set par 1 3; fix 1; mini; rel 1; mini', ...)
- Note -
The '-c' option switch implicitly implies '-b' and is always effective;
the global variable commands is only effective when batch mode is
explicitly set via the '-b' (or 'b') option switch.
..., '-n', ParameterNames, ... option switch followed by a string
argument. Parameters Names.
If the string constant '-n' followed by a string argument ParameterNames
is passed, within Minuit, the variational parameters are given the names
specified in ParameterNames, instead of the default values 'par #1',
'par #2', ... . Defining a global non-empty string variable named
'parnames' in the caller workspace controls the parameter names in the
same way (backward compatibility; setting name via the command line should
be preferred).
If both the '-n' option and the global variable parnames are present, the
latter is overridden.
Parameter names are separated by commas or blanks, and must be as many as
the variational parameters. If the number of names does not match the
number of parameters, this option is neglected.
..., '-s', StepBounds, .... option switch followed by a matrix.
Definition of parameter steps and limits.
A string constant '-s' in the argument list, followed by a 2-, 3-, or
4-column matrix, defines new parameter steps and/or parameter limits.
Defining a global non-empty matrix variable named 'stepbounds' in the
caller workspace controls parameter steps and/or limits in the same way
(backward compatibility).
The '-c' switch always overrides the global variable stepbounds.
Format of StepBounds is the following.
Each line is referred to a variational parameter.
The first column contains indices of the parameters whose steps and/or
limits are to be modified.
- If StepBounds is a two- or four-column matrix, the second column
is interpreted as the new steps, i.e. StepBounds(k,2) is the step of
the StepBounds(k,1)-th parameter;
- If StepBounds is a three-column matrix, columns 2 and 3 are interpreted
as upper and lower limits, i.e. the StepBounds(k,1)-th parameter is
constrained between StepBounds(k,2) and StepBounds(k,3).
- If StepBounds is a four-column matrix, columns 2 is interpreted as new
steps, column 3 and 4 as parameter limits.

- Note -
step = 0 means that the parameter is kept fixed;
a negative step signifies usage of the default step definition.
RETURN VALUES
BestPars vector (same size as InitGuess): the optimized parameters
return by Fminuit.
Errs vector (same size as InitGuess): the linearized (parabolic)
errors associated to the optimized parameters BestPars.
Chi2 scalar: value of the user function for the variational
parameters BestPars, divided by ConstError^2. Usually, this is the
chi-square value corresponding to the fitting parameters returned by Fminuit
and the experimental data.
ErrMatrix the error matrix, proportional to the inverse Hessian matrix.
ErrMatrix is a (length(BestPars), length(BestPars)) matrix, corresponding
row-by-row, column-by-column to the best fit parameters.
Fixed parameters correspond to rows and column padded with zeros.
UNSUPPORTED
Nonlinear errors be can calculated by issuing a MINOS Minuit commands.
Their values however are not interfaced to Matlab (Scilab).
Copyright (C) G. Allodi 1996 - 2002