Simultaneously fit a function to an array of sqw objects.
Optionally allows background functions that vary independently for each sqw object.
Simultaneously fit several objects to a given function:
>> [wout, fitdata] = multifit_func (w, func, pin) % all parameters free
>> [wout, fitdata] = multifit_func (w, func, pin, pfree) % selected parameters free to fit
>> [wout, fitdata] = multifit_func (w, func, pin, pfree, pbind) % binding of various parameters in fixed ratios
With optional 'background' functions added to the global function, one per object
>> [wout, fitdata] = multifit_func (..., bkdfunc, bpin)
>> [wout, fitdata] = multifit_func (..., bkdfunc, bpin, bpfree)
>> [wout, fitdata] = multifit_func (..., bkdfunc, bpin, bpfree, bpbind)
If unable to fit, then the program will halt and display an error message.
To return if unable to fit, call with additional arguments that return status and error message:
>> [wout, fitdata, ok, mess] = multifit_func (...)
Additional keywords controlling which ranges to keep, remove from objects, control fitting algorithm etc.
>> [wout, fitdata] = multifit_func (..., keyword, value, ...)
Keywords are:
'keep' range of x values to keep
'remove' range of x values to remove
'mask' logical mask array (true for those points to keep)
'select' if present, calculate output function only at the points retained for fitting
'list' indicates verbosity of output during fitting
'fit' alter convergence critera for the fit etc.
'evaluate' evaluate function at initial parameter values only, with argument check as well
'chisqr' evaluate chi-squared at the initial parameter values (ignored if 'evaluate' not set)
Example:
>> [wout, fitdata] = multifit_func (..., 'keep', xkeep, 'list', 0)
Input:
======
win sqw object or array of sqw objects to be fitted
func_handle
Function handle to function to be fitted e.g. @gauss
Must have form:
y = my_function (x1,x2,... ,xn,p)
or, more generally:
y = my_function (x1,x2,... ,xn,p,c1,c2,...)
- x1,x2,.xn Arrays of x coordinates along each of the n dimensions
- p a vector of numeric parameters that can be fitted
- c1,c2,... any further arguments needed by the function e.g.
they could be the filenames of lookup tables for
resolution effects)
e.g. Two dimensional Gaussian:
function y = gauss2d(x1,x2,p)
y = p(1).*exp(-0.5*(((x1 - p(2))/p(4)).^2+((x2 - p(3))/p(5)).^2);
pin Initial function parameter values
- If the function my_function takes just a numeric array of parameters, p, then this
contains the initial values [pin(1), pin(2)...]
- If further parameters are needed by the function, then wrap as a cell array
{[pin(1), pin(2)...], c1, c2, ...}
pfree [Optional] Indicates which are the free parameters in the fit
e.g. [1,0,1,0,0] indicates first and third are free
Default: all are free
pbind [Optional] Cell array that indicates which of the free parameters are bound to other parameters
in a fixed ratio determined by the initial parameter values contained in pin:
pbind={1,3} parameter 1 is bound to parameter 3.
pbind={{1,3},{4,3},{5,6}} parameter 1 bound to 3, 4 bound to 3, and 5 bound to 6
In this case, parmaeters 1,3,4,5,6 must all be free in pfree.
To explicity give the ratio, ignoring that determined from pin:
pbind=(1,3,0,7.4) parameter 1 is bound to parameter 3, ratio 7.4 (the extra '0' is required)
pbind={{1,3,0,7.4},{4,3,0,0.023},{5,6}}
To bind to background parameters (see below), give the index of the background function
handle in the cell array bkdfunc defined below.
pbind={1,3,[7,2]} parameter 1 bound to background parameter 3 of background
function handle bkdfunc{7,2}.
pbind={1,3,[7,2],3.14} Give explicit binding ratio.
Optional background function(s):
--------------------------------
It is sometimes convenient to fit a global function, func, to the collection of objects w,
but have a function that might be used to provide an object-specific background. For example,
we may want to fit S(Q,w) to several one-dimensional cuts, but have an independent linear
background for each cut. These are defined similarly to the above
bkdfunc Cell array of background function handles
- If contains a single handle, then the same function applies to every object
in the collection of objects, w, that is to be fitted.
(Internally, is expanded into a cell array of the same size as w. If wish to bind
background parameters to a particular background handle, then refer to the
corresponding index of that expanded array).
- Otherwise, the size of the cell array must match the size of w, and there
will be a one-to-one correspondence of the background function handles to the elements of w.
The form required for the functions is identical to that for func above, i.e.
y = my_function (x1,x2,... ,xn,p)
or, more generally:
y = my_function (x1,x2,... ,xn,p,c1,c2,...)
- x1,x2,.xn Arrays of x coordinates along each of the n dimensions
- p a vector of numeric parameters that can be fitted
- c1,c2,... any further arguments needed by the function e.g.
they could be the filenames of lookup tables for
resolution effects)
e.g. Two dimensional Gaussian:
function y = gauss2d(x1,x2,p)
y = p(1).*exp(-0.5*(((x1 - p(2))/p(4)).^2+((x2 - p(3))/p(5)).^2);
bpin Cell array of initial parameter values for the background function(s), following the
same definitions and conventions as pin
- If a single element in the cell array, then will be used for every background function;
- Otherwise, the size of the cell array must match the size of w, and there will be a
one-to-one correspondence of the elements to the background initial parameters
bpfree Array, or cell array of arrays indicating which parameters are free to vary.
- If single array, or cell array with a single array, then applies to every background function
- Otherwise, the size of the cell array must match the size of w, and there will be a
one-to-one correspondence of the elements to indicate the free background parameters
bpbind Cell array of binding cell arrays of the form defined for pbind. Indicates how background
parameters are bound. Take care here: as the object that defined the binding is itself
a cell array of cell arrays, it is easy to get confused as to whether the binding description
applies to all background functions, or if it is a set of distinct binding descriptions, one
for each background function. The size of the outermost cell array dictates which:
- If a single element in the cell array, then will be used for every background function;
- Otherwise, the size of the cell array must match the size of w, and there will be a
one-to-one correspondence of the elements to the background functions.
Examples of a single binding description:
{1,4} Background parameter (bp) 1 is bound to bp 3, with the fixed
ratio determined by the initial values
{2,3,[7,2]} Bp 2 bound to bp 3 of background function handle bkdfunc{7,2}
{5,11,0} Bp 5 bound to parameter 11 of the global fitting function, func
{{1,4}, {2,3,[7,2]}, {5,11,0}} Several bindings defined together
{5,11,0,0.013} Explicit ratio for binding bp 5 to parameter 11 of the global fitting function
{1,4,[7,2],14.15} Explicit ratio for binding bp 1 to bp 4 of background function [7,2]
In a call to multifit: as an example of the need to take care:
bpbind = {{1,4}, {2,3,[7,2]}, {5,11,0}} binding description for three separate backgrounds
bpbind = { {{1,4}, {2,3,[7,2]}, {5,11,0}} } Binding description for all background functions
Optional keywords:
------------------
'list' Numeric code to control output to Matlab command window to monitor
status of fit
=0 for no printing to command window
=1 prints iteration summary to command window
=2 additionally prints parameter values at each iteration
'fit' Array of fit control parameters
fcp(1) relative step length for calculation of partial derivatives
fcp(2) maximum number of iterations
fcp(3) Stopping criterion: relative change in chi-squared
i.e. stops if chisqr_new-chisqr_old < fcp(3)*chisqr_old
'keep' Array or cell array of arrays giving ranges of x to retain for fitting.
- single array: applies to all elements of w
- a cell array of arrays must have length the same as w, and describes the
keep ranges for those elements one-by-one.
A range is specified by an arrayv of numbers which define a hypercube.
For example in case of two dimensions:
[xlo, xhi, ylo, yhi]
or in the case of n-dimensions:
[x1_lo, x1_hi, x2_lo, x2_hi,..., xn_lo, xn_hi]
More than one range can be defined in rows,
[Range_1; Range_2; Range_3;...; Range_m]
where each of the ranges are given in the format above.
'remove' Ranges to remove from fitting. Follows the same format as 'keep'.
If a point appears within both xkeep and xremove, then it will
be removed from the fit i.e. xremove takes precendence over xkeep.
'mask' Array, or cell array of arrays, of ones and zeros, with the same number
of elements as the data arrays in the input object(s) in w. Indicates which
of the data points are to be retained for fitting.
'select' Calculates the returned function values, wout, only at the points
that were selected for fitting by 'keep', 'remove', 'mask' and were
not eliminated for having zero error bar etc; this is useful for plotting the output, as
only those points that contributed to the fit will be plotted.
A final useful pair of keyword is:
'evaluate' Evaluate the fitting function at the initial parameter values only. Useful for
checking the validity of starting parameters.
'chisqr' If 'evaulate' is set, then if this option keyword is present the reduced
chi-squared is evaluated. Otherewise, chi-squared is set to zero.
Output:
=======
wout Array or cell array of the objects evaluated at the fitted parameter values
If there was a problem i.e. ok==false, wout=[]
fitdata Result of fit for each dataset
fitdata.p - parameter values
fitdata.sig - estimated errors of global parameters (=0 for fixed parameters)
fitdata.bp - background parameter values
fitdata.bsig - estimated errors of background (=0 for fixed parameters)
fitdata.corr - correlation matrix for free parameters
fitdata.chisq - reduced Chi^2 of fit (i.e. divided by
(no. of data points) - (no. free parameters))
fitdata.pnames - parameter names
fitdata.bpnames- background parameter names
If there was a problem i.e. ok==false, fitdata=[]
ok True if all ok, false if problem fitting.
mess Character string contaoning error message if ~ok; '' if ok
EXAMPLES:
Fit a 2D Gaussian, allowing only height and position to vary:
>> ht=100; x0=1; y0=3; sigx=2; sigy=1.5;
>> [wfit, fdata] = fit(w, @gauss2d, [ht,x0,y0,sigx,0,sigy], [1,1,1,0,0,0])
Allow all parameters to vary, but remove two rectangles from the data and allow
independent planar background for each plane in the units of the x and y axes:
>> ht=100; x0=1; y0=3; sigx=2; sigy=1.5;
>> const=0; slope_x=0; slope_y=0;
>> [wfit, fdata] = fit(w, @gauss2d, [ht,x0,y0,sigx,0,sigy], @planar, [const,slope_x,slope_y], ...
'remove',[0.2,0.5,2,0.7; 1,2,1.4,3])
% Overloaded methods:
sqw/multifit_func
sqw/multifit_func
d4d/multifit_func
d3d/multifit_func
d2d/multifit_func
d1d/multifit_func
d0d/multifit_func