;+
; NAME:
; MPFIT2DFUN
;
; AUTHOR:
; Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
; craigm@lheamail.gsfc.nasa.gov
; UPDATED VERSIONs can be found on my WEB PAGE:
; http://cow.physics.wisc.edu/~craigm/idl/idl.html
;
; PURPOSE:
; Perform Levenberg-Marquardt least-squares fit to a 2-D IDL function
;
; MAJOR TOPICS:
; Curve and Surface Fitting
;
; CALLING SEQUENCE:
; parms = MPFIT2DFUN(MYFUNCT, X, Y, Z, ERR, start_parms, ...)
;
; DESCRIPTION:
;
; MPFIT2DFUN fits a user-supplied model -- in the form of an IDL
; function -- to a set of user-supplied data. MPFIT2DFUN calls
; MPFIT, the MINPACK-1 least-squares minimizer, to do the main
; work. MPFIT2DFUN is a specialized version for two-dimensional
; data.
;
; Given the data and their uncertainties, MPFIT2DFUN finds the best set
; of model parameters which match the data (in a least-squares
; sense) and returns them in an array.
;
; The user must supply the following items:
; - Two arrays of independent variable values ("X", "Y").
; - An array of "measured" *dependent* variable values ("Z").
; - An array of "measured" 1-sigma uncertainty values ("ERR").
; - The name of an IDL function which computes Z given (X,Y) ("MYFUNCT").
; - Starting guesses for all of the parameters ("START_PARAMS").
;
; There are very few restrictions placed on X, Y, Z, or MYFUNCT.
; Simply put, MYFUNCT must map the (X,Y) values into Z values given
; the model parameters. The (X,Y) values are usually the independent
; X and Y coordinate positions in the two dimensional plane, but need
; not be.
;
; MPFIT2DFUN carefully avoids passing large arrays where possible to
; improve performance.
;
; See below for an example of usage.
;
; USER FUNCTION
;
; The user must define a function which returns the model value. For
; applications which use finite-difference derivatives -- the default
; -- the user function should be declared in the following way:
;
; FUNCTION MYFUNCT, X, Y, P
; ; The independent variables are X and Y
; ; Parameter values are passed in "P"
; ZMOD = ... computed model values at (X,Y) ...
; return, ZMOD
; END
;
; The returned array YMOD must have the same dimensions and type as
; the "measured" Z values.
;
; User functions may also indicate a fatal error condition
; using the ERROR_CODE common block variable, as described
; below under the MPFIT_ERROR common block definition.
;
; See the discussion under "ANALYTIC DERIVATIVES" and AUTODERIVATIVE
; in MPFIT.PRO if you wish to compute the derivatives for yourself.
; AUTODERIVATIVE is accepted and passed directly to MPFIT. The user
; function must accept one additional parameter, DP, which contains
; the derivative of the user function with respect to each parameter
; at each data point, as described in MPFIT.PRO.
;
; CREATING APPROPRIATELY DIMENSIONED INDEPENDENT VARIABLES
;
; The user must supply appropriate independent variables to
; MPFIT2DFUN. For image fitting applications, this variable should
; be two-dimensional *arrays* describing the X and Y positions of
; every *pixel*. [ Thus any two dimensional sampling is permitted,
; including irregular sampling. ]
;
; If the sampling is regular, then the x coordinates are the same for
; each row, and the y coordinates are the same for each column. Call
; the x-row and y-column coordinates XR and YC respectively. You can
; then compute X and Y as follows:
;
; X = XR # (YC*0 + 1) eqn. 1
; Y = (XR*0 + 1) # YC eqn. 2
;
; For example, if XR and YC have the following values:
;
; XR = [ 1, 2, 3, 4, 5,] ;; X positions of one row of pixels
; YC = [ 15,16,17 ] ;; Y positions of one column of
; pixels
;
; Then using equations 1 and 2 above will give these values to X and
; Y:
;
; X : 1 2 3 4 5 ;; X positions of all pixels
; 1 2 3 4 5
; 1 2 3 4 5
;
; Y : 15 15 15 15 15 ;; Y positions of all pixels
; 16 16 16 16 16
; 17 17 17 17 17
;
; Using the above technique is suggested, but *not* required. You
; can do anything you wish with the X and Y values. This technique
; only makes it easier to compute your model function values.
;
; CONSTRAINING PARAMETER VALUES WITH THE PARINFO KEYWORD
;
; The behavior of MPFIT can be modified with respect to each
; parameter to be fitted. A parameter value can be fixed; simple
; boundary constraints can be imposed; limitations on the parameter
; changes can be imposed; properties of the automatic derivative can
; be modified; and parameters can be tied to one another.
;
; These properties are governed by the PARINFO structure, which is
; passed as a keyword parameter to MPFIT.
;
; PARINFO should be an array of structures, one for each parameter.
; Each parameter is associated with one element of the array, in
; numerical order. The structure can have the following entries
; (none are required):
;
; .VALUE - the starting parameter value (but see the START_PARAMS
; parameter for more information).
;
; .FIXED - a boolean value, whether the parameter is to be held
; fixed or not. Fixed parameters are not varied by
; MPFIT, but are passed on to MYFUNCT for evaluation.
;
; .LIMITED - a two-element boolean array. If the first/second
; element is set, then the parameter is bounded on the
; lower/upper side. A parameter can be bounded on both
; sides. Both LIMITED and LIMITS must be given
; together.
;
; .LIMITS - a two-element float or double array. Gives the
; parameter limits on the lower and upper sides,
; respectively. Zero, one or two of these values can be
; set, depending on the values of LIMITED. Both LIMITED
; and LIMITS must be given together.
;
; .PARNAME - a string, giving the name of the parameter. The
; fitting code of MPFIT does not use this tag in any
; way. However, the default ITERPROC will print the
; parameter name if available.
;
; .STEP - the step size to be used in calculating the numerical
; derivatives. If set to zero, then the step size is
; computed automatically. Ignored when AUTODERIVATIVE=0.
; This value is superceded by the RELSTEP value.
;
; .RELSTEP - the *relative* step size to be used in calculating
; the numerical derivatives. This number is the
; fractional size of the step, compared to the
; parameter value. This value supercedes the STEP
; setting. If the parameter is zero, then a default
; step size is chosen.
;
; .MPSIDE - the sidedness of the finite difference when computing
; numerical derivatives. This field can take four
; values:
;
; 0 - one-sided derivative computed automatically
; 1 - one-sided derivative (f(x+h) - f(x) )/h
; -1 - one-sided derivative (f(x) - f(x-h))/h
; 2 - two-sided derivative (f(x+h) - f(x-h))/(2*h)
;
; Where H is the STEP parameter described above. The
; "automatic" one-sided derivative method will chose a
; direction for the finite difference which does not
; violate any constraints. The other methods do not
; perform this check. The two-sided method is in
; principle more precise, but requires twice as many
; function evaluations. Default: 0.
;
; .MPMINSTEP - the minimum change to be made in the parameter
; value. During the fitting process, the parameter
; will be changed by multiples of this value. The
; actual step is computed as:
;
; DELTA1 = MPMINSTEP*ROUND(DELTA0/MPMINSTEP)
;
; where DELTA0 and DELTA1 are the estimated parameter
; changes before and after this constraint is
; applied. Note that this constraint should be used
; with care since it may cause non-converging,
; oscillating solutions.
;
; A value of 0 indicates no minimum. Default: 0.
;
; .MPMAXSTEP - the maximum change to be made in the parameter
; value. During the fitting process, the parameter
; will never be changed by more than this value.
;
; A value of 0 indicates no maximum. Default: 0.
;
; .TIED - a string expression which "ties" the parameter to other
; free or fixed parameters. Any expression involving
; constants and the parameter array P are permitted.
; Example: if parameter 2 is always to be twice parameter
; 1 then use the following: parinfo[2].tied = '2 * P[1]'.
; Since they are totally constrained, tied parameters are
; considered to be fixed; no errors are computed for them.
; [ NOTE: the PARNAME can't be used in expressions. ]
;
; Future modifications to the PARINFO structure, if any, will involve
; adding structure tags beginning with the two letters "MP".
; Therefore programmers are urged to avoid using tags starting with
; the same letters; otherwise they are free to include their own
; fields within the PARINFO structure, and they will be ignored.
;
; PARINFO Example:
; parinfo = replicate({value:0.D, fixed:0, limited:[0,0], $
; limits:[0.D,0]}, 5)
; parinfo[0].fixed = 1
; parinfo[4].limited(0) = 1
; parinfo[4].limits(0) = 50.D
; parinfo[*].value = [5.7D, 2.2, 500., 1.5, 2000.]
;
; A total of 5 parameters, with starting values of 5.7,
; 2.2, 500, 1.5, and 2000 are given. The first parameter
; is fixed at a value of 5.7, and the last parameter is
; constrained to be above 50.
;
;
; COMPATIBILITY
;
; This function is designed to work with IDL 5.0 or greater.
;
; Because TIED parameters rely on the EXECUTE() function, they cannot
; be used with the free version of the IDL Virtual Machine.
;
;
; INPUTS:
; MYFUNCT - a string variable containing the name of an IDL
; function. This function computes the "model" Z values
; given the X,Y values and model parameters, as described above.
;
; X - Array of "X" independent variable values, as described above.
; These values are passed directly to the fitting function
; unmodified.
;
; Y - Array of "Y" independent variable values, as described
; above. X and Y should have the same data type.
;
; Z - Array of "measured" dependent variable values. Z should have
; the same data type as X and Y. The function MYFUNCT should
; map (X,Y)->Z.
;
; ERR - Array of "measured" 1-sigma uncertainties. ERR should have
; the same data type as Z. ERR is ignored if the WEIGHTS
; keyword is specified.
;
; START_PARAMS - An array of starting values for each of the
; parameters of the model. The number of parameters
; should be fewer than the number of measurements.
; Also, the parameters should have the same data type
; as the measurements (double is preferred).
;
; This parameter is optional if the PARINFO keyword
; is used (see MPFIT). The PARINFO keyword provides
; a mechanism to fix or constrain individual
; parameters. If both START_PARAMS and PARINFO are
; passed, then the starting *value* is taken from
; START_PARAMS, but the *constraints* are taken from
; PARINFO.
;
; RETURNS:
;
; Returns the array of best-fit parameters.
;
; KEYWORD PARAMETERS:
;
; BESTNORM - the value of the summed, squared, weighted residuals
; for the returned parameter values, i.e. the chi-square value.
;
; COVAR - the covariance matrix for the set of parameters returned
; by MPFIT. The matrix is NxN where N is the number of
; parameters. The square root of the diagonal elements
; gives the formal 1-sigma statistical errors on the
; parameters IF errors were treated "properly" in MYFUNC.
; Parameter errors are also returned in PERROR.
;
; To compute the correlation matrix, PCOR, use this example:
; PCOR = COV * 0
; FOR i = 0, n-1 DO FOR j = 0, n-1 DO $
; PCOR[i,j] = COV[i,j]/sqrt(COV[i,i]*COV[j,j])
; or equivalently, in vector notation,
; PCOR = COV / (PERROR # PERROR)
;
; If NOCOVAR is set or MPFIT terminated abnormally, then
; COVAR is set to a scalar with value !VALUES.D_NAN.
;
; DOF - number of degrees of freedom, computed as
; DOF = N_ELEMENTS(DEVIATES) - NFREE
; Note that this doesn't account for pegged parameters (see
; NPEGGED).
;
; ERRMSG - a string error or warning message is returned.
;
; FTOL - a nonnegative input variable. Termination occurs when both
; the actual and predicted relative reductions in the sum of
; squares are at most FTOL (and STATUS is accordingly set to
; 1 or 3). Therefore, FTOL measures the relative error
; desired in the sum of squares. Default: 1D-10
;
; FUNCTARGS - A structure which contains the parameters to be passed
; to the user-supplied function specified by MYFUNCT via
; the _EXTRA mechanism. This is the way you can pass
; additional data to your user-supplied function without
; using common blocks.
;
; By default, no extra parameters are passed to the
; user-supplied function.
;
; GTOL - a nonnegative input variable. Termination occurs when the
; cosine of the angle between fvec and any column of the
; jacobian is at most GTOL in absolute value (and STATUS is
; accordingly set to 4). Therefore, GTOL measures the
; orthogonality desired between the function vector and the
; columns of the jacobian. Default: 1D-10
;
; ITERARGS - The keyword arguments to be passed to ITERPROC via the
; _EXTRA mechanism. This should be a structure, and is
; similar in operation to FUNCTARGS.
; Default: no arguments are passed.
;
; ITERPROC - The name of a procedure to be called upon each NPRINT
; iteration of the MPFIT routine. It should be declared
; in the following way:
;
; PRO ITERPROC, MYFUNCT, p, iter, fnorm, FUNCTARGS=fcnargs, $
; PARINFO=parinfo, QUIET=quiet, ...
; ; perform custom iteration update
; END
;
; ITERPROC must either accept all three keyword
; parameters (FUNCTARGS, PARINFO and QUIET), or at least
; accept them via the _EXTRA keyword.
;
; MYFUNCT is the user-supplied function to be minimized,
; P is the current set of model parameters, ITER is the
; iteration number, and FUNCTARGS are the arguments to be
; passed to MYFUNCT. FNORM should be the
; chi-squared value. QUIET is set when no textual output
; should be printed. See below for documentation of
; PARINFO.
;
; In implementation, ITERPROC can perform updates to the
; terminal or graphical user interface, to provide
; feedback while the fit proceeds. If the fit is to be
; stopped for any reason, then ITERPROC should set the
; common block variable ERROR_CODE to negative value (see
; MPFIT_ERROR common block below). In principle,
; ITERPROC should probably not modify the parameter
; values, because it may interfere with the algorithm's
; stability. In practice it is allowed.
;
; Default: an internal routine is used to print the
; parameter values.
;
; MAXITER - The maximum number of iterations to perform. If the
; number is exceeded, then the STATUS value is set to 5
; and MPFIT returns.
; Default: 200 iterations
;
; NFEV - the number of MYFUNCT function evaluations performed.
;
; NITER - the number of iterations completed.
;
; NOCOVAR - set this keyword to prevent the calculation of the
; covariance matrix before returning (see COVAR)
;
; NPRINT - The frequency with which ITERPROC is called. A value of
; 1 indicates that ITERPROC is called with every iteration,
; while 2 indicates every other iteration, etc. Note that
; several Levenberg-Marquardt attempts can be made in a
; single iteration.
; Default value: 1
;
; PARINFO - Provides a mechanism for more sophisticated constraints
; to be placed on parameter values. When PARINFO is not
; passed, then it is assumed that all parameters are free
; and unconstrained. Values in PARINFO are never
; modified during a call to MPFIT.
;
; See description above for the structure of PARINFO.
;
; Default value: all parameters are free and unconstrained.
;
; PERROR - The formal 1-sigma errors in each parameter, computed
; from the covariance matrix. If a parameter is held
; fixed, or if it touches a boundary, then the error is
; reported as zero.
;
; If the fit is unweighted (i.e. no errors were given, or
; the weights were uniformly set to unity), then PERROR
; will probably not represent the true parameter
; uncertainties. *If* you can assume that the true reduced
; chi-squared value is unity -- meaning that the fit is
; implicitly assumed to be of good quality -- then the
; estimated parameter uncertainties can be computed by
; scaling PERROR by the measured chi-squared value.
;
; DOF = N_ELEMENTS(Z) - N_ELEMENTS(PARMS) ; deg of freedom
; PCERROR = PERROR * SQRT(BESTNORM / DOF) ; scaled uncertainties
;
; QUIET - set this keyword when no textual output should be printed
; by MPFIT
;
; STATUS - an integer status code is returned. All values other
; than zero can represent success. It can have one of the
; following values:
;
; 0 improper input parameters.
;
; 1 both actual and predicted relative reductions
; in the sum of squares are at most FTOL.
;
; 2 relative error between two consecutive iterates
; is at most XTOL
;
; 3 conditions for STATUS = 1 and STATUS = 2 both hold.
;
; 4 the cosine of the angle between fvec and any
; column of the jacobian is at most GTOL in
; absolute value.
;
; 5 the maximum number of iterations has been reached
;
; 6 FTOL is too small. no further reduction in
; the sum of squares is possible.
;
; 7 XTOL is too small. no further improvement in
; the approximate solution x is possible.
;
; 8 GTOL is too small. fvec is orthogonal to the
; columns of the jacobian to machine precision.
;
; WEIGHTS - Array of weights to be used in calculating the
; chi-squared value. If WEIGHTS is specified then the ERR
; parameter is ignored. The chi-squared value is computed
; as follows:
;
; CHISQ = TOTAL( (Z-MYFUNCT(X,Y,P))^2 * ABS(WEIGHTS) )
;
; Here are common values of WEIGHTS:
;
; 1D/ERR^2 - Normal weighting (ERR is the measurement error)
; 1D/Z - Poisson weighting (counting statistics)
; 1D - Unweighted
;
; XTOL - a nonnegative input variable. Termination occurs when the
; relative error between two consecutive iterates is at most
; XTOL (and STATUS is accordingly set to 2 or 3). Therefore,
; XTOL measures the relative error desired in the approximate
; solution. Default: 1D-10
;
; YFIT - the best-fit model function, as returned by MYFUNCT.
;
; EXAMPLE:
;
; p = [2.2D, -0.7D, 1.4D, 3000.D]
; x = (dindgen(200)*0.1 - 10.) # (dblarr(200) + 1)
; y = (dblarr(200) + 1) # (dindgen(200)*0.1 - 10.)
; zi = gauss2(x, y, p)
; sz = sqrt(zi>1)
; z = zi + randomn(seed, 200, 200) * sz
;
; p0 = [0D, 0D, 1D, 10D]
; p = mpfit2dfun('GAUSS2', x, y, z, sz, p0)
;
; Generates a synthetic data set with a Gaussian peak, and Poisson
; statistical uncertainty. Then the same function (but different
; starting parameters) is fitted to the data to see how close we can
; get.
;
; It is especially worthy to notice that the X and Y values are
; created as full images, so that a coordinate is attached to each
; pixel independently. This is the format that GAUSS2 accepts, and
; the easiest for you to use in your own functions.
;
;
; COMMON BLOCKS:
;
; COMMON MPFIT_ERROR, ERROR_CODE
;
; User routines may stop the fitting process at any time by
; setting an error condition. This condition may be set in either
; the user's model computation routine (MYFUNCT), or in the
; iteration procedure (ITERPROC).
;
; To stop the fitting, the above common block must be declared,
; and ERROR_CODE must be set to a negative number. After the user
; procedure or function returns, MPFIT checks the value of this
; common block variable and exits immediately if the error
; condition has been set. By default the value of ERROR_CODE is
; zero, indicating a successful function/procedure call.
;
;
; REFERENCES:
;
; MINPACK-1, Jorge More', available from netlib (www.netlib.org).
; "Optimization Software Guide," Jorge More' and Stephen Wright,
; SIAM, *Frontiers in Applied Mathematics*, Number 14.
;
; MODIFICATION HISTORY:
; Written, transformed from MPFITFUN, 26 Sep 1999, CM
; Alphabetized documented keywords, 02 Oct 1999, CM
; Added example, 02 Oct 1999, CM
; Tried to clarify definitions of X and Y, 29 Oct 1999, CM
; Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM
; Check to be sure that X, Y and Z are present, 02 Nov 1999, CM
; Documented PERROR for unweighted fits, 03 Nov 1999, CM
; Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
; Copying permission terms have been liberalized, 26 Mar 2000, CM
; Propagated improvements from MPFIT, 17 Dec 2000, CM
; Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
; Add DOF keyword to return degrees of freedom, CM, 23 June 2003
; Minor documentation adjustment, 03 Feb 2004, CM
; Fix the example to prevent zero errorbars, 28 Mar 2005, CM
; Defend against users supplying strangely dimensioned X and Y, 29
; Jun 2005, CM
; Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
; Move STRICTARR compile option inside each function/procedure, 9 Oct 2006
; Add COMPATIBILITY section, CM, 13 Dec 2007
;
; $Id: mpfit2dfun.pro,v 1.11 2010/04/09 04:58:35 craigm Exp $
;-
; Copyright (C) 1997-2000, 2002, 2003, 2004, 2005, Craig Markwardt
; This software is provided as is without any warranty whatsoever.
; Permission to use, copy, modify, and distribute modified or
; unmodified copies is granted, provided this copyright and disclaimer
; are included unchanged.
;-
FORWARD_FUNCTION mpfit2dfun_eval, mpfit2dfun, mpfit
; This is the call-back function for MPFIT. It evaluates the
; function, subtracts the data, and returns the residuals.
function mpfit2dfun_eval, p, dp, _EXTRA=extra
COMPILE_OPT strictarr
common mpfit2dfun_common, fcn, x, y, z, err, wts, f, fcnargs
;; The function is evaluated here. There are four choices,
;; depending on whether (a) FUNCTARGS was passed to MPFIT2DFUN, which
;; is passed to this function as "hf"; or (b) the derivative
;; parameter "dp" is passed, meaning that derivatives should be
;; calculated analytically by the function itself.
if n_elements(fcnargs) GT 0 then begin
if n_params() GT 1 then f = call_function(fcn,x,y,p, dp, _EXTRA=fcnargs)$
else f = call_function(fcn,x,y,p, _EXTRA=fcnargs)
endif else begin
if n_params() GT 1 then f = call_function(fcn,x,y,p, dp) $
else f = call_function(fcn,x,y,p)
endelse
;; Compute the deviates, applying either errors or weights
if n_elements(err) GT 0 then begin
result = (z-f)/err
endif else if n_elements(wts) GT 0 then begin
result = (z-f)*wts
endif else begin
result = (z-f)
endelse
;; Make sure the returned result is one-dimensional.
result = reform(result, n_elements(result), /overwrite)
return, result
end
function mpfit2dfun, fcn, x, y, z, err, p, WEIGHTS=wts, FUNCTARGS=fa, $
BESTNORM=bestnorm, nfev=nfev, STATUS=status, $
parinfo=parinfo, query=query, $
npegged=npegged, nfree=nfree, dof=dof, $
covar=covar, perror=perror, niter=iter, yfit=yfit, $
quiet=quiet, ERRMSG=errmsg, _EXTRA=extra
COMPILE_OPT strictarr
status = 0L
errmsg = ''
;; Detect MPFIT and crash if it was not found
catch, catcherror
if catcherror NE 0 then begin
MPFIT_NOTFOUND:
catch, /cancel
message, 'ERROR: the required function MPFIT must be in your IDL path', /info
return, !values.d_nan
endif
if mpfit(/query) NE 1 then goto, MPFIT_NOTFOUND
catch, /cancel
if keyword_set(query) then return, 1
if n_params() EQ 0 then begin
message, "USAGE: PARMS = MPFIT2DFUN('MYFUNCT', X, Y, ERR, "+ $
"START_PARAMS, ... )", /info
return, !values.d_nan
endif
if n_elements(x) EQ 0 OR n_elements(y) EQ 0 OR n_elements(z) EQ 0 then begin
message, 'ERROR: X, Y and Z must be defined', /info
return, !values.d_nan
endif
;; Use common block to pass data back and forth
common mpfit2dfun_common, fc, xc, yc, zc, ec, wc, mc, ac
fc = fcn & xc = x & yc = y & zc = z & mc = 0L
;; These optional parameters must be undefined first
ac = 0 & dummy = size(temporary(ac))
ec = 0 & dummy = size(temporary(ec))
wc = 0 & dummy = size(temporary(wc))
if n_elements(fa) GT 0 then ac = fa
if n_elements(wts) GT 0 then begin
wc = sqrt(abs(wts))
endif else if n_elements(err) GT 0 then begin
wh = where(err EQ 0, ct)
if ct GT 0 then begin
message, 'ERROR: ERROR value must not be zero. Use WEIGHTS.', $
/info
return, !values.d_nan
endif
ec = err
endif
result = mpfit('mpfit2dfun_eval', p, $
parinfo=parinfo, STATUS=status, nfev=nfev, BESTNORM=bestnorm,$
covar=covar, perror=perror, niter=iter, $
nfree=nfree, npegged=npegged, dof=dof, $
ERRMSG=errmsg, quiet=quiet, _EXTRA=extra)
;; Retrieve the fit value
yfit = temporary(mc)
;; Some cleanup
xc = 0 & yc = 0 & zc = 0 & wc = 0 & ec = 0 & mc = 0 & ac = 0
;; Print error message if there is one.
if NOT keyword_set(quiet) AND errmsg NE '' then $
message, errmsg, /info
return, result
end