;+ ; 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