;+ ; NAME: ; MPFITFUN ; ; 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 IDL function ; ; MAJOR TOPICS: ; Curve and Surface Fitting ; ; CALLING SEQUENCE: ; parms = MPFITFUN(MYFUNCT, X, Y, ERR, start_params, ...) ; ; DESCRIPTION: ; ; MPFITFUN fits a user-supplied model -- in the form of an IDL ; function -- to a set of user-supplied data. MPFITFUN calls ; MPFIT, the MINPACK-1 least-squares minimizer, to do the main ; work. ; ; Given the data and their uncertainties, MPFITFUN 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: ; - An array of independent variable values ("X"). ; - An array of "measured" *dependent* variable values ("Y"). ; - An array of "measured" 1-sigma uncertainty values ("ERR"). ; - The name of an IDL function which computes Y given X ("MYFUNCT"). ; - Starting guesses for all of the parameters ("START_PARAMS"). ; ; There are very few restrictions placed on X, Y or MYFUNCT. Simply ; put, MYFUNCT must map the "X" values into "Y" values given the ; model parameters. The "X" values may represent any independent ; variable (not just Cartesian X), and indeed may be multidimensional ; themselves. For example, in the application of image fitting, X ; may be a 2xN array of image positions. ; ; MPFITFUN 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, P ; ; The independent variable is X ; ; Parameter values are passed in "P" ; YMOD = ... computed model values at X ... ; return, YMOD ; END ; ; The returned array YMOD must have the same dimensions and type as ; the "measured" Y 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 by MPFITFUN 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. ; ; 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. ; ; INPUTS: ; MYFUNCT - a string variable containing the name of an IDL function. ; This function computes the "model" Y values given the ; X values and model parameters, as desribed above. ; ; X - Array of independent variable values. ; ; Y - Array of "measured" dependent variable values. Y should have ; the same data type as X. The function MYFUNCT should map ; X->Y. ; ; ERR - Array of "measured" 1-sigma uncertainties. ERR should have ; the same data type as Y. 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 residuals for the ; returned parameter values. ; ; 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: ; IDL> PCOR = COV * 0 ; IDL> 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)) ; ; If NOCOVAR is set or MPFIT terminated abnormally, then ; COVAR is set to a scalar with value !VALUES.D_NAN. ; ; CASH - when set, the fit statistic is changed to a derivative of ; the CASH statistic. The model function must be strictly ; positive. ; ; 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. ; ; NFREE - the number of free parameters in the fit. This includes ; parameters which are not FIXED and not TIED, but it does ; include parameters which are pegged at LIMITS. ; ; NITER - the number of iterations completed. ; ; NOCOVAR - set this keyword to prevent the calculation of the ; covariance matrix before returning (see COVAR) ; ; NPEGGED - the number of free parameters which are pegged at a ; LIMIT. ; ; 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(X) - 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( (Y-MYFUNCT(X,P))^2 * ABS(WEIGHTS) ) ; ; Here are common values of WEIGHTS: ; ; 1D/ERR^2 - Normal weighting (ERR is the measurement error) ; 1D/Y - 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: ; ; ; First, generate some synthetic data ; npts = 200 ; x = dindgen(npts) * 0.1 - 10. ; Independent variable ; yi = gauss1(x, [2.2D, 1.4, 3000.]) ; "Ideal" Y variable ; y = yi + randomn(seed, npts) * sqrt(1000. + yi); Measured, w/ noise ; sy = sqrt(1000.D + y) ; Poisson errors ; ; ; Now fit a Gaussian to see how well we can recover ; p0 = [1.D, 1., 1000.] ; Initial guess (cent, width, area) ; p = mpfitfun('GAUSS1', x, y, sy, p0) ; Fit a function ; print, p ; ; Generates a synthetic data set with a Gaussian peak, and Poisson ; statistical uncertainty. Then the same function is fitted to the ; data (with different starting parameters) to see how close we can ; get. ; ; ; 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, Apr-Jul 1998, CM ; Added PERROR keyword, 04 Aug 1998, CM ; Added COVAR keyword, 20 Aug 1998, CM ; Added ITER output keyword, 05 Oct 1998 ; D.L Windt, Bell Labs, windt@bell-labs.com; ; Added ability to return model function in YFIT, 09 Nov 1998 ; Analytical derivatives allowed via AUTODERIVATIVE keyword, 09 Nov 1998 ; Parameter values can be tied to others, 09 Nov 1998 ; Cosmetic documentation updates, 16 Apr 1999, CM ; More cosmetic documentation updates, 14 May 1999, CM ; Made sure to update STATUS, 25 Sep 1999, CM ; Added WEIGHTS keyword, 25 Sep 1999, CM ; Changed from handles to common blocks, 25 Sep 1999, CM ; - commons seem much cleaner and more logical in this case. ; Alphabetized documented keywords, 02 Oct 1999, CM ; Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM ; Corrected EXAMPLE (offset of 1000), 30 Oct 1999, CM ; Check to be sure that X and Y 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 ; Corrected errors in EXAMPLE, 26 Mar 2000, CM ; Copying permission terms have been liberalized, 26 Mar 2000, CM ; Propagated improvements from MPFIT, 17 Dec 2000, CM ; Added CASH statistic, 10 Jan 2001 ; Added NFREE and NPEGGED keywords, 11 Sep 2002, CM ; Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002 ; Add DOF keyword to return degrees of freedom, CM, 23 June 2003 ; ; $Id: mpfitfun.pro,v 1.6 2003/06/30 21:47:36 craigm Exp $ ;- ; Copyright (C) 1997-2002, 2003, 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 mpfitfun_eval, mpfitfun, mpfit ; This is the call-back function for MPFIT. It evaluates the ; function, subtracts the data, and returns the residuals. function mpfitfun_eval, p, dp, _EXTRA=extra common mpfitfun_common, fcn, x, y, err, wts, f, fcnargs ;; The function is evaluated here. There are four choices, ;; depending on whether (a) FUNCTARGS was passed to MPFITFUN, 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, p, dp, _EXTRA=fcnargs)$ else f = call_function(fcn, x, p, _EXTRA=fcnargs) endif else begin if n_params() GT 1 then f = call_function(fcn, x, p, dp) $ else f = call_function(fcn, x, p) endelse ;; Compute the deviates, applying either errors or weights if n_elements(err) GT 0 then begin result = (y-f)/err endif else if n_elements(wts) GT 0 then begin result = (y-f)*wts endif else begin result = (y-f) endelse ;; Make sure the returned result is one-dimensional. result = reform(result, n_elements(result), /overwrite) return, result end ;; Implement residual and gradient scaling according to the ;; prescription of Cash (ApJ, 228, 939) pro mpfitfun_cash, resid, dresid common mpfitfun_common, fcn, x, y, err, wts, f, fcnargs sz = size(dresid) m = sz(1) n = sz(2) ;; Do rudimentary dimensions checks, so we don't do something stupid if n_elements(y) NE m OR n_elements(f) NE m OR n_elements(resid) NE m then begin DIM_ERROR: message, 'ERROR: dimensions of Y, F, RESID or DRESID are not consistent' endif ;; Scale gradient by sqrt(y)/f gfact = temporary(dresid) * rebin(reform(sqrt(y)/f,m,1),m,n) dresid = reform(dresid, m, n, /overwrite) ;; Scale residuals by 1/sqrt(y) resid = temporary(resid)/sqrt(y) return end function mpfitfun, fcn, x, y, err, p, WEIGHTS=wts, FUNCTARGS=fa, $ BESTNORM=bestnorm, nfev=nfev, STATUS=status, $ parinfo=parinfo, query=query, CASH=cash, $ covar=covar, perror=perror, yfit=yfit, $ niter=niter, nfree=nfree, npegged=npegged, dof=dof, $ quiet=quiet, ERRMSG=errmsg, _EXTRA=extra 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 = MPFITFUN('MYFUNCT', X, Y, ERR, "+ $ "START_PARAMS, ... )", /info return, !values.d_nan endif if n_elements(x) EQ 0 OR n_elements(y) EQ 0 then begin message, 'ERROR: X and Y must be defined', /info return, !values.d_nan endif if n_elements(err) GT 0 OR n_elements(wts) GT 0 AND keyword_set(cash) then begin message, 'ERROR: WEIGHTS or ERROR cannot be specified with CASH', /info return, !values.d_nan endif if keyword_set(cash) then begin scalfcn = 'mpfitfun_cash' endif ;; Use common block to pass data back and forth common mpfitfun_common, fc, xc, yc, ec, wc, mc, ac fc = fcn & xc = x & yc = y & 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('mpfitfun_eval', p, SCALE_FCN=scalfcn, $ parinfo=parinfo, STATUS=status, nfev=nfev, BESTNORM=bestnorm,$ covar=covar, perror=perror, $ niter=niter, 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 & 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