;+ ; NAME: ; MPCURVEFIT ; ; 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 (replaces CURVEFIT) ; ; MAJOR TOPICS: ; Curve and Surface Fitting ; ; CALLING SEQUENCE: ; YFIT = MPCURVEFIT(X, Y, WEIGHTS, P, [SIGMA,] FUNCTION_NAME=FUNC, ; ITER=iter, ITMAX=itmax, ; CHISQ=chisq, NFREE=nfree, DOF=dof, ; NFEV=nfev, COVAR=covar, [/NOCOVAR, ] [/NODERIVATIVE, ] ; FUNCTARGS=functargs, PARINFO=parinfo, ; FTOL=ftol, XTOL=xtol, GTOL=gtol, TOL=tol, ; ITERPROC=iterproc, ITERARGS=iterargs, ; NPRINT=nprint, QUIET=quiet, ; ERRMSG=errmsg, STATUS=status) ; ; DESCRIPTION: ; ; MPCURVEFIT fits a user-supplied model -- in the form of an IDL ; function -- to a set of user-supplied data. MPCURVEFIT calls ; MPFIT, the MINPACK-1 least-squares minimizer, to do the main ; work. ; ; Given the data and their uncertainties, MPCURVEFIT finds the best ; set of model parameters which match the data (in a least-squares ; sense) and returns them in the parameter P. ; ; MPCURVEFIT returns the best fit function. ; ; 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 weighting values ("WEIGHTS"). ; - The name of an IDL function which computes Y given X ("FUNC"). ; - Starting guesses for all of the parameters ("P"). ; ; There are very few restrictions placed on X, Y or FUNCT. Simply ; put, FUNCT 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. ; ; MPCURVEFIT 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: ; ; ; MYFUNCT - example user function ; ; X - input independent variable (vector same size as data) ; ; P - input parameter values (N-element array) ; ; YMOD - upon return, user function values ; ; DP - upon return, the user function must return ; ; an ARRAY(M,N) of derivatives in this parameter ; ; ; PRO MYFUNCT, x, p, ymod, dp ; ymod = F(x, p) ;; Model function ; ; if n_params() GE 4 then begin ; ; Create derivative and compute derivative array ; dp = make_array(n_elements(x), n_elements(p), value=x[0]*0) ; ; ; Compute derivative if requested by caller ; for i = 0, n_elements(p)-1 do dp(*,i) = FGRAD(x, p, i) ; endif ; END ; ; where FGRAD(x, p, i) is a model function which computes the ; derivative of the model F(x,p) with respect to parameter P(i) at X. ; The returned array YMOD must have the same dimensions and type as ; the "measured" Y values. The returned array DP[i,j] is the ; derivative of the ith function value with respect to the jth ; parameter. ; ; 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. ; ; If NODERIVATIVE=1, then MPCURVEFIT will never request explicit ; derivatives from the user function, and instead will user numerical ; estimates (i.e. by calling the user function multiple times). ; ; 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. ; ; .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 in ; one iteration. ; ; 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. ] ; ; .MPPRINT - if set to 1, then the default ITERPROC will print the ; parameter value. If set to 0, the parameter value ; will not be printed. This tag can be used to ; selectively print only a few parameter values out of ; many. Default: 1 (all parameters printed) ; ; ; 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: ; X - Array of independent variable values. ; ; Y - Array of "measured" dependent variable values. Y should have ; the same data type as X. The function FUNCT should map ; X->Y. ; ; 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-FUNCT(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 ; ; P - 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). ; ; Upon successful completion the new parameter values are ; returned in P. ; ; If both START_PARAMS and PARINFO are passed, then the starting ; *value* is taken from START_PARAMS, but the *constraints* are ; taken from PARINFO. ; ; SIGMA - 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 SIGMA 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 SIGMA ; by the measured chi-squared value. ; ; DOF = N_ELEMENTS(X) - N_ELEMENTS(P) ; deg of freedom ; CSIGMA = SIGMA * SQRT(CHISQ / DOF) ; scaled uncertainties ; ; RETURNS: ; ; Returns the array containing the best-fitting function. ; ; KEYWORD PARAMETERS: ; ; CHISQ - 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: ; 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. ; ; 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 ; ; FUNCTION_NAME - a scalar string containing the name of an IDL ; procedure to compute the user model values, as ; described above in the "USER MODEL" section. ; ; FUNCTARGS - A structure which contains the parameters to be passed ; to the user-supplied function specified by FUNCT 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 ; ; ITER - the number of iterations completed. ; ; 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, FUNCT, 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. ; ; FUNCT 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 FUNCT. 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. ; ; ITMAX - 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 FUNCT 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. ; ; NOCOVAR - set this keyword to prevent the calculation of the ; covariance matrix before returning (see COVAR) ; ; NODERIVATIVE - if set, then the user function will not be queried ; for analytical derivatives, and instead the ; derivatives will be computed by finite differences ; (and according to the PARINFO derivative settings; ; see above for a description). ; ; 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. ; ; 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. ; ; TOL - synonym for FTOL. Use FTOL instead. ; ; 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 ; ; YERROR - upon return, the root-mean-square variance of the ; residuals. ; ; ; 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 ; yfit = mpcurvefit(x, y, 1/sy^2, p0, $ ; Fit a function ; FUNCTION_NAME='GAUSS1P',/autoderivative) ; print, p ; ; Generates a synthetic data set with a Gaussian peak, and Poisson ; statistical uncertainty. Then the same function is fitted to the ; data to see how close we can get. GAUSS1 and GAUSS1P are ; available from the same web page. ; ; ; 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: ; Translated from MPFITFUN, 25 Sep 1999, CM ; Alphabetized documented keywords, 02 Oct 1999, CM ; Added QUERY keyword and query checking of MPFIT, 29 Oct 1999, CM ; Check to be sure that X and Y are present, 02 Nov 1999, CM ; Documented SIGMA 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 ; Corrected behavior of NODERIVATIVE, 13 May 2002, CM ; Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002 ; Make more consistent with comparable IDL routines, 30 Jun 2003, CM ; Minor documentation adjustment, 03 Feb 2004, CM ; Fix error in documentation, 26 Aug 2005, CM ; Convert to IDL 5 array syntax (!), 16 Jul 2006, CM ; Move STRICTARR compile option inside each function/procedure, 9 Oct 2006 ; Fix bug in handling of explicit derivatives with errors/weights ; (the weights were not being applied), CM, 2012-07-22 ; Add more documentation on calling interface for user function and ; parameter derivatives, CM, 2012-07-22 ; ; $Id: mpcurvefit.pro,v 1.11 2012/07/22 21:08:58 cmarkwar Exp $ ;- ; Copyright (C) 1997-2000, 2002, 2003, 2004, 2005, 2012, 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 mpcurvefit_eval, mpcurvefit, mpfit ; This is the call-back function for MPFIT. It evaluates the ; function, subtracts the data, and returns the residuals. function mpcurvefit_eval, p, dp, _EXTRA=extra COMPILE_OPT strictarr common mpcurvefit_common, fcn, x, y, wts, f, fcnargs ;; The function is evaluated here. There are four choices, ;; depending on whether (a) FUNCTARGS was passed to MPCURVEFIT, 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 call_procedure, fcn, x, p, f, dp,_EXTRA=fcnargs $ else call_procedure, fcn, x, p, f, _EXTRA=fcnargs endif else begin if n_params() GT 1 then call_procedure, fcn, x, p, f, dp $ else call_procedure, fcn, x, p, f endelse ;; Compute the deviates, applying the weights result = (y-f)*wts ;; Apply weights to derivative quantities if n_params() GT 1 then begin np = n_elements(p) nf = n_elements(f) for j = 0L, np-1 do dp[j*nf] = dp[j*nf:j*nf+nf-1] * wts endif ;; Make sure the returned result is one-dimensional. result = reform(result, n_elements(result), /overwrite) return, result end function mpcurvefit, x, y, wts, p, perror, function_name=fcn, $ iter=iter, itmax=maxiter, $ chisq=bestnorm, nfree=nfree, dof=dof, $ nfev=nfev, covar=covar, nocovar=nocovar, yerror=yerror, $ noderivative=noderivative, tol=tol, ftol=ftol, $ FUNCTARGS=fa, parinfo=parinfo, $ errmsg=errmsg, STATUS=status, QUIET=quiet, $ query=query, _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: YFIT = MPCURVEFIT(X, Y, WTS, P, DP)", /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(fcn) EQ 0 then fcn = 'funct' if n_elements(noderivative) EQ 0 then noderivative = 0 common mpcurvefit_common, fc, xc, yc, wc, mc, ac fc = fcn & xc = x & yc = y & wc = sqrt(abs(wts)) & mc = 0L ac = 0 & dummy = size(temporary(ac)) if n_elements(fa) GT 0 then ac = fa if n_elements(tol) GT 0 then ftol = tol result = mpfit('mpcurvefit_eval', p, maxiter=maxiter, $ autoderivative=noderivative, ftol=ftol, $ parinfo=parinfo, STATUS=status, nfev=nfev, BESTNORM=bestnorm,$ covar=covar, perror=perror, niter=iter, nfree=nfree, dof=dof,$ ERRMSG=errmsg, quiet=quiet, _EXTRA=extra) ;; Retrieve the fit value yfit = temporary(mc) ;; Now do some clean-up xc = 0 & yc = 0 & wc = 0 & mc = 0 & ac = 0 if NOT keyword_set(quiet) AND errmsg NE '' then $ message, errmsg, /info $ else $ p = result yerror = p[0]*0 if n_elements(dof) GT 0 AND dof[0] GT 0 then begin yerror[0] = sqrt( total( (y-yfit)^2 ) / dof[0] ) endif return, yfit end