;+
; 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 greater
; than zero can represent success (however STATUS EQ 5 may
; indicate failure to converge). 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
; Better documentation for STATUS, CM, 2016-04-29
;
; $Id: mpcurvefit.pro,v 1.12 2016/05/19 16:08:49 cmarkwar Exp $
;-
; Copyright (C) 1997-2000, 2002, 2003, 2004, 2005, 2012, 2016 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