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mpcurvefit.pro


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: PRO MYFUNCT, X, P, YMOD ; The independent variable is X ; Parameter values are passed in "P" YMOD = ... computed model values at X ... 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 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. .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, $ FUNCTION_NAME='GAUSS1P') ; 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 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 $Id: mpcurvefit.pro,v 1.6 2004/02/26 04:18:16 craigm Exp $

Routines

result = mpcurvefit_eval(p, dp, _EXTRA=_EXTRA)
result = mpcurvefit(x, y, wts, p, perror, function_name=function_name, iter=iter, itmax=itmax, chisq=chisq, nfree=nfree, dof=dof, nfev=nfev, covar=covar, nocovar=nocovar, yerror=yerror, noderivative=noderivative, tol=tol, ftol=ftol, FUNCTARGS=FUNCTARGS, parinfo=parinfo, errmsg=errmsg, STATUS=STATUS, QUIET=QUIET, query=query, _EXTRA=_EXTRA)

Routine details

top mpcurvefit_eval

result = mpcurvefit_eval(p, dp, _EXTRA=_EXTRA)

Parameters

p
dp

Keywords

_EXTRA

top mpcurvefit

result = mpcurvefit(x, y, wts, p, perror, function_name=function_name, iter=iter, itmax=itmax, chisq=chisq, nfree=nfree, dof=dof, nfev=nfev, covar=covar, nocovar=nocovar, yerror=yerror, noderivative=noderivative, tol=tol, ftol=ftol, FUNCTARGS=FUNCTARGS, parinfo=parinfo, errmsg=errmsg, STATUS=STATUS, QUIET=QUIET, query=query, _EXTRA=_EXTRA)

Parameters

x
y
wts
p
perror

Keywords

function_name
iter
itmax
chisq
nfree
dof
nfev
covar
nocovar
yerror
noderivative
tol
ftol
FUNCTARGS
parinfo
errmsg
STATUS
QUIET
query
_EXTRA

File attributes

Modification date: Tue Oct 1 14:36:01 2013
Lines: 586