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  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  UPDATED VERSIONs can be found on my WEB PAGE:


  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, ...)


  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
  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:
    ; The independent variables are X and Y
    ; Parameter values are passed in "P"
    ZMOD = ... computed model values at (X,Y) ...
    return, ZMOD
  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.
  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
  Then using equations 1 and 2 above will give these values to X and
    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
    .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
                  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:
                  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.


  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.


  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
  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


  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
        Note that this doesn't account for pegged parameters (see
  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
              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
              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.


  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
  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

    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.


  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

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