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MPFITEXPR

MPFITEXPR

## 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 arbitrary expression

## Major Topics

Curve and Surface Fitting

## Calling Sequence

MYFUNCT = 'X*(1-X)+3'
parms = MPFITEXPR(MYFUNCT, XVAL, YVAL, ERR, start_parms, ...)

## Description

MPFITEXPR fits a user-supplied model -- in the form of an arbitrary IDL
expression -- to a set of user-supplied data. MPFITEXPR calls
MPFIT, the MINPACK-1 least-squares minimizer, to do the main
work.
Given the data and their uncertainties, MPFITEXPR 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").
- A text IDL expression which computes Y given X.
- Starting guesses for all of the parameters ("START_PARAMS").
There are very few restrictions placed on X, Y or the expression of
the model. Simply put, the expression 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.
Some rules must be obeyed in constructing the expression. First,
the independent variable name *MUST* be "X" in the expression,
regardless of the name of the variable being passed to MPFITEXPR.
This is demonstrated in the above calling sequence, where the X
variable passed in is called "XVAL" but the expression still refers
to "X". Second, parameter values must be referred to as an array
named "P".
If you do not pass in starting values for the model parameters,
MPFITEXPR will attempt to determine the number of parameters you
intend to have (it does this by looking for references to the array
variable named "P"). When no starting values are passed in, the
values are assumed to start at zero.
MPFITEXPR carefully avoids passing large arrays where possible to
improve performance.
See below for an example of usage.
EVALUATING EXPRESSIONS
This source module also provides a function called MPEVALEXPR. You
can use this function to evaluate your expression, given a list of
parameters. This is one of the easier ways to compute the model
once the best-fit parameters have been found. Here is an example:
YMOD = MPEVALEXPR(MYFUNCT, XVAL, PARMS)
where MYFUNCT is the expression (see MYFUNCT below), XVAL is the
list of "X" values, and PARMS is an array of parameters. The
returned array YMOD contains the expression MYFUNCT evaluated at
each point in XVAL.
PASSING PRIVATE DATA TO AN EXPRESSION
The most complicated optimization problems typically involve other
external parameters, in addition to the fitted parameters. While
it is extremely easy to rewrite an expression dynamically, in case
one of the external parameters changes, the other possibility is to
use the PRIVATE data structure.
The user merely passes a structure to the FUNCTARGS keyword. The
user expression receives this value as the variable PRIVATE.
MPFITEXPR nevers accesses this variable so it can contain any
desired values. Usually it would be an IDL structure so that any
named external parameters can be passed to the expression.

## 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 as an equality constraint. 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 a TIED expression. ]
.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)
.MPFORMAT - IDL format string to print the parameter within
ITERPROC. Default: '(G20.6)' (An empty string will
also use the default.)
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
"MP", but otherwise they are free to include their own fields
within the PARINFO structure, which will be ignored by MPFIT.

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.

## Compatibility

This function is designed to work with IDL 5.0 or greater. Because
this function uses the IDL EXECUTE() function, it will not work
with the free version of the IDL Virtual machine.

## Inputs

MYFUNCT - a string variable containing an IDL expression. The
only restriction is that the independent variable *must*
be referred to as "X" and model parameters *must* be
referred to as an array called "P". Do not use symbol
names beginning with the underscore, "_".
The expression should calculate "model" Y values given
the X values and model parameters. Using the vector
notation of IDL, this can be quite easy to do. If your
expression gets complicated, you may wish to make an IDL
function which will improve performance and readibility.
The resulting array should be of the same size and
dimensions as the "measured" Y values.
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.
If no parameters are given, then MPFITEXPR attempts
to determine the number of parameters by scanning
the expression. Parameters determined this way are
initialized to zero. This technique is not fully
reliable, so users are advised to pass explicit
parameter starting values.

## Returns

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:
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
FUNCTARGS - passed directly to the expression as the variable
PRIVATE. Any user-private data can be communicated to
the user expression using this keyword.
Default: PRIVATE is undefined in user expression
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
x = dindgen(200) * 0.1 - 10. ; Independent variable
yi = gauss1(x, [2.2D, 1.4, 3000.]) + 1000 ; "Ideal" Y variable
y = yi + randomn(seed, 200) * sqrt(yi) ; Measured, w/ noise
sy = sqrt(y) ; Poisson errors
; Now fit a Gaussian to see how well we can recover
expr = 'P[0] + GAUSS1(X, P[1:3])' ; fitting function
p0 = [800.D, 1.D, 1., 500.] ; Initial guess
p = mpfitexpr(expr, x, y, sy, p0) ; Fit the expression
print, p
plot, x, y ; Plot data
oplot, x, mpevalexpr(expr, x, p) ; Plot model
Generates a synthetic data set with a Gaussian peak, and Poisson
statistical uncertainty. Then a model consisting of a constant
plus Gaussian is fit to the data.

## 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 NITER 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
Parameter values can be tied to others, 09 Nov 1998
Added MPEVALEXPR utility function, 09 Dec 1998
Cosmetic documentation updates, 16 Apr 1999, CM
More cosmetic documentation updates, 14 May 1999, CM
Made sure to update STATUS value, 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
Check to be sure that X and Y are present, 02 Nov 1999, CM
Documented PERROR for unweighted fits, 03 Nov 1999, CM
Removed ITERPROC='' when quiet EQ 1, 10 Jan 2000, CM
Changed to ERROR_CODE for error condition, 28 Jan 2000, CM
Updated the EXAMPLE, 26 Mar 2000, CM
Copying permission terms have been liberalized, 26 Mar 2000, CM
Propagated improvements from MPFIT, 17 Dec 2000, CM
Correct reference to _WTS in MPFITEXPR_EVAL, 25 May 2001, CM
(thanks to Mark Fardal)
Added useful FUNCTARGS behavior (as yet undocumented), 04 Jul
2002, CM
Documented FUNCTARGS/PRIVATE behavior, 22 Jul 2002, CM
Added NFREE and NPEGGED keywords, 13 Sep 2002, CM
Documented RELSTEP field of PARINFO (!!), CM, 25 Oct 2002
Add DOF keyword, CM, 31 Jul 2003
Add FUNCTARGS keyword to MPEVALEXPR, CM 08 Aug 2003
Minor documentation adjustment, 03 Feb 2004, CM
Move STRICTARR compile option inside each function/procedure, 9 Oct 2006
Clarify documentation on user-function, derivatives, and PARINFO,
27 May 2007
Add COMPATIBILITY section, CM, 13 Dec 2007
\$Id: mpfitexpr.pro,v 1.15 2007/12/15 14:28:27 craigm Exp \$

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