;+
; NAME:
; MPFITCOVAR
;
; 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 with full covariance matrix
;
; MAJOR TOPICS:
; Curve and Surface Fitting
;
; CALLING SEQUENCE:
; parms = MPFITCOVAR(MYFUNCT, X, Y, YCOVAR, start_params, ...)
;
; DESCRIPTION:
;
; MPFITCOVAR fits a user-supplied model -- in the form of an IDL
; function -- to a set of user-supplied data, assuming that the
; measured data points have the given covariance matrix. MPFITCOVAR
; calls MPFIT, the MINPACK-1 least-squares minimizer, to do the main
; work.
;
; This function is similar to MPFITFUN() except that the user must
; pass a matrix YCOVAR, which specifies the covariance between Y
; values. I.e. YCOVAR[i,j] gives the covariance between Y[i] and
; Y[j]. YCOVAR is necessarily positive definite and symmetric.
;
; MPFITCOVAR uses a singular value decomposition of YCOVAR in order
; to construct uncorrelated deviates. MPFITCOVAR only keeps the
; largest singular values from the decomposition. The threshold
; between kept and excluded singular values is determined by the
; keyword parameter SINGULAR_VALUE_THRESH. This is equivalent to
; keeping only the independent measurements and discarding the
; correlated ones. By setting a smaller value for the threshold, one
; one effectively keeps more data points, at the risk of introducing
; more correlation and noise in the fit. By setting a larger value
; for the threshold, one effectively deletes correlated data points,
; at the risk of ignoring slightly uncorrelated points. Upon return
; the NSING keyword contains the number of retained singular values.
;
; Because MPFITCOVAR requires the full measurement covariance matrix
; and uses singular value decomposition, it can be used for
; modest-sized problems, but will certainly run into difficulties
; with large input covariance matrices.
;
; Given the data and their uncertainties, MPFITCOVAR 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 N "measured" *dependent* variable values ("Y").
; - An NxN measurement covariance matrix ("YCOVAR").
; - The name of an IDL function which computes Y given X ("MYFUNCT").
; - Starting guesses for all of the parameters ("START_PARAMS").
;
; There are very few restrictions placed on X, Y or MYFUNCT. Simply
; put, MYFUNCT 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.
;
; MPFITFUN 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:
;
; FUNCTION MYFUNCT, X, P
; ; The independent variable is X
; ; Parameter values are passed in "P"
; YMOD = ... computed model values at X ...
; return, YMOD
; 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 "EXPLICIT DERIVATIVES" and AUTODERIVATIVE
; in MPFIT.PRO if you wish to compute the derivatives for yourself.
; AUTODERIVATIVE is accepted by MPFITFUN 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 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 TIED parameters rely on the EXECUTE() function, they cannot
; be used with the free version of the IDL Virtual Machine.
;
;
; INPUTS:
; MYFUNCT - a string variable containing the name of an IDL function.
; This function computes the "model" Y values given the
; X values and model parameters, as desribed above.
;
; 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.
; NOTE: the following special cases apply:
; * if Y is NaN or Infinite, and the NAN keyword is
; set, then the corresponding data point is ignored
;
; YCOVAR - a matrix of dimension ARRAY(NY,NY) where
; NY=N_ELEMENTS(Y). This matrix gives the covariance
; between the elements of Y, as described above. YCOVAR
; must be symmetric. YCOVAR is assumed to be constant
; during the fit (i.e. it cannot be re-computed each
; iteration).
;
; 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.
;
;
; RETURNS:
;
; Returns the array of best-fit parameters.
;
;
; KEYWORD PARAMETERS:
;
; BESTNORM - the value of the summed squared residuals for the
; returned parameter values.
;
; COVAR - the output parameter 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 - 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
; 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
;
; MAX_SINGULAR_VALUE - Upon return, the maximum singular value found
; while computing the decomposition of YCOVAR.
;
; 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
;
; NSING - upon return, the number of effectively independent values
; in data array Y.
;
; 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.
;
; SINGULAR_VALUE_THRESHOLD - the minumum acceptable singular value from
; the decomposition of YCOVAR, relative to the maximum
; value. I.e. singular values larger than
; MAX_SINGULAR_VALUE*SINGULAR_VALUE_THRESHOLD are kept.
;
; SINGULAR_VALUES - upon return, an array of the singular values
; found based on the covariance matrix.
;
; 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.
;
; 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
; 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 (cent, width, area)
; p = mpfitfun('GAUSS1', x, y, sy, p0) ; 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 (with different starting parameters) to see how close we can
; get.
;
;
; 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, May 2009, CM
; Additional documentation, 2016-05-19, CM
;
;
; $Id: mpfitcovar.pro,v 1.2 2017/01/03 18:53:29 cmarkwar Exp $
;-
; Copyright (C) 2009, 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 mpfitcovar_eval, mpfitcovar, mpfit
; This is the call-back function for MPFIT. It evaluates the
; function, subtracts the data, and returns the residuals.
function mpfitcovar_eval, p, dp1, _EXTRA=extra
COMPILE_OPT strictarr
common mpfitcovar_common, fcn, x, y, f, fcnargs, sbdt, nsing
;; Save the original DP matrix for later use
if n_params() GT 1 then if n_elements(dp) GT 0 then dp0 = dp
;; The function is evaluated here. There are four choices,
;; depending on whether (a) FUNCTARGS was passed to MPFITFUN, 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 f = call_function(fcn, x, p, dp, _EXTRA=fcnargs)$
else f = call_function(fcn, x, p, _EXTRA=fcnargs)
endif else begin
if n_params() GT 1 then f = call_function(fcn, x, p, dp) $
else f = call_function(fcn, x, p)
endelse
np0 = n_elements(p)
nf0 = n_elements(f)
;; Actual residuals (correlated)
result = (y-f)
;; Transformed residuals (uncorrelated)
res1 = sbdt ## result
;; Transformed Jacobian (uncorrelated)
if n_elements(dp0) GT 0 AND n_elements(dp) EQ np0*nf0 then begin
dp1 = dp[0:nsing-1,*]
for j = 0L, np0-1 do dp1[j*nsing] = sbdt ## dp[j*nf:j*nf+nf-1]
endif
;; Make sure the returned result is one-dimensional.
res1 = reform(res1, n_elements(res1), /overwrite)
return, res1
end
function mpfitcovar, fcn, x, y, ycovar, p, FUNCTARGS=fa, $
BESTNORM=bestnorm, nfev=nfev, STATUS=status, $
parinfo=parinfo, query=query, $
covar=covar, perror=perror, yfit=yfit, $
niter=niter, nfree=nfree, npegged=npegged, dof=dof, $
nsing=nsing, max_singular_value=bmax, $
singular_values=b, $
singular_value_threshold=bthresh0, $
quiet=quiet, ERRMSG=errmsg, NAN=NAN, _EXTRA=extra
COMPILE_OPT strictarr
status = 0L
errmsg = ''
b = 0 & dummy = temporary(b) ;; Erase the SINGULAR_VALUES keyword
;; 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: PARMS = MPFITCOVAR('MYFUNCT', X, Y, YCOVAR, "+ $
"START_PARAMS, ... )", /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(bthresh0) EQ 0 then begin
bthresh = 1d-5
endif else begin
bthresh = bthresh0[0]
endelse
;; Use common block to pass data back and forth
common mpfitcovar_common, fc, xc, yc, mc, ac, $
sbdt, nsing1
fc = fcn & xc = x & yc = y & mc = 0L
;; Singular value decomposition of covariance matrix into
;; singular values B, left vectors C and right vectors D^T.
;; Since YCOVAR is symmetric and square, C and D should be
;; equal.
svdc, ycovar, b, c, d
bmax = max(b)
c = 0 ;; Don't need C
if bmax EQ 0 then begin
errmsg = 'ERROR: all singular values of covariance matrix are zero'
message, errmsg, /info
return, !values.d_nan
endif
;; Zero out the values less than threshold
wh = where(b/bmax LT bthresh, nzero)
if nzero GT 0 then b[wh] = 0
;; Number of significant singular values.
wh = where(b/bmax GE bthresh, nsing)
;; compute B^(-1/2) D^T - this array is rectangular because we
;; ignore the transformed values with singular values LT BTHRESH
sbdt = transpose(d[wh,*])
for i = 0L, nsing-1 do begin
sbdt[*,i] = sbdt[*,i] / sqrt(b[wh[i]])
endfor
nsing1 = nsing
d = 0 ;; No longer need D
;; FUNCTARGS
ac = 0 & dummy = size(temporary(ac))
if n_elements(fa) GT 0 then ac = fa
result = mpfit('mpfitcovar_eval', p, $
parinfo=parinfo, STATUS=status, nfev=nfev, BESTNORM=bestnorm,$
covar=covar, perror=perror, $
niter=niter, nfree=nfree, npegged=npegged, dof=dof, $
ERRMSG=errmsg, quiet=quiet, _EXTRA=extra)
;; Retrieve the fit value
yfit = temporary(mc)
;; Some cleanup
xc = 0 & yc = 0 & & mc = 0 & ac = 0 & sbdt = 0 & nsing1 = 0
;; Print error message if there is one.
if NOT keyword_set(quiet) AND errmsg NE '' then $
message, errmsg, /info
return, result
end