GAUSSFIT

The GAUSSFIT function computes a non-linear least-squares fit to a function f (x) with from three to six unknown parameters. f (x) is a linear combination of a Gaussian and a quadratic; the number of terms is controlled by the keyword parameter NTERMS.

This routine is written in the IDL language. Its source code can be found in the file gaussfit.pro in the lib subdirectory of the IDL distribution.

Syntax

Result = GAUSSFIT( X, Y [, A] [, CHISQ=variable] [, ESTIMATES=array] [, MEASURE_ERRORS=vector] [, NTERMS=integer{3 to 6}] [, SIGMA=variable] [, YERROR=variable])

Return Value

Returns three to six values that are the non-linear least squares fit to a function f (x).

Arguments

X

An n-element vector of independent variables.

Y

A vector of dependent variables, the same length as X.

A

A named variable that will contain the coefficients A of the fit.

Keywords

CHISQ

Set this keyword to a named variable that will contain the value of the chi-square goodness-of-fit.

ESTIMATES

Set this keyword equal to an array of starting estimates for the parameters of the equation. If the NTERMS keyword is specified, the ESTIMATES array should have NTERMS elements. If NTERMS is not specified, the ESTIMATES array should have six elements. If the ESTIMATES array is not specified, estimates are calculated by first subtracting a constant (if NTERMS >= 4) or a linear term (if NTERMS >= 5), and then forming a simple estimate of the Gaussian coefficients.

MEASURE_ERRORS

Set this keyword to a vector containing standard measurement errors for each point Y[i]. This vector must be the same length as X and Y.

NTERMS

Set this keyword to an integer value between three and six to specify the function to be used for the fit. The values correspond to the functions shown below. In all cases:

NTERMS=6

NTERMS=5

NTERMS=4

NTERMS=3

NTERMS=6 is the default setting. Here, A0 is the height of the Gaussian, A1 is the center of the Gaussian, A2 is the width (the standard deviation) of the Gaussian, A3 is the constant term, A4 is the linear term, and A5 is the quadratic term.

SIGMA

Set this keyword to a named variable that will contain the 1-sigma error estimates of the returned parameters.

YERROR

Set this keyword to a named variable that will contain the standard error between YFIT and Y.

Examples

The following example shows how to use GAUSSFIT to fit to four different functions, with NTERMS=3,4,5,6. To simulate actual data, random noise has been added to each function.

pro ex_gaussfit 
 
   ; Define the independent variable. 
   n = 101 
   x = (FINDGEN(n)-(n/2))/4 
 
   ; Define the coefficients. 
   a = [4.0, 1.0, 2.0, 1.0, 0.25, 0.01] 
   print, 'Expected: ', a 
   z = (x - a[1])/a[2]    ; Gaussian variable 
   !P.MULTI = [0,2,2]     ; set up 2x2 plot window 
   seed = 123321          ; Pick a starting seed value 
 
   for nterms=3,6 do begin 
      ; Define the dependent variables. Start with random noise. 
      y = 0.4*RANDOMN(seed, n) 
 
      ; Use a switch statement so we fall through to each term. 
      switch nterms of 
         6: y = y + a[5]*x^2 
         5: y = y + a[4]*x 
         4: y = y + a[3] 
         3: y = y + a[0]*exp(-z^2/2) 
      endswitch 
 
      ; Fit the data to the function, storing coefficients in 
      ; coeff: 
      yfit = GAUSSFIT(x, y, coeff, NTERMS=nterms) 
      print, 'Result:   ', coeff[0:nterms-1] 
      ; Plot the original data and the fitted curve: 
      PLOT, x, y, TITLE='nterms='+STRTRIM(nterms,2) 
      OPLOT, x, yfit, THICK=2 
   endfor 
end 

When this program is compiled and executed, IDL prints the following results:

IDL> ex_gaussfit 
Expected: 4.00000 1.00000 2.00000 1.00000 0.250000 0.0500000 
Result: 3.95437 1.03176 2.07216 
Result: 4.38669 0.948479 1.84426 0.909676 
Result: 3.93348 0.839296 2.02112 1.05237 0.249002 
Result: 3.80389 0.993375 2.07302 1.16684 0.249051 0.0484357 

Version History

Introduced: Original

CHISQ, SIGMA, and YERROR keywords added: 5.6

See Also

COMFIT, CURVEFIT, GAUSS2DFIT, POLY_FIT, REGRESS, SFIT, SVDFIT