Curve and Surface Fitting

The problem of curve fitting may be stated as follows:

Given a tabulated set of data values {x_i, y_i} and the general form of a mathematical model (a function f(x) with unspecified parameters), determine the parameters of the model that minimize an error criterion. The problem of surface fitting involves tabulated data of the form {xi, yi, zi} and a function f(x, y) of two spatial dimensions.

For example, we can use the CURVEFIT routine to determine the parameters A and B of a user-supplied function f(x), such that the sums of the squares of the residuals between the tabulated data {xi, yi} and function are minimized. We will use the following function and data:

f (x) = a (1 -e^-bx)

x_i = [0.25, 0.75, 1.25, 1.75, 2.25]

y_i = [0.28, 0.57, 0.68, 0.74, 0.79]

First we must provide a procedure written in IDL to evaluate the function, f, and its partial derivatives with respect to the parameters a₀ and a₁:

PRO funct, X, A, F, PDER 
   F = A[0] * (1.0 - EXP(-A[1] * X)) 
   ; If the function is called with four parameters, 
   ; calculate the partial derivatives: 
   IF N_PARAMS() GE 4 THEN BEGIN 
      ; PDER's column dimension is equal to the number of 
      ; elements in xi and its row dimension is equal to  
      ; the number of parameters in the function F: 
      pder = FLTARR(N_ELEMENTS(X), 2) 
      ; Compute the partial derivatives with respect to 
      ; a0 and place in the first row of PDER: 
      pder[*, 0] = 1.0 - EXP(-A[1] * X) 
      ; Compute the partial derivatives with respect to 
      ; a1 and place in the second row of PDER: 
      pder[*, 1] = A[0] * x * EXP(-A[1] * X) 
  ENDIF 
END 

Next, we can use the following IDL commands to find the function's parameters:

;Define the vectors of tabulated: 
X = [0.25, 0.75, 1.25, 1.75, 2.25] 
;data values: 
Y = [0.28, 0.57, 0.68, 0.74, 0.79] 
;Define a vector of weights: 
W = 1.0 / Y 
;Provide an initial guess of the function's parameters: 
A = [1.0, 1.0] 
;Compute the parameters a0 and a1: 
yfit = CURVEFIT(X, Y, W, A, SIGMA_A, FUNCTION_NAME = 'funct') 
;Print the parameters, which are returned in A: 
PRINT, A 

IDL prints:

  0.787386  1.71602 

Thus the nonlinear function that best fits the data is:

f (x) = 0.787386 ( 1 --e-1.71602x )

Routines for Curve and Surface Fitting

Below is a brief description of IDL routines for curve and surface fitting.