;+ ; NAME: ; FITEXY ; PURPOSE: ; Best straight-line fit to data with errors in both coordinates ; EXPLANATION: ; Linear Least-squares approximation in one-dimension (y = a + b*x), ; when both x and y data have errors Users might be interested in ; Michael Williams MPFITEXY routines which include a number of ; enhancements to FITEXY. ; ( https://github.com/williamsmj/mpfitexy ) ; ; ; CALLING EXAMPLE: ; FITEXY, x, y, A, B, X_SIG= , Y_SIG= , [sigma_A_B, chi_sq, q, TOL=] ; ; INPUTS: ; x = array of values for independent variable. ; y = array of data values assumed to be linearly dependent on x. ; ; REQUIRED INPUT KEYWORDS: ; X_SIGMA = scalar or array specifying the standard deviation of x data. ; Y_SIGMA = scalar or array specifying the standard deviation of y data. ; ; OPTIONAL INPUT KEYWORD: ; TOLERANCE = desired accuracy of minimum & zero location, default=1.e-3. ; ; OUTPUTS: ; A_intercept = constant parameter result of linear fit, ; B_slope = slope parameter, so that: ; ( A_intercept + B_slope * x ) approximates the y data. ; OPTIONAL OUTPUT: ; sigma_A_B = two element array giving standard deviation of ; A_intercept and B_slope parameters, respectively. ; The standard deviations are not meaningful if (i) the ; fit is poor (see parameter q), or (ii) b is so large that ; the data are consistent with a vertical (infinite b) line. ; If the data are consistent with *all* values of b, then ; sigma_A_B = [1e33,e33] ; chi_sq = resulting minimum Chi-Square of Linear fit, scalar ; q - chi-sq probability, scalar (0-1) giving the probability that ; a correct model would give a value equal or larger than the ; observed chi squared. A small value of q indicates a poor ; fit, perhaps because the errors are underestimated. As ; discussed by Tremaine et al. (2002, ApJ, 574, 740) an ; underestimate of the errors (e.g. due to an intrinsic dispersion) ; can lead to a bias in the derived slope, and it may be worth ; enlarging the error bars to get a reduced chi_sq ~ 1 ; ; COMMON: ; common fitexy, communicates the data for computation of chi-square. ; ; PROCEDURE CALLS: ; CHISQ_FITEXY() ;Included in this file ; MINF_BRACKET, MINF_PARABOLIC, ZBRENT ;In IDL Astronomy Library ; MOMENT(), CHISQR_PDF() ;In standard IDL distribution ; ; PROCEDURE: ; From "Numerical Recipes" column by Press and Teukolsky: ; in "Computer in Physics", May, 1992 Vol.6 No.3 ; Also see the 2nd edition of the book "Numerical Recipes" by Press et al. ; ; In order to avoid problems with data sets where X and Y are of very ; different order of magnitude the data are normalized before the fitting ; process is started. The following normalization is used: ; xx = (x - xm) / xs and sigx = x_sigma / xs ; where xm = MEAN(x) and xs = STDDEV(x) ; yy = (y - ym) / ys and sigy = y_sigma / ys ; where ym = MEAN(y) and ys = STDDEV(y) ; ; ; MODIFICATION HISTORY: ; Written, Frank Varosi NASA/GSFC September 1992. ; Now returns q rather than 1-q W. Landsman December 1992 ; Use CHISQR_PDF, MOMENT instead of STDEV,CHI_SQR1 W. Landsman April 1998 ; Fixed typo for initial guess of slope, this error was nearly ; always insignificant W. Landsman March 2000 ; Normalize X,Y before calculation (from F. Holland) W. Landsman Nov 2006 ;- function chisq_fitexy, B_angle ; ; NAME: ; chisq_fitexy ; PURPOSE: ; Function minimized by fitexy (computes chi-square of linear fit). ; It is called by minimization procedures during execution of fitexy. ; CALLING SEQUENCE: ; chisq = chisq_fitexy( B_angle ) ; INPUTS: ; B_angle = arc-tangent of B_slope of linear fit. ; OUTPUTS: ; Result of function = chi_square - offs (offs is in COMMON). ; COMMON: ; common fitexy, communicates the data from pro fitexy. ; PROCEDURE: ; From "Numerical Recipes" column: Computer in Physics Vol.6 No.3 ; MODIFICATION HISTORY: ; Written, Frank Varosi NASA/GSFC 1992. common fitexy, xx, yy, sigx, sigy, ww, Ai, offs B_slope = tan( B_angle ) ww = 1/( ( (B_slope * sigx)^2 + sigy^2 ) > 1.e-30 ) if N_elements( ww ) EQ 1 then sumw = ww * N_elements( xx ) \$ else sumw = total( ww ) y_Bx = yy - B_slope * xx Ai = total( ww * y_Bx )/sumw return, total( ww * (y_Bx - Ai)^2 ) - offs end ;------------------------------------------------------------------------------- pro fitexy, x, y, A_intercept, B_slope, sigma_A_B, chi_sq, q, TOLERANCE=Tol, \$ X_SIGMA=x_sigma, Y_SIGMA=y_sigma compile_opt idl2 common fitexy, xx, yy, sigx, sigy, ww, Ai, offs if N_params() LT 4 then begin print,'Syntax - fitexy, x, y, A, B, X_SIG=sigx, Y_SIG=sigy,' print,' [sigma_A_B, chi_sq, q, TOLERANCE = ]' return endif ; Normalize data before running fitexy xm = (MOMENT(x, SDEV = xs, /DOUBLE))[0] ym = (MOMENT(y, SDEV = ys, /DOUBLE))[0] xx = (x - xm) / xs yy = (y - ym) / ys sigx = x_sigma / xs sigy = y_sigma / ys ;Compute first guess for B_slope using standard 1-D Linear Least-squares fit, ; where the non-linear term involving errors in x are ignored. ; (note that Tx is a transform to reduce roundoff errors) ww = sigx^2 + sigy^2 if N_elements( ww ) EQ 1 then sumw = ww * N_elements( xx ) \$ else sumw = total( ww ) Sx = total( xx * ww ) Tx = xx - Sx/sumw B = total( ww * yy * Tx ) / total( ww * Tx^2 ) ;Find the minimum chi-sq while including the non-linear term (B * sigx)^2 ; involving variance in x data (computed by function chisq_fitexy): ; using minf_bracket (=MNBRAK) and minf_parabolic (=BRENT) offs = 0 ang = [ 0, atan( B ), 1.571 ] chi = fltarr( 3 ) for j=0,2 do chi[j] = chisq_fitexy( ang[j] ) ;this is for later... if N_elements( Tol ) NE 1 then Tol=1.e-3 a0 = ang[0] a1 = ang[1] minf_bracket, a0,a1,a2, c0,c1,c2, FUNC="chisq_fitexy" minf_parabolic, a0,a1,a2, Bang, chi_sq, FUNC="chisq_fitexy", TOL=Tol if N_params() EQ 7 then q = 1 - chisqr_pdf( chi_sq, N_elements(x) - 2 ) A_intercept = Ai ;computed in function chisq_fitexy ang = [a0,a1,a2,ang] chi = [c0,c1,c2,chi] ;Now compute the variances of estimated parameters, ; by finding roots of ( (chi_sq + 1) - chisq_fitexy ). ;Note: ww, Ai are computed in function chisq_fitexy. offs = chi_sq + 1 wc = where( chi GT offs, nc ) if (nc GT 0) then begin angw = [ang[wc]] d1 = abs( angw - Bang ) MOD !PI d2 = !PI - d1 wa = where( angw LT Bang, na ) if (na GT 0) then begin d = d1[wa] d1[wa] = d2[wa] d2[wa] = d endif Bmax = zbrent( Bang,Bang+max(d1),F="chisq_fitexy",T=Tol ) -Bang Amax = Ai - A_intercept Bmin = zbrent( Bang,Bang-min(d2),F="chisq_fitexy",T=Tol ) -Bang Amin = Ai - A_intercept if N_elements( ww ) EQ 1 then r2 = 2/( ww * N_elements( x ) ) \$ else r2 = 2/total( ww ) sigma_A_B = [ Amin^2 + Amax^2 + r2 , Bmin^2 + Bmax^2 ] sig_A_B = sqrt( sigma_A_B/2 ) / ([1,cos(Bang)^2]) endif ;Finally, transform parameters back to orignal units. B_slope = tan( Bang ) *ys /xs A_intercept = A_intercept*ys - tan(Bang) * ys / xs *xm + ym if Nc GT 0 then sigma_A_B = [SQRT( (sig_A_B[0] * ys)^2 + \$ (sig_A_B[1] * ys / xs * xm)^2 ), sig_A_B[1] * ys / xs] \$ else sigma_A_B = [1.e33,1.e33] return end