Notes on Weighted Least Squares Straight line Fit Passing Through The Origin Amarjeet Bhullar November 14, 2008

Data Set For given {x i, y i } find line through them; i.e., find a and b in y = a+bxFor given {x i, y i } find line through them; i.e., find a and b in y = a+bx (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) (x 4,y 4 ) (x 5,y 5 ) (x 6,y 6 ) (x 7,y 7 )

Least Squares Universal formulation of fitting: minimize squares of differences between data and functionUniversal formulation of fitting: minimize squares of differences between data and function – Example: for fitting a line, minimize Using appropriate a and b – General solution: take derivatives w.r.t. unknown variables, set equal to zero

Linear Least Squares: Equal Weighting

Data Reduction and Error Analysis for the Physical Sciences by Philip R Bevington (1969)

Uncertainties or Estimation of Errors: In a & b Using the propagation of errors:Using the propagation of errors:

Uncertainty or Estimation of Error: In Calculated a The uncertainty in parameter a

Uncertainty or Estimation of Error: In Calculated b The uncertainty in parameter b

Uncertainties or Estimation of Errors: In Calculated a & b Intercept Uncertainty or Error Slope Uncertainty or Error Where &

Linear Least Squares fit : Linear least squares fitting and error of a straight line which MUST go through the origin (0, 0).Linear least squares fitting and error of a straight line which MUST go through the origin (0, 0). Partial derivative w. r. t. b is zeroPartial derivative w. r. t. b is zero

Uncertainty or Estimation of Error in b Where

Weighted Least Squares Straight Line Fitting

Uncertainties in a and b: Unequal Weighting Intercept Uncertainty or Error Slope Uncertainty or Error Where

Weighted Least Squares Straight Line Fit: Eq (6) in draft should be Where Eq (7) in draft should be

Uncertainty in b: Unequal Weighting Eq (8) in draft should be

Conclusion Eq (6) Eq (7) Eq (8)