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

2. Data Set
For given {xi, yi } find line through them; i.e., find a and b in y = a+bx
(x6,y6)
(x3,y3)
(x5,y5)
(x1,y1)
(x7,y7)
(x4,y4)
(x2,y2)

3. Least Squares
Universal 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

4. Linear Least Squares: Equal Weighting

5. Linear Least Squares: Equal Weighting
Data Reduction and Error Analysis for the Physical Sciences by Philip R Bevington (1969)

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

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

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

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

10. Linear Least Squares fit :
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 zero

11. Uncertainty or Estimation of Error in b
Where

12. Weighted Least Squares Straight Line Fitting

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

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

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

16. Conclusion
Eq (6)
Eq (7)
Eq (8)