Notes onNotes onWeighted Least Squares Straight Weighted Least Squares Straight

line Fitline FitPassing Through The OriginPassing Through The Origin

Amarjeet BhullarAmarjeet Bhullar

November 14, 2008November 14, 2008

Data SetData Set

• For given {xFor given {xii, y, yi i } find line through } find line through them;them;i.e., find a and b in y = a+bxi.e., find a and b in y = a+bx

(x(x11,y,y11))

(x(x22,y,y22))

(x(x33,y,y33))

(x(x44,y,y44))

(x(x55,y,y55))(x(x66,y,y66))

(x(x77,y,y77))

bxay

Least SquaresLeast Squares

• Universal formulation of fitting:Universal formulation of fitting:minimize squares of differences betweenminimize squares of differences betweendata and functiondata and function– Example: for fitting a line, minimizeExample: for fitting a line, minimize

Using appropriate a and bUsing appropriate a and b– General solution: take derivatives w.r.t. General solution: take derivatives w.r.t.

unknown variables, set equal to zerounknown variables, set equal to zero

iii

i

bxay 22

2 1

Linear Least Squares: Equal Linear Least Squares: Equal WeightingWeighting

iii

iii

i

bxaybxayaa

0)(212

22

2

iiii

iii

i

bxayxbxaybb

0)(212

22

2

i

ii

iii

i xbaNxbay ]1[

i

ii

ii

ii xbxayx 2

ba

xx

xN

yx

y

ii

ii

ii

iii

ii

2

iii

ii

ii

ii

ii

yx

y

xx

xN

ba

1

2

iii

ii

ii

ii

ii

yx

y

Nx

xx

ba

2

1

Linear Least Squares: Equal Linear Least Squares: Equal WeightingWeighting

i i i iiiiii yxxyxIntercepta 21

i i iiiii yxyxNSlopeb 1

22

ii

ii xxN

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

Uncertainties or Estimation of Uncertainties or Estimation of Errors: In a & bErrors: In a & b

• Using the propagation of errors:Using the propagation of errors:

2

22

22

22

iyvuz y

zvz

uz

i

22

2

1

1

ii

ii

iij

j

i iiji

j

xxN

xNxyb

xxxya

Uncertainty or Estimation of Error: Uncertainty or Estimation of Error: In Calculated aIn Calculated a

ii

ii

ii

ii

i iii

ii

ii

ii

i i iijiij

ii

N

j

j ja

xxxNx

xxxxxN

xxxxxx

ya

222

222

2

222

222

2

2

222

22

12

2

2

22

2

2

The uncertainty in parameter aThe uncertainty in parameter a

Uncertainty or Estimation of Error: Uncertainty or Estimation of Error: In Calculated bIn Calculated b

The uncertainty in parameter bThe uncertainty in parameter b

222

2

2

2222

2

2

222

12

2

2

22

2

2

NxxNN

xNxNxN

xxNxxN

ya

ii

ii

ii

ii

ii

i iiijj

N

j

j jb

Uncertainties or Estimation of Uncertainties or Estimation of Errors: Errors:

In Calculated a & bIn Calculated a & bIntercept Uncertainty or ErrorIntercept Uncertainty or Error

iia x2

22

Slope Uncertainty or ErrorSlope Uncertainty or Error

2

2 Nb

22 )(2

1

i

ii bxayN

WhereWhere

&& 22

ii

ii xxN

bxay

?bxy

Linear Least Squares fit : Linear Least Squares fit :

• Linear least squares fitting and error of Linear least squares fitting and error of a straight line which MUST go through a straight line which MUST go through the origin (0, 0).the origin (0, 0).

• Partial derivative w. r. t. b is zeroPartial derivative w. r. t. b is zero

222 1

iii i

bxy

bxy

ii

iii

iii i

i

xbyx

bxyxb

2

2

2

02

ii

iii

x

yxSlopeb 2

bxy Uncertainty or Estimation of Error Uncertainty or Estimation of Error in b in b

22

2

2

22

22

2

22

)(

ii

ii

ji

i

jj

j jjb

x

x

xx

yb

ii

iii

x

yxb 2

ii

j

j xx

yb

2

ii

b x22

2

22 )(1

1

i

ii bxyN

WhereWhere

Weighted Least Squares Straight Weighted Least Squares Straight Line FittingLine Fitting

iii

iiii

i

bxaybxayaa

0)(1212

22

iii

i

i

iii

i

bxayxbxaybb

0)(212

22

2222

21

i

ii

i

i

i

i

i

i yxxyxa

2222

11

i

i

i

i

i

ii

i

yxyxb

2

22

2

2

1

i

i

i

i

i

xx

bxay

Uncertainties in a and b: Unequal Uncertainties in a and b: Unequal Weighting Weighting

Intercept Uncertainty or ErrorIntercept Uncertainty or Error

i i

ia

x2

22 1

Slope Uncertainty or ErrorSlope Uncertainty or Error

i ib 22 11

WhereWhere2

22

2

2

1

i

i

i

i

i

xx

bxay

?bxy

Weighted Least Squares Straight Weighted Least Squares Straight Line Fit:Line Fit:

bxy

iii

i

i

iii

i

bxyxbxybb

0)(212

22

2

i i

i

i i

ii

x

yx

b

2

2

2

i i

i

i i

ii xbyx2

2

2

Eq (6) in draft should beEq (6) in draft should be

0

850

5202 log11

IIbADC i

ii i

WhereWhere2

850

5202

1i

i i

b

Eq (7) in draft should beEq (7) in draft should be

Uncertainty in b: Unequal Uncertainty in b: Unequal WeightingWeighting

2

2

2

2

1

,1

j

j

j

i i

i

i i

ii

xyb

xwithyxb

i i

i

i

i

i

j j

jj

j jjb

xx

x

yb

2

2

22

2

2

4

2

22

2

22

11

1

i i

ii i

ib x

x

2

22

2

22 11

Eq (8) in draft should beEq (8) in draft should be

850

5202

22 1

i i

iADC b

ConclusionConclusion

0

850

5202 log11

IIbADC i

ii i

2850

5202

1i

i i

b

850

5202

22 1

i i

iADC b

Eq (6) Eq (6)

Eq (7) Eq (7)

Eq (8) Eq (8)