linear regression line of best fit. gradient = intercept =

Linear Regression

Line of Best Fit

22 )( xxnyxxyn

a

22

2

)( xxnxyxxy

b

Gradient =

Intercept =

Consider the following graph

d1

d2d3

d4

d6

d5

d8d7

We want a Line where d1 - d7 has the minimum distance

d1

d2d3

d4

d6

d5

d8d7

Just adding will not do it

A better method is to square the error

S = d21 + d2

2 + d23 + d2

5+ d24 + d2

6

We now need to find when ‘S’ is a ‘minimum’

S = d2i

= y – (ax + b)2

= y – ax - b2

= y – ax - b2

S

Ignoring the summation sign

=dsda

2(y – ax – b) . (-x)

=dsdb

2(y – ax – b) . (-1)

We need to find when these are = zero

=dsda

2(y – ax – b) . (-x)

0 = (y – ax – b) . (-x)

0 = (-yx + ax2 + bx) .

We need to find when these are = zero

=dsdb

2(y – ax – b) . (-1)

0 = (y – ax – b) . (-1)

0 = (- y + ax + b) .

This this gives us two equations

0 = (- y + ax + b)

0 = (-yx + ax2 + bx)

Rearranging gives

y = + ax + b

yx = + ax2 + bx

This is a set of simultaneous equations and can be solved for ‘a’ and ‘b’

Put back the Summation signs

y = ax + b

yx = ax2 + bx

This can be rearranged

yx = a. x2 + b.x

y = a. x + bn

Now solve for ‘a’ and ‘b’

22 )( xxnyxxyn

a

22

2

)( xxnxyxxy

b

Gradient =

Intercept =

Easy

Try an Example

n x y

Freq   Inductive reactance

1 50   30

2 100   65

3 150   90

4 200   130

5 250   150

6 300   190

7 350   200 0

50

100

150

200

250

0 100 200 300 400

FrequencyIn

duct

ive

Rea

ctan

ce

Plot your data

Consider the following data

Not very straight

Make two new columns

Use Method of Least Squares

xy x2

1500 2500

6500 10000

13500 22500

26000 40000

37500 62500

57000 90000

70000 122500

n x y

Freq   Inductive reactance

1 50   30

2 100   65

3 150   90

4 200   130

5 250   150

6 300   190

7 350   200

1400 855 212000 350000

22 )( xxnyxxyn

a

22

2

)( xxnxyxxy

b

Now for y = a.x + b

where

Use Method of Least Squares

xy x2

1500 2500

6500 10000

13500 22500

26000 40000

37500 62500

57000 90000

70000 122500

n x y

Freq   Inductive reactance

1 50   30

2 100   65

3 150   90

4 200   130

5 250   150

6 300   190

7 350   200

1400 855 212000 350000

22 )( xxnyxxyn

a

Find ‘ a ‘

a = 7 x 212000 - 1400 x 855

7 x 350000

- (1400)2

a = 0.5857

Use Method of Least Squares

xy x2

1500 2500

6500 10000

13500 22500

26000 40000

37500 62500

57000 90000

70000 122500

n x y

Freq   Inductive reactance

1 50   30

2 100   65

3 150   90

4 200   130

5 250   150

6 300   190

7 350   200

1400 855 212000 350000

Find ‘ b ‘

b = 855 x 350000 - 1400 x 212000

7 x 350000

- (1400)2

b = 5

22

2

)( xxnxyxxy

b

Use Method of Least Squares

xy x2

1500 2500

6500 10000

13500 22500

26000 40000

37500 62500

57000 90000

70000 122500

n x y

Freq   Inductive reactance

1 50   30

2 100   65

3 150   90

4 200   130

5 250   150

6 300   190

7 350   200

1400 855 212000 350000

Line of best fit

50 34.29

100 63.57

150 92.86

200 122.14

250 151.43

300 180.71

350 210.00

Make two more columns

y = a.x + b

New values for ‘y’ are found from

Plot this new data on the original graph

0

50

100

150

200

250

0 50 100 150 200 250 300 350 400

Easy