Finding the Best Fit Line Using the Method of Least Squares

[Helpeme]

Homework Statement

number of years(x): [1,2,3,4,5,6,]
number of accidents(y): [12,6,5,4,2,2]

or

(1,12),(2,6),(3,5),(4,4),(5,2),(6,2)

Homework Equations

the relation between x and y is on the form: y = a + b(1/x)
by changing (1/x) to z, the model become linear instead of inverse.
use this to 'find' a and b

The Attempt at a Solution

y = a+b(1/x)
z = 1/x
y = a+bz

matrix A = [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)]
vector y = [12,6,5,4,2,2]
vector x = [a,b]

finds a solution to A^TAx = A^Ty

6a + 21b = 31
21a + 91b = 77

a = 11.4557
b = -1.8

y = 11.4667 - 1.8/x

but this is wrong the right answer is:
y = 0.3914 + 11.6945/x

i think my problem is how to deal with z and 1/x.

 

[Ray Vickson]

Homework Statement

number of years(x): [1,2,3,4,5,6,]
number of accidents(y): [12,6,5,4,2,2]

or

(1,12),(2,6),(3,5),(4,4),(5,2),(6,2)

Homework Equations

the relation between x and y is on the form: y = a + b(1/x)
by changing (1/x) to z, the model become linear instead of inverse.
use this to 'find' a and b

The Attempt at a Solution

y = a+b(1/x)
z = 1/x
y = a+bz

matrix A = [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)]
vector y = [12,6,5,4,2,2]
vector x = [a,b]

finds a solution to A^TAx = A^Ty

6a + 21b = 31
21a + 91b = 77

a = 11.4557
b = -1.8

y = 11.4667 - 1.8/x

but this is wrong the right answer is:
y = 0.3914 + 11.6945/x

i think my problem is how to deal with z and 1/x.

Go back and look at what you wrote: you use the matrix A = [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)], so you are fitting the equation y = a + bx.

RGV

 

[Helpeme]

do u mean the matrix should be [(1,1),(1,(1/2)),(1,(1/3)),(1,(1/4)),(1,(1/5)),(1,(1/6))]

 

[Ray Vickson]

do u mean the matrix should be [(1,1),(1,(1/2)),(1,(1/3)),(1,(1/4)),(1,(1/5)),(1,(1/6))]

What do YOU think?

RGV

 

[Helpeme]

i think that's not right at all and its not funny to trial and error matrix calculations.. they are pretty intense and tiresome

can you please give me some more hints.. if you know how to do this. because i have no clue anymore.

EDIT:

alright it worked..a little miss calculation. thanks for the replies :)