8. The owner of the rapidly expanding business finds that for the first five months of the year the sales (in thousands) are $4.0, $4.4, $5.2, $6.4 and $8.0. The owner plots these figures on the graph and conjectures that for the rest of the year, the sales curve can be approximated by a quadratic polynomial. Find the least squares quadratic fit to the sales curve, and use it to project the sale for the twelfth month of the year.

Solution:

The graph of the sales of the first 5 months:

Let's write the above information as points.

We know the sales is about the the first 5 months from month 1, 2,3, 4 and 5 and then we have the sales figures of $4.0, $4.4, $5.2, $6.4 and $8.0. Writing them in (x,y) point form we have:

(1, 4), (2, 4.4), (3, 5.2), (4, 6.4) and (5, 8).

clearly we see that it is a parabola which is a polynomial in the form `y=a_0+a_1x+a_2x^2` and we want to find the `a_0, a_1 and a_2`.

our equation for finding `a_0, a_1 and a_2` is v*`=(M^TM)^-1M^Ty` so we are going to setup the matrix M and y

`M=[[1,x_1,x_1^2],[1,x_2,x_2^2],[1,x_3,x_3^2],[1,x_4,x_4^2],[1,x_5,x_5^2]]`

we can use the x values for x1, x2, x3, x4 and x5 and square them for the `x_1^2, x_2^2, x_3^2,x_4^2 and x_5^2` as follow:

`M=[[1,1,1],[1,2,4],[1,3,9],[1,4,16],[1,5,25]]`

set up the y matrix with all y values from the points (1, 4), (2, 4.4), (3, 5.2), (4, 6.4) and (5, 8).

`y=[[4],[4.4],[5.2],[6.4],[8]]`

Using our y and M we can solve the v*`=(M^TM)^-1M^Ty` as follow:

First the M transpose `M^T=[[1,1,1,1,1],[1,2,3,4,5],[1,4,9,16,25]]` and multiply it by M

`M^TM=[[1,1,1,1,1],[1,2,3,4,5],[1,4,9,16,25]]*[[1,1,1],[1,2,4],[1,3,9],[1,4,16],[1,5,25]]=[[5,15,55],[15,55,225],[55,225,979]]`

To then inverse of `[[5,15,55],[15,55,225],[55,225,979]]^-1=[[23/5, -33/10, 1/2],[-33/10, 187/70, -3/7],[1/2,-3/7,1/14]]`

Now got the inverse, the last part is to multiply it by `M^Ty` which we write :

We already have `M^T` to be `M^T=[[1,1,1,1,1],[1,2,3,4,5],[1,4,9,16,25]]`

*`=(M^TM)^-1M^Ty=[[23/5, -33/10, 1/2],[-33/10, 187/70, -3/7],[1/2,-3/7,1/14]] * [[1,1,1,1,1],[1,2,3,4,5],[1,4,9,16,25]] * [[1,1,1],[1,2,4],[1,3,9],[1,4,16],[1,5,25]]*[[4],[4.4],[5.2],[6.4],[8]]=`

using MATLAB we can solve this:

- Code: Select all

>> M=[1 1 1; 1 2 4; 1 3 9; 1 4 16; 1 5 25]

M =

1 1 1

1 2 4

1 3 9

1 4 16

1 5 25

y=[4, 4.4, 5.2, 6.4, 8]

>> rats( inv( transpose(M)*M ) * transpose(M) * y)

ans =

4

-1/5

1/5

so we got a vectot `[[a_0],[ a_1],[ a_2]]=[[4],[ -1/5],[ 1/5]]`

`:. a_0 = 4, a_1=-1/5 and a_2=1/5` and our polynomial equation is `y=4 -1/5x + 1/5x^2`

Now if for the purpose of test, graph the 12 month sales