How would I go about setting up the equation? I understand how to do linear leas

Question

How would I go about setting up the equation? I understand how to do linear least squares.

Let lis consider a simplified version of algorithm that goes into ranking college football teams for the Bowl Championship Series. This formula for computer ranking the teams was devised by Kenneth Massey^1 and constitutes a part of the overall scheme that goes into determining the rankings that come out regularly during football season. Suppose, we have 4 teams, which we will call T_1 through T_4 for simplicity. Assume that the following outcomes occur: We can use this to construct a linear system by assigning ranking points r_i to team T_i via: r_1 - r_2 = 4 r_3 r_1 = 9 r_1 - r_4 = 6 r_3 - r_4 = 3 r_2 - r_4 = 7 This is an overdetermined system and does not even have a unique least squares solution because we could always add a constant c to any solution [r_1 + c,r_2 + c,r_3 + c,r_4 +c]^T and still satisfy all the equations equally well. This can be fixed by adding another equation like, r_1 + r_2 + r_3 + r_4 = 20. Given these equations, use the linear least squares to rank the 4 teams.

Explanation / Answer

See the reference at http://eeweb.poly.edu/iselesni/lecture_notes/least_squares/least_squares_SP.pdf, especially equation (7) on page 2. In terms of the specification given in your problem, we have the H matrix, entered in R, as

> tt
V1 V2 V3 V4
[1,] 1 -1 0 0
[2,] -1 0 1 0
[3,] 1 0 0 -1
[4,] 0 0 1 -1
[5,] 0 1 0 -1
[6,] 1 1 1 1

Now, your y, the quantities on the right side are entered as b in R:

> b
[1] 4 9 6 3 7 20

By Equation (7) of the reference, the problem is solved in R with

> solve(t(tt)%*%tt)%*%t(tt)%*%b
[,1]
V1 5.250
V2 4.625
V3 9.125
V4 1.000

See that sum of the quantities is

> sum(solve(t(tt)%*%tt)%*%t(tt)%*%b)
[1] 20

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