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Ranking Sports Teams Using Linear Algebra

J.S.
May 11th, 2022

Introduction

Rankings are very important for all sports including local sports!

I will be doing my own ranking of the top 5 seeded teams in the RMAC basketball Tournament for 2022.

  • Round Robin (Every competitor plays everyone once)
  • Construct a matrix using 1 and -1 to signify wins and loses
  • Use the method of Least squares to put matrix into a solvable form

\[ \begin{aligned} A & = b \\ A*A' & = b*A' \end{aligned} \]

  • Use Homework 6 code for Guass-Jordan Elimination

Description of Methods

I have identified the top 5 ranked teams in the RMAC for the 2022 Men's Basketball Tournament

  1. Black Hills State
  2. Colorado Mesa University
  3. Fort Lewis University
  4. University of Colorado Colorado Springs
  5. Colorado School of Minds

I then looked up the 10 games these teams played against each other to obtain my data and put it in a matrix.

To give the class a visual I have done a smaller example.

Description of Methods

A is the matrix representing wins and loses and b is a matrix representing the point difference in each game.

A = [-1,0,1;0,1,-1;-1,1,0]

b = [5;20;8]

A =

  -1   0   1
   0   1  -1
  -1   1   0

b =

    5
   20
    8

Description of Methods

The Next step is to use the Method of Least Squares to put this matrix into a solvable form.

  • Because this was not a class topic I will not go into detail about it here.

The primary method I will be using from class is Guass-Jordan Elimination

  • This involves using elementary row operations to find an augmented matrix's rref

Octave Code

The octave code for the Method of Least Squares is very simple in octave.

A = [-1,0,1;0,1,-1;-1,1,0];
b = [5;20;8];
T = A'


T =

  -1   0  -1
   0   1   1
   1  -1   0

Octave Code

Final Matrix to Run Gauss-Jordan Elimination on.

A = [-1,0,1;0,1,-1;-1,1,0];
b = [2;7;10];
T = A';
S = T*A;
P = T*b;
F = [S,P]

F =

    2   -1   -1  -12
   -1    2   -1   17
   -1   -1    2   -5

Commands and Outputs

Run Guass-Jordan Elimination

  • Input:

c = [S,P]

  • Output:

\[ \begin{aligned} A = \begin{bmatrix} 1 & 0 & 0 & -5.5\\ 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 1 \end{bmatrix} \end{aligned} \]

Discussion of Results

  • Now we can interpret our results

\[ \begin{aligned} A = \begin{bmatrix} 1 & 0 & 0 & -5.5\\ 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 1 \end{bmatrix} \end{aligned} \]

  • Notice that because we transposed our original matrix that the rows now correspond to each team and their x value is their rank.

  • For example

Team 1 is the lowest ranking team, and Team 2 is the Highest rank Team.

Discussion of Results

\[ \begin{aligned} A = \begin{bmatrix} 1 & 0 & 0 & 0& 0 & 5.6\\ 0 & 1 & 0 & 0 & 0 & -1.6\\ 0 & 0 & 1 & 0 & 0 & 4\\ 0 & 0 & 0 & 1 & 0 & -7 \\ 0 & 0 & 0 & 0 & 1 & 2 \end{bmatrix} \end{aligned} \]

  1. Black Hill State as 1
  2. Colorado Mesa University as 4
  3. Fort Lewis University as 2
  4. University of Colorado Colorado Springs as 5
  5. Colorado School of Minds as 3

References