is it a good time to be a mariners fan? ranking baseball teams using linear algebra
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Is it a Good Time to be a Mariners Fan? Ranking Baseball Teams Using Linear Algebra. By Melissa Joy and Lauren Asher. How are sports teams usually ranked?. Winning Percentage system : The team with the highest percentage of wins is ranked first. Problems:. - PowerPoint PPT PresentationTRANSCRIPT
Is it a Good Time to be a Mariners Fan?
Ranking Baseball Teams Using Linear Algebra
By Melissa Joy and Lauren Asher
How are sports teams usually ranked?
Winning Percentage system: The team with the highest percentage of wins is ranked first.
Problems:
•If all the teams do not play all the other teams then your winning percentage depends on how good the teams you play are.
•Possibility of ties
Solution: Linear Algebra…
MLB 2006 Regular Season(through April 17th)
Los Angeles Angels (A)
Seattle Mariners (B)
Oakland Athletics (C)
Texas Rangers (D)
A vs. B:
W 5-4
L 8-10
L 4-6
A vs. D:
W 5-2
W 5-4
L 3-11
B vs. C:
W 6-2
L 0-5
L 0-3
C vs. D:
L 3-6
W 5-4
L 3-5
Sum of Points Scored in the 3 games
A vs. B: 17-20
A vs. D: 13-17
B vs. C: 6-10
C vs. D: 11-15
How to find the ranking vector
According to Charles Redmond, the vector yielding the ranking has this formula:
1/3 1/3 0 1/31/3 1/3 1/3 00 1/3 1/3 1/31/3 0 1/3 1/3
Sum of rows represents the number of games played
Sum of the rows =1
Making an Adjacency Matrix
-7
-1
0
8
Sum of Points Scored in the 3 games
A vs. B: 17-20
A vs. D: 13-17
B vs. C: 6-10
C vs. D: 11-15
A: -3 + -4 = -7
B: 3 + -4 = -1
C: 4 + -4 = 0
D: 4 + 4 = 8
Finding an S vector
1/3 1/3 0 1/31/3 1/3 1/3 00 1/3 1/3 1/31/3 0 1/3 1/3
Eigenvalues:
= 1
= -1/3
= 1/3
Eigenvectors:
1111
1-11-1
10-10
010-1
Normalized Eigenvectors:
½½½½
½-½½-½
1/√20-1 /√20
0
1/√2
0
-1 /√2
Solving for Eigenvectors
A Linear Decomposition of S
-7
-1
0
8
-7
-1
0
8
-7
-1
0
8
-7
-1
0
8
1/2
1/2
1/2
1/2
1/2
-1/2
1/2
-1/2
1/√2
0
-1/ √2
0
0
1/√2
0
-1/ √2
0
-5
-7/ √2
-9/ √2
-5/2
5/2
-5/2
5/2
-7/2
0
7/2
0
0
-9/2
0
9/2
Plugging S into the Limit
The limit can be expanded into the decomposed form of S
The eigenvalues are substituted in for M/3
The limit becomes:
The Final Ranking
-7/207/20
0-9/209/2
-5/25/2-5/25/2
-1.7501.750
0-2.2502.25
-.625.625-.625.625
-2.375-1.6251.1252.875
And the winner is…
-2.375-1.6251.1252.875
1. Texas Rangers (D)2. Oakland Athletics (C)3. Seattle Mariners (B)4. Los Angeles Angels (A)
This ranking is based on points.
It is a better early season predictor because:
• Measures skill rather than simply wins and losses
• Eliminates ties