|
At right are the
rankings of all NCAA Division 1-A college football teams thru the games
of January 7, 2008 (i.e., all bowl games) using MinV, a model
developed by Jay Coleman of the University of North Florida. MinV
generates a ranking that minimizes the number of game score violations --
that is, the number of times a game's winner is ranked behind the team it
defeated. In other words, MinV guarantees the ranking with the best
retrodictive accuracy.
The ranking at right
results in only 78 violations out of 718 Division 1-A
games, or a minimum violation percentage of 10.86%.
In addition to minimizing the total number of violations, the
ranking this week minimizes the total weighted violations, where
each violation is weighted by the victory margin. (In other words, the
ranking shown violates the games in which the scores were as close as
possible, as opposed to violating games in which the victory margins were
larger.) The total weight (i.e., victory margins) of the violated games
this week is 625 points. Finally, the ranking shown at right
at least approximately matches the game score differences (the victory
margins) as closely as mathematically possible, while exceeding neither
the minimum number of violations nor the minimum weighted
violations. Due to the size of the problem (the number of games
played thus far), MinV was not able to guarantee that the ranking shown
is the one that optimally matches the victory margins. However, the
ranking shown is likely a reasonably close approximation.
A minimum violations
ranking has never before been presented for college football (due in part
to the extreme computational difficulty involved for a problem with 120
teams). However, there are literally trillions of different
rankings at any given point in time that would yield the same minimum
number of violations; the ranking shown is only one of those.
"Minimizing Game
Score Violations in College Football Rankings," an article
describing MinV and its application to the 1994 through 2004 college
football seasons, appears in the November-December 2005 issue of Interfaces,
a journal of the Institute for Operations Research and the Management
Sciences (INFORMS).
The final MinV
ranking for the 2006 college football season can be found here.
The final MinV ranking
for the 2005 college football season can be found here.
The final MinV ranking
for the 2004 college football season can be found here.
The final MinV
pre-NCAA Tournament ranking for college basketball in 2005 can be
found here.
|