Hubbry Logo
Colley MatrixColley MatrixMain
Open search
Colley Matrix
Community hub
Colley Matrix
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Colley Matrix
Colley Matrix
Colley Matrix
View on Wikipedia
from Wikipedia

The Colley Matrix is a computer-generated sports rating system designed by Dr. Wesley Colley. It is one of more than 40 polls, rankings, and formulas recognized by the NCAA in its list of national champion selectors in college football.[1]

Methodology

[edit]

In his initial paper at Princeton University, Colley states, "The method is based on very simple statistical principles, and uses only Div. I-A[a] wins and losses as input — margin of victory does not matter. The scheme adjusts effectively for strength of schedule, in a way that is free of bias toward conference, tradition, or region."[2] Colley claims that his method is bias-free for estimating the ranking of a team given a particular schedule, though his claim that the formula adjusts effectively for strength of schedule has been disputed.[2][3][4] The resulting values for each team are identified as a ranking.[3]

The formula was adjusted in 2007 to account for games against FCS teams.[5]

Colley Matrix is a special case of the Generalized row sum method, a parametric family of ranking methods developed by P. Yu. Chebotarev (1989).[6]: 4 

National champions

[edit]

As an NCAA-designated major selector, the NCAA regards the following teams as Colley's national champion selection, however these selections are listed under the "Final National Poll Leaders" section of the NCAA's record book rather than the "National Champions" section.[7] Unlike most of the NCAA's major selectors, the Colley Matrix does not award a physical trophy to its national champion.

In four years (2011, 2012, 2016, 2017) the Colley Matrix selected a national champion that did not win the BCS or CFP national championship game. In each of the years, the Colley Matrix was the only NCAA-designated "major selector" to select that champion.[8]: 117–118 

"2017 National Champions" signage at the home stadium of the UCF Knights football program; Colley Matrix ranked UCF first for the 2017 NCAA Division I FBS football season
Season Champion Record Ref.
1998 Tennessee 13–0 [9]
1999 Florida State 12–0 [10]
2000 Oklahoma 13–0 [11]
2001 Miami (FL) 12–0 [12]
2002 Ohio State 14–0 [13]
2003 LSU 13–1 [14]
2004 USC 13–0[b] [15]
2005 Texas 13–0 [16]
2006 Florida 13–1 [17]
2007 LSU 12–2 [18]
2008 Florida 13–1 [19]
2009 Alabama 14–0 [20]
2010 Auburn 14–0 [21]
2011 † Oklahoma State 12–1 [22]
2012 † Notre Dame 12–1 [23]
2013 Florida State 14–0 [24]
2014 Ohio State 14–1 [25]
2015 Alabama 14–1 [26]
2016 † Alabama 14–1 [27]
2017 † UCF 13–0 [28]
2018 Clemson 15–0 [29]
2019 LSU 15–0 [30]
2020 Alabama 13–0 [31]
2021 Georgia 14–1 [32]
2022 Georgia 15–0 [33]
2023 Michigan 15–0 [34]
2024 Ohio State 14–2 [35]

† Years in which Colley Matrix selection did not win BCS or CFP national championship game.

History

[edit]

The NCAA record book indicates that the Colley Matrix has been active since 1992, however this appears to be an error and no Colley selections are listed for 1992–1997.[8]: 112  The season rankings on Colley's own website begin in 1998.[36] The Colley Matrix was one of the computer rankings used during Bowl Championship Series (BCS) system of determining national championship game participants starting in the 2001 season. The Peter Wolfe and Wes Colley/Atlanta Journal-Constitution computer rankings were used in place of The New York Times and Dunkel rankings. The change was made because the BCS wanted computer rankings that did not depend heavily on margin of victory.[37]

The Colley Matrix has chosen a different national champion from the Bowl Championship Series or College Football Playoff champion four times:[8]: 117–118 

In each of the above instances, the Colley Matrix was the only NCAA recognized selector to choose a champion other than the BCS or CFP winner.[8]: 117–118 

Criticism and controversies

[edit]

The methodology of the rankings have been questioned by others on the grounds of subjectivity and specifics of the statistical math.[3][41] It has also been criticized for placing too much weight on a team's win-loss record and not correctly emphasizing a team's strength of schedule, strength of record, margin of victory, and head-to-head results, as well as for problems with the formula used for the calculation.[3] Dr. Ed Feng, a mathematician at Stanford University, criticized the system because it does not consider specific game results, stating that "[t]he method does not care who a team loses to in ranking them. It considers the win loss record of each team and the number of games played between each pair of teams. However, the specifics of who won each game are not an input to Colley’s method".[42]

In the final BCS rankings for the 2010 season, LSU was incorrectly ranked ahead of Boise State, at No. 10 instead of No. 11. The error was a result of Colley failing to input an FCS playoff game (Appalachian State vs. Western Illinois) correctly, a mistake that affected an order that helped determine bowl pairings that season.[43]

The ranking system was widely criticized after ranking Notre Dame ahead of Alabama for the 2012 season following the 2013 BCS National Championship Game, in which Alabama defeated Notre Dame 42–14.[39][44][45] It was similarly criticized after ranking Alabama first and Clemson second for the 2016 season following the 2017 College Football Playoff National Championship, in which Clemson defeated Alabama 35–31.

The Colley Matrix is most well known for ranking Central Florida ahead of Alabama in 2017 despite Alabama's victory in that season's College Football Playoff.[46][47] Central Florida later proclaimed themselves as co-national champions because of the ranking, becoming the only school to claim a national championship based solely on the Colley Matrix.[48][49][50] UCF was not selected for the College Football Playoff that season.

In 2018, the Mountain West Conference moved away from using four polls, one being Colley Matrix, to determine the host site for its conference championship game in football, due to "a shift to place a priority on head-to-head competition."[51]

Since its creation, the four instances in which the Colley Matrix has chosen a different national champion from the BCS/CFP winner are the most of any NCAA recognized selector in that timeframe.[8]: 117–118  Colley Matrix is also the only NCAA recognized selector to ever choose a different champion than the CFP (in use since the 2014 season), which it has done twice.[8]: 117–118 

In college basketball

[edit]

Colley's formula is also used for men's college basketball rankings (though this serves no purpose in terms of awarding a national champion and is not used by the March Madness selection committee).[52] These rankings are updated daily rather than weekly like they are for football or the college basketball AP Poll and were first used in the 2002–03 season.[52] Like with the football rankings, the Colley Matrix has often ranked a team that did not win that season's NCAA tournament first, doing so on 15 occasions[c] in 21 seasons as of the 2024–25 championship (not including the 2019–20 season, in which the NCAA tournament was canceled due to the COVID-19 pandemic).

Notes

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Colley Matrix

View on Grokipedia
from Grokipedia
The Colley Matrix is a bias-free, computer-generated ranking system for American college football teams, developed by astrophysicist Wesley N. Colley in the mid-1990s while at Princeton University, which employs linear algebra to compute team ratings solely from game outcomes (wins and losses) while adjusting for strength of schedule. Colley's method originated from his dissatisfaction with existing rankings that incorporated subjective factors or complex metrics like margin of victory; instead, it prioritizes simplicity and objectivity, assuming that the probability of team i defeating team j is 12+ri−rj\frac{1}{2} + r_i - r_j, where rir_i represents team i's rating (a measure of strength bounded between 0 and 1). This probabilistic model leads to a system of linear equations solved via matrix inversion: Cr=w\mathbf{C} \mathbf{r} = \mathbf{w}, where C\mathbf{C} is the symmetric Colley Matrix (with diagonal entries equal to the number of games played by each team plus 2, and off-diagonal entries of -1 for each game between teams), r\mathbf{r} is the vector of team ratings, and w\mathbf{w} is a vector derived from each team's wins minus losses (adjusted by adding 1 to incorporate a prior assumption of mediocrity). The addition of 2 to the diagonals acts as regularization to ensure the matrix is positive definite and invertible, preventing instability in rankings even for teams with few games. First posted online in during Colley's time at Harvard, the system gained prominence when it was selected as one of six official computer rankings for the (BCS) starting after the 2000 season, contributing one-third to the BCS formula for determining participants until the BCS era ended in 2013. Its transparency—publishing the full methodology and allowing users to adjust rankings by simulating game changes—distinguished it from other BCS models, and it aligned closely with human polls like the rankings. Post-BCS, the Colley Matrix continues to produce weekly rankings for football (now including games against FCS teams) and has been extended to , remaining an NCAA-recognized selector as of 2025. It produces rankings for the 2025 college football and basketball seasons. Key features include its indifference to non-performance factors such as conference affiliation, team tradition, or regional biases, making it a "deservedness" metric rather than a predictive tool; ratings tend to cluster around 0.5 (mediocrity), with standard errors estimated via to quantify uncertainty (typically around 0.04). While modifications have been proposed to refine it—such as incorporating beta distributions for win probabilities—the original formulation endures for its mathematical elegance and empirical robustness in handling imbalanced schedules among approximately 130 Division I teams.

Introduction

Overview

The Colley Matrix is a computer-generated developed by astrophysicist Wesley N. Colley to rank Football Bowl Subdivision (FBS) teams. Designed to eliminate subjective biases such as conference affiliations or historical traditions, the system relies exclusively on game outcomes to generate objective evaluations. At its core, the Colley Matrix seeks to produce unbiased rankings by accounting for each team's wins and losses while adjusting for the relative strength of their opponents' schedules. This approach ensures that victories against stronger teams carry more weight than those against weaker ones, providing a fair measure of performance across the division. The output consists of a numerical rating assigned to each team, with higher scores reflecting superior overall play; the top-rated team is designated as the national champion. It is one of 43 major selectors recognized by the NCAA for identifying college football national champions. Historically, the Colley Matrix served as one of six computer rankings in the Bowl Championship Series from 2001 to 2013.

Significance

The Colley Matrix gained formal recognition from the (NCAA) as a major selector for national champions beginning with the 1998 season, establishing it as a credible tool for evaluating team performance, and remains recognized as of 2025. From 2001 to 2013, it was integrated as one of six computer components in the (BCS) formula, which determined automatic qualification for the national championship game and influenced assignments. This inclusion underscored its reliability in a system designed to balance human polls with objective data-driven assessments. A core aspect of the Colley Matrix's significance lies in its "bias-free" design, which deliberately excludes subjective human judgments, margin-of-victory considerations, and preseason predictions to focus exclusively on win-loss outcomes adjusted for . This approach aimed to enhance objectivity in determinations, countering the perceived flaws in traditional polls that could favor popularity or biases over pure performance metrics. By prioritizing mathematical neutrality, the method promoted a more equitable framework for ranking teams across divisions. Through its BCS tenure, the Colley Matrix indirectly shaped the evolution of postseason selection, contributing to the foundational legacy that informed the (CFP) introduced in 2014. As of 2025, it continues to produce weekly rankings for the season. Its transparent methodology, fully detailed in public documentation unlike some proprietary systems, fueled broader debates on ranking rigor during the BCS era, emphasizing the advantages of verifiable, equation-based models in resolving controversies over undefeated teams or strength-of-schedule disputes. This influence highlighted the potential for computer rankings to drive reforms toward greater fairness in college football governance.

History

Origins and Development

The Colley Matrix was developed by Wesley N. Colley, an astrophysicist who earned his Ph.D. in Astrophysical Sciences from in 1998. As a lifelong enthusiast, Colley sought to address the perceived biases inherent in traditional ranking polls, such as the (AP) and (UPI) systems, which relied heavily on subjective human judgments and could favor teams with easier schedules or larger victory margins. His approach emphasized objectivity by focusing exclusively on binary game outcomes—wins and losses—while adjusting for opponents' through a mathematical framework, thereby eliminating subjective factors like team tradition or conference prestige. Colley began experimenting with ranking models in the mid-1990s during his graduate studies at Princeton, but the system's formal inception occurred in 1998 amid the introduction of computer-assisted rankings in the newly formed (BCS). The first public iteration of the Colley Matrix rankings was released that fall, with weekly updates posted on a personal website hosted at , where Colley served as a ; these rankings quickly gained traction, attracting significant online traffic. Although NCAA records list the Colley Matrix as an active selector since 1992, no verifiable rankings or selections from Colley exist prior to the 1998 season, suggesting the earlier date may reflect an administrative designation rather than actual publication. Early validation of the method involved retrospective applications to prior seasons, which confirmed its consistency and robustness in producing stable rankings without to specific years' data. For instance, when tested on historical outcomes, the matrix reliably identified national champions in line with established polls while highlighting schedule-adjusted performance differences. This testing phase underscored the system's reliance on linear algebra to derive unbiased ratings, setting the stage for its broader evaluation.

Adoption and Evolution

The Colley Matrix was officially adopted as one of six computer rankings in the (BCS) formula beginning with the 2001 season, providing a mathematical component to the selection process for the game. This inclusion marked its transition from an independent ranking system to a formalized element of FBS postseason determinations, where it contributed equally alongside other computer models and human polls to generate average rankings. The method remained in use throughout the BCS era, which spanned from 1998 to 2013, without significant alterations until a targeted adjustment in to address games against Football Championship Subdivision (FCS, formerly Division I-AA) opponents. This update treated FCS games as non-Division I for ranking purposes, effectively excluding FCS teams from the primary FBS matrix while incorporating their outcomes in a manner that preserved the system's bias-free principles and accounted for schedule expansions that increased such matchups. No major methodological overhauls followed this change, allowing the core linear algebra framework to persist unchanged into the post-BCS period. Following the BCS's dissolution after the 2013 season and the advent of the (CFP), the Colley Matrix exerted indirect influence through its ongoing recognition by the NCAA as a valid national champion selector, enabling teams to claim titles based on its final rankings in the absence of a unanimous human committee decision. Maintained independently by its creator, Wesley Colley, the system continues to produce weekly rankings for the FBS season, published on colleyrankings.com, with archives extending through the 2025 campaign and no integration into the CFP's primary selection criteria.

Methodology

Core Principles

The Colley Matrix ranking system is designed to produce unbiased evaluations of teams by relying exclusively on win-loss records from Division I Football Bowl Subdivision (FBS) games, deliberately excluding factors such as margins of victory, home-field advantage, or conference strength biases to ensure objectivity and simplicity. This approach avoids subjective adjustments that could introduce or favoritism, focusing instead on the binary outcome of each contest as the sole input for determining . By treating victories and defeats equally regardless of score differential or venue, the method emphasizes fairness in assessing a team's record without penalizing or rewarding stylistic elements of play. A key principle is the incorporation of strength-of-schedule adjustments through an iterative process that evaluates teams relative to their opponents' overall performance, rewarding those who succeed against tougher competition without relying on complex external metrics. This adjustment is inherently bias-free, as it emerges directly from the interconnected win-loss data across all teams, allowing the system to account for schedule difficulty in a transparent manner. All Division I FBS games are weighted equally in this framework, while contests against non-FBS opponents are systematically excluded to maintain consistency and prevent dilution of the dataset with incomparable matchups. To initiate the ranking process, the system applies a neutral prior assumption to every team, assigning an initial rating equivalent to a 1-1 record, which draws from Laplace's to avoid overemphasizing early-season results or unplayed games. This starting point ensures that no team enters with an advantage or disadvantage based on reputation or preseason hype, promoting a level playing field where final rankings reflect season-long outcomes adjusted for the prior. The iterative solving of the system then refines these priors based on actual results, as detailed in subsequent formulations.

Mathematical Formulation

The Colley Matrix method determines team ratings rir_i for each ii by solving the Cr⃗=b⃗C \vec{r} = \vec{b}
Add your contribution
Related Hubs
User Avatar
No comments yet.