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by Carl D. Meyer, Amy N. Langville
Who's #1?
Cover
Half title
Title
Copyright
Contents
Preface
Purpose
Audience
Prerequisites
Teaching from This Book
Acknowledgments
Chapter 1. Introduction to Ranking
Social Choice and Arrow’s Impossibility Theorem
Arrow’s Impossibility Theorem
Small Running Example
Chapter 2. Massey’s Method
Initial Massey Rating Method
Massey’s Main Idea
The Running Example Using the Massey Rating Method
Advanced Features of the Massey Rating Method
The Running Example: Advanced Massey Rating Method
Summary of the Massey Rating Method
Chapter 3. Colley’s Method
The Running Example
Summary of the Colley Rating Method
Connection between Massey and Colley Methods
Chapter 4. Keener’s Method
Strength and Rating Stipulations
Selecting Strength Attributes
Laplace’s Rule of Succession
To Skew or Not to Skew?
Normalization
Chicken or Egg?
Ratings
Strength
The Keystone Equation
Constraints
Perron–Frobenius
Important Properties
Computing the Ratings Vector
Forcing Irreducibility and Primitivity
Summary
The 2009–2010 NFL Season
Jim Keener vs. Bill James
Back to the Future
Can Keener Make You Rich?
Conclusion
Chapter 5. Elo’s System
Elegant Wisdom
The K-Factor
The Logistic Parameter ξ
Constant Sums
Elo in the NFL
Hindsight Accuracy
Foresight Accuracy
Incorporating Game Scores
Hindsight and Foresight with ξ = 1000, K = 32, H = 15
Using Variable K-Factors with NFL Scores
Hindsight and Foresight Using Scores and Variable K-Factors
Game-by-Game Analysis
Conclusion
Chapter 6. The Markov Method
The Markov Method
Voting with Losses
Losers Vote with Point Differentials
Winners and Losers Vote with Points
Beyond Game Scores
Handling Undefeated Teams
Summary of the Markov Rating Method
Connection between the Markov and Massey Methods
Chapter 7. The Offense–Defense Rating Method
OD Objective
OD Premise
But Which Comes First?
Alternating Refinement Process
The Divorce
Combining the OD Ratings
Our Recurring Example
Scoring vs. Yardage
The 2009–2010 NFL OD Ratings
Mathematical Analysis of the OD Method
Diagonals
Sinkhorn–Knopp
OD Matrices
The OD Ratings and Sinkhorn–Knopp
Cheating a Bit
Chapter 8. Ranking by Reordering Methods
Rank Differentials
The Running Example
Solving the Optimization Problem
The Relaxed Problem
An Evolutionary Approach
Advanced Rank-Differential Models
Summary of the Rank-Differential Method
Properties of the Rank-Differential Method
Rating Differentials
The Running Example
Solving the Reordering Problem
Summary of the Rating-Differential Method
Chapter 9. Point Spreads
What It Is (and Isn’t)
The Vig (or Juice)
Why Not Just Offer Odds?
How Spread Betting Works
Beating the Spread
Over/Under Betting
Why Is It Difficult for Ratings to Predict Spreads?
Using Spreads to Build Ratings (to Predict Spreads?)
NFL 2009–2010 Spread Ratings
Some Shootouts
Other Pair-wise Comparisons
Conclusion
Chapter 10. User Preference Ratings
Direct Comparisons
Direct Comparisons, Preference Graphs, and Markov Chains
Centroids vs. Markov Chains
Conclusion
Chapter 11. Handling Ties
Input Ties vs. Output Ties
Incorporating Ties
The Colley Method
The Massey Method
The Markov Method
The OD, Keener, and Elo Methods
Theoretical Results from Perturbation Analysis
Results from Real Datasets
Ranking Movies
Ranking NHL Hockey Teams
Induced Ties
Summary
Chapter 12. Incorporating Weights
Four Basic Weighting Schemes
Weighted Massey
Weighted Colley
Weighted Keener
Weighted Elo
Weighted Markov
Weighted OD
Weighted Differential Methods
Chapter 13. “What If . . .” Scenarios and Sensitivity
The Impact of a Rank-One Update
Sensitivity
Chapter 14. Rank Aggregation–Part 1
Arrow’s Criteria Revisited
Rank-Aggregation Methods
Borda Count
Average Rank
Simulated Game Data
Graph Theory Method of Rank Aggregation
A Refinement Step after Rank Aggregation
Rating Aggregation
Producing Rating Vectors from Rating Aggregation-Matrices
Summary of Aggregation Methods
Chapter 15. Rank Aggregation–Part 2
The Running Example
Solving the BILP
Multiple Optimal Solutions for the BILP
The LP Relaxation of the BILP
Constraint Relaxation
Sensitivity Analysis
Bounding
Summary of the Rank-Aggregation (by Optimization) Method
Revisiting the Rating-Differential Method
Rating Differential vs. Rank Aggregation
The Running Example
Chapter 16. Methods of Comparison
Qualitative Deviation between Two Ranked Lists
Kendall’s Tau
Kendall’s Tau on Full Lists
Kendall’s Tau on Partial Lists
Spearman’s Weighted Footrule on Full Lists
Spearman’s Weighted Footrule on Partial Lists
Partial Lists of Varying Length
Yardsticks: Comparing to a Known Standard
Yardsticks: Comparing to an Aggregated List
Retroactive Scoring
Future Predictions
Learning Curve
Distance to Hillside Form
Chapter 17. Data
Massey’s Sports Data Server
Pomeroy’s College Basketball Data
Scraping Your Own Data
Creating Pair-wise Comparison Matrices
Chapter 18. Epilogue
Analytic Hierarchy Process (AHP)
The Redmond Method
The Park-Newman Method
Logistic Regression/Markov Chain Method (LRMC)
Hochbaum Methods
Monte Carlo Simulations
Hard Core Statistical Analysis
And So Many Others
Glossary
Bibliography
Index
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