linear algebra /Starting Pitcher Rotation in the Chinese Professional Baseball LeagueSaying that – Eigenvalues do a lot is sort of like noticing that there are hammers on every construction site. I like to play around with ranking schema – ranking college teams, ranking race car drivers, ranking pizza. Matrix algebra, and the mighty eigen, plays a major role in ranking.
I found a fun little paper that outlines a method to help managers put their pitchers in the correct order. To accomplish this task, you first need to identify the criteria that matters in ranking pitchers: this article used in Innings Pitched, Earned Run Average, Walks/Hits per Inning, Strike Outs per Inning. The choice of criteria should fit your ability to find the historical data.
Now we have the data:
Pitcher’s name IPG ERA WHIP K/9
Huang, Qin-Zhi 5.00 3.50 1.21 3.40
Ken Ray 6.20 2.32 1.25 7.10
Wang, Fong-Sin 3.10 2.60 1.32 5.57
How should we rank them? The authors created a questionnaire and submitted it to a series of coaches, managers and players – a nine point ranking system for the criteria. They took the responses and created a lamda-max eignenvalue/eigenvector designed to weigh each of the criteria.
K/9 = .214
WHIP = .361
The most important factor for ranking a pitcher is WHIP, then IPG, then K/9 with ERA last. Oddly enough, none of these other terms mean anything to the baseball novice. But ask a non-fan about ERAs and they will know it is something from baseball. Also of interest for me, notice how close the values for WHIP are: 1.21, 1.25 and 1.34.
Apply the eigen to the historical stats and we see that: Ken is my best pitcher and Huang is my worst.
Eigenvalues and Eigenvectors and Structural Engineering
The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes (www.quora.com). Many structures from buildings and bridges have a natural frequency of vibration. Meaning, all these structures have their own system of eigenvibrations andeigenfrequencies. Factors such as wind and earthquakes can cause these structures to vibrate. Sometimes these types of vibrations can cause a structure to break or collapse. One famously known as the Tacoma Narrows Bridge (1940) collapsed due to aeroelastic flutter.
Eigenvalues can also be used to test for cracks or deformities in structural components used for construction. For example, when a beam is struck, its natural frequencies (eigenvalues) can be heard or measured. If a team is flawed it will result in a dull sound because the flaw causes the eigenvalues to change.
The eigenvalues can also be used to determine if a structure has deformed under the application of a particular force. Eigenvalues for the structure are measured before and after the application of force. If it changes that means the structure has undergone deformation.