1. Triangle Partitioning and Linear Optimization of Forward Lines Chris Kang shinwook@uw.edu University of Washington NESSIS 2015

2. Motivation Objective of a coach is to “win a hockey game” by: 1.Finding “chemistry” between players 2.Finding “balance” among the lines (allocating appropriate Time on Ice) 3.Matching up against opposing lines Question: How can you analytically answer all these questions at once? Proposed Solution: A variation of team level With or Without You (WoWY) analysis (Triangle Partitioning)

3. Idea Traditional idea of finding chemistry between two players is to apply WoWY analysis Works well in the case of finding defensive pairs: Compare and contrast all possible defensive pairings What about forward lines? We can try David Johnson’s SuperWoWY* However, most teammates do not get to play on all possible lines, thus, sample size is too small for any analysis *www.puckanalytics.com

4. Triangle Partitioning Key Concept: Use implicit WoWY stats – triangle of players Instead of looking for data where three players (3-tuple) are on the ice at the same time, use a chain of WoWY between two players Example: Line1 = {Sidney Crosby, Patric Hornqvist, David Perron} Crosby & Hornqvist - 54.9 CF% Crosby & Perron – 58.8 CF% Hornqvist & Perron – 56.8 CF% => Implicit CF% of Line1 - 56.83 CF%

5. Triangle Partitioning Generalization for all 4 Lines: Let V = {player 1, player 2, player 3, ….., player 12 } Then, E={e 1, e 2, e 3, …., e } where e n is some statistical measurement between two players (i.e. CF%) Then, consider all partitioning of the lines into 4 triangles.

6. Triangle Partitioning

8. Data

9. Triangle Partitioning

10. Average Line Rank – “Gellability” PlayerAverage Rank 87 SIDNEY CROSBY1.846753 57 DAVID PERRON2.003247 26 DANIEL WINNIK2.311688 17 BLAKE COMEAU2.329221 19 BEAU BENNETT2.41039 72 PATRIC HORNQVIST2.45974 71 EVGENI MALKIN2.462338 14 CHRIS KUNITZ2.576623 13 NICK SPALING2.677922 16 BRANDON SUTTER2.733117 23 STEVE DOWNIE2.894805 40 MAXIM LAPIERRE3.292208

11. Application : Optimization With the implicit line data available for all possible configurations, we compute the optimal Time on Ice (TOI) to determine the “best” balanced lines. Linear Programming Setup: Maximize Team_CF% = t 1 L 1 + t 2 L 2 + t 3 L 3 + t 4 L 4 With Constraints : t 1 + t 2 + t 3 + t 4 = 1 0 < t 1 ≤ ⅓ 0 < t 2 ≤ ⅓ 0 < t 3 ≤ ⅓ 0 < t 4 ≤ ⅓

12. Distribution of Optimal Configuration

13. Distribution of Not-So-Optimal Configuration

14. Distribution…

15. Summary With the Triangle Partitioning idea, we can assess the chemistry of the entire lineup We can analyze the upper and the lower limit of a team CF% We can analyze the optimal allocation of Time on Ice for the 4 forward lines Brandon Sutter is not a good player

16. Thank You! Email: shinwook@uw.edu