Post on 04-Jul-2020
SALO Intro MARKOV The Matrix Results
MARKOVThe New Road to WAR
Gordon Arsenoff
2018-08-10
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
SALO 2.0: Bayesian ordered logit for SoG and penalties
Pr(away shot) = Λ(αsa − Xβs − Zγs)Pr(home shot) = Λ(αsh + Xβs + Zγs)
Pr(away pen) = Λ(αpa − Xβp − Zγp)Pr(home pen) = Λ(αph + Xβp + Zγp)
(likelihood)
γs ∼ N(0, σs) γp ∼ N(0, σp) (prior)
GP ∼ BB(82, αG + γsβG , φ) ([name?])
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
WAR: combining multiple abilities into an overall value
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
cWAR linearly maps micro ability to macro results
Ability Events Goals Wins
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Wins (added) aren’t linear in goals (added)G <- (0:100) / 100W <- skellam::pskellam(-1, (1 - G) * 6, G * 6)W <- W / (1 - skellam::dskellam(0, (1 - G) * 6, G * 6))
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00
GF%
W%
I bWAR 2.x accounts for thisGordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Goals (added) aren’t (globally) linear in ability, either!
Ability Events
Goals
Score Effects
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
A Markov chain randomly changes state in discrete timeaccording to a matrix of transition probabilities
M =[
Mjj MjkMkj Mkk
]
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Markov property: Distribution of Yt given Y0 equals M t
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
SALO defines a game state space and transition matrix
Period
Lead
Strength
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Transition matrices defined separately for each period
M∗ = (M1)2400 × (M2)2400 × (M3)2400
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Lead change prob. is SALO shot prob. × (constant) Sh%
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Penalties, PPG, and expiration cause strength changes
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Assumptions are added for finiteness and completeness
I M is cropped to 135 × 135 (15 leads, 9 strength states)I Average ice time: forwards 15:00, defensemen 20:00I Model-based replacement player
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Top players mostly D (ice time assumption is severe)
season player pos mean sd
20152016 KRIS LETANG D 4.055 1.55020152016 PAVEL DATSYUK C 3.916 1.26720132014 MARK GIORDANO D 3.880 1.57220152016 JAKE MUZZIN D 3.741 1.58220152016 HAMPUS LINDHOLM D 3.740 1.57320152016 COLTON PARAYKO D 3.727 1.57420142015 PATRICE BERGERON C 3.701 1.25820172018 RADKO GUDAS D 3.449 1.60220152016 OLIVER EKMAN-LARSSON D 3.382 1.54720152016 CARL HAGELIN L 3.356 1.248
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
WAR is almost linear in shots and unrelated to penaltiesgamma_p gamma_s
−0.08 −0.04 0.00 0.04 −0.10 −0.05 0.00 0.05 0.10 0.15−2
0
2
4
SALO Ability estimate
MA
RK
OV
WA
R/8
2
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
Corsica WAR is not quite linear in MARKOV WAR
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−2.5
0.0
2.5
5.0
−2 0 2 4
MARKOV WAR / 82
Cor
sica
WA
R R
ates
+ P
en /
82
Gordon ArsenoffMARKOV
SALO Intro MARKOV The Matrix Results
https://www.salohockey.net
\begin{appendix}Gordon ArsenoffMARKOV