A Multiresolution Stochastic Process Model for Basketball Possession Outcomes Dan Cervone, Alex D’Amour, Luke Bornn, Kirk Goldsberry Harvard Statistics Department August 11, 2015 Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
NBA optical tracking data Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
NBA optical tracking data Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
NBA optical tracking data ( x , y ) locations for all 10 players (5 on each team) at 25Hz. ( x , y , z ) locations for the ball at 25Hz. Event annotations (shots, passes, fouls, etc.). 1230 games from 2013-14 NBA, each 48 minutes, featuring 461 players in total. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Expected Possession Value (EPV) 1.6 RASHARD James NORRIS COLE POSSESSION LEWIS LEBRON JAMES POSSESSION shot POSS. 1.4 Splits the defense; clear path to basket Accelerates 1.2 EPV towards basket Accelerates into the Pass paint Pass 1.0 Runs behind { basket and { Slight dip in EPV after defenders close Slight dip in EPV after crossing 3 point line EPV constant while crossing 3 point line pass is en route 0.8 0 3 6 9 12 15 18 time Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . T ( ω ): possession length Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . T ( ω ): possession length Z t ( ω ) , 0 ≤ t ≤ T ( ω ): time series of optical tracking data. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . T ( ω ): possession length Z t ( ω ) , 0 ≤ t ≤ T ( ω ): time series of optical tracking data. F ( Z ) = σ ( { Z − 1 : 0 ≤ s ≤ t } ): natural filtration. t s Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . T ( ω ): possession length Z t ( ω ) , 0 ≤ t ≤ T ( ω ): time series of optical tracking data. F ( Z ) = σ ( { Z − 1 : 0 ≤ s ≤ t } ): natural filtration. t s Definition The expected possession value (EPV) at time t ≥ 0 during a possession is ν t = E [ X |F ( Z ) ]. t Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
EPV definition Let Ω be the space of all possible basketball possessions. For ω ∈ Ω X ( ω ) ∈ { 0 , 2 , 3 } : point value of possession ω . T ( ω ): possession length Z t ( ω ) , 0 ≤ t ≤ T ( ω ): time series of optical tracking data. F ( Z ) = σ ( { Z − 1 : 0 ≤ s ≤ t } ): natural filtration. t s Definition The expected possession value (EPV) at time t ≥ 0 during a possession is ν t = E [ X |F ( Z ) ]. t EPV provides an instantaneous snapshot of the possession’s value, given its full spatiotemporal history. ν t is a Martingale: E [ ν t + h |F ( Z ) ] = ν t for all h > 0. t Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Calculating EPV ν t = E [ X |F ( Z ) ] t Regression-type prediction methods: – Data are not traditional input/output pairs. – No guarantee of stochastic consistency. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Calculating EPV ν t = E [ X |F ( Z ) ] t Regression-type prediction methods: – Data are not traditional input/output pairs. – No guarantee of stochastic consistency. Markov chains: + Stochastically consistent. – Information is lost through discretization. – Many rare transitions. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Calculating EPV ν t = E [ X |F ( Z ) ] t Regression-type prediction methods: – Data are not traditional input/output pairs. – No guarantee of stochastic consistency. Markov chains: + Stochastically consistent. – Information is lost through discretization. – Many rare transitions. Brute force, “God model” for basketball. + Allows Monte Carlo calculation of ν t by simulating future possession paths. – Z t is high dimensional and includes discrete events (passes, shots, turnovers). Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
A coarsened process Finite collection of states C = C poss ∪ C end ∪ C trans . C poss : Ball possession states { player } × { region } × { defender within 5 feet } Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
A coarsened process Finite collection of states C = C poss ∪ C end ∪ C trans . C poss : Ball possession states C end : End states { made 2, made 3, turnover } { player } × { region } × { defender within 5 feet } Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
A coarsened process Finite collection of states C = C poss ∪ C end ∪ C trans . C poss : Ball possession states C end : End states { made 2, made 3, turnover } { player } × { region } × { defender within 5 feet } C trans : Transition states {{ pass linking c , c ′ ∈ C poss } , { shot attempt from c ∈ C poss } , turnover in progress, rebound in progress } . Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
A coarsened process Finite collection of states C = C poss ∪ C end ∪ C trans . C poss : Ball possession states C end : End states { made 2, made 3, turnover } { player } × { region } × { defender within 5 feet } C trans : Transition states {{ pass linking c , c ′ ∈ C poss } , { shot attempt from c ∈ C poss } , turnover in progress, rebound in progress } . C t ∈ C : state of the possession at time t . C (0) , C (1) , . . . , C ( K ) : discrete sequence of distinct states. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Possible paths for C t Player 1 Player 2 Player 3 Player 4 Player 5 Pass to P2 Pass to P3 Pass to P4 Pass to P5 C poss C trans Turnover Shot Rebound C end Made 2pt Made 3pt End of possession Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Possible paths for C t Player 1 Player 2 Player 3 Player 4 Player 5 Pass to P2 Pass to P3 Pass to P4 Pass to P5 C poss C trans Turnover Shot Rebound C end Made 2pt Made 3pt End of possession � min { s : s > t , C s ∈ C trans } if C t ∈ C poss τ t = t if C t �∈ C poss δ t = min { s : s ≥ τ t , C s �∈ C trans } . Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Stopping times for switching resolutions � min { s : s > t , C s ∈ C trans } if C t ∈ C poss τ t = t if C t �∈ C poss δ t = min { s : s ≥ τ t , C s �∈ C trans } . Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Stopping times for switching resolutions � min { s : s > t , C s ∈ C trans } if C t ∈ C poss τ t = t if C t �∈ C poss δ t = min { s : s ≥ τ t , C s �∈ C trans } . Key assumptions: A1 C is marginally semi-Markov. Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Stopping times for switching resolutions � min { s : s > t , C s ∈ C trans } if C t ∈ C poss τ t = t if C t �∈ C poss δ t = min { s : s ≥ τ t , C s �∈ C trans } . Key assumptions: A1 C is marginally semi-Markov. A2 For all s > δ t , P ( C s | C δ t , F ( Z ) ) = P ( C s | C δ t ). t Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Stopping times for switching resolutions � min { s : s > t , C s ∈ C trans } if C t ∈ C poss τ t = t if C t �∈ C poss δ t = min { s : s ≥ τ t , C s �∈ C trans } . Key assumptions: A1 C is marginally semi-Markov. A2 For all s > δ t , P ( C s | C δ t , F ( Z ) ) = P ( C s | C δ t ). t Theorem Assume (A1)–(A2), then for all 0 ≤ t < T, � E [ X | C δ t = c ] P ( C δ t = c |F ( Z ) ν t = ) . t c ∈{C trans ∪C end } Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
Multiresolution models EPV: E [ X | C δ t = c ] P ( C δ t = c |F ( Z ) � ν t = ) . t c ∈{C trans ∪C end } Dan Cervone (Harvard) Multiresolution Basketball Modeling August 11, 2015
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