STAMP: A shot-type-aware areal multilevel Poisson model for league-wide comparison of basketball shot charts

ST AMP: A shot-t yp e-a w are areal m ultilev el P oisson mo del for league-wide comparison of bask etball shot c harts Kazuhiro Y amada 1 and Keisuk e F ujii 1,2* 1 Graduate Sc ho ol of Informatics, Nago ya Univ ersit y , Nagoy a, Japan. 2 Cen ter for Adv anced Intelligence Pro ject, RIKEN, Osak a, Japan. *Corresp onding author(s). E-mail(s): fujii@i.nago y a-u.ac.jp ; Con tributing authors: yamada.k azuhiro.t5@s.mail.nagoy a-u.ac.jp ; Abstract Sho oting lo cation is a core indicator of offensive style in in v asion sp orts. Exist- ing basketball shot-chart analyses often use spatial information for descriptive visualization, lo cation-based efficiency modeling, or clustering play ers in to sho ot- ing archet yp es, yet few studies provide a unified framework for fair comparison of shot-type-sp ecific tendencies. W e prop ose the shot-type-aw are areal multilev el P oisson (ST AMP) mo del, whic h join tly models team-lev el field-goal attempts across predefined court regions, seasons, and shot types using a P oisson likeli- ho od with a p ossession-based exp osure offset. The hierarc hical random-effects structure combines team, area, team-area, and team-side random effects with shot-t yp e-sp ecific random slop es for key shot categories. W e fit the mo del using appro ximate Bay esian inference via the Integrated Nested Laplace Approxima- tion (INLA), enabling efficient analysis of more than 3 × 10 5 shots from tw o seasons of B.LEA GUE (the men’s professional bask etball league in Japan). The ST AMP model ac hieves better out-of-sample predictiv e p erformance than simpler baselines, yielding interpretable relative-rate maps and left-right bias summaries. Case studies illustrate how the mo del reveals team-sp ecific spatial tendencies for comparativ e analysis, and we discuss its limitations and p oten tial extensions. 1 1 In tro duction In inv asion games, sho oting is one of the most crucial actions. Beyond its practical v alue for p erformance ev aluation and scouting, shot data ha ve also b ecome a testb ed for methodological innov ation. This type of work is esp ecially prev alen t in soccer and bask etball; how ever, because their purp oses and tasks differ, the mo deling strategies adopted in each field often diverge. In so ccer, the relativ e rarit y of goals, which are the most critical even t, has caused rep eated attention to the v alue of shots. The most w ell-known metric for shots in so ccer is the Exp ected Goal (xG), whic h represen ts the probability that a given shot results in a goal ( Anzer & Bauer , 2021 ; Lucey , Bialko wski, Monfort, Carr, & Matthews , 2015 ; Rathk e , 2017 ; Xu, Bretzner, W ang, & Maki , 2025 ). Using trac king data (time- series data that con tin uously record play er or ball positions/velociti es during a game) and even t data (discrete data that record action types and co ordinates during a game, suc h as passes), situation-dep enden t goal probabilities are mo deled and quan tified. The models used include logistic regression ( Brechot & Flepp , 2020 ; Eggels, V an Elk, & Pec henizkiy , 2016 ; F airchild, Pelec hrinis, & Kokk o dis , 2018 ; Pardo , 2020 ), gradi- en t bo osting trees suc h as X GBo ost ( Chen & Guestrin , 2016 )( Cavus & Biecek , 2022 ; Hewitt & Karakuş , 2023 ; Mead, O’Hare, & McMenemy , 2023 ; Pardo , 2020 ), and in some cases (Bay esian) hierarchical mo dels ( Scholtes & Karakuş , 2024 ; T ureen & Olthof , 2022 ). By con trast, in basketball, while research attempting to capture shot quality do es exist ( Kambhamettu, Shriv astav a, & Gwilliam , 2024 ; Schmid, Schöpf, & Kolbinger , 2025 ), it is relativ ely scarce, partly because pla yer p osture data, although critical, are not widely a v ailable. Instead, w ork that visualizes or quan tifies sho oting efficiency or tendencies is more common ( Y amada & F ujii , 2025 ). Previous studies ( Cao, Y ang, & Hu , 2025 ; Ho , 2025 ; Miller, Bornn, A dams, & Goldsb erry , 2014 ; Reich, Ho dges, Carlin, & Reich , 2006 ; Scrucca & Karlis , 2025 ; W ong-T oi, Y ang, Shen, & Hu , 2023 ; Zuccolotto, Sandri, & Manisera , 2021 , 2023 ) ha ve centered on mo deling spatial information using shot charts, whic h record the lo cations of shots taken. This literature can broadly b e divided into t wo strands: studies fo cusing on sho oting efficiency b y lo cation and studies fo cusing on spatial sho oting tendencies. The former mainly addresses the question of where shots should b e taken, with an eye tow ard applications in optimizing shot strategy ( Ehrlich & Sanders , 2024 ; Jiao, Hu, & Y an , 2021 ; Sandholtz, Mortensen, & Bornn , 2020 ). The latter is primarily concerned with understanding play ers’ or teams’ spatial preferences, sometimes as a basis for clustering them into archet yp es. F or this second strand, the modeling approac h most frequen tly applied to shot c harts has b een the Log-Gaussian Cox Pro cess (LGCP; Møller, Syversv een, and W aagep etersen ( 1998 )) ( Cao, Cai, et al. , 2025 ; Hu, Y ang, & Xue , 2021 ; Miller et al. , 2014 ; Sandholtz et al. , 2020 ; Yin, Hu, & Shen , 2023 ). LGCP mo dels the log-intensit y ov er contin uous space as a Gaussian pro cess, t yp- ically estimated v ia a discretized, sparsit y-inducing represen tation. Ho wev er, when marks, that is, categorical lab els asso ciated with each ev ent (e.g., shot type: jump shot, la yup, etc.), are introduced, the contin uous field can absorb v ariation that should instead b e attributed to those categories. F or comparativ e inference across pla yers or teams, one natural approach is to introduce random effects indexed b y grouping 2 factors (suc h as team or play er) and to imp ose cen tering or sum-to-zero constraints so that these effects are iden tifiable and interpretable as deviations from a common baseline. These additions, how ever, increase coupling among parameters, reduce spar- sit y , and w orsen the conditioning of the Hessian, leading to numerical instabilit y and higher computational cost. In practice, LGCPs can therefore b e cumbersome for fair comparisons b et ween pla yers or teams, esp ecially when mark information is required. A ccordingly , w e in tro duce the shot-type-aw are areal m ultilevel Poisson (ST AMP) mo del, a P oisson generalized linear mixed model (GLMM) for shot counts ov er prede- fined court regions. By incorp orating p ossession counts as an offset term and random effects for team, area, team × area, team × side and shot-t yp e-dep enden t team × area slop es, ST AMP yields exp osure-adjusted comparisons across teams. Mathematically , the sp ecification is a laten t Gaussian mo del, which enables fast Bay esian inference via the Integrated Nested Laplace Approximation (INLA; Rue, Martino, and Chopin ( 2009 )). This structure allows us to fit league-scale, shot-type-aw are spatial mo dels in min utes; thus, the metho d is practical for routine performance analysis. Similarly , shot counts ov er predefined areas hav e b een mo deled using a Poisson GLMM ( W ong-T oi et al. , 2023 ). Their goal, how ever, is play er categorization while accoun ting for shot success, whereas our fo cus is comparative inference for play ers and teams. In particular, w e emphasize team-level shot c harts as a primary unit of analysis, whic h has b een less frequen tly addressed in prior spatial shot-chart work. F urthermore, b y using p ossession counts as an offset, w e mitigate concerns regarding comparability based on exp osure, a challenge in that study , enabling rate-based comparisons ev en when exposure levels differ. In parallel, marked spatial p oin t process approaches hav e b een proposed ( Jiao et al. , 2021 ; Y eung, Sit, & F ujii , 2025 ). In contrast, our model is more interpretable, as it adopts a m ultiplicatively separable, log-linear decomp osition on the rate scale, and it is more scalable b ecause inference relies on sparse GLM- M/INLA machinery rather than Marko v Chain Monte Carlo (MCMC) or deep neural net work-based models. Our contributions are threefold: • Mo del: reusable template for marked spatial count data. W e form ulate the ST AMP mo del as a latent Gaussian Poisson GLMM for spatially structured coun t data with multiplicativ ely separable, log-linear rate comp onen ts indexed by team, season, area, side, and shot type. Shot type plays the role of a categorical mark, with mark-specific random slop es on team-b y-area effects; all random effects (team, area, team × area, team × side, and mark-sp ecific slop es) are mean-centered and constrained to sum to zero, ensuring iden tifiability and in terpretable log-rate con trasts. While our empirical study focuses on bask etball, the same template can b e applied to other inv asion sp orts or spatially aggregated ev ent pro cesses defined o ver pre-specified regions and categorical marks. • Inference & scalability: fast and stable. The ST AMP mo del remains within the laten t Gaussian class and is fitted via INLA with Penalized Complexity (PC) priors on v ariance and correlation h yp erparameters ( Simpson, Rue, Riebler, Martins, & Sørb ye , 2017 ), a voiding MCMC and heavy deep arc hitectures while still providing stable approximate Bay esian inference for league-scale data (on the order of 3 × 10 5 shots) in roughly a min ute p er mo del on a standard CPU. 3 • Utilit y: league-wide in terpretabilit y and fair comparisons. The mo del pro vides interpretable log-rate contrasts, supp orts exp osure-fair team-to-team com- parisons by incorp orating p ossession counts as an offset. In the exp erimen ts, we demonstrate the predictiv e v alidity and practical application in the B.LEAGUE (the men’s professional basketball league in Japan). In brief, the ST AMP mo del is a Poisson GLMM for areal shot coun ts with a p os- session offset and fixed effects for shot-type, season, and court side. It includes four k ey random-effects comp onen ts, tw o interaction terms (team × area and team × side) and tw o shot-t yp e-specific team × area slop es, that capture team- and location-sp ecific deviations in shot tendencies within a latent Gaussian INLA framework with PC pri- ors on v ariance and correlation hyperparameters. Using only priors that pass prior predictiv e chec ks, we then systematically ev aluate out-of-sample predictiv e perfor- mance under alternative random-effects sp ecifications obtained b y switching on and off these four comp onents. F ull details of the mo del sp ecification, prior choices, and exp erimen tal design are given in Metho ds 4 . Guided b y this specification, our empiri- cal analysis is designed to address three questions. First, we quan tify how m uch eac h of the four random-effects comp onen ts contributes to out-of-sample predictive p er- formance by comparing the sixteen candidate structures obtained by switc hing them on and off (Section 2.2 ). Second, we inv estigate what spatial and left-right patterns the fully sp ecified ST AMP mo del reveals at the league and team levels, fo cusing on represen tative B.LEA GUE teams (Section 2.3 ). Third, we examine whether adding further team-level slop es for shot-type preferences yields meaningful gains in predictiv e p erformance beyond the core ST AMP structure (Section 2.4 ). 2 Results 2.1 Dataset In this study , we use play-b y-play data for a total of 2,352 games from the 2023–24 and 2024–25 B.LEA GUE seasons. Play-b y-play data are time-stamp ed logs that record on- court actions such as shots and reb ounds, as well as game even ts such as substitutions. Note that B.LEAGUE play-b y-play data is av ailable for viewing on the official website for each game, but it is not provided as op en data in a machine-readable format for bulk acquisition for researc h purp oses. This study utilized play-b y-play data with shot lo cation provided for research purp oses by Data Stadium Inc., the official data supplier for B.LEAGUE. In the B.LEAGUE, each team plays against an opp onen t in rounds consisting of one or tw o consecutive games; during the regular-season, B1 teams play 60 games ov er 36 rounds, and B2 teams play 60 games ov er 32 rounds. Afterwards, the top eight teams in each division adv ance to a p ost-season tournament pla y ed in a best-of-three (t wo-win) format. F or modeling, w e use only the num b er of field-goal attempts in eac h court area for eac h shot-type, after standardizing the offensiv e direction, together with the num b er of p ossessions for each team. The shot-t yp e is determined from the action type field in the pla y-by-pla y data, and the court area divisions follo w the definitions commonly used 4 Fig. 1 Half-court area divisions; 1: under basket, 2: in the paint, 3: inside right wing, 4: inside right, 5: inside center, 6: inside left, 7: inside left wing, 8: outside right wing, 9: outside right, 10: outside center, 11: outside left, 12: out side leftwing, 13: back court. in the B.LEAGUE (Figure 1 ). How ever, left and righ t symmetric areas are merged in to a single area (for example, outside left and outside right are merged into outside ). W e also merge several similar shot-type lab els to ensure sufficient counts in each category: fadea wa y and turnaround jump shot; la yup and driving la yup; dunk, tip-in, alley- o op, and ho ok shot; and step-bac k jump shot and pull-up jump shot, for a total of 6 categories. The dataset con tains 306,444 field-goal attempts in total, of which jump shots (denoted jump_shot ) account for 156,423 attempts, step-back / pull-up jump shots (denoted step_pull ) accoun t for 50,454 attempts, layup group (denoted lay_up ) accoun t for 48,550 attempts, floating jump shots (denoted floater ) account for 19,000 attempts, dunk / tip-in / alley-o op / ho ok shots (denoted rim_finishes ) acoun t for 17,819 attempts, and fadeaw ay / turnaround jump shots (denoted fade_turn ) accoun t for 14,198 attempts. The mean num b er of field-goal attempts p er team p er season is 4,032.16, with a standard deviation of 251.80. Note that each court area is also assigned to a coarse side category (Left, Cen ter, Righ t) based on Figure 1 : zones 6, 7, 11, and 12 are treated as Left; zones 1, 2, 5, 10, and 13 as Cen ter; and zones 3, 4, 8, and 9 as Right. 2.2 Predictiv e p erformance of ST AMP mo del W e first compare predictive p erformance across the sixteen structures obtained by switc hing on or off each group of random-effect components: team × area effects ( use_ta ), team × side effects ( use_ts ), shot-type-sp ecific random slop es for jump shots ( use_jump ), and shot-type-sp ecific random slop es for step-back/pull-up shots ( use_step ). All candidates share the same fixed-effects structure; only the presence or absence of these random-effect blo c ks differs (see Metho ds 4 for details). F or each random-effects structure, w e fit the ST AMP mo del to regular-season data and ev aluate out-of-sample predictiv e performance on the p ost-season data. Our 5 T able 1 Mo del comparison for core structures. The columns use_ta , use_ts , use_jump , and use_step indicate whether each comp onen t is included ( ✓ ) or excluded (–). The primary criterion is the Monte Carlo estimate of the out-of-sample exp ected log predictive density d elpd MC post , with LPML regular reported as a secondary , fitting-data criterion. Larger v alues (less negative) indicate b etter predictive performance. All v alues are computed under the prior setting U sd = 1 . 5 and U slope = 1 . 0 describ ed in Section 4.2 . Boldface en tries denote the b est v alue among all 16 candidate structures, including those not sho wn in this table. use_ta use_ts use_jump use_step d elp d MC post ↑ LPML reg ↑ – – – – − 6 , 581 − 130 , 617 ✓ – – – − 6 , 422 − 115 , 261 – – ✓ ✓ − 3 , 499 − 50 , 141 ✓ – ✓ ✓ − 3 , 385 − 37 , 835 ✓ ✓ ✓ ✓ − 3 , 383 − 49 , 967 primary criterion is the Monte Carlo approximation d elp d MC post of the exp ected log pre- dictiv e density on p ost-season p ossessions, computed by drawing p osterior samples of the latent Gaussian field and av eraging the log-likelihoo d con tributions cell-wise. As a complemen tary fitting-data measure, we also report the log pseu do-marginal likeli- ho od on the regular-season data, LPML reg . T able 1 summarizes d elp d MC post and LPML reg for selected core structures under a representativ e prior configuration. The simplest baseline that omits all random effects ( use_ta = 0 , use_ts = 0 , use_jump = 0 , use_step = 0 ) exhibits the worst predictive p erformance for post-season data. Adding only team × area or team × side in teractions yields mo dest impro vemen ts, while the largest gains arise when both interactions and shot-type-sp ecific slop es are included. The full mo del with all four comp onen ts switched on ( use_ta = 1 , use_ts = 1 , use_jump = 1 , use_step = 1 ) ac hieves the b est predictive score for p ost-season, outp er- forming the baseline by roughly ∆ d elp d MC post ≈ 3 . 2 × 10 3 while also attaining the highest LPML reg among the structures with use_ta = 1 , use_jump = 1 , and use_step = 1 . How- ev er, the gain from including the team × side effects is very small in out-of-sample terms: the mo del with use_ta = 1 , use_ts = 0 , use_jump = 1 , use_step = 1 attains an d elp d MC post only tw o units w orse than the full mo del, while ac hieving a substantially b et- ter LPML reg . This suggests that most of the predictiv e gain comes from the team × area in teractions and jump/step slopes, and that team × side effects pro vide, at best, a marginal refinement. Despite its ric her random-effect structure, the full mo del remains computationally feasible: a single INLA fit requires on the order of one min ute of w all-clo c k time on our hardware (see Section 4.3 ). T o assess the sensitivit y of this comparison to the prior h yp erparameters, we rep eated the sixteen-model comparison across all prior configurations that passed the prior predictiv e chec ks in Section 4.2 . F or each random-effects structure, we a veraged d elp d MC post and LPML reg o ver these prior settings; the full mo del remained the b est p er- former in terms of mean d elp d MC post , and the improv ement ov er the baseline stay ed on 6 Fig. 2 Left-righ t asymmetry caterpillar plot. The horizontal axis sho ws the logarithm of the ratio of relative occurrence rates b etw een right and left (aggregated for each team using the geometric mean ov er the season; v alues closer to the left indicate a bias tow ard the left side). The dashed line indicates no left-right bias, the points represent the estimated v alues, and the horizontal bars represen t the approximate 95% confidence interv al. the order of 3 . 2 × 10 3 . Moreov er, for every prior configuration, the highest d elp d MC post w as alwa ys attained by one of the t wo richest structures, use_ta = 1 , use_jump = 1 , use_step = 1 with use_ts either 0 or 1 ; no sparser sp ecification ever ranked first. The difference betw een these tw o top mo dels was alwa ys small (on the order of a few units in d elp d MC post ), reinforcing the view that team × side effects play a secondary role compared with team × area interactions and shot-type-sp ecific slop es. The ranking of the sixteen structures was also highly stable across prior configurations: Kendall’s τ rank correlations b et ween the model rankings were ab o ve 0.97. T aken together, these results indicate that the qualitative adv antage of including team × area interactions and shot-t yp e-specific random slopes (with or without team × side effects) is robust to reasonable changes in the prior sp ecification. 2.3 Spatial patterns and interpretation T o illustrate how the ST AMP model with all random effects can b e used to interpret spatial tendencies in practice, we first summarize league-wide left-righ t biases and then present detailed examples for tw o representativ e B.LEAGUE teams, Utsunomiya Brex and Nagoy a Diamond Dolphins. Figure 2 shows a caterpillar plot of the team-by-side effects z ids , summarized as the log ratio log(Right / Left) of the p osterior multipliers for the right and left sides, aggregated ov er seasons via a geometric mean. V alues greater than zero indicate a preference for shots on the righ t side, whereas negativ e v alues indicate a left-side 7 Fig. 3 Area-sp ecific relativ e app earance trends for the Utsunomiya Brex in the 2024-25 season (Left: jump_shot , Right: step_pull ); side scaling applied. Colors and lab els represent team-level p ercen tiles (pXX), with larger v alues and brighter colors indicating relatively more frequent o ccurrences. Fig. 4 Area-sp ecific relative app earance trends for the Nagoy a Diamond Dolphins in the tw o seasons (left: 2023–24 season, right: 2024–25 season); side scaling applied. Colors and lab els represent team- level percentiles (pXX), with larger v alues and brighter colors indicating relatively more frequent occurrences. preference; horizontal bars indicate approximate 95% credible interv als. Most teams exhibit mo derate left-right asymmetries whose interv als o verlap the symmetric case log(Right / Left) = 0 , but a few teams sho w clearly identifiable side-to-side biases. This type of summary is directly in terpretable for coaches and analysts: it highlights whether a team tends to fav or one side of the court ov er the other, after adjusting for o verall shot volume. W e then fo cus on tw o B.LEA GUE teams, Utsunomiy a Brex and Nagoy a Diamond Dolphins, to provide concrete examples of the spatial patterns captured by the ST AMP mo del. F or eac h team, we combine the estimated shot-type-sp ecific random slopes with the team × side effects and conv ert the resulting p osterior mean rates into p er- cen tiles relative to the league-wide distribution for each area. Intuitiv ely , these maps summarize where a team sho ots more or less frequently than the league baseline, after accoun ting for o verall shooting v olume. 8 Figure 3 sho ws side-adjusted shot charts for Utsunomiy a Brex in the 2024-25 sea- son, for jump_shot (left panel) and step_pull (right panel). Each tile corresp onds to one of the predefined court regions, and its color enco des the team-lev el p ercen tile of the p osterior rate among all teams in the same season (brighter colors indicate higher p ercen tiles), while o verlaid lab els (pXX) rep ort the corresp onding p ercen tile in n umerical form. By construction, the figure highlights hot and cold regions in terms of relativ e attempt frequency , rather than raw shot counts, making it easy to iden tify where Utsunomiya is particularly active or inactive compared with league-wide ten- dencies. The c hart indicates that Utsunomiya is not only a team that attempts many three-p oin t shots, but also shows a relativ ely strong preference for taking ab o ve the break three-p oin ters when sho oting off the dribble. This qualitative agreemen t with Utsunomiy a’s reputation as a three-p oin t-oriented team ( Onuma , 2025 ) offers a use- ful sanit y chec k on the ST AMP estimates, suggesting that the mo del captures salien t asp ects of their offensive profile. T o illustrate temp oral comparisons, Figure 4 presents side-adjusted step_pull c harts for Nagoy a Diamond Dolphins in the 2023-24 and 2024-25 seasons, display ed side b y side. Because the color scale and p ercentile definition are shared across panels, c hanges in shading and lab els directly reflect shifts in the team’s relativ e use of each region from season to season. In this wa y , ST AMP-based maps provide an in terpretable league-normalized summary of ho w spatial tendencies of a team remain stable or ev olve. 2.4 Extended team-lev el slop es T o examine whether further team-level heterogeneity in shot-t yp e preferences improv es predictiv e p erformance, w e conducted an additional exp erimen t that augments the ST AMP mo del with extra team-level slopes. Fixing use_ta = 1 , use_jump = 1 , and use_step = 1 , we considered all 2 × 2 4 = 32 model configurations obtained by toggling the inclusion of team-by-side effects ( use_ts ∈ { 0 , 1 } ) and four additional team-level, season-sp ecific slop es for the remaining shot-type categories, which hav e small spa- tial v ariation: lay_up , floater , rim_finishes , and fade_turn (indicators use_lay , use_float , use_rim , use_fade ). All models w ere fitted under the same PC prior configuration as in the main comparison ( U sd = 1 . 5 , U slope = 1 . 0 ; Section 4.2 ) and ev aluated using d elp d MC post and LPML reg . Selected results are summarized in T able 2 , whic h lists the top 10 extended-slop e configurations ranked b y d elp d MC post . As sho wn in T able 2 , some highly flexible sp ecifications that turn on most of the additional slopes achiev e slightly larger d elp d MC post than the original full ST AMP mo del, but at the cost of extremely po or LPML reg (e.g., on the order of − 5 × 10 5 ), suggest- ing o verfitting and unstable fit to the regular-season data. F o cusing on configurations that simultaneously p erform well on b oth criteria, the b est trade-off w as obtained b y the mo del w ith use_ta = 1 , use_ts = 0 , use_jump = 1 , use_step = 1 , and additional team-level slop es only for floater and fade_turn ( use_lay = use_rim =0, use_float = use_fade =1; T able 2 ). This configuration improv ed the p ostseason cri- terion to d elp d MC post ≈ − 3 . 35 × 10 3 , ab out +3 . 0 × 10 1 relativ e to the original full mo del 9 T able 2 T op 10 extended team-level slop e configurations, ranked by d elpd MC post (larger values indicate b etter out-of-sample predictive p erformance). All mo dels include team × area interactions and jump/step team × area × season slop es ( use_ta = ✓ , use_jump = ✓ , use_step = ✓ ); the table rep orts which additional components are switched on ( ✓ ) or off (–). The prior scales are fixed at U sd = 1 . 5 and U slope = 1 . 0 . Boldface entries denote the b est v alue among all 32 candidate structures, including those not shown in this table. use_ts use_lay use_float use_rim use_fade d elp d MC post ↑ LPML reg ↑ ✓ ✓ ✓ – ✓ − 3 , 284 − 510 , 493 ✓ ✓ – ✓ ✓ − 3 , 308 − 355 , 010 ✓ ✓ – – ✓ − 3 , 341 − 73 , 155 – ✓ ✓ – ✓ − 3 , 347 − 37 , 184 – ✓ – – ✓ − 3 , 347 − 37 , 404 ✓ ✓ ✓ ✓ – − 3 , 347 − 50 , 299 – ✓ ✓ ✓ ✓ − 3 , 348 − 37 , 297 – – ✓ ✓ ✓ − 3 , 349 − 37 , 232 – ✓ ✓ ✓ – − 3 , 349 − 37 , 453 – – ✓ – ✓ − 3 , 352 − 36 , 806 ( use_ta = use_ts = use_jump = use_step = 1 ), and sim ultaneously increased LPML reg b y roughly 1 . 0 × 10 3 . Ov erall, these results indicate that allo wing additional team-level slop es for a subset of shot t yp es can yield incremental gains, but the marginal improv ement in predictiv e p erformance ov er the original ST AMP sp ecification is mo dest compared to the gains already obtained from including team × area interactions and shot-type- sp ecific jump/step slop es. F or parsimon y and in terpretability , w e therefore adopt the full ST AMP model without the extra team-level slop es on lay_up , floater , rim_finishes , and fade_turn as our primary sp ecification, and regard this extended- slop e analysis as a robustness chec k. In applications where the main ob jectiv e is to obtain a concise, stable description of team-level spatial tendencies, such additional slop es may not justify the extra mo del complexity , whereas they can b e selectively in tro duced when one wishes to sharp en relative comparisons for a particular shot-type category of interest. 3 Discussion 3.1 Summary and practical implications The ST AMP model extends conv entional shot-c hart analyses b y join tly mo delling team-lev el field-goal attempts across predefined court regions, seasons, and shot types, while remaining computationally feasible for league-scale datasets. The resulting relativ e-rate maps and left–right bias summaries can pro vide coac hes and analysts with interpretable diagnostics of where and how a team tends to generate shots rela- tiv e to the rest of the league, as illustrated by the examples for Utsunomiya Brex and Nago ya Diamond Dolphins (Section 2.3 ). Bey ond the case studies in this pap er, w e believe that the ST AMP mo del has the p oten tial to serve as a routine monitoring and scouting to ol. Because the mo del only 10 requires play-b y-play-lev el shot records and p ossession coun ts, it can b e refit as new games accumulate and embedded into existing analytics dash b oards without additional trac king infrastructure. The p er-possession rate formulation facilitates comparisons fair across teams with differen t paces, while the p ercen tile-based maps and left-righ t bias summaries can b e ov erlaid on conv entional efficiency statistics (e.g., field-goal p ercen tage by region) to jointly assess “ho w often” and “ho w well” teams sho ot from differen t lo cations. In practice, such a to ol could help coaching staff quic kly iden tify st ylistic tendencies and p oten tial blind sp ots. 3.2 Metho dological implica tions and robustness Metho dologically , the ST AMP mo del is sp ecified as a Poisson linear mixed mo del with Gaussian random effects, whic h places it squarely within the class of latent Gaussian mo dels targeted b y INLA. W orking with predefined court regions further helps keep the dimensionalit y of the latent field manageable and the resulting components inter- pretable, enabling fitting and inference at league scale without resorting to M CMC or hea vy architectures. Our systematic comparison of structures and prior configurations suggests that the main substantiv e gains in predictiv e performance arise from including team × area effects together with shot-type-sp ecific slop es for jump shots and step-bac k/pull-up jump shots (Section 2.2 ). This conclusion was robust across all PC prior scale c hoices that passed the prior predictive c hecks, with the richest mo dels consistently ranking at or near the top. The extended exp erimen t with additional team-level slop es for lay_up , floater , rim_finishes , and fade_turn show ed only modest incremen tal impro vemen ts in d elp d MC post relativ e to the original ST AMP sp ecification. These findings suggest that, unless a sp ecific application calls for detailed re lativ e comparisons within those particular shot-t yp e categories, the simpler slop e structure app ears to retain most of the predictive and explanatory p o wer while keeping the mo del more parsi- monious. In a similar spirit, side effects are handled in a delib erately simple wa y in the presen t sp ecification, through additive team × side in teractions only . If an analysis required more detailed side-sp ecific b eha viour, the same framew ork could b e extended to allo w shot-t yp e-dep enden t team × side slop es or other side-v arying comp onen ts, at the cost of additional complexit y and data requiremen ts. F rom a modeling persp ectiv e, the ST AMP model can b e viewed as a prag- matic compromise betw een fully contin uous (marked) p oint-process form ulations (e.g., LGCP) and purely descriptive shot-chart summaries. By aggregating shots to pre- defined court regions and mo delling region-level attempt rates with a log-linear hierarc hical structure, the ST AMP mo del no longer tracks the exact ( x, y ) lo cation of eac h shot. Consequently , it cannot represent very fine-grained within-region pref- erences (for example, a tendency to fav or certain sp ots in the paint). In return, the mo del w orks with a muc h low er-dimensional laten t field, enjoys far more replication p er parameter, and can b e easier to fit and interpret at the league scale. Com bined with the partial po oling structure across teams and seasons, this often makes the ST AMP mo del more data-efficien t than con tinuous-space LGCP approaches when applied at the league scale. 11 The choice of predefined regions is also motiv ated by practice. W e adopt a prede- fined partition of the half-court that closely matches the zone la yout used in official B.League shot charts, so that the regions remain coarse enough to hav e a clear tac- tical interpretation while still b eing fine enough to reveal substantial differences in shot selection. In applied analysis and scouting, shot charts are often summarized at this regional resolution, so w orking with these predefined areas preserv es practical in terpretability while enabling efficient partial p ooling. 3.3 Limitations and future directions This study has sev eral limitations regarding scop e and generalizabilit y . All analyses are based on t wo seasons of B.LEAGUE play and are carried out at the team level within a single men’s league. W e do not examine longer historical windows, women’s com- p etitions, lo wer divisions, or multi-league settings, nor do we fit play er-level mo dels. Consequen tly , it remains an op en question how well the present ST AMP sp ecifica- tion and its fitted spatial patterns w ould transfer to other contexts, and applying the mo del elsewhere would likely require recalibrating priors and p ossibly adapting random effects. Second, the current work fo cuses exclusively on shot frequency rather than shot outcomes or efficiency . W e do not mo del made/missed outcomes, effectiv e field-goal p ercen tage, or exp ected points, so the ST AMP mo del as sp ecified here is best viewed as a to ol for comparing where teams tend to generate shots, not ho w efficien t they are from those lo cations. Nev ertheless, the same hierarc hical framework can b e extended to join tly mo del shot volume and outcomes. A natural next step would b e to com bine the current P oisson model for attempt coun ts with an additional binomial (or logisti c) submodel for made/missed outcomes, thereb y defining b oth an exp ected attempt rate (xR) and exp ected p oin ts (xP) at the team-region-shot-t yp e lev el. Suc h an extension would retain the interpretabilit y of the present ST AMP mo del sp ecification, while offering a unified view of spatial shooting tendencies and scoring efficiency . Third, by construction, the ST AMP model is an areal model: shots are aggregated to predefined court regions and summarized via a log-linear P oisson hierarch y . As a result, the mo del cannot resolve fine-grained v ariation in contin uous shot lo cations within a region, nor do es it explicitly accoun t for defender p ositions, play t yp e, or other con textual factors. Such information is av eraged within regions or indirectly absorb ed in to the random effects, which is appropriate for team-lev el tendency comparisons, but ma y b e limiting for more detailed tactical analysis. If richer mark information, such as pla y-type lab els (e.g., those provided b y commercial tagging services), is av ailable, the same areal framework could b e extended to incorp orate these marks as additional fixed or random effects, or ev en as alternative resp onses, thereb y enabling complementary mo dels that fo cus on offensiv e style or play-con text patterns. An additional mo deling choice concerns the shot-type taxonomy used in the presen t analysis. F or mo deling conv enience and to secure sufficient counts p er cell, w e man ually merged several pla y-by-pla y action lab els into six broader shot-type cate- gories (e.g., grouping layups and driving layups, or fadeaw ay and turnaround jump shots). Although these groupings are defensible from a basketball p erspective, they 12 are inevitably somewhat ad ho c. If richer contextual information, such as sho oting p osture, ball tra jectory , or play er tracking data, were a v ailable, the indicator functions that gate the shot-t yp e-specific random slop es could, in principle, b e replaced b y fea- tures or laten t represen tations learned from those data. This w ould open the do or to more data-driven and potentially more nuanced shot-t yp e partitions within the same hierarc hical framework, while preserving the o verall structure of ST AMP . Bey ond team-lev el applications, extending the ST AMP mo del to play er-level com- parisons would b e a natural next step. A straightforw ard approach w ould b e to embed additional random effects for pla yers nested within teams, allowing the same regional and shot-type structure to describ e within-team v ariation. F or cross-team compar- isons, ho wev er, it may often b e more meaningful to compare play ers who o ccup y similar tactical roles (e.g., ball handlers, sho oters, bigs). In such settings, one could either fit separate models by role or include role indicators and role-specific random effects so that differences in spatial tendencies are interpreted relative to pla yers with com- parable resp onsibilities and usage, pro vided that each play er accumulates sufficient shot volume. Practically , adding such play er- and role-lev el components also increases the num b er of h yp erparameters and the size of the laten t field, which in turn raises the computational cost of INLA. If play er comparison is the only ob jectiv e, a simpler pla yer-lev el shot-chart mo del may therefore be a more computationally con v enient c hoice. More broadly , the ST AMP framework is p oten tially applicable to other inv asion sp orts in which shot actions are spatially structured, such as handball or ice ho c key . In con trast, for sp orts like so ccer, where shots are relatively rare and muc h of the tactical ric hness lies in the build-up pla y , it may b e more practically useful to adapt the same areal Poisson idea to mo del passes, entries in to dangerous zones, or other in termediate actions using an appropriate exposure offset rather than shots themselv es. In this sense, the ST AMP mo del can b e regarded as a general template for spatially structured coun t data that can b e repurp osed according to the most informative action t yp e in eac h sp ort. 4 Metho ds 4.1 Prop osed mo del: shot-type-aw are areal m ultilev el P oisson (ST AMP) mo del In this study , for team i = 1 , . . . , I , season s = 1 , . . . , S , area a = 1 , . . . , A , side d ∈ { Left , Right , Center } , and shot-type k ∈ K , we mo del the num b er of field goal attempts in the cell ( i, s, a, d, k ) , denoted by Y isadk , using the team × season p ossession coun t E is as an exp osure offset. Throughout, we adopt a shot-terminated definition of possession: a p ossession ends at each field goal attempt, and an offensiv e reb ound b egins a new p ossession. 13 4.1.1 Lik eliho o d and linear predictor The likelihoo d and linear predictor are giv en by Y isadk ∼ Poisson( µ isadk ) , log µ isadk = log E is + η isadk , (1) where η isadk denotes the linear predictor. W e decomp ose η isadk as η isadk = β 0 + β (season) s + β (side) d + β (shot _ type) k | {z } fixed effects + u i |{z} team (iid) + v a |{z} area (iid) + w ias | {z} team × area (seasonally correlated) + z ids |{z} team × side (season-sp ecific) + M X m =1 γ m ( k ) r ( m ) ias | {z} shot-type-specific team × area (seasonally correlated) , (2) sub ject to the sum-to-zero constraints I X i =1 u i = 0 , (3) A X a =1 v a = 0 , (4) I X i =1 A X a =1 S X s =1 w ias = 0 , (5) X d ∈{ Left , Right , Cen ter } z ids = 0 ( ∀ i, s ) , (6) I X i =1 A X a =1 S X s =1 r ( m ) ias = 0 ( ∀ m ) , (7) where, E is is the num b er of p ossessions for team i in season s (offset term); u i is an i.i.d. random effect for team i ; and v a is an i.i.d. random effect for area a . The terms w ias are seasonally correlated team × area effects; for eac h ( i, a ) we assume an equicorrelation structure corr( w ias , w ias ′ ) = ρ ta for s  = s ′ . F or team × side, we use season-sp ecific i.i.d. random effects z ids with zero-sum constrain ts P d z ids = 0 for eac h ( i, s ) , but do not imp ose an explicit temp oral correlation across seasons . Since side is a coarse factor (Left/Righ t/Center) and the corresp onding coun ts are relatively abundan t, we mainly in tro duce seasonal correlation for the more gran ular and data-sparse team × area and shot-t yp e-sp ecific effects. 14 The collection { r ( m ) ias } M m =1 denotes shot-type-dep endent random slop es for team × area; for eac h m and ( i, a ) these are also mo deled as seasonally correlated across s , analogously to w ias . The function γ m ( k ) is a known co ding function for shot-type k . As the simplest example, using the indicator function based on the family of shot-type sets {K m } M m =1 , γ m ( k ) = 1 { k ∈ K m } , (8) ( 2 ) can b e written as η isadk = · · · + M X m =1 1 { k ∈ K m } r ( m ) ias , (9) and this is the sp ecification adopted in our exp eriments. Note that each fixed effect is treated as a deviation from a reference category ( N (0 , 100 2 ) is used as a nearly uninformed prior). Because including a uniform interaction for all shot-types would cause ov er- parameterization and destabilize cells ( i, s, a, d, k ) with few shots, w e assign random slop es only to a limited set of shot-types and treat the remaining shot-type effects as fixed effects. This design keeps the mo del additiv e on the log scale and stabilizes computation; together with sum-to-zero cons train ts that fix the ov erall baseline and regularizing h yp erpriors (described below) that shrink the v ariances and temp oral correlations of the seasonally correlated effects tow ards their base mo dels, it yields an appropriate degree of partial p ooling. F urthermore, the equicorrelation structure across seasons for team × area and shot-type-sp ecific effects allows us to capture b oth p ersistence and year-to-y ear v ariation, while the season-sp ecific team × side effects flex- ibly absorb coarse directional biases within eac h season. Note that imposing, for eac h season and area, a sum-to-zero constraint across teams on the season-correlated effects could yield more discriminativ e (i.e., more relative) team differences, but we did not pursue this due to implemen tation difficulties. F rom a lik eliho od p erspective, we retain a P oisson sp ecification with a possession- based offset rather than switching to a negative-binomial lik eliho od with an additional disp ersion parameter. In our setting, the dominan t sources of ov erdisp ersion are het- erogeneit y across teams, regions, and shot types, which are already mo deled through the random effects; adding a separate disp ersion parameter at the lik eliho o d level w ould effectively la yer extra o verdispersion on top of these comp onen ts, complicat- ing in terpretation and iden tifiability . Prior predictiv e chec ks (Section 4.2 ) did not indicate substan tial lack of fit at the aggregated cell level, supp orting the adequacy of the P oisson form ulation. F or similar reasons, we do not imp ose additional spa- tial smo othing priors (e.g., conditional autoregressive structure) across neighboring regions. Instead, predefined court regions are treated as discrete areal categories with indep enden t random effects (i.e., we do not imp ose any spatial smo othing across neigh- b oring regions): this preserv es sharp team-specific hot and cold sp ots, a voids blurring practically meaningful contrasts b et w een regions, and keeps the laten t Gaussian field sufficien tly low-dimensional for efficient INLA-based inference. 15 4.1.2 Priors for random effects and hyperparameters The random effects are modeled as u i iid ∼ N (0 , σ 2 team ) , (10) v a iid ∼ N (0 , σ 2 area ) , (11) w ia := ( w ia 1 , . . . , w iaS ) ⊤ ∼ N  0 , σ 2 ta R S ( ρ ta )  , (12) z ids iid ∼ N (0 , σ 2 ts ) , (13) r ( m ) ia :=  r ( m ) ia 1 , . . . , r ( m ) iaS  ⊤ ∼ N  0 , σ 2 m R S ( ρ m )  , m = 1 , . . . , M , (14) where R S ( ρ ) denotes the S × S equicorrelation matrix with ones on the diagonal and off-diagonal elements equal to ρ . F or the v ariance and correlation h yp erparameters, we use PC priors. In particular, for given thresholds U • , V • and tail probabilities α prec , α cor , we specify Pr( σ • > U • ) = α prec , Pr( | ρ • | > V • ) = α cor , (15) so that large standard deviations and correlations close to ± 1 are a priori p enalized, while mo derate v alues remain plausible. In all analyses we fix α prec = 0 . 05 , so that there is only 5% prior probability that an y standard deviation exceeds its scale param- eter U • . F or the season-correlation parameters ρ ta and ρ m w e use PC-cor priors with V • = 0 . 7 and α cor = 0 . 7 , that is, Pr  | ρ • | > 0 . 7  = 0 . 7 , (16) whic h enco des a prior preference for mo derate-to-strong correlations while still shrinking tow ards ρ • = 0 . The detailed calibration of other thresholds using prior predictive c hec ks is describ ed in Section 4.2 . 4.2 Prior sp ecification and prior predictive c hecks As describ ed ab o ve, we place PC priors on the standard deviations and seasonal corre- lation parameters of the random effects. In practice, we tune the scale parameters U • that control the t ypical size of the standard deviations, and we divide them into tw o groups: a common scale U sd for the main random effects (team, area, and team × area) and a separate scale U slope for the shot-type-sp ecific random slop es r ( m ) ias . W e follow a Bay esian workflo w that emphasizes prior predictive chec ks to cali- brate these hyperparameters and detect implausible mo del b eha vior ( Gabry , Simpson, V ehtari, Betancourt, & Gelman , 2019 ). Concretely , for eac h candidate’s c hoice of PC prior scales ( U sd , U slope ) , w e sim ulate from the prior predictive distribution of the ST AMP mo del, and compare summaries of the simulated shot charts to simple empirical b enc hmarks derived from the B.LEAGUE data, as describ ed below. In particular, to assess whether a given pair ( U sd , U slope ) pro duces realistic prior predictiv e b eha vior, we consider the summary statistics of the replicated data. Let 16 T cell , 95 denote the 95 th p ercen tile of the empirical cell-level rates Y isadk /E is across all cells c = ( i, s, a, d, k ) , and let T tot , 95 denote the 95 th p ercen tile of the team-by- season total coun ts P a,d,k Y isadk o ver ( i, s ) . F or each configuration of ( U sd , U slope ) w e generate 800 prior predictive replicates and compute tw o-sided p -v alues p cell , 95 = 2 min  Pr( T rep cell , 95 ≤ T obs cell , 95 ) , Pr( T rep cell , 95 ≥ T obs cell , 95 )  , (17) and analogously p tot , 95 for T tot , 95 . The probabilities are approximated by empirical prop ortions ov er the 800 replicates. F ollowing a conv en tional prior predictive c heck, w e regard a configuration as acceptable if b oth p cell , 95 and p tot , 95 exceed a nominal threshold (here 0 . 05 ). W e p erformed a grid searc h ov er U sd ∈ { 0 . 5 , 0 . 8 , 1 . 0 , 1 . 2 , 1 . 5 } , U slope ∈ { 0 . 5 , 0 . 8 , 1 . 0 , 1 . 2 , 1 . 5 } , (18) resulting in 25 candidate configurations. Among these, 7 pairs satisfied the prior predictiv e criterion ( p cell , 95 > 0 . 05 and p tot , 95 > 0 . 05 ): ( U sd , U slope ) ∈ { (1 . 5 , 1 . 5) , (1 . 5 , 1 . 2) , (1 . 5 , 1 . 0) , (1 . 5 , 0 . 8) , (1 . 5 , 0 . 5) , (1 . 2 , 1 . 2) , (1 . 2 , 0 . 5) } . (19) W e regard these seven settings as admissible priors and use all of them in a subse- quen t prior sensitivit y analysis. F or brevit y , we rep ort model-comparison results for a represen tative configuration from this admissible set. 4.3 Mo del fittin g and ev aluation All mo dels are fitted in a Bay esian framework using the latent Gaussian struc- ture describ ed in Section 4.1 and the PC priors and prior predictiv e calibration in Section 4.2 . In what follows, we distinguish b et w een r e gular-se ason p ossessions, whic h are used for mo del fitting, and p ost-se ason p ossessions, which are used only for out-of-sample ev aluation (see Section 2.1 for details of the dataset construction). 4.3.1 Data split (fitting and ev aluation) W e train all mo dels on regular-season p ossessions and ev aluate their out-of-sample p erformance on p ost-season p ossessions, reflecting the intended deploymen t scenario in whic h a model fitted on past seasons is used to assess p ost-season tendencies. In total, the dataset contains 306 , 444 field-goal attempts, of which 296 , 768 shots ( 96 . 84% ) o ccur in the regular season and 9 , 676 shots ( 3 . 16% ) o ccur in the p ost-season. Regular-season shots con tribute to the likelihoo d used for parameter estimation, whereas post-season shots are held out and used only for out-of-sample predictiv e ev aluation. 4.3.2 Mo del configurations W e compare a family of 16 candidate sp ecifications obtained by switching on or off four groups of random effects. Concretely , we in tro duce binary indicators 17 use_ta , use_ts , use_jump , use_step ∈ { 0 , 1 } con trolling the inclusion of team × area effects w ias ( use_ta ), team × side effects z ids ( use_ts ), and shot-type-sp ecific random slop es for jump shots ( use_jump ) and step-back / pull-up jump shots ( use_step ). W e then consider all combinations { use_ta , use_ts , use_jump , use_step } ∈ { 0 , 1 } 4 , (20) so that the most complex sp ecification uses all four components and the simplest one includes only the fixed effects and the team- and area-lev el main effects. Random slop es are introduced only for the jump_shot and step_pull categories. These shot-t yp es ha ve sufficien tly many attempts per team to supp ort additional parameters, and they exhibit substantially larger spatial v ariability than other cate- gories, making them natural candidates for capturing team-sp ecific spatial tendencies. F or the remaining shot-t yp es w e retain only fixed effects, whic h helps av oid o ver- parameterization and instability in cells c with few observ ations. All candidate mo dels share the same fixed-effect structure and the same family of h yp erpriors; only the presence or absence of these random-effect blo c ks differs. 4.3.3 Fitting via INLA F or each mo del configuration, we aggregate the play-b y-pla y data to the cell level c and form regular-season counts Y reg c , p ost-season counts Y post c , and team-by-season exp osure terms E reg is and E post is . The mo del in ( 2 ) is then fitted to the regular-season data by treating Y reg c as the resp onse and log E reg is as the offset. W e use a Poisson lik eliho o d with a log link and exploit the latent Gaussian structure to p erform approx- imate Bay esian inference via INLA, as implemented in the R-INLA pac k age. F or each fit w e request p osterior summaries for all fixed and random effects and for the lin- ear predictors η c , and enable configuration sampling ( config = TRUE ) so that we can dra w p osterior samples from the latent Gaussian field using inla.posterior.sample . These p osterior draws are later used to ev aluate out-of-sample predictive p erformance on the p ost-season data via Monte Carlo integration of the log predictive densit y . All random-effect blo cks are implemented as indep enden t Gaussian effects with model="iid" in R-INLA . Seasonally correlated team × area effects w ias and shot- t yp e-sp ecific slop es r ( m ) ias are represented using the group mechanism with group = season and an exc hangeable correlation structure across seasons, corresp onding to the equicorrelation matrix R S ( ρ ) in Section 4.1 . The team × side effects z ids are modeled as season-specific i.i.d. effects without an explicit temp oral correlation, with zero- sum constraints ov er d for eac h ( i, s ) imp osed via the extraconstr argument. T o impro ve n umerical stability we add a small ridge term to the fixed effects (through control.fixed ) and use strategy="adaptive" and int.strategy="ccd" for the INLA internal appro ximations. 4.3.4 Criterion on fitting (regular-season) data On the regular-season data used for fitting, we monitor the log pseudo-marginal lik e- liho od (LPML; Geisser and Eddy ( 1979 ); Held, Schrödle, and Rue ( 2009 )). LPML is 18 defined as LPML reg = X c log  CPO c  , (21) where CPO c denotes the conditional predictive ordinate for cell c (regular-season observ ation), as returned by INLA. Larger v alues of LPML reg corresp ond to b etter in- sample predictive fit. In the result tables w e rep ort LPML reg as a secondary diagnostic to c heck that the mo dels selected by the p ost-season criterion are also reasonable on the fitting data. 4.3.5 Out-of-sample ev aluation on the p ost-season Our primary mo del-comparison criterion is the exp ected log predictiv e densit y on the p ost-season data. F or each fitted mo del we keep the p osterior obtained from regular- season data and ev aluate predictive performance on the held-out p ost-season counts Y post c with exp osure E post is . Let θ denote the collection of fixed and random effects and h yp erparameters; for each cell c with E post is > 0 we consider the log predictiv e densit y log p  Y post c | θ  , (22) where the Poisson mean is µ post c = E post is exp( η c ) . F rom each INLA fit, we dra w J p osterior samples θ (1) , . . . , θ ( J ) of the latent Gaus- sian field (here J = 400 ), and approximate the p oin twise log predictive densities by Mon te Carlo, b ℓ c = log 1 J J X j =1 p  Y post c | θ ( j )  ! . (23) Summing o ver all p ost-season cells yields the Monte Carlo estimate of the exp ected log predictive densit y , d elp d MC post = X c b ℓ c . (24) W e use d elp d MC post as the primary selection criterion, with larger v alues indicating better out-of-sample predictive p erformance on p ost-season shot c harts. In the result tables w e rep ort d elp d MC post together with LPML reg to jointly assess extrap olativ e performance and fit to the regular-season data. All computations are carried out in R using a single BLAS thread for repro- ducibilit y . With the presen t dataset (on the order of 3 × 10 5 shots aggregated to team-season-area-side-shot-t yp e cells), each INLA fit completes w ithin roughly a min ute on a standard workstation-class CPU. 19 4.3.6 A dditional team-level shot-t yp e slop es In addition to the main family of 16 candidate mo dels based on team × area and shot-t yp e-sp ecific slopes for jump and step-bac k/pull-up shots (Section 4.3 ), we also considered an extended set of mo dels that in tro duce team-level shot-type slop es for the remaining categories ( lay_up , floater , rim_finishes , fade_turn ). Concretely , w e kept the team × area slop es for jump_shot and step_pull unchanged and switched on/off the presence of team-level slop es for each of the four additional shot types, com bined with use_ts ∈ { 0 , 1 } , yielding 2 × 2 4 = 32 additional candidates. The prior h yp erparameters were fixed to the v alues calibrated in Section 4.2 , with U sd = 1 . 5 , U slope = 1 . 0 , α prec = 0 . 05 and V = 0 . 7 , α cor = 0 . 7 . W e ev aluated these extended mo d- els in the same w ay as the main candidates, using d elp d MC post on p ost-season data and LPML reg on regular-season data. Comp eting in terests The authors hav e no comp eting interests to declare that are relev ant to the conten t of this article. Author Con tributions K.Y. contributed to the study conception and design. Data preparation, mo deling, and analysis were p erformed by K.Y. The first draft of the manuscript was con tributed b y K.Y. and K.F. All authors read and approv ed the final man uscript. Data a v ailability The data of the researc h is obtained by participating in a comp etition hosted by the academic organizations. The central idea of this study was indep enden t of the comp etition (not restricted by the comp etition). Data acquisition was based on the con tract betw een the basketball league and Data Stadium, Inc., but not b et w een the pla yers/teams and us. F unding Information K. Y amada gratefully ackno wledges supp ort from the “THERS Make New Standards Program for the Next Generation Researchers”. This w ork was financially supported b y JSPS KAKENHI Gran t Num b er 23H03282. A ckno wledgments. The data used in this study w ere pro vided by the Insti- tute of Statistical Mathematics, Academ y for Statistics and Data Science, Research Organization of Information and Systems, and Data Stadium Inc. 20 References Anzer, G., & Bauer, P . (2021). A goal scoring probability mo del for shots based on sync hronized p ositional and ev ent data in football (so ccer). F r ontiers in Sp orts and A ctive Living , V olume 3 - 2021 , , https://doi.org/10.3389/fspor .2021.624475 Brec hot, M., & Flepp, R. (2020). 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