class: center, middle background-image: url(prp_qr.png) background-size: 10 % <br> # Curve Clustering Methods for Analyzing Pass Rush in the NFL .large[Robert Bajons | Joint work with Kurt Hornik | 23 Sept. 2023] <img src="Nessis.png" width="300px"/> --- # Personal Motivation - In Europe we're mostly playing a different kind of football (soccer). -- - Austria (a small country in central Europe) is not very good at it. -- - For European standards: Austria is quite good at American football: - Austria is playing for the European title (against Finland) this year. - Last season the Vienna Vikings won the European League of Football (the European pendant of the NFL). - NFL: Bernhard Raimann (Indianapolis Colts), Bernhard Seikovits (Arizona Cardinals, practice squad). -- - About me: sports enthusiast, interested in statistical modelling of various sports (mainly soccer). -- - Kurt Hornik (my supervisor): "Why don't we model real football?" --- # Motivation Routes in the NFL: focus on **pass rush**. <img src="nessis_RB_files/figure-html/plot1.5-1.png" width="1350px" height="433px" style="display: block; margin: auto;" /> --- # Motivation Pass rush in the NFL. <img src="nessis_RB_files/figure-html/plot2-1.png" width="576" style="display: block; margin: auto;" /> --- name: prwslide # Motivation Pass rush in the NFL: add pressure weights. ([Details](#wm)) <img src="nessis_RB_files/figure-html/plot2.5-1.png" width="576" style="display: block; margin: auto;" /> --- # Motivation - Assign weights to pass rush routes via machine learning model. - Analyse weighted curves instead of regular curves. - Weights helpful to distinguish routes according to their **worth**. - **Clustering algorithm for weighted curves needed.** - **Note:** Not only useful for pass rush but any type of route that may be assign weights. --- class: inverse, center, middle # Methodology --- # Formal Description of the Problem - Observed data `\(\boldsymbol{Y} = \{\boldsymbol{y}_1,\dots,\boldsymbol{y}_n\}\)` of `\(n\)` curves. - Each `\(\boldsymbol{y_i}\)` is `\(m_i \times 3\)` dimensional matrix, `\((x,y)\)`-coordinates and weights `\(w\)`. - `\(m_i\)` not fixed but varies for each observation. (E.g.: routes in NFL dependent on length of play.) -- **Goal:** Cluster routes, i.e. `\((x,y)\)`-coordinates, by accounting for the information from the weights `\(w\)`. -- - Initial idea: Use a `\(K\)`-means type of algorithm for clustering. - Adjust usual procedure by deriving a weighted `\(K\)`-means method. - Data preprocessing necessary to use `\(K\)`-means algorithm. --- # Bézier Curves .pull-left[ - Parametric curves defined on `\(t \in [0,1]\)` by control points `\(\boldsymbol{\theta} = (\theta_0,\dots,\theta_p)\)` using Bernstein polynomials `\(b_p^P(t) = \binom{P}{p}t^p(1-t)^{P-p}\)`, `\(p = 0,\dots,P\)`. `$$B(t,\boldsymbol{\theta}) = \sum_{p = 0}^P \theta_p b_p^P(t), \quad t \in [0,1]$$` - Control points form a polygon and Bézier curve defined by them lies in convex hull of the points. ] .pull-right[ <img src="nessis_RB_files/figure-html/plotBC-1.png" width="576" style="display: block; margin: auto;" /> ] --- # Preprocessing Data - The trajectory of each observation `\(\boldsymbol{y}_i\)`, is represented by a 2-dimensional Bézier curve: `$$\boldsymbol{y}_i(t) = \sum_{i = 1}^{m_i} b_p^P(t)(x_i,y_i)$$` - Control vector `\(\boldsymbol{\theta}\)` is given by `\((x,y)\)` points observed from `\(\boldsymbol{y}_i\)`. .pull-left[ - In theory **any form** of interpolation between the points could be used. **Advantages** of Bézier curves: - Natural starting ( `\(t = 0\)` ) and end point ( `\(t = 1\)` ). - `\(\Rightarrow\)` Bézier curves allow for a very general smooth representation of the curve (helpful in the context of football routes). - Flexible usage for curves of different lengths. ] .pull-right[ - **Usage** for `\(K\)`-means clustering: - Use prespecified number of `\(M\)` equidistant points on above curve to represent `\(\boldsymbol{y}_i\)`, such that each observation is of dimension `\(M \times 2\)`. - Aggregate weights to get `\(M\)` new weights by considering proximity and order of points. ] --- # From `\(K\)`-means to weighted `\(K\)`-means .panelset[ .panel[.panel-name[K-means] - Given a set of observations `\(\boldsymbol{Y}\)`, find a partition into `\(K\)` clusters by minimizing the within cluster sum of squares `$$S = \sum_{k = 1}^K \sum_{i:g_i = k} (x_i-p_k)^2$$` - Prototypes (centers) `\(p_k\)` are given as cluster means: `$$p_k = \frac{1}{N_k}\sum_{i:g_i = k} x_i$$` - Basic implementation: Use an iterative refinement procedure, which alternates between an assignment step and cluster update step. ] .panel[.panel-name[weighted K-means] - Given a set of observations `\(\boldsymbol{Y}\)`, where each `\(y_i \in \boldsymbol{Y}\)` can be assigned weights `\(w_i\)`, find a partition into `\(K\)` clusters by minimizing the weighted within cluster sum of squares `$$S_w = \sum_{k = 1}^K \sum_{i:g_i = k} w_i(x_i-p_k)^2$$` - Prototypes (centers) `\(p_k\)` are given as weighted cluster means: `$$p_k = \frac{\sum_{i:g_i = k} w_ix_i}{\sum_{i:g_i = k} w_i}$$` - Adapt iterative refinement procedure to case of weighted observation. ] ] --- # Weighted `\(K\)`-means for Curves - Transform observation from preprocessing: `\(\boldsymbol{y}_i \in \mathbb{R}^{M \times 3}\)` is split up in vectors `\(\boldsymbol{z}_i, \boldsymbol{w}_i \in \mathbb{R}^{2M}\)`. `$$\boldsymbol{z}_i = (x_1,\dots,x_M,y_1,\dots,y_M)$$` `$$\boldsymbol{w}_i = (w_1,\dots,w_M,w_1,\dots,w_M)$$` - Find clusters and prototypes such that: `$$\min_{(p_{jk}),(g_i)}\sum_{k = 1}^K \sum_{i:g_i = k} \sum_{j = 1}^{\tilde{M}} w_{i,j}(z_{i,j}-p_{k,j})^2.$$` - Find clusters by iteratively alternating between finding the optimal prototypes for a given cluster and finding the optimal cluster assignments given prototypes. --- # Weighted `\(K\)`-means Algorithm 1. Initialize appropriate starting assignments of clusters. 2. For given cluster assignment find the optimal prototypes: `$$p_{k,j} = \frac{\sum_{i:g_i = k} w_{i,j}z_{i,j}}{\sum_{i:g_i = k} w_{i,j}}$$` 3. Given prototypes find optimal cluster assignments by minimizing `$$\sum_{j = 1}^M w_{i,j}(z_{i,j}-p_{k,j})^2$$` over `\(k\)`. 4. Repeat steps 2-3 until convergence, i.e. until change in function to optimize is below some tolerance level. --- # Weighted `\(K\)`-means Recap - Weighted `\(K\)`-means algorithm provides clustering of curves, i.e. `\((x,y)\)`-coordinates, by accounting for weights `\(w\)`. - Spatial distance of curves is **less** important, when weights are small and **more** important, when weights are high. - **Irrelevant** routes are likely to be clustered together, even if spatially far apart. - Implementation of algorithm in R-programming language. --- class: inverse, center, middle # Application --- name: res1 # Clustering Results <img src="nessis_RB_files/figure-html/plot_clusters-1.png" width="1008" style="display: block; margin: auto;" /> [More comparisons](#res_pr_only1) --- # Clustering Results <img src="nessis_RB_files/figure-html/plot_clusters2-1.png" width="1008" style="display: block; margin: auto;" /> --- # Analysing Teams .panelset[ .panel[.panel-name[Defense] <img src="nessis_RB_files/figure-html/plot_teams-1.png" width="1008" style="display: block; margin: auto;" /> ] .panel[.panel-name[Offense] <img src="nessis_RB_files/figure-html/plot_teams_def_more-1.png" width="1008" style="display: block; margin: auto;" /> ] ] --- # Analysing Players .panelset[ .panel[.panel-name[Profiles] <img src="nessis_RB_files/figure-html/plot_players-1.png" width="1008" style="display: block; margin: auto;" /> ] .panel[.panel-name[Radars] <img src="nessis_RB_files/figure-html/plot_radars-1.png" width="1008" style="display: block; margin: auto;" /> ] ] --- # Play Database for Pass Rush .panelset[ .panel[.panel-name[Idea] .pull-left[ Easy filtering for pass rush plays based on a number of information: - Team - Players - Routes/Clusters - Play specific (Yard line, down, pressure, etc) Useful for: - Preparation for game - Analysis of strength and weaknesses - Scouting ] .pull-right[ Example: - Plays with Myles Garrett from the right and Jadaveon Clowney from the left. - Both on routes of high pressures (cluster 10 (right) and 7 (left)). - Pressure exerted on the QB. ] ] .panel[.panel-name[Example] <div class="reactable html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-7973f1737f8cc9070e5f" style="width:auto;height:auto;"></div> <script type="application/json" data-for="htmlwidget-7973f1737f8cc9070e5f">{"x":{"tag":{"name":"Reactable","attribs":{"data":{"team":["CLE","CLE","CLE","CLE","CLE"],"uId":["2021091209_2293","2021091209_3639","2021092601_693","2021101708_1650","2021103103_1080"],"possessionTeam":["KC","KC","CHI","ARI","PIT"],"quarter":[3,4,1,2,2],"down":[3,3,1,3,2],"gameClock":["10:31","03:41","04:18","10:12","10:39"],"plays":[{"nflId":[38667,41227,42816,44813,44847,44903,46073,46122,47829,53455,53481],"cl":[1,7,3,10,1,2,3,2,3,4,8],"pressure":[0,1,0,0,0,0,0,0,0,0,0],"coln":["X7591","X7592","X7593","X7594","X7595","X7596","X7597","X7598","X7599","X7600","X7601"],"positionGroup":["DL","DL","Backs","DL","DL","S","Backs","S","Backs","Backs","LB"],"pff_role_orig":["Pass Rush","Pass Rush","Coverage","Pass Rush","Pass Rush","Coverage","Coverage","Coverage","Coverage","Coverage","Pass Rush"],"displayName":["Malik Jackson","Jadeveon Clowney","Troy Hill","Myles Garrett","Malik McDowell","John Johnson","Denzel Ward","M.J. 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Contact: [robert.bajons@wu.ac.at](mailto:robert.bajons@wu.ac.at) --- class: inverse, center, middle # Appendix --- name: wm # Modeling Pressure Weights .pull-left[ Modelling: - Extreme gradient boosted trees. - Hyperparameter tuning via 5-fold cross validation on a grid. - Model trained by optimizing over logloss. - Evaluation on hold out set. ] .pull-right[ Features: + Distance to the Quarterback: distance of defensive player, average distance of offense (total and nearest 6), average distance of defense (total and nearest 6, considered player excluded). + Distance to opponents: Nearest opponent, 2nd nearest opponent, 3rd nearest opponent, average distance to nearest 6 opponents. + Speed: QB speed and defender speed. + Acceleration. + Distance traveled since snap. + Pressure on play (hit, hurry, sack). ] [hop back](#prwslide) --- name:res_pr_only1 # Pass Rush Routes only .panelset[ .panel[.panel-name[Routes] <img src="nessis_RB_files/figure-html/plot_clusters_pr_only-1.png" width="1008" style="display: block; margin: auto;" /> ] .panel[.panel-name[Pressure] <img src="nessis_RB_files/figure-html/plot_clusters_pr_only2-1.png" width="1008" style="display: block; margin: auto;" /> ] ] [Back](#res1) --- name:res_pr_only2 # Pass Rush Routes only - Analyse predictive performance of clusters in terms of correctly classifying pressure on a play. - Are the clusters helping in predicting pressure? - Fit a glm to predict pressure of each individual route (with only the cluster as predictor). - Table below shows results based on different kind of metrics. 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