Chapter 17 Exercises

Section A: Conceptual Questions (17.1--17.2)

Exercise 17.1 --- Explain in your own words why possession percentage is a poor proxy for territorial control. Give a concrete tactical example involving two teams with identical possession but different spatial profiles.

Exercise 17.2 --- A Voronoi diagram partitions the pitch into dominant regions based on Euclidean distance. List three factors present in real soccer that violate the Euclidean-distance assumption, and for each, explain whether it would cause the Voronoi model to overestimate or underestimate a player's true dominant region.

Exercise 17.3 --- The Delaunay triangulation is the dual of the Voronoi diagram. Explain what a Delaunay edge between two players represents in tactical terms. Why might an analyst use Delaunay edges rather than the full set of pairwise connections?

Exercise 17.4 --- A team's average Voronoi cell area for its four defenders is 320 m$^2$, while the opponent's four defenders average 180 m$^2$. Interpret this difference in the context of pressing intensity and defensive compactness.

Exercise 17.5 --- Calculate the total number of data points produced by a 90-minute match with 25 Hz tracking data for 22 players (each with $x$ and $y$ coordinates) plus the ball (with $x$, $y$, and $z$ coordinates). Show your working.

Section B: Voronoi Diagrams (17.2)

Exercise 17.6 --- Using SciPy, compute the Voronoi diagram for the following set of 6 player positions (in metres):

Plot the result on a 105 x 68 m pitch. Clip unbounded cells to the pitch boundaries.

Exercise 17.7 --- For the six players in Exercise 17.6, compute the area of each player's Voronoi cell (after clipping). Which player controls the most space? Which controls the least?

Exercise 17.8 --- Add two "ghost" players at $(0, 34)$ and $(105, 34)$ to represent the goalkeepers. How does this change the Voronoi diagram and the dominant-region areas of players near the goal lines?

Exercise 17.9 --- Write a function that takes a set of player positions and returns the team compactness index, defined as the standard deviation of the Voronoi cell areas within the team. Test it with a compact formation (e.g., 4-4-2 in a low block) and a stretched formation (e.g., 3-5-2 in transition).

Exercise 17.10 --- Implement a weighted Voronoi diagram where the distance from a point $x$ to player $i$ is scaled by the inverse of the player's current speed: $d_i(x) = |x - p_i| / (1 + \alpha |v_i|)$. Plot the result for $\alpha = 0.3$ and compare with the unweighted version.

Section C: Pitch Control Models (17.3)

Exercise 17.11 --- Derive the influence function $I_i(x)$ for a player at position $(30, 40)$ with velocity $(3, 1)$ m/s. Use $\Delta t = 0.7$ s, $\sigma_{\parallel} = 10$ m, and $\sigma_{\perp} = 5$ m. Compute $I_i(x)$ at the point $(38, 43)$.

Exercise 17.12 --- Explain the role of the look-ahead time $\Delta t$ in the Fernandez--Bornn model. What happens if $\Delta t$ is set to (a) 0 and (b) a very large value like 5 s?

Exercise 17.13 --- Implement the Fernandez--Bornn pitch control model for a simplified scenario with 4 players (2 per team). Evaluate the pitch control surface on a $53 \times 34$ grid and produce a heat-map visualisation.

Exercise 17.14 --- Compare the pitch control surfaces generated by the Fernandez--Bornn (Gaussian influence) model and the Spearman (time-to-intercept) model for the same 4-player scenario from Exercise 17.13. Discuss the visual and quantitative differences.

Exercise 17.15 --- A striker is running at 10.5 m/s toward goal. A centre-back 12 m away is stationary. Using the time-to-intercept formula from Section 17.3.3, compute which player arrives first at a point 6 m ahead of the centre-back, directly between the two players. Assume $a_{\max} = 4.0$ m/s$^2$ for both players.

Exercise 17.16 --- The computational cost of evaluating pitch control on an $M \times N$ grid for $P$ players is $O(MNP)$ per frame. For a 90-minute match at 25 Hz with 22 players and a $105 \times 68$ grid, calculate the total number of influence evaluations. Propose two strategies for reducing this cost without significantly degrading the model's accuracy.

Exercise 17.17 --- Implement a pitch control model that accepts a ball position and adjusts each player's influence based on distance from the ball. Players closer to the ball should have slightly expanded influence (reflecting urgency to contest) while distant players should have standard influence. Define a reasonable distance-decay function and justify your choice.

Section D: Space Creation and Exploitation (17.4)

Exercise 17.18 --- Define "space creation" in your own words. Explain why a player can create space for a team-mate without ever touching the ball.

Exercise 17.19 --- A centre-forward drops from position $(85, 34)$ to $(70, 34)$ over 3 seconds, pulling a centre-back with them. The centre-back moves from $(82, 34)$ to $(72, 34)$. Meanwhile, a winger's Voronoi cell area increases from 120 m$^2$ to 195 m$^2$. Compute the space created by the centre-forward for the winger.

Exercise 17.20 --- Implement the counterfactual space-creation method described in Section 17.4.1. Given tracking data for 11 attacking players across 50 frames, freeze one designated player at their initial position and recompute Voronoi areas for the remaining team-mates. Report $\Delta A$ for each team-mate.

Exercise 17.21 --- Design a metric called Space Exploitation Efficiency (SEE) that rewards players who receive the ball in space that was recently created by a team-mate. Define the metric formally, state your assumptions, and suggest a reasonable time window.

Exercise 17.22 --- Using synthetic data, simulate a team with five attackers and five defenders. One attacker makes a diagonal run into the channel. Compute and visualise (a) the Voronoi diagram before and after the run, and (b) the $\Delta A$ for each team-mate.

Section E: Off-Ball Movement (17.5)

Exercise 17.23 --- Classify the following movements using the taxonomy from Section 17.5.2: (a) A striker runs from the centre circle toward the corner flag. (b) A central midfielder drops 15 m toward their own goal to receive. (c) A left-winger cuts inside while the left-back overlaps. (d) A striker makes a run in behind but the ball goes to the opposite flank.

Exercise 17.24 --- Implement the detect_penetrating_runs function from Section 17.5.3. Generate synthetic tracking data for a single attacker making three runs in behind over a 60-second period. Verify that your implementation detects all three runs.

Exercise 17.25 --- Extend the run-detection algorithm to also classify lateral runs (movement perpendicular to the goal direction). Define an appropriate velocity threshold and a minimum duration. Test on synthetic data.

Exercise 17.26 --- Compute the run quality score $Q_{\text{run}}$ for the following run: - Depth gained: 18 m - Space created for team-mates: 85 m$^2$ - Defenders engaged: 2 - Pass received: No - $\Delta xT$: 0.04

Use weights $w_1 = 0.01$, $w_2 = 0.005$, $w_3 = 0.15$, $w_4 = 7.0$. Discuss whether the weights seem reasonable and how you might calibrate them.

Section F: Dangerous Space and Tactical Applications (17.6--17.7)

Exercise 17.27 --- Compute the Dangerous Space Matrix (DSM) for a simplified scenario. Place 4 defenders at known positions and define an xT grid (you may use a simple distance-based approximation). Produce a heat-map of the DSM and identify the most dangerous undefended zone.

Exercise 17.28 --- A team's spatial entropy in the final third is $H = 1.8$ nats (using $K = 6$ zones). Compute the maximum possible entropy for $K = 6$ zones. What fraction of maximum entropy does this team achieve? Interpret the result.

Exercise 17.29 --- Design a pressing trigger detector using pitch control. Define a pressing trigger as a moment when the opponent's pitch control in their own defensive third drops below a threshold $\tau$. Implement this detector on synthetic tracking data and evaluate how the threshold $\tau$ affects the number of triggers detected.

Exercise 17.30 --- A club's analytics department has computed that their team creates an average of 35 m$^2$ of dangerous space in the final third per possession, while the league average is 28 m$^2$. Their dangerous-space exploitation rate is 12 %, while the league average is 18 %. Write a briefing memo (200--300 words) to the coaching staff interpreting these numbers and suggesting an actionable improvement.

Exercise 17.31 --- Implement the spatial value added (SVA) metric described in Section 17.7.5. For a synthetic pass from $(50, 34)$ to $(75, 45)$, compute the SVA by differencing the pitch-control- weighted xT before and after the pass. Visualise both pitch control surfaces and annotate the SVA value.

Exercise 17.32 --- Using the concepts from this chapter, design a complete analytical pipeline for evaluating a team's performance in a single match. Your pipeline should include: (1) Voronoi-based compactness over time, (2) pitch control surfaces at key moments, (3) space creation in the final third, (4) dangerous space conceded. Write pseudocode for the full pipeline and discuss how you would present the results to a coaching staff.

Section G: Integration and Synthesis

Exercise 17.33 --- Compare and contrast the Voronoi-based and pitch- control-based approaches to spatial analysis. Under what circumstances would you recommend each?

Exercise 17.34 --- A recruitment analyst wants to identify centre-forwards who create significant space for team-mates despite modest goal-scoring records. Design a screening methodology using the spatial metrics from this chapter. Which metrics would you prioritise, and what thresholds would you set?

Exercise 17.35 --- Discuss two ethical or practical limitations of using tracking-data-based spatial models in professional soccer. For each, propose a mitigation strategy.