Chapter 17: Key Takeaways
Core Concepts
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Soccer is a spatial contest. The majority of tactical value is created through movement and positioning, not on-ball actions. Spatial analytics provides the tools to measure what was previously invisible.
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Voronoi diagrams are the foundation. By partitioning the pitch into dominant regions based on nearest-player proximity, Voronoi tessellations give a simple yet powerful snapshot of spatial ownership. They are fast to compute, easy to interpret, and serve as a baseline for more sophisticated models.
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Voronoi has limitations. Because standard Voronoi diagrams rely on Euclidean distance alone, they ignore player velocity, acceleration, body orientation, and physical differences. These limitations motivate probabilistic approaches.
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Pitch control models replace binary ownership with probability. The Fernandez--Bornn framework models each player's spatial influence as a velocity-adjusted bivariate Gaussian. Summing influences across teams and normalising produces a continuous control surface. Spearman's time-to-intercept model offers a physics-based alternative.
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Space creation is measurable. By comparing a player's team-mates' Voronoi areas with a counterfactual (freezing the space-creating player), we can quantify how much space a player generates for others. This captures contributions invisible to event data.
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Exploitation completes the picture. Space creation is only valuable if the team plays the ball into the newly created zone. Combining space-creation volume with expected threat at the reception point yields a single metric for the value of off-ball movement.
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Off-ball runs can be detected and classified algorithmically. Velocity projections toward goal, perpendicular to goal, and away from goal allow automated classification of penetrating, lateral, dropping, and decoy runs from tracking data.
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Dangerous space is the intersection of control, value, and vulnerability. The Dangerous Space Matrix (DSM) multiplies pitch control by expected threat and a defender-absence term, identifying zones that are both valuable and poorly defended.
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Spatial entropy measures attacking unpredictability. Teams with diverse final-third entry patterns are harder to defend and tend to generate higher-quality chances.
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Spatial value added (SVA) unifies pitch control and expected threat. By differencing pitch-control-weighted xT before and after an action, SVA provides a spatially-aware measure of how much an action advances the team's attacking prospects.
Practical Guidelines
- Grid resolution matters. Use 2 m cells for real-time analysis and 1 m cells for offline or publication-quality work.
- Always clip Voronoi cells to the pitch boundary. Unbounded cells produce misleading area calculations.
- Validate pitch control models against observed outcomes (e.g., loose-ball contests) before using them for tactical inference.
- Freeze only the space-creating player in counterfactual analysis. Freezing multiple players creates unrealistic scenarios.
- Combine spatial metrics with event data for the richest picture. Neither data source is sufficient alone.
Common Pitfalls
| Pitfall | Consequence | Mitigation |
|---|---|---|
| Using raw Voronoi areas without clipping | Unbounded cells inflate area | Clip to pitch rectangle |
| Ignoring velocity in spatial models | Overestimate stationary players' control | Use pitch control models |
| Setting $\Delta t$ too large | Unrealistic projected positions | Keep $\Delta t \in [0.5, 1.5]$ s |
| Computing DSM without defender density | Overvalue zones near defenders | Include $D_{\text{def}}$ term |
| Averaging spatial metrics without context | Mix offensive and defensive phases | Condition on possession state |
Key Equations
| Concept | Equation |
|---|---|
| Voronoi cell | $V_i = \{x : \|x - p_i\| \le \|x - p_j\|\ \forall j \ne i\}$ |
| Player influence | $I_i(x) = \frac{1}{2\pi|\Sigma_i|^{1/2}} \exp(-\frac{1}{2}(x-\mu_i)^T \Sigma_i^{-1}(x-\mu_i))$ |
| Pitch control | $\mathrm{PC}_A(x) = I_A(x) / (I_A(x) + I_B(x))$ |
| Space creation | $\Delta A_j(t) = A_j^{\text{actual}}(t) - A_j^{\text{counterfactual}}(t)$ |
| DSM | $\mathrm{DSM}(x,y) = \mathrm{PC}_A(x,y) \cdot xT(x,y) \cdot (1 - D_{\text{def}}(x,y))$ |
| Spatial entropy | $H = -\sum_k p_k \log p_k$ |
| SVA | $\text{SVA} = \text{PC-xT}_{A,\text{after}} - \text{PC-xT}_{A,\text{before}}$ |
What Comes Next
Chapter 18 builds on the spatial foundations established here by introducing network analysis and passing models, where the spatial context of passes (origin, destination, and the pitch control landscape at the moment of the pass) is used to evaluate passing quality beyond simple completion rates.