Why is routing all coordinate conversions through a canonical system (e.g., standard metres) preferred over writing pairwise converters? It is faster computationally It reduces the number of conversion functions from to It avoids floating-point errors It is required by FIFA regulations

(b) It reduces the number of conversion functions from to . With providers, pairwise converters require functions, while routing through a hub needs only "to-hub" and "from-hub" functions.

The formula normalizes a coordinate to which range?

(c) . Dividing by the pitch length maps the coordinate from to .

The concept of "Space Value" combines pitch control with: Player salaries Expected Threat (xT) -- a spatial value model The number of fans in the stadium The referee's position

(b) Expected Threat (xT) -- a spatial value model. Space Value = , integrating the product of spatial dominance and the value of each location.

Chapter 6 Self-Assessment Quiz

Q: What are the FIFA-recommended dimensions for an international soccer pitch? 100 m x 64 m 110 m x 75 m 105 m x 68 m 120 m x 80 m

(c) 105 m x 68 m. This is the standard adopted by FIFA for international matches and used as the reference frame by most data providers.

Q: In the StatsBomb coordinate system, the pitch dimensions are: 100 x 100 105 x 68 120 x 80 120 x 68

(c) 120 x 80. StatsBomb uses arbitrary units with the pitch spanning 0-120 in the x-direction and 0-80 in the y-direction.

Q: Which data provider uses percentage-based coordinates with the origin at the bottom-left and the y-axis pointing upward? StatsBomb Opta Wyscout FIFA EPTS

(b) Opta. Opta uses a 100 x 100 coordinate system with the origin at the bottom-left corner and y increasing upward.

Q: In the FIFA EPTS tracking data coordinate system, the origin is located at: The top-left corner of the pitch The bottom-left corner of the pitch The centre of the pitch The centre of the home team's goal

(c) The centre of the pitch. FIFA EPTS uses a centre-origin system with coordinates in actual metres, ranging from -52.5 to +52.5 in x and -34.0 to +34.0 in y.

Q: To convert a StatsBomb x-coordinate to standard metres (105 m pitch), you multiply by: 105 / 100 120 / 105 105 / 120 100 / 105

(c) 105 / 120 = 0.875. StatsBomb x ranges from 0 to 120; standard metres range from 0 to 105. So the scale factor is 105/120.

Q: Converting from Wyscout to standard metres requires flipping the y-axis because: Wyscout uses metres, not percentages Wyscout's y-axis points downward (top = 0) Wyscout's x-axis is reversed Wyscout's origin is at the centre of the pitch

(b) Wyscout's y-axis points downward (top = 0). Standard metre coordinates have the y-axis pointing upward, so a reflection is needed: .

Q: In the standard thirds model, the "final third" (attacking third) spans which x-range on a 105 m pitch? 0.0 to 35.0 m 35.0 to 70.0 m 70.0 to 105.0 m 88.5 to 105.0 m

(c) 70.0 to 105.0 m. The pitch is divided into three equal 35 m segments. The attacking third (final third) is the segment closest to the opposition goal.

Q: Zone 14 is located: Inside the penalty area In the centre of the pitch Just outside the penalty area in the central channel On the wing near the corner flag

(c) Just outside the penalty area in the central channel. Zone 14 is the central area between the edge of the penalty area and the two-thirds line, approximately and in standard metres.

Test your understanding of the soccer pitch as a coordinate system. Aim for a score of at least 80% before moving on to Chapter 7.

Q1. What are the FIFA-recommended dimensions for an international soccer pitch?

(a) 100 m x 64 m
(b) 110 m x 75 m
(c) 105 m x 68 m
(d) 120 m x 80 m

Answer

**(c) 105 m x 68 m.** This is the standard adopted by FIFA for international matches and used as the reference frame by most data providers.

Q2. In the StatsBomb coordinate system, the pitch dimensions are:

(a) 100 x 100
(b) 105 x 68
(c) 120 x 80
(d) 120 x 68

Answer

**(c) 120 x 80.** StatsBomb uses arbitrary units with the pitch spanning 0-120 in the x-direction and 0-80 in the y-direction.

Q3. Which data provider uses percentage-based coordinates with the origin at the bottom-left and the y-axis pointing upward?

(a) StatsBomb
(b) Opta
(c) Wyscout
(d) FIFA EPTS

Answer

**(b) Opta.** Opta uses a 100 x 100 coordinate system with the origin at the bottom-left corner and y increasing upward.

Q4. In the FIFA EPTS tracking data coordinate system, the origin is located at:

(a) The top-left corner of the pitch
(b) The bottom-left corner of the pitch
(c) The centre of the pitch
(d) The centre of the home team's goal

Answer

**(c) The centre of the pitch.** FIFA EPTS uses a centre-origin system with coordinates in actual metres, ranging from -52.5 to +52.5 in x and -34.0 to +34.0 in y.

Q5. To convert a StatsBomb x-coordinate to standard metres (105 m pitch), you multiply by:

(a) 105 / 100
(b) 120 / 105
(c) 105 / 120
(d) 100 / 105

Answer

**(c) 105 / 120 = 0.875.** StatsBomb x ranges from 0 to 120; standard metres range from 0 to 105. So the scale factor is 105/120.

Q6. Converting from Wyscout to standard metres requires flipping the y-axis because:

(a) Wyscout uses metres, not percentages
(b) Wyscout's y-axis points downward (top = 0)
(c) Wyscout's x-axis is reversed
(d) Wyscout's origin is at the centre of the pitch

Answer

**(b) Wyscout's y-axis points downward (top = 0).** Standard metre coordinates have the y-axis pointing upward, so a reflection is needed: $y' = 68 - y \cdot (68/100)$.

Q7. An affine transformation in 2D consists of:

(a) Only rotation and translation
(b) A linear transformation (scaling, rotation, reflection) plus a translation
(c) Only scaling
(d) A non-linear warping function

Answer

**(b) A linear transformation (scaling, rotation, reflection) plus a translation.** The general form is $\mathbf{x}' = A\mathbf{x} + \mathbf{t}$, where $A$ is a 2x2 matrix and $\mathbf{t}$ is a translation vector.

Q8. Why is routing all coordinate conversions through a canonical system (e.g., standard metres) preferred over writing pairwise converters?

(a) It is faster computationally
(b) It reduces the number of conversion functions from $N^2$ to $2N$
(c) It avoids floating-point errors
(d) It is required by FIFA regulations

Answer

**(b) It reduces the number of conversion functions from $N^2$ to $2N$.** With $N$ providers, pairwise converters require $N(N-1)$ functions, while routing through a hub needs only $N$ "to-hub" and $N$ "from-hub" functions.

Q9. In the standard thirds model, the "final third" (attacking third) spans which x-range on a 105 m pitch?

(a) 0.0 to 35.0 m
(b) 35.0 to 70.0 m
(c) 70.0 to 105.0 m
(d) 88.5 to 105.0 m

Answer

**(c) 70.0 to 105.0 m.** The pitch is divided into three equal 35 m segments. The attacking third (final third) is the segment closest to the opposition goal.

Q10. Zone 14 is located:

(a) Inside the penalty area
(b) In the centre of the pitch
(c) Just outside the penalty area in the central channel
(d) On the wing near the corner flag

Answer

**(c) Just outside the penalty area in the central channel.** Zone 14 is the central area between the edge of the penalty area and the two-thirds line, approximately $70 \leq x \leq 88.5$ and $20.5 \leq y \leq 47.5$ in standard metres.

Q11. The five-channel model divides the pitch into wings, half-spaces, and a central channel. The half-spaces are considered tactically important because:

(a) They have the most grass
(b) They sit between centre-backs and full-backs, offering diagonal passing and shooting angles
(c) They are closest to the goal
(d) They are the widest channels

Answer

**(b) They sit between centre-backs and full-backs, offering diagonal passing and shooting angles.** Actions in the half-spaces can wrong-foot defensive lines and create high-quality chances.

Q12. The penalty area on a standard pitch (105 x 68 m) has approximate dimensions of:

(a) 16.5 m deep and 40.32 m wide
(b) 11 m deep and 7.32 m wide
(c) 5.5 m deep and 18.32 m wide
(d) 16.5 m deep and 16.5 m wide

Answer

**(a) 16.5 m deep and 40.32 m wide.** The penalty area extends 16.5 m from each goal post (total width = 7.32 + 2 x 16.5 = 40.32 m) and 16.5 m into the pitch.

Q13. What percentage of the total pitch area does the penalty area cover (approximately)?

(a) 4.7%
(b) 9.3%
(c) 15.0%
(d) 20.0%

Answer

**(b) 9.3%.** Penalty area = 16.5 x 40.32 = 665.3 m$^2$. Total pitch = 105 x 68 = 7140 m$^2$. Percentage = 665.3 / 7140 = 9.3%.

Q14. In kernel density estimation, the "bandwidth" parameter controls:

(a) The color of the heat map
(b) The number of data points used
(c) The amount of smoothing applied to the density estimate
(d) The size of the pitch

Answer

**(c) The amount of smoothing applied to the density estimate.** A larger bandwidth produces a smoother surface (more bias, less variance); a smaller bandwidth produces a spikier surface (less bias, more variance).

Q15. Silverman's rule of thumb for bandwidth selection gives approximately:

(a) $h = N$
(b) $h \approx 1.06 \sigma N^{-1/5}$
(c) $h = \sigma / N$
(d) $h = \sqrt{N}$

Answer

**(b) $h \approx 1.06 \sigma N^{-1/5}$.** This is a common automatic bandwidth selector, where $\sigma$ is the standard deviation of the data and $N$ is the sample size.

Q16. A binned heat map with very few bins (e.g., 2 x 2) is problematic because:

(a) It uses too much memory
(b) It masks spatial detail and may obscure important patterns
(c) It violates pitch regulations
(d) It is impossible to compute

Answer

**(b) It masks spatial detail and may obscure important patterns.** With only four zones, events from very different locations are grouped together, making it impossible to distinguish meaningful spatial variation.

Q17. A Voronoi tessellation assigns each point on the pitch to the:

(a) Fastest player
(b) Nearest player
(c) Tallest player
(d) Player with the ball

Answer

**(b) Nearest player.** In a Voronoi diagram, each point belongs to the cell of the player whose Euclidean distance to that point is smallest.

Q18. The main limitation of using Voronoi tessellation for pitch control is:

(a) It is computationally expensive
(b) It only works for one team
(c) It ignores player speed, direction of movement, and ball trajectory
(d) It requires 3D coordinates

Answer

**(c) It ignores player speed, direction of movement, and ball trajectory.** Voronoi diagrams assume all players are equally fast and stationary, which is unrealistic.

Q19. In a velocity-aware pitch control model, distance is replaced by:

(a) Player height
(b) Time-to-reach
(c) Pass accuracy
(d) Shot angle

Answer

**(b) Time-to-reach.** Velocity-aware models estimate how long each player needs to reach a given point, accounting for current speed and direction.

Q20. The pitch control value $PC_A(x, y) = 0.8$ at a given point means:

(a) 80% of the pitch is controlled by Team A
(b) Team A has an 80% probability of gaining possession of a ball played to $(x, y)$
(c) Team A has 8 players nearby
(d) The point is 80 metres from Team A's goal

Answer

**(b) Team A has an 80% probability of gaining possession of a ball played to $(x, y)$.** Pitch control is a pointwise probability that indicates which team would likely win a 50-50 ball at that location.

Q21. When comparing heat maps of two players, a recruitment analyst should be cautious because:

(a) Heat maps are always wrong
(b) Differences in team tactics, pitch dimensions, and league context may explain spatial differences more than individual ability
(c) Heat maps do not use coordinates
(d) Only goalkeepers have heat maps

Answer

**(b) Differences in team tactics, pitch dimensions, and league context may explain spatial differences more than individual ability.** A player's spatial profile is heavily influenced by their team's system, not just personal tendencies.

Q22. The formula $x_{\text{norm}} = x / L$ normalizes a coordinate to which range?

(a) $[-1, 1]$
(b) $[0, L]$
(c) $[0, 1]$
(d) $[0, 100]$

Answer

**(c) $[0, 1]$.** Dividing by the pitch length $L$ maps the coordinate from $[0, L]$ to $[0, 1]$.

Q23. If you see a striker's heat map concentrated in their own defensive third, the most likely explanation is:

(a) The striker is playing a deep-lying role
(b) The direction-of-play has not been correctly aligned
(c) The data is from a different sport
(d) Either (a) or (b); you should check the direction-of-play flag before drawing conclusions

Answer

**(d) Either (a) or (b); you should check the direction-of-play flag before drawing conclusions.** While unusual positioning is possible, a direction-of-play error is a common data pipeline mistake that should be ruled out first.

Q24. The Gaussian kernel used in 2D KDE has the form:

(a) $K(u) = |u|$
(b) $K(u) = \frac{1}{2\pi|H|^{1/2}} \exp(-\frac{1}{2} u^T H^{-1} u)$
(c) $K(u) = 1$ if $|u| < h$, else $0$
(d) $K(u) = u^2$

Answer

**(b) $K(u) = \frac{1}{2\pi|H|^{1/2}} \exp(-\frac{1}{2} u^T H^{-1} u)$.** This is the standard bivariate Gaussian kernel with bandwidth matrix $H$.

Q25. The concept of "Space Value" combines pitch control with:

(a) Player salaries
(b) Expected Threat (xT) -- a spatial value model
(c) The number of fans in the stadium
(d) The referee's position

Answer

**(b) Expected Threat (xT) -- a spatial value model.** Space Value = $\iint PC_A(x,y) \cdot xT(x,y) \, dx \, dy$, integrating the product of spatial dominance and the value of each location.