Limit of Voronoi diagrams in soccer analysis, 1, essence

My claim in this article is that "do not rely on Voronoi diagrams unlimitedly in soccer."

I am concerned about criticism to players by using Voronoi diagrams under blind reliance.

As a fact, when 22 players do not move, a Voronoi diagram indicates regions that the players can cover.

On the other hand, when some players move fast, a Voronoi diagram is far from regions that the players can cover.

I think, Voronoi diagrams are not compatible with soccer.

This is because the essence of soccer is in the movement of players.



The reason why Voronoi diagrams are used in soccer analysis is because "likely regions are obtained only from players' positions."



In this article, I roughly explain what the Voronoi diagram is.

In the next article, I will show difficulties in applying Voronoi diagrams when players move, with simple numerical calculation.



The contents of this article are as follows.

• Why are Voronoi diagrams used?
• Questionable scenes.
• Scenes where Voronoi diagrams can be applied, during a game.
• Scenes where Voronoi diagrams can be applied, in our lives.

Why are Voronoi diagrams used?

A Voronoi diagram calculated with a python module. The dashed lines mean lines heading toward infinity.

Voronoi diagram (Wikipedia)

The description in Wikipedia is very difficult for me.

However, important here is that "boundaries in the Voronoi diagrams are part of bisectors calculated from positions of points."

In analyzing soccer matches, Voronoi diagrams would be applied in various ways.

In these analysis, we have a common recognition to the Voronoi diagram.

It is that "Voroni cells (the area indicated by the Voronoi diagram) can be regarded as an area that players can cover without being disturbed by the other players."

I guess that this recognition is born from the following stages.

1. "Boundary lines in a Voronoi diagram are bisector."
2. "It takes the same time from neighboring points to a point on their bisector."
3. "Voronoi cells are regions where players can occupy (players can move without being disturbed by the other players)"

It may be no problem, at first glance. However, this recognition includes the following two assumptions.

1. All player do not move at that time (stationary state).
2. Their velocity profile (their running ability) are the same.

Under these two assumptions, Voronoi cells can become players' occupied regions.

"Players' same running ability" is an acceptable assumption for simplification.

On the other hand, "Players' stationary state" is quite different from its essence of soccer (players continue to move during a match).

Therefore, the Voronoi diagram is not suitable for soccer analysis.

Questionable scenes

The following figure is an image of the video of Emil Hansen's Voronoi diagram.

In this figure (at 25 sec), the yellow small circle indicates the ball, and the red and blue circles indicate players.

Near the left end, the red ball holder runs to the opponent goal, and the blue goalkeeper approach to the ball holder.

This Voronoi diagram is obtained from such situation, and indicates that the blue goalkeeper can cover most of the left-side penalty area.

Is this correct?

My answer is NO.

This is because the Voronoi diagram assumes the stationary state of objects.

Scenes where Voronoi diagrams are likely to be used during the game

As described in the above, in some soccer scenes, the Voronoi diagram is not appropriate for analysis.

However, in other soccer scene, it may be appropriate.

For example, I give a scene when the ball holder team just start their offence against the organized defense in front of a penalty area.

At this moment, movements of all players will be stagnant.

A Voronoi diagram is useful for estimating an area that the defense-side players can cover.

Before starting set plays, movements of players are also stagnant.

However, a Voronoi diagram will be useless in this situation, because offense players change their positions after starting a set play.


In sports other than football, Voronoi diagrams are appropriate for estimating defense areas in baseball.

Scenes where Voronoi diagrams can be applied, in our lives

I guess, Voronoi diagrams can be used to estimate regions of multiple base points that do not move.

These "multiple  base points" are, for example, police stations, fire departments, schools, delivery stations, and bases of rescue teams.

To strictly set regions (districts) of these base points, several factors should be considered.

Regions (boundary lines) are determined by repeating corrections while taking these factors into consideration.

Voronoi diagrams can be used as initial values ​​while repeating such corrections.

On the other hand, if regions are for air travel using helicopters or airplanes, Voronoi cells can be used as region themselves.



In the next article, via simple numerical simulation, it is shown that Voronoi diagrams cannot be applied when players' velocities are considered.