A midfielder's match in five numbers. Vectors
A player's performance can be written as a list of numbers in a fixed order. That list is a vector, and it's the first step to comparing players, finding replacements and feeding football into machine learning.
Beginner Part 1 of Linear Algebra Through Football
Contents
The football question
A central midfielder has just played a full match. How do you describe his performance in a way a computer can compare with every other midfielder?
Match reports use words: "tidy", "busy", "got about the pitch". Words are hard to compare. Numbers aren't.
The concept
A vector is an ordered list of numbers. Each position in the list always means the same thing, so two vectors can be compared position by position.
Here is our midfielder's match:
| Passes | Tackles | Shots | Chances created | Distance (km) |
|---|---|---|---|---|
| 68 | 7 | 3 | 5 | 11.2 |
Written as a vector:
$$\mathbf{v} = (68,\ 7,\ 3,\ 5,\ 11.2)$$
In plain football
- Bold v is the name of the vector: this midfielder, this match.
- Five numbers means five dimensions. The first is always passes, the second always tackles, and so on.
- Order matters. (7, 68, …) would be a completely different player: seven passes and 68 tackles.
That's all a vector is. The power comes from what you can do with it.
Things you can do with a vector
Scale it: per 90 minutes
Suppose he was substituted after 75 minutes. To compare him fairly with players who played the full 90, scale everything up by 90 ÷ 75 = 1.2:
$$1.2 \times \mathbf{v} = (81.6,\ 8.4,\ 3.6,\ 6,\ 13.4)$$
In plain football
- Multiplying a vector by a single number multiplies every entry by it. This is called scalar multiplication.
- It's exactly how "per 90" stats are made: at this rate, over a full match, he'd have made about 82 passes and 8 tackles.
- Per-90 figures can flatter players who only play short spells, often late on when the game is stretched, so treat them with care.
Add them: a run of matches
His next match looks like this: 54 passes, 4 tackles, 1 shot, 3 chances, 10.6 km. Add the two vectors entry by entry to get his totals:
$$\begin{aligned} &(68,\ 7,\ 3,\ 5,\ 11.2) \\ +\; &(54,\ 4,\ 1,\ 3,\ 10.6) \\ =\; &(122,\ 11,\ 4,\ 8,\ 21.8) \end{aligned}$$
In plain football
- Vector addition adds passes to passes, tackles to tackles, and so on. It never mixes them.
- Divide the total by 2 and you have his average match: (61, 5.5, 2, 4, 10.9).
- A season is the same idea with 38 vectors instead of two.
Subtract them: the difference between two players
Another midfielder has (52, 3, 1, 2, 10.1). Subtract his vector from ours:
$$\begin{aligned} &(68,\ 7,\ 3,\ 5,\ 11.2) \\ -\; &(52,\ 3,\ 1,\ 2,\ 10.1) \\ =\; &(16,\ 4,\ 2,\ 3,\ 1.1) \end{aligned}$$
In plain football
- Each entry is how much more of that thing our midfielder did.
- 16 more passes looks like the biggest gap. It isn't. That's 31% more passes, but 4 more tackles is more than twice as many.
- The numbers are on different scales: a midfielder makes dozens of passes and a handful of tackles. Comparing raw gaps lets the biggest numbers shout loudest.
That last point matters for everything that follows in this series. Before comparing players, analysts usually put every entry on the same scale, often by turning each into a z-score: how many standard deviations above or below average it is, as in the Normal distribution. Then a big pass count and a big tackle count count the same.
Show the mathsThe general definition, and why the operations work entry by entry. Optional.
A vector with n entries is written
$$\mathbf{v} = (v_1,\ v_2,\ \ldots,\ v_n)$$
and lives in n-dimensional space, \(\mathbb{R}^n\). Our midfielder is a point in \(\mathbb{R}^5\). Vectors are also often written standing up, as a column:
$$\mathbf{v} = \begin{pmatrix} 68 \\ 7 \\ 3 \\ 5 \\ 11.2 \end{pmatrix}$$
Addition and scalar multiplication work entry by entry:
$$\mathbf{a} + \mathbf{b} = (a_1 + b_1,\ \ldots,\ a_n + b_n)$$
$$c\,\mathbf{v} = (c\,v_1,\ \ldots,\ c\,v_n)$$
The average of m vectors is their sum times \(1/m\), which is why a season average is just addition followed by scaling.
Why it matters
Once a performance is a vector, a computer can do things a scout can't do by eye across thousands of players:
- Compare two players number by number.
- Measure similarity: which midfielders play most like ours?
- Find replacements: who is closest to the player we're about to sell?
- Group players by style: ball-winners, creators, box-to-box runners.
- Feed machine learning models, which expect every example as a vector of numbers.
All of those need a way to measure how far apart two vectors are, which is where this series goes next.
Limitations
- A vector only knows what you put in it. Five numbers say nothing about positioning, decision-making or leadership. Choose the entries carefully.
- Counts depend on context. A midfielder in a side that has the ball all game will make more passes, whatever his quality.
- Units and scales differ. Kilometres and pass counts can't be compared raw, as the subtraction showed.
Try it yourself
Pick two midfielders from your team and write down the same five numbers for each from their last match: passes, tackles, shots, chances created and distance. Scale each to 90 minutes, then subtract one vector from the other. Which gap looks biggest, and which is biggest once you think in percentages?
Further reading
- Vectors, what even are they?, 3Blue1Brown. The best visual introduction there is; the first chapter of Essence of linear algebra.
- Vectors and spaces, Khan Academy. Short lessons with practice exercises.
- Vectors and matrices, StatLect. The formal definitions, in the notation used in statistics.