Glossary

Reading the symbols · The terms

Reading the symbols

\(p\)

The chance of one thing happening, written as a number from 0 to 1. A penalty taker with p = 0.75 scores three in four. Goal or miss? The Bernoulli distribution

\(P(\,\dots)\) P of

The probability that whatever is in the brackets happens. P(X = 2) is the chance of exactly two goals. Five penalties, how many go in? The Binomial distribution

\(P(A \mid B)\) P of A given B

The chance of A once you know B has happened. P(win | lead) is the chance of winning for a side that's already leading. 1–0 up at half-time. Will they hold on? Bayes' theorem

\(\approx\) approximately

The number has been rounded. 19 ÷ 24 ≈ 79%. Four from four. How good is he really? Bayesian thinking

\(\le \;\; \ge\) at most, at least

P(X ≤ 3) is the chance of three or fewer; P(X ≥ 3) of three or more. How many shots until he scores? The Geometric distribution

\(\sqrt{\;\;}\) square root

The number that, multiplied by itself, gives what's inside, so √9 = 3. It turns squared gaps back into ordinary units, as in the distance between two players' stats and in standard deviations. Who plays most like him? Measuring player similarity with distance

\(\Sigma\) sigma, or sum

Add up what follows, once for each item underneath it, such as every match or every stat. Same style, different volume? The dot product and cosine similarity

\(n!\) n factorial

Every whole number from 1 up to n multiplied together, so 4! = 24. It counts the ways things can be ordered, and it appears in the Poisson formula. How many goals will we score? The Poisson distribution

\(\tbinom{n}{k}\) n choose k

How many ways there are to pick k things from n. There are 10 ways to score exactly 2 penalties from 5. Five penalties, how many go in? The Binomial distribution

\(e\)

A fixed number, about 2.718, that appears whenever something grows or fades at a steady rate. In the Poisson model, e to the power −λ is the chance of no goals at all. How many goals will we score? The Poisson distribution

\(\ln \;\; \log\) natural log

The logarithm that undoes e. Log loss uses it to punish a confident wrong forecast far more than a cautious one. Is accuracy the right score? Evaluating a model

\(\lambda\) lambda

A rate. In most articles, the goals a side is expected to score in a match (Poisson), or per minute (Exponential, Gamma). In Dixon-Coles, the home side's expected goals. One exception, in PCA, it's an eigenvalue, how much of the spread one direction carries. How many goals will we score? The Poisson distribution

\(\mu\) mew

An average, the centre of a Normal curve, such as 30 km/h for the sprint speeds. In Dixon-Coles, the away side's expected goals. How fast is fast? The Normal distribution

\(\sigma\) sigma

The standard deviation, how spread out values are around the average, such as 2 km/h for the sprint speeds. σ² is the variance. In the Log-Normal, it's the spread of the logs. How fast is fast? The Normal distribution

\(\theta\) theta

The true value you're trying to learn, such as a penalty taker's real conversion rate. In the vector articles it's an angle instead. Four from four. How good is he really? Bayesian thinking

\(\alpha,\ \beta\) alpha, beta

The two settings of a prior. For a Beta prior, read them as successes and failures; for a Gamma prior, as goals and games. The bigger they are, the more evidence the prior is worth. Eleven goals in five games. Can they keep it up? Updating a team's scoring rate

\(\sim\) is distributed as

λ ~ Gamma(α, β) means our belief about λ follows a Gamma curve. Eleven goals in five games. Can they keep it up? Updating a team's scoring rate

\(\propto\) is proportional to

Equal up to a fixed multiplier. Bayes' rule is often written this way, with the multiplier filled in at the end so the chances add up to 1. Four from four. How good is he really? Bayesian thinking

\(\rho\) row

How much Dixon-Coles adjusts the four lowest scores, 0–0, 1–0, 0–1 and 1–1. A negative ρ means a few more low-scoring draws. Every team its own attack and defence. Dixon-Coles from scratch

\(\xi\) ksai, or zai

How fast old matches fade in Dixon-Coles. At 0.003 a day, a match counts half as much after 231 days. Every team its own attack and defence. Dixon-Coles from scratch

\(\eta\) eeta

The step size, or learning rate, in gradient descent. Too small and the model crawls; too big and it overshoots. How does a model learn? Rolling downhill with gradient descent

\(\tau\) tow, rhymes with cow

A time constant. In the made-up sprint, how quickly a winger gets up to speed, 1.2 seconds. Still accelerating, or at full speed? The derivative

\(\pi\) pie

About 3.14159, the circle number, which turns up in the Normal curve's formula. In passing networks it means something else, each player's long-run share of the ball. How fast is fast? The Normal distribution

\(\frac{dv}{dt}\) d v by d t

A derivative, how fast v changes as t changes, at one instant. Acceleration is the derivative of speed. Still accelerating, or at full speed? The derivative

\(\int\) integral

Adding up very many thin slices, the area under a curve. Distance run is the integral of speed. We know how fast he was running. How far did he get? Integration

\(\hat{y}\) y hat

An estimate or prediction of y, as against the real thing. ŷ is the model's guess at the result. (In the movement article, a hat marks a direction of length 1.) What are we trying to predict? Features and targets

\(\bar{x}\) x bar

The average of x. p̄ is the average pass. Forward or sideways? Passing as a vector

\(\mathbf{v},\ \lVert \mathbf{v} \rVert\) v, the length of v

A bold letter is a vector, a list of numbers such as a player's stats or a run's forward and sideways parts. The double bars give its length, such as the distance a run covers. A midfielder's match in five numbers. Vectors

\(\mathbb{R}^n\) R n

Every possible list of n numbers. A player described by five stats is a point in R⁵. A midfielder's match in five numbers. Vectors

\(E[\,\dots]\) expected value of

The long-run average of what's in the brackets. Not to be confused with the E in Elo, which is the home side's expectancy. Stubborn or jumpy? Bias and variance

The terms

Probability

How likely something is, from 0 (never) to 1 (certain), often written as a percentage. Goal or miss? The Bernoulli distribution

Distribution

Every possible outcome with its chance, such as the chance of 0, 1, 2, 3 or more goals in a match. How many goals will we score? The Poisson distribution

Poisson distribution

The chances of 0, 1, 2 or more goals when goals arrive at a steady average rate. The starting point for predicting scorelines. How many goals will we score? The Poisson distribution

Expected goals (xG)

How many goals a side's chances were worth on average, adding up the chance of scoring from each shot. Goal or miss? The Bernoulli distribution

Mean and median

The mean is the ordinary average; the median is the middle value. When a few values are huge, like transfer fees, the mean is dragged above the median. Why the average transfer fee misleads. The Log-Normal distribution

Standard deviation and variance

How spread out values are around the average. The variance is the standard deviation squared. How fast is fast? The Normal distribution

z-score

How many standard deviations a value is above or below the average, which lets you compare things measured in different units. How fast is fast? The Normal distribution

Correlation

How closely two things move together, from 0 (no link) to 1 (a perfect straight line). Shots and goals over a season have a correlation of 0.78. Do more shots mean a better attack?

Luck margin (standard error)

How much a figure could move by chance alone, given how many matches it rests on. A gap smaller than this could be luck. One good season or a good model? Cross-validation

Regression to the mean

Extreme seasons tend to be followed by more ordinary ones, because part of what made them extreme was luck. How does a model learn? Rolling downhill with gradient descent

Base rates

How often each result happens overall, such as home wins about 44% of the time. The forecast of someone who knows nothing about the teams. Is accuracy the right score? Evaluating a model

Prior and posterior

What you believe before seeing the evidence, and what you believe after combining it with the evidence. Four from four. How good is he really? Bayesian thinking

Bayes' theorem

The rule for updating a chance when new evidence arrives, such as a pre-match chance updated by the half-time score. 1–0 up at half-time. Will they hold on? Bayes' theorem

Likelihood and maximum likelihood

How probable the results we actually saw are under a given set of numbers. Maximum likelihood picks the numbers that make them most probable. Every team its own attack and defence. Dixon-Coles from scratch

Partial pooling

Pulling each team's own figure part of the way towards the league average, further when its own figure is noisier. Which ground is hardest to visit? Partial pooling

Features and target

Features are what a model knows before kick-off; the target is what it's trying to predict. What are we trying to predict? Features and targets

Training and test data

The matches a model learns from, and separate matches kept back to check it honestly. Has the model learned, or just memorised? Training data and test data

Held-back seasons (validation)

Part of the training data set aside to choose a model's settings, so the test seasons stay unseen. Brilliant in training, gone on matchday. Overfitting

Overfitting

Learning the training matches too well, luck included, so the model does worse on matches it hasn't seen. Brilliant in training, gone on matchday. Overfitting

Cross-validation

Testing a model again and again on different slices of the data; for football, walking forward season by season. One good season or a good model? Cross-validation

Bias and variance

A model can be wrong by being too stubborn (bias) or too jumpy (variance). Most choices trade one against the other. Stubborn or jumpy? Bias and variance

Log loss

The main score used here for probability forecasts. It punishes confident wrong forecasts hardest. Lower is better. Is accuracy the right score? Evaluating a model

Brier score

Another score for probability forecasts, the squared gap between the chances given and what happened. Lower is better. Is accuracy the right score? Evaluating a model

Calibration and resolution

Calibration is whether a forecaster's 30% happens 30% of the time. Resolution is how well it tells matches apart. Sure of itself, or just better informed? Calibration in depth

Gradient descent

How most models learn. Measure how the error changes as you nudge a setting, step downhill, and repeat. How does a model learn? Rolling downhill with gradient descent

Derivative

The rate something is changing at one instant, such as acceleration, the rate speed changes. Still accelerating, or at full speed? The derivative

Integral

A total built from many small pieces, such as the distance run from speed, second by second. We know how fast he was running. How far did he get? Integration

Vector

A list of numbers treated as one thing, such as a player's stats or a pass's forward and sideways length. A midfielder's match in five numbers. Vectors

Matrix

A table of numbers, such as a squad's stats with one row per player, that can be worked on all at once. Four players, four numbers each. The whole team becomes a matrix

Logistic regression

A model that turns a weighted sum of features into a chance for each result. Home, draw or away, with probabilities. A first real model, logistic regression

Decision tree

A model that predicts with a string of yes-or-no questions, learned from past matches. Three questions and a prediction. Decision trees

Random forest

Many decision trees, each grown on slightly different matches, averaged together. Ask a hundred pundits. Random forests

Gradient boosting

Small trees added one after another, each fixing some of what the model so far gets wrong. Learning from its mistakes. Gradient boosting

Elo rating

One number per team that rises and falls with results, by more when the result is a surprise. Tuning Elo honestly. Five dials and a moving target

Dixon-Coles

A football model that rates every team's attack and defence from goals, with a small correction for low scores. Every team its own attack and defence. Dixon-Coles from scratch