Glossary: symbols and terms | Football Data Science

\(p\)

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

\(\Delta\) delta

A change, the figure after minus the figure before. A league whose yellow cards went from 3.66 to 3.71 a game has a Δ of +0.05. Did VAR change Scottish football? Difference-in-differences

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

Freeze frame

A record of where the keeper and the nearby players stood at the moment a shot was struck. It lets an expected-goals model see who was in the way. Expected goals from scratch. Where everyone stood

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?

Goals per game (an average)

An average over many matches, not a score anyone gets, like the 2.4 children in the average family. A side averaging 1.6 a game scores about 61 goals over a 38-game season, draws a blank in about one match in five and scores two or more in about half. How many goals will we score? The Poisson distribution

Natural side (penalties)

The side a penalty taker finds easier, across his body. For a right-footer it's his left, the keeper's right; for a left-footer the mirror image. Takers of either foot put just over half their penalties there. Are left-footed penalty takers worse?

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. Better finishers, or luck? The standard error and the luck margin

Confidence interval

A figure with its luck margin either side, such as 34.7% plus or minus 6.9 points. A 95% interval is two standard errors each way and catches the true figure about 95 times in 100. Better finishers, or luck? The standard error and the luck margin

Null hypothesis

The dull starting assumption a test tries to knock down, such as "kick-off time makes no difference to goals". Are TV games duller? Hypothesis tests and p-values

p-value

How often luck alone would give a gap at least as big as the real one, if the null hypothesis were true. It isn't the chance the gap is luck, and it says nothing about why. Are TV games duller? Hypothesis tests and p-values

Permutation (shuffle) test

Deals the group labels out again at random thousands of times, keeping the results where they are, to see what luck alone can do with the same games. Are TV games duller? Hypothesis tests and p-values

Statistical significance

A p-value under an agreed line, usually 0.05. It means too big to put down to luck with any comfort, not big or important. Are TV games duller? Hypothesis tests and p-values

Correlation and causation

Two things moving together doesn't mean one causes the other. Teams that foul more take fewer points, but mostly because dominant teams rarely need to foul. Do football stats mislead? Five correlations put to the test

Confounder

A third thing that drives two others, so they move together without one causing the other. Dominance drives both fewer fouls and more points. Do football stats mislead? Five correlations put to the test

Mediator

Something in the middle of a chain of causes, such as the extra shots on target that home advantage works through. Hold it fixed and the effect you wanted to measure disappears. Does home advantage vanish once you count the shots? Causal diagrams

Causal diagram

A drawing of what causes what, with an arrow from each cause to what it affects. It decides what to hold fixed before you look at the numbers. Does home advantage vanish once you count the shots? Causal diagrams

Natural experiment

When the world changes one thing for you, close to a real test. In 2020/21 every Scottish match was played behind closed doors, a test of what crowds do. Do football stats mislead? Five correlations put to the test

Difference-in-differences

The change in a group that got something, minus the change in a similar group that didn't over the same time. VAR's effect would be the Premiership's change beyond the Championship's. Did VAR change Scottish football? Difference-in-differences

Parallel trends

The assumption behind difference-in-differences, that the two groups would have moved together without the change. Check it season by season before trusting the answer. Did VAR change Scottish football? Difference-in-differences

Placebo test

Running the same comparison at a date when nothing happened. If it still finds an effect, the real result can't be trusted either. Did VAR change Scottish football? Difference-in-differences

Regression discontinuity

Comparing the cases just either side of a sharp line, such as sixth and seventh at the split, which are nearly alike apart from the line itself. Does finishing sixth rather than seventh matter? Regression discontinuity

Bandwidth

How far either side of the line a regression discontinuity looks. Too wide and it compares better teams with worse ones instead of measuring the line. Does finishing sixth rather than seventh matter? Regression discontinuity

Randomised experiment

A test where chance, such as a coin toss, decides who gets the treatment, so the two groups are alike on everything else. The shootout toss is one. Does kicking first win shootouts? Randomised experiments, and how many you need

Statistical power

The chance a study of a given size spots an effect that's really there. Thirty-six shootouts spot a real 60% edge only about one time in six. Does kicking first win shootouts? Randomised experiments, and how many you need

Event study

Lining up every case by the moment something happened, such as a manager leaving, and looking at the games before and after. The games before show why it happened. Sacked after a bad run, poached after a good one. Event studies

Matching

Comparing each case that got the treatment only with cases alike on everything you measured, such as leaders and level sides with the same pre-match odds. It can't fix what you didn't measure. Is a half-time lead worth what it looks like? Matching like with like

Home bias (referees)

Referees treating home sides more kindly. In Scotland away players are booked 10 to 20% more often per foul than home players doing the same thing. Do referees favour the home side?

Competitive balance

How evenly matched a league's teams are. Measured here as how spread out the table is compared with a league of equal teams where luck decides every match. Scotland's table is 2.3 times as spread out as luck alone would make it, but only 1.5 times without Celtic and Rangers. Is Scottish football less competitive than the rest of Europe?

Shortened and drifted (odds)

A price shortens when the odds fall, so the bookmaker now thinks it more likely, and drifts when they lengthen. A side that drifts from evens to 3.0 has gone from about a 50% chance to about 33%. Does late money know something? Friday's odds against kick-off

Closing price (kick-off price)

The last price a bookmaker offers before kick-off, after the team news is out. In the Scottish Championship it is clearly more accurate than the price on the Friday before; in the Premiership it isn't. Does late money know something? Friday's odds against kick-off

Selection bias

When the cases you study picked themselves, so they differ from the rest before anything happens. Managers make changes straight after a bad spell, so the pick-up afterwards looks like the change working. Do substitutions work? Changes made while losing, like for like

Counterfactual

What would have happened otherwise, such as a club's season had it kept its manager. It can never be watched, only estimated from a fair comparison. Bottom at Christmas, new manager by March. Did the change make it worse? Counterfactuals

Potential outcomes

The two results a club could have, one with a change and one without. Only one is ever seen; the effect of the change is the gap between them. Bottom at Christmas, new manager by March. Did the change make it worse? Counterfactuals

Collider bias

A false link made by looking only at cases picked on two things at once, or by controlling for something both things cause. An academy that picks on technique plus athleticism ends up with skilful players who seem less athletic, though the two weren't linked. Are gifted players less athletic? Collider bias and the academy

Berkson's paradox

The form of collider bias where the data were selected on the collider, named after Joseph Berkson, who found it among hospital patients in 1946. Are gifted players less athletic? Collider bias and the academy

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

K-nearest neighbours (KNN)

Forecast a match from the k past matches most like it, by counting what happened in them. Too few neighbours is mostly luck; too many stops being about this match. Matches like this one. K-nearest neighbours

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

Linear regression (least squares)

The straight line through the data whose squared misses add up to the least. Its slope says what one more of something is worth, such as about 0.3 goals for each shot on target. How many shots is a goal worth? Linear regression

R² (share explained)

How much of the differences between teams a model accounts for, from 0% to 100%. Shots on target explain about 72% of the differences in goals between team-seasons. How many shots is a goal worth? Linear regression

Naive Bayes

Start from how often each result happens and let each clue shift the odds in turn, treating every clue as separate news. Overlapping clues get counted twice, which makes it overconfident. Clue by clue. Naive Bayes, and the danger of counting twice

Unsupervised learning

Finding structure in data that comes with no right answers to learn from, such as grouping teams by how they play. Four ways to play. K-means and team styles

K-means clustering

Sorts things into k groups so each sits as close as possible to its group's average. On eight team stats it finds four styles, from dominant to outshot. Four ways to play. K-means and team styles

Hierarchical clustering

Joins the two closest teams or groups, again and again, into a family tree. You don't have to choose the number of groups first. Four ways to play. K-means and team styles

Time series

Numbers in time order, such as a club's points a game season after season. The question is how much each value remembers the ones before. What happens next season? Time series and ARIMA

Autocorrelation

How closely a series matches itself a step or more earlier. A season's points match the season before at 0.83 across all clubs, 0.46 without Celtic and Rangers. What happens next season? Time series and ARIMA

ARIMA

A family of forecasting models built from past values (AR), changes rather than levels (I) and past surprises (MA). Exponential smoothing is one member. What happens next season? Time series and ARIMA

Exponential smoothing

Keep a running level and move it part of the way towards each new value, so recent seasons count most. With 70% a season it forecast next season best. What happens next season? Time series and ARIMA

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

Payoff matrix

A table of what each combination of choices is worth, such as the chance a penalty goes in for each place the taker aims and each way the keeper dives. Where should he put it? Penalties and the Nash equilibrium

Zero-sum game

A contest where one side's gain is exactly the other's loss. Every penalty scored is one the keeper didn't save. Where should he put it? Penalties and the Nash equilibrium

Mixed strategy

Choosing at random in set proportions, such as going to your weaker side four times in ten, so the other side can't read you. Where should he put it? Penalties and the Nash equilibrium

Nash equilibrium

A pair of plans where neither side can do better by changing only its own. At a penalty, the mix of kicks and dives that leaves nothing to exploit. Where should he put it? Penalties and the Nash equilibrium

Minimax

Choosing the plan whose worst case is as good as possible. In a zero-sum game, both sides doing this lands on the same result, the equilibrium. Why don't keepers just stand still? The middle, the Panenka and three choices

Dominated strategy

A choice that does worse than another whatever the other side does, so it's never worth using. A choice can avoid this and still get none of the best mix. Why don't keepers just stand still? The middle, the Panenka and three choices

Action bias

Preferring to do something over doing nothing, even when doing nothing works better, such as a keeper diving when staying put would save more. Why don't keepers just stand still? The middle, the Panenka and three choices

Prisoner's dilemma

A game where each side does best for itself by breaking a deal, so both end up worse off than if they'd kept it, such as two clubs both raising wages. Why Gijón wasn't a prisoner's dilemma

Stag hunt

A game where everyone does best by cooperating, but only if they're sure the others will, such as a back four stepping up together for the offside trap. Why must the whole back four step up together? The stag hunt and the offside trap

Coordination game

A game with more than one stable outcome, where what matters is everyone choosing the same one, such as the whole line stepping up or the whole line dropping off. Why must the whole back four step up together? The stag hunt and the offside trap

Winner's curse

When several bidders guess the worth of the same thing, the winner is usually the one who guessed highest, so it tends to overpay, such as the club that wins the race for a striker. Why does the club that wins the transfer race so often lose? The winner's curse

Common value

Something worth roughly the same to every bidder, though nobody knows exactly how much, such as a striker's goals to any club in the same league. Why does the club that wins the transfer race so often lose? The winner's curse

Bid shading

Offering less than your own estimate, to allow for the winner's curse. The more clubs in the race, the more you shade. Why does the club that wins the transfer race so often lose? The winner's curse

Dominant strategy

A choice that's best whatever the other side does, such as West Germany sitting back at 1–0 in Gijón. Why Gijón wasn't a prisoner's dilemma

Hawk-dove game

A game where the best move is the opposite of the other player's, such as going in hard or pulling out of a 50/50. Nobody should do the same thing every time. Go in hard or pull out? Hawks, doves and the 50/50 challenge

Evolutionarily stable strategy

A mix that sticks because anyone who strays from it does worse, so the whole group drifts back to it, such as going in hard in 40% of 50/50s in the made-up example. Go in hard or pull out? Hawks, doves and the 50/50 challenge

Nash bargaining solution

Where two sides who both gain from a deal should settle. With money only, split the difference, so each gets its walk-away plus half of what the deal adds. Who wins the contract talks? Bargaining, walk-away offers and deadline day

Walk-away (disagreement point)

What each side gets if there's no deal, such as a player's best offer from another club. The two walk-aways set the deal zone. Who wins the contract talks? Bargaining, walk-away offers and deadline day

Bayesian game

A game where one side knows something the other doesn't, such as whether a player is really hurt. The side in the dark updates its beliefs with Bayes' theorem. Is he really hurt? Signals, types and the rule that keeps players honest

Signalling (pooling and separating)

Sending an action the other side can see, such as going down injured. If everyone sends it, it's pooling and tells you little; if only one type can afford it, it's separating and can be trusted. Is he really hurt? Signals, types and the rule that keeps players honest

Bluffing

Sending a false signal, such as a dummy run, just often enough that the other side can't tell it from the real thing. Why make the dummy run? Bluffing, decoys and how often to fake

Game tree

A drawing of a game where players move in turn, branching at each choice and ending with what each path is worth, such as one manager's substitution and the other's reply. Who should change first? Substitutions, game trees and backward induction

Backward induction

Solving a game tree from the end, working out the last choice first, then each earlier one knowing what will follow. Who should change first? Substitutions, game trees and backward induction

Subgame-perfect equilibrium

A plan that's the best choice at every point in a game, not just at the start, so it ignores threats and promises nobody would carry out. Who should change first? Substitutions, game trees and backward induction

Repeated game

The same game played again and again by the same players, so that what you do now can be answered later, such as two clubs who meet every season. Why do teams give the ball back? Repeated games and the shadow of the future

Tit-for-tat

A strategy for a repeated game that cooperates first, then copies whatever the other side did last time. Why do teams give the ball back? Repeated games and the shadow of the future

Gambler's fallacy

Believing a run must end, such as a keeper diving the other way because the last three penalties went to the same side. Why do teams give the ball back? Repeated games and the shadow of the future

Mechanism design

Choosing the rules and payoffs of a game so that each player's own best move gives the outcome the designer wants, such as three points for a win to stop teams settling for draws. Who designs the game? Three points for a win and the rules behind the football

Decision variable

Something you control in an optimisation problem, such as whether or not to sign a particular player. Who's the best signing? It depends what you're optimising

Objective function

The single number an optimisation tries to make as big (or as small) as possible, such as points added by the season's signings. Who's the best signing? It depends what you're optimising

Constraint

A rule the answer has to obey, such as the transfer budget or "we must sign a centre-back". Who's the best signing? It depends what you're optimising

Feasible region

Every choice that obeys all the constraints, such as every set of signings the budget allows. The best answer is found inside it. Who's the best signing? It depends what you're optimising

Binding constraint

A rule that changes the best answer, because the best choice without it would break it. A rule the best answer obeys anyway doesn't bind and costs nothing. How far forward should we push? Attack, defence and the price of a rule

Shadow price

What loosening a rule by one unit would be worth, such as the points a season one more goal conceded would buy. Also called the Lagrange multiplier. How far forward should we push? Attack, defence and the price of a rule

Pareto front

The options nobody beats on every measure at once, such as the strikers no one else outdoes on both fee and goals. Everything else can be ruled out. Who's the best striker? The Pareto front

Linear programming

Finding the best answer when the objective and every rule are straight-line sums, such as sharing a season's minutes among a squad. The best answer is always at a corner of the feasible region. Who plays how much? Squad minutes as a linear program

Simplex method

The classic way to solve a linear program, moving from corner to better corner of the feasible region until no move gains anything. Inside the solver: the simplex method by hand

Pivot

One step of the simplex method, moving from one corner to the next, such as giving a player minutes until a rule runs out. Inside the solver: the simplex method by hand

Integer programming

A linear program where some answers must be whole numbers, such as signing a player (1) or not (0), with no part-signings allowed. £30m and five gaps: why you can't sign 0.83 of a centre-back

LP relaxation

An integer program with the whole-number rule dropped, so fractions are allowed. Its answer is a ceiling the real answer can't beat. £30m and five gaps: why you can't sign 0.83 of a centre-back

Branch and bound

Solving an integer program by splitting it into smaller problems (sign him, or don't) and throwing away any branch whose best possible score can't beat the best answer found so far. £30m and five gaps: why you can't sign 0.83 of a centre-back

Assignment problem

Pairing each of one group with exactly one of another at the lowest total cost, such as each marker with one attacker at a corner. Who marks whom? The assignment problem at set pieces

Hungarian method

A classic way to solve the assignment problem without trying every plan, by taking each row's and column's smallest cost away until a full set of zero-cost pairings appears. Who marks whom? The assignment problem at set pieces

Network (graph)

Points joined by links, such as players joined by passing lanes. Each link can carry a number, like the chance a pass gets through. Playing out under a press: the safest route isn't the shortest

Shortest path

The best route through a network from one point to another. "Shortest" means best by whatever the links measure, such as the highest chance of getting the ball to the striker. Playing out under a press: the safest route isn't the shortest

Scheduling problem

Deciding what happens when, with rules that link one moment to the next, such as which players start which games when nobody can start three in a row. Seven games in 21 days: who plays when?

Expected utility

The expected value of what outcomes are really worth to you, such as points rather than goals. A goal is worth more points at some scores than others. Shoot, cross or keep the ball? It depends on the score

Risk-averse and risk-seeking

A risk-averse side turns down gambles that look good on average because the downside hurts too much, like a side one up. A risk-seeking side takes them because it has little to lose, like a side one down. Shoot, cross or keep the ball? It depends on the score

Monte Carlo optimisation

Choosing between options by simulating each one many times and picking the one that reaches the goal most often, such as the signing that keeps a side off the bottom in the most simulated seasons. Striker or centre-back? Let the simulation decide

Common random numbers

Running every option through the same simulated seasons, with the same luck, so the only difference between them is the decision. It makes small gaps far easier to see. Striker or centre-back? Let the simulation decide

Optimiser's curse

The option that looks best after a noisy comparison usually had the best luck too, so its estimate is too good. Re-run the winner before trusting its number. Striker or centre-back? Let the simulation decide

Dynamic programming

Solving a chain of decisions by starting at the last one and working backwards, so each earlier choice knows the best that can follow. Used here to plan how to play each game of a relegation run-in. Go for it or play it steady? Planning the run-in backwards

Bellman equation

The step dynamic programming repeats. The best chance from a situation is the best, over your choices, of the weighted average of the best chances from wherever the next match could leave you. Go for it or play it steady? Planning the run-in backwards

Policy

A rule giving the right choice for every situation you could be in, such as how to play with three games left and two points behind, rather than one fixed plan. Go for it or play it steady? Planning the run-in backwards

State

Everything about the situation that matters for the decisions still to come, such as games left and points behind, and nothing else. Go for it or play it steady? Planning the run-in backwards

Predictive distribution

What a model says the scores should look like, every score and how often, rather than a single best guess. Allowing for doubt about each side's strength makes it wider. Was 7–0 a freak? Checking a model against the scores it predicts

Posterior predictive check

Testing a model by comparing the scores it says should happen with the scores that did, such as how many sevens a league should see in 25 seasons against how many it saw. Was 7–0 a freak? Checking a model against the scores it predicts

Chance one is better

Comparing two uncertain rates by drawing a plausible value for each from its belief, many times, and counting how often one beats the other. Four wins from six against one from seven came out at about nine in ten, not certainty. Four from six. Is the new manager really better?

Control group

Cases like the ones being studied in every way that matters except the thing being tested, such as sides with the same bad run that kept their manager. What they did next is what you'd expect without the change. Does sacking the manager bring a bounce?

Censoring

When we stop watching before the story ends, so we only know it lasted at least that long. A manager still in the job at the end of the data is censored; counting him as finished, or leaving him out, makes tenures look shorter than they are. How long does a manager last? Survival analysis

Kaplan-Meier survival curve

The share still going after each length of time, built by working through the departures in order and asking what share of those still being watched survived each one. Half of Scottish top-flight managers are gone within 465 days. How long does a manager last? Survival analysis

Hazard (rate)

How fast those still going are leaving, such as departures per manager per year in the job. A flat hazard means the danger doesn't fade with time; for Scottish managers it stays near 0.56 a year. How long does a manager last? Survival analysis

Log-rank test

Compares two survival curves by asking how many in one group left against how many would have if both groups faced the same hazard. Old Firm managers seem to last longer, but the test puts the gap within luck. How long does a manager last? Survival analysis

Landmark analysis

Judging everyone at the same point, such as a manager's 10th league game, only counting those who got there and starting the clock again from it, so nobody is credited with time they had already survived. How long does a manager last? Survival analysis

Expected points

What a match is worth on average before it's played, from the odds' chances, 3 times the chance of a win plus the chance of a draw. A side given 50% to win and 25% to draw expects 1.75 points. Bad run or bad manager? Spotting a real change with CUSUM

CUSUM

A running tally of how far results fall short of what was expected, forgiving a little each game and never going below zero, with an alarm when it gets too big. Any setting that caught most sacked managers also went off on most managers who were exactly par. Bad run or bad manager? Spotting a real change with CUSUM

False alarm

A warning that goes off when nothing has really changed. In football, luck alone sets off most alarms built from a run of results. Bad run or bad manager? Spotting a real change with CUSUM

Arrival time

How long a player needs to get to a spot, his reaction time plus the run, starting from where he'll be once he's reacted. With the published defaults, 0.7 seconds to react, then 5 metres a second. Who gets there first? Timing a pass, and the cost of waiting

Margin (of a pass)

How many seconds the quickest defender is too late to cut out a pass, at the best point he can reach. Above zero the pass gets there; within 0.3 seconds either way it's too close to call. Who gets there first? Timing a pass, and the cost of waiting

Pitch control

A map of which team can get to each spot on the pitch first, worked out from where every player is standing and running. The full model turns that into a chance of winning the ball there. Who gets there first? Timing a pass, and the cost of waiting

Segment analysis (subgroup analysis)

Splitting things into groups that share a trait and comparing the groups, such as midfields with two holding players against midfields with one. Who ends up in each group is rarely random, so compare like with like. Best eleven or best combination? Cohort analysis and the midfield partnership

Cohort analysis

Grouping things by a shared starting point and following each group over time, such as every midfield pair that first started together in the autumn, or every customer who signed up in January. Laid out as a table with one row per starting point, it thins out to the right. Best eleven or best combination? Cohort analysis and the midfield partnership

Survivorship bias

Judging only the ones still going. Midfield partnerships look better the longer they last because managers break up the poor ones, not because any pair improves; follow the same pairs and the rise disappears. Best eleven or best combination? Cohort analysis and the midfield partnership

Possession-adjusted

A count scaled to what it would be if the ball were shared evenly. A midfielder whose side rarely has the ball gets more chances to tackle, so his tackles are scaled down by how much of the ball the other side had. Best eleven or best combination? Cohort analysis and the midfield partnership

Glossary

Reading the symbols