Generalised additive models (GAMs) explained for betting

In one sentence

A GAM is a GLM where each input's effect is a flexible smooth curve learned from data rather than a fixed straight line.

How it works

Many effects in sport are curved: goal rates rise through a match and spike near the end, and the value of rest days climbs, then levels off. A straight-line model forces these into a slope; a GAM lets the data draw the shape.

Each input gets its own smooth curve, built from flexible pieces called splines. The model adds these curves together (hence "additive"), then applies a link, just like a GLM. A penalty on wiggliness stops the curves chasing noise.

The big selling point is that you can still plot and inspect each effect. That makes GAMs a good middle ground between simple regression and black-box machine learning.

The maths

g\big(E[Y]\big) = \beta_0 + f_1(x_1) + f_2(x_2) + \dots + f_k(x_k)

\text{Fit by minimising:} \quad -\text{log-likelihood} + \sum_{j} \lambda_j \int f_j''(x)^2 \, dx

g: the link function (log for goal rates, logit for probabilities).
f_1 to f_k: smooth curves, one per input.
λ_j: smoothing penalties; bigger values force straighter curves.
f″: the curvature of each curve, which the penalty discourages.

In words: add up a smooth curve for each input, and balance fitting the data against keeping the curves gentle.

Worked betting example

An in-play goal-rate GAM: log goal rate per minute = log(0.030) + f(minute). The fitted curve gives f ≈ 0.05 for minutes 71 to 80 and f ≈ 0.20 for minutes 81 to 90 plus about 3 minutes of stoppage time.

The match is 0-0 after 70 minutes.

Minutes 71 to 80: rate 0.030 × e^0.05 ≈ 0.0315 per minute, times 10 minutes ≈ 0.315 expected goals.
Minutes 81 to 93: rate 0.030 × e^0.20 ≈ 0.0366 per minute, times 13 minutes ≈ 0.476 expected goals.
Total expected goals remaining ≈ 0.792. Treating goals as a Poisson process, P(at least one goal) = 1 − e^(−0.792) ≈ 54.7%. Fair price for over 0.5 ≈ 1.83.
Betfair prices: over 0.5 at 1.70, under 0.5 at 2.40.
Backing under 0.5 with £10 at 2% commission: a win pays £14 × 0.98 = £13.72. EV ≈ 0.453 × £13.72 − 0.547 × £10 ≈ +£0.75.
Backing over at 1.70 has EV ≈ −£0.78.

The baseline 0.030 should come from these two teams' pre-match expectations, not a league average, or the edge is illusory.

Where it's good

In-play goal and point rates that change through a match.
Effects with a natural curve: age curves for players, days since last run in racing, travel distance.
Weather effects, such as temperature on scoring, where extremes matter.
Replacing hand-made bands (for example, "0 to 7 days, 8 to 14 days") with a smooth fit.
Explaining a model to yourself: each plotted curve is a clear story.

Limitations and pitfalls

Curves can still overfit where data is thin, especially at the edges of the input range.
Interactions (for example, the minute effect depending on the score) must be added explicitly as two-way smooths, which need much more data.
Choosing the number of spline pieces and smoothing method changes results; check sensitivity.
In-play, markets react within seconds; a model that is right but slow cannot get matched at the price it wants.
Extrapolating beyond the data, such as unusual stoppage times, is unreliable.
Betfair in-play markets have a delay and suspend on events, which limits how you can act on model output.

How to build it

R: mgcv is the gold standard. Python: pygam, or statsmodels GLMGam.
Data: for in-play rates, minute-by-minute records with goals and match state.
Let the software choose smoothness by REML or cross-validation rather than setting it by hand.
Tip: plot every fitted curve with its confidence band before using the model; wild wiggles are a warning sign.

Generalised linear models: the straight-line version.
Linear regression: the simplest member of the family.
Survival analysis: another way to model time to the next goal.
Regularisation: the same penalty idea applied to coefficients.
Poisson processes: the model for goals arriving over time.