Chapter 17 · Worked solutions
Chapter 17 workings: Patching the Machine: Dixon and Coles
Chapter seventeen worked: the Dixon-Coles patch applied by hand to a fresh match, and how to argue for a faster forgetting rate after a change of manager.
The patch multiplies four cells of the plain Poisson grid by a factor τ and leaves the rest alone. With a negative ρ, 0-0 and 1-1 go up and 1-0 and 0-1 come down by exactly the same total, which is why the grid still adds up to one.
The Poisson predictor applies the same patch: set ρ and compare.
The exercises
Patching λ = 1.8, μ = 1.4 with ρ = −0.08
For a match with λ = 1.8 and μ = 1.4, apply the Dixon-Coles correction at ρ = −0.08 and give the four patched probabilities.
Plain Poisson first
P(0-0) = e−3.2 = 4.08 per cent. The others multiply that by λ, by μ and by λμ: 1-0 = 7.34 per cent, 0-1 = 5.71 per cent, 1-1 = 10.27 per cent.
Work out each τ
λμρ = 1.8 × 1.4 × −0.08 = −0.2016, so τ(0,0) = 1.2016. μρ = −0.112, so τ(1,0) = 0.8880. λρ = −0.144, so τ(0,1) = 0.8560. And τ(1,1) = 1 + 0.08 = 1.0800.
Multiply
Each naive cell times its factor gives the patched column in the table.
Check the books balance
The four cells total 27.39 per cent before and 27.39 per cent after. Probability has moved, none has been created.
| Score | Factor | Value | Plain Poisson | Patched | Change |
|---|---|---|---|---|---|
| 0-0 | 1 − λμρ | 1.2016 | 4.08% | 4.90% | +0.82 pts |
| 1-0 | 1 + μρ | 0.8880 | 7.34% | 6.52% | −0.82 pts |
| 0-1 | 1 + λρ | 0.8560 | 5.71% | 4.88% | −0.82 pts |
| 1-1 | 1 − ρ | 1.0800 | 10.27% | 11.09% | +0.82 pts |
The answer
0-0 4.90 per cent, 1-0 6.52 per cent, 0-1 4.88 per cent, 1-1 11.09 per cent; the four-cell total is unchanged.
Choosing ξ after a change of manager
Pick a forgetting rate ξ for a side that has just changed manager mid-season, and justify it against the original 0.0065.
Know what the dial means
A match t half-weeks old keeps a weight of e−ξt. The half-life, the age at which a match counts half, is ln 2 ÷ ξ.
Put numbers on the options
The table shows three settings and what they leave of a match from eight weeks ago (16 half-weeks) and one from a year ago (104 half-weeks).
Make the argument
A new manager usually means a new shape, new selections and new set pieces, so results from under the old one describe a different side. I would at least double ξ for this club, to around 0.013, which halves the half-life to about half a season and lets the new regime take over the ratings by spring.
Say what it costs
A faster dial throws away information and chases noise in the first few matches. That is why a raised rate is best treated as temporary, eased back once the new side has a dozen or so results of its own. Chapter 18 has the same trade-off as the Elo K-factor.
| Setting | ξ | Half-life | Weight at 8 weeks | Weight at 1 year |
|---|---|---|---|---|
| Dixon and Coles, 1997 | 0.0065 | 106.6 half-weeks (53.3 weeks) | 0.901 | 0.509 |
| Twice as fast | 0.0130 | 53.3 half-weeks (26.7 weeks) | 0.812 | 0.259 |
| Three times as fast | 0.0195 | 35.5 half-weeks (17.8 weeks) | 0.732 | 0.132 |
The answer
Raise ξ above 0.0065, for example to about 0.013, so the past fades in months rather than a year; ease it back once the new side has its own record.
These are the workings. The argument for them, and the exercises themselves, are in chapter 17 of the book, Patching the Machine: Dixon and Coles.
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