elo PackageThe elo package includes functions to address all kinds of Elo calculations.
library(elo)Most functions begin with the prefix “elo.”, for easy autocompletion.
Vectors or scalars of Elo scores are denoted elo.A or elo.B.
Vectors or scalars of wins by team A are denoted by wins.A.
Vectors or scalars of win probabilities are denoted by p.A.
Vectors of team names are denoted team.A or team.B.
To calculate the probability team.A beats team.B, use elo.prob():
## [1] 0.500000 0.359935To calculate the score update after the two teams play, use elo.update():
wins.A <- c(1, 0)
elo.update(wins.A, elo.A, elo.B, k = 20)## [1] 10.0000 -7.1987To calculate the new Elo scores after the update, use elo.calc():
elo.calc(wins.A, elo.A, elo.B, k = 20)## elo.A elo.B
## 1 1510.000 1490.000
## 2 1492.801 1607.199elo.run() functionTo calculate a series of Elo updates, use elo.run(). This function has a formula = and data = interface. We first load the dataset tournament.
## 'data.frame': 56 obs. of 6 variables:
## $ team.Home : chr "Blundering Baboons" "Defense-less Dogs" "Fabulous Frogs" "Helpless Hyenas" ...
## $ team.Visitor : chr "Athletic Armadillos" "Cunning Cats" "Elegant Emus" "Gallivanting Gorillas" ...
## $ points.Home : num 14 21 15 13 22 18 20 23 25 23 ...
## $ points.Visitor: num 22 18 11 15 13 20 22 10 16 18 ...
## $ week : num 1 1 1 1 2 2 2 2 3 3 ...
## $ half : chr "First Half of Season" "First Half of Season" "First Half of Season" "First Half of Season" ...formula = should be in the format of wins.A ~ team.A + team.B. The score() function will help to calculate winners on the fly (1 = win, 0.5 = tie, 0 = loss).
tournament$wins.A <- tournament$points.Home > tournament$points.Visitor
elo.run(wins.A ~ team.Home + team.Visitor, data = tournament, k = 20)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.For more complicated Elo updates, you can include the special function k() in the formula = argument. Here we’re taking the log of the win margin as part of our update.
elo.run(score(points.Home, points.Visitor) ~ team.Home + team.Visitor +
k(20*log(abs(points.Home - points.Visitor) + 1)), data = tournament)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.You can also adjust the home and visitor teams with different k’s:
k1 <- 20*log(abs(tournament$points.Home - tournament$points.Visitor) + 1)
elo.run(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + k(k1, k1/2), data = tournament)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.It’s also possible to adjust one team’s Elo for a variety of factors (e.g., home-field advantage). The adjust() special function will take as its second argument a vector or a constant.
elo.run(score(points.Home, points.Visitor) ~ adjust(team.Home, 10) + team.Visitor,
data = tournament, k = 20)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.elo.run() also recognizes if the second column is numeric, and interprets that as a fixed-Elo opponent.
tournament$elo.Visitor <- 1500
elo.run(score(points.Home, points.Visitor) ~ team.Home + elo.Visitor,
data = tournament, k = 20)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.The special function regress() can be used to regress Elos back to a fixed value after certain matches. Giving a logical vector identifies these matches after which to regress back to the mean. Giving any other kind of vector regresses after the appropriate groupings (see, e.g., duplicated(..., fromLast = TRUE)). The other three arguments determine what Elo to regress to (to =, which could be a different value for different teams), by how much to regress toward that value (by =), and whether to regress teams which aren’t actively playing (regress.unused =).
tournament$elo.Visitor <- 1500
elo.run(score(points.Home, points.Visitor) ~ team.Home + elo.Visitor +
regress(half, 1500, 0.2),
data = tournament, k = 20)##
## An object of class 'elo.run.regressed', containing information on 8 teams and 56 matches, with 2 regressions.The special function group() tells elo.run() when to update Elos. It also determines matches to group together in as.matrix().
er <- elo.run(score(points.Home, points.Visitor) ~ team.Home + team.Visitor +
group(week),
data = tournament, k = 20)
as.matrix(er)## Athletic Armadillos Blundering Baboons Cunning Cats Defense-less Dogs
## [1,] 1510.000 1490.000 1490.000 1510.000
## [2,] 1499.425 1500.575 1500.575 1499.425
## [3,] 1489.425 1490.575 1510.575 1509.425
## [4,] 1499.458 1480.542 1520.542 1499.458
## [5,] 1489.458 1470.542 1520.542 1489.458
## [6,] 1490.033 1461.971 1509.681 1480.033
## [7,] 1501.153 1453.313 1499.945 1470.647
## [8,] 1509.785 1444.682 1509.104 1461.489
## [9,] 1519.765 1455.165 1499.124 1451.005
## [10,] 1527.812 1446.423 1507.865 1442.959
## [11,] 1537.372 1458.979 1508.123 1434.819
## [12,] 1547.007 1450.239 1518.364 1437.130
## [13,] 1556.067 1461.837 1528.173 1429.335
## [14,] 1564.318 1453.079 1518.019 1421.394
## Elegant Emus Fabulous Frogs Gallivanting Gorillas Helpless Hyenas
## [1,] 1490.000 1510.000 1510.000 1490.000
## [2,] 1500.575 1499.425 1499.425 1500.575
## [3,] 1490.575 1489.425 1509.425 1510.575
## [4,] 1480.542 1499.458 1499.458 1520.542
## [5,] 1490.542 1509.458 1509.458 1520.542
## [6,] 1501.403 1518.883 1508.883 1529.113
## [7,] 1510.789 1528.618 1517.541 1517.993
## [8,] 1501.302 1538.106 1527.554 1507.980
## [9,] 1502.056 1527.241 1526.800 1518.844
## [10,] 1512.539 1537.228 1516.813 1508.362
## [11,] 1502.978 1524.672 1516.555 1516.501
## [12,] 1511.718 1515.038 1514.245 1506.260
## [13,] 1501.910 1522.832 1505.185 1494.661
## [14,] 1509.851 1532.986 1513.944 1486.411This can be useful in situations when using the Elo framework for games which aren’t explicitly head-to-head (e.g., golf, swimming). For those situations, the person who won can be considered as having beaten (head-to-head) every other person. The person who came in second “beat” everyone but the first. However, we wouldn’t want to update Elos after every “head-to-head”; rather, they should all be considered together in updating Elo.
An example might help clarify. Suppose participants 1-3 go head-to-head in a game, with participant 2 coming in first, participant 1 coming in second, and participant 3 coming in last. Then we might have a dataset like
## team1 team2 won
## 1 Part 2 Part 1 1
## 2 Part 2 Part 3 1
## 3 Part 1 Part 3 1We would want to consider all three of these matches at the same time, so we add a grouping variable and run elo.run():
d$group <- 1
final.elos(elo.run(won ~ team1 + team2 + group(group), data = d, k = 20))## Part 1 Part 2 Part 3
## 1500 1520 1480There are several helper functions that are useful to use when interacting with objects of class "elo.run".
summary.elo.run() reports some summary statistics.
e <- elo.run(score(points.Home, points.Visitor) ~ team.Home + team.Visitor,
data = tournament, k = 20)
summary(e)##
## An object of class 'elo.run', containing information on 8 teams and 56 matches.
##
## Mean Square Error: 0.2195
## AUC: 0.6304
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 0.5 1
## TRUE 6 1 16
## (tie) 2 1 9
## FALSE 8 3 10rank.teams(e)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1 7 3
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 8 5 2
## Gallivanting Gorillas Helpless Hyenas
## 4 6as.matrix.elo.run() creates a matrix of running Elos.
## Athletic Armadillos Blundering Baboons Cunning Cats Defense-less Dogs
## [1,] 1510.000 1490.000 1500.000 1500.000
## [2,] 1510.000 1490.000 1490.000 1510.000
## [3,] 1510.000 1490.000 1490.000 1510.000
## [4,] 1510.000 1490.000 1490.000 1510.000
## [5,] 1499.425 1490.000 1500.575 1510.000
## [6,] 1499.425 1500.575 1500.575 1499.425
## Elegant Emus Fabulous Frogs Gallivanting Gorillas Helpless Hyenas
## [1,] 1500 1500 1500 1500
## [2,] 1500 1500 1500 1500
## [3,] 1490 1510 1500 1500
## [4,] 1490 1510 1510 1490
## [5,] 1490 1510 1510 1490
## [6,] 1490 1510 1510 1490as.data.frame.elo.run() gives the long version (perfect, for, e.g., ggplot2).
str(as.data.frame(e))## 'data.frame': 56 obs. of 8 variables:
## $ team.A : Factor w/ 8 levels "Athletic Armadillos",..: 2 4 6 8 3 4 7 8 4 3 ...
## $ team.B : Factor w/ 8 levels "Athletic Armadillos",..: 1 3 5 7 1 2 5 6 1 2 ...
## $ p.A : num 0.5 0.5 0.5 0.5 0.471 ...
## $ wins.A : num 0 1 1 0 1 0 0 1 1 1 ...
## $ update.A: num -10 10 10 -10 10.6 ...
## $ update.B: num 10 -10 -10 10 -10.6 ...
## $ elo.A : num 1490 1510 1510 1490 1501 ...
## $ elo.B : num 1510 1490 1490 1510 1499 ...Finally, final.elos() will extract the final Elos per team.
final.elos(e)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1564.318 1453.079 1518.019
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 1421.394 1509.851 1532.986
## Gallivanting Gorillas Helpless Hyenas
## 1513.944 1486.411It is also possible to use the Elos calculated by elo.run() to make predictions on future match-ups.
results <- elo.run(score(points.Home, points.Visitor) ~ adjust(team.Home, 10) + team.Visitor,
data = tournament, k = 20)
newdat <- data.frame(
team.Home = "Athletic Armadillos",
team.Visitor = "Blundering Baboons"
)
predict(results, newdata = newdat)## [1] 0.6676045We now get to elo.run() when custom probability calculations and Elo updates are needed. Note that these use cases are coded in R instead of C++ and may run as much as 50x slower than the default.
For instance, suppose you want to change the adjustment based on team A’s current Elo:
custom_update <- function(wins.A, elo.A, elo.B, k, adjust.A, adjust.B, ...)
{
k*(wins.A - elo.prob(elo.A, elo.B, adjust.B = adjust.B,
adjust.A = ifelse(elo.A > 1500, adjust.A / 2, adjust.A)))
}
custom_prob <- function(elo.A, elo.B, adjust.A, adjust.B)
{
1/(1 + 10^(((elo.B + adjust.B) - (elo.A + ifelse(elo.A > 1500, adjust.A / 2, adjust.A)))/400.0))
}
er2 <- elo.run(score(points.Home, points.Visitor) ~ adjust(team.Home, 10) + team.Visitor,
data = tournament, k = 20, prob.fun = custom_prob, update.fun = custom_update)## Using R instead of C++final.elos(er2)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1564.660 1452.278 1518.165
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 1420.876 1510.015 1533.209
## Gallivanting Gorillas Helpless Hyenas
## 1514.233 1486.564Compare this to the results from the default:
er3 <- elo.run(score(points.Home, points.Visitor) ~ adjust(team.Home, 10) + team.Visitor,
data = tournament, k = 20)
final.elos(er3)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1563.967 1452.822 1517.847
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 1421.358 1509.906 1533.127
## Gallivanting Gorillas Helpless Hyenas
## 1514.205 1486.768This example is a bit contrived, as it’d be easier just to use adjust() (actually, this is tested for in the tests), but the point remains.
All three of the “basic” functions accept formulas as input, just like elo.run().
dat <- data.frame(elo.A = c(1500, 1500), elo.B = c(1500, 1600),
wins.A = c(1, 0), k = 20)
form <- wins.A ~ elo.A + elo.B + k(k)
elo.prob(form, data = dat)## [1] 0.500000 0.359935elo.update(form, data = dat)## [1] 10.0000 -7.1987elo.calc(form, data = dat)## elo.A elo.B
## 1 1510.000 1490.000
## 2 1492.801 1607.199Note that for elo.prob(), formula = can be more succinct:
elo.prob(~ elo.A + elo.B, data = dat)## [1] 0.500000 0.359935We can even adjust the Elos:
## elo.A elo.B
## 1 1509.712 1490.288
## 2 1492.534 1607.466The first model computes teams’ win percentages, and feeds the differences of percentages into a regression. Including an adjustment using adjust() in the formula also includes that in the model. You could also adjust the intercept for games played on neutral fields by using the neutral() function.
e.winpct <- elo.winpct(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor) # to get rid of ties for now
summary(e.winpct)##
## An object of class 'elo.winpct', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1566
## AUC: 0.8339
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 7 32
## (tie) 0 0
## FALSE 9 3rank.teams(e.winpct)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1 7 4
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 8 5 2
## Gallivanting Gorillas Helpless Hyenas
## 3 6predict(e.winpct, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE))## 1
## 0.9690678tournament$neutral <- replace(rep(0, nrow(tournament)), 30:35, 1)
summary(elo.winpct(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + neutral(neutral) + group(week),
data = tournament, subset = points.Home != points.Visitor))##
## An object of class 'elo.winpct', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1565
## AUC: 0.825
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 6 32
## (tie) 0 0
## FALSE 10 3The models can be built “running”, where predictions for the next group of games are made based on past data. Consider using the skip= argument to skip the first few groups (otherwise the model might have trouble converging).
Note that predictions from this object use a model fit on all the data.
e.winpct <- elo.winpct(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor, running = TRUE, skip = 5)
summary(e.winpct)##
## An object of class 'elo.winpct', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1769
## AUC: 0.8339
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 5 21
## (tie) 6 13
## FALSE 5 1predict(e.winpct, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE)) # the same thing## 1
## 0.9690678It’s also possible to compare teams’ skills using logistic regression. A matrix of dummy variables is constructed, one for each team, where a value of 1 indicates a home team and -1 indicates a visiting team. The intercept then indicates a home-field advantage. To denote games played in a neutral setting (that is, without home-field advantage), use the neutral() function. In short, the intercept will then be set to 1 - neutral(). Including an adjustment using adjust() in the formula also includes that in the model.
results <- elo.glm(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor) # to get rid of ties for now
summary(results)##
## Call:
## stats::glm(formula = wins.A ~ . - 1, family = family, data = dat,
## weights = wts, subset = NULL, na.action = stats::na.pass)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.0108 -0.8255 0.4050 0.6560 2.1217
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## home.field 1.0307 0.3871 2.663 0.00775 **
## `Athletic Armadillos` 1.4289 0.9546 1.497 0.13442
## `Blundering Baboons` -0.9637 0.9043 -1.066 0.28659
## `Cunning Cats` 0.5377 0.9483 0.567 0.57074
## `Defense-less Dogs` -1.7413 1.0356 -1.681 0.09268 .
## `Elegant Emus` 0.3931 0.8818 0.446 0.65576
## `Fabulous Frogs` 0.8489 0.8807 0.964 0.33509
## `Gallivanting Gorillas` 0.3994 0.9500 0.420 0.67417
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 70.701 on 51 degrees of freedom
## Residual deviance: 48.037 on 43 degrees of freedom
## AIC: 64.037
##
## Number of Fisher Scoring iterations: 5
##
## Mean Square Error: 0.1566
## AUC: 0.8375
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 7 32
## (tie) 0 0
## FALSE 9 3rank.teams(results)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1 7 3
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 8 5 2
## Gallivanting Gorillas Helpless Hyenas
## 4 6predict(results, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE))## 1
## 0.9684256summary(elo.glm(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + neutral(neutral) + group(week),
data = tournament, subset = points.Home != points.Visitor))##
## Call:
## stats::glm(formula = wins.A ~ . - 1, family = family, data = dat,
## weights = wts, subset = NULL, na.action = stats::na.pass)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.0349 -0.7789 0.3933 0.7618 2.2148
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## home.field 1.0886 0.4229 2.574 0.0101 *
## `Athletic Armadillos` 1.6006 0.9750 1.642 0.1007
## `Blundering Baboons` -0.8541 0.8930 -0.956 0.3389
## `Cunning Cats` 0.5801 0.9446 0.614 0.5391
## `Defense-less Dogs` -1.8507 1.0449 -1.771 0.0765 .
## `Elegant Emus` 0.5762 0.8994 0.641 0.5218
## `Fabulous Frogs` 0.8470 0.8804 0.962 0.3360
## `Gallivanting Gorillas` 0.5777 0.9279 0.623 0.5335
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 70.701 on 51 degrees of freedom
## Residual deviance: 48.405 on 43 degrees of freedom
## AIC: 64.405
##
## Number of Fisher Scoring iterations: 5
##
## Mean Square Error: 0.1556
## AUC: 0.8375
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 7 32
## (tie) 0 0
## FALSE 9 3The models can be built “running”, where predictions for the next group of games are made based on past data. Consider using the skip= argument to skip the first few groups (otherwise the model might have trouble converging).
Note that predictions from this object use a model fit on all the data.
results <- elo.glm(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor, running = TRUE, skip = 5)
summary(results)##
## Call:
## stats::glm(formula = wins.A ~ . - 1, family = family, data = dat,
## weights = wts, subset = NULL, na.action = stats::na.pass)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.0108 -0.8255 0.4050 0.6560 2.1217
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## home.field 1.0307 0.3871 2.663 0.00775 **
## `Athletic Armadillos` 1.4289 0.9546 1.497 0.13442
## `Blundering Baboons` -0.9637 0.9043 -1.066 0.28659
## `Cunning Cats` 0.5377 0.9483 0.567 0.57074
## `Defense-less Dogs` -1.7413 1.0356 -1.681 0.09268 .
## `Elegant Emus` 0.3931 0.8818 0.446 0.65576
## `Fabulous Frogs` 0.8489 0.8807 0.964 0.33509
## `Gallivanting Gorillas` 0.3994 0.9500 0.420 0.67417
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 70.701 on 51 degrees of freedom
## Residual deviance: 48.037 on 43 degrees of freedom
## AIC: 64.037
##
## Number of Fisher Scoring iterations: 5
##
## Mean Square Error: 0.2098
## AUC: 0.8375
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 6 19
## (tie) 6 13
## FALSE 4 3predict(results, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE)) # the same thing## 1
## 0.9684256It’s also possible to compare teams’ skills using a Markov-chain-based model, as outlined in Kvam and Sokol (2006). In short, imagine a judge who randomly picks one of two teams in a matchup, where the winner gets chosen with probability p (here, for convenience, ‘k’) and the loser with probability 1-p (1-k). In other words, we assume that the probability that the winning team is better than the losing team given that it won is k, and the probability that the losing team is better than the winning team given that it lost is (1-k). This forms a transition matrix, whose stationary distribution gives a ranking of teams. The differences in ranking are then fed into a logistic regession model to predict win status. Any adjustments made using adjust() are also included in this logistic regression. You could also adjust the intercept for games played on neutral fields by using the neutral() function.
mc <- elo.markovchain(score(points.Home, points.Visitor) ~ team.Home + team.Visitor, data = tournament,
subset = points.Home != points.Visitor, k = 0.7)
summary(mc)##
## An object of class 'elo.markovchain', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1688
## AUC: 0.8
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 10 29
## (tie) 0 0
## FALSE 6 6rank.teams(mc)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1 7 3
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 8 4 2
## Gallivanting Gorillas Helpless Hyenas
## 6 5predict(mc, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE))## 1
## 0.9594476summary(elo.markovchain(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + neutral(neutral),
data = tournament, subset = points.Home != points.Visitor, k = 0.7))##
## An object of class 'elo.markovchain', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1732
## AUC: 0.7857
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 10 28
## (tie) 0 0
## FALSE 6 7These models can also be built “running”, where predictions for the next group of games are made based on past data. Consider using the skip= argument to skip the first few groups (otherwise the model might have trouble converging).
Note that predictions from this object use a model fit on all the data.
mc <- elo.markovchain(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor, k = 0.7, running = TRUE, skip = 5)
summary(mc)##
## An object of class 'elo.markovchain', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.229
## AUC: 0.8
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 6 19
## (tie) 6 13
## FALSE 4 3predict(mc, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE)) # the same thing## 1
## 0.9594476Note that by assigning probabilities in the right way, this function emits the Logistic Regression Markov Chain model (LRMC). Use the in-formula function k() for this. IMPORTANT: note that k() denotes the probability assigned to the winning team, not the home team (for instance). If rH(x) denotes the probability that the home team is better given that they scored x points more than the visiting team (allowing for x to be negative), then an LRMC model might look something like this:
elo.markovchain(floor(wins.home) ~ team.home + team.visitor + k(ifelse(x > 0, rH(x), 1 - rH(x))))Why do we use floor() here? This takes care of the odd case where teams tie. In this case, rH(x) < 0.5 because we expected the home team to win by virtue of being home. By default, elo.markovchain() will split any ties down the middle (i.e., 0.5 and 0.5 instead of p and 1-p), which isn’t what we want; we want the visiting team to get a larger share than the home team. Telling elo.markovchain() that the visiting team “won” gives the visiting team its whole share of p.
Alternatively, if h denotes a home-field advantage (in terms of score), the model becomes:
elo.markovchain(ifelse(home.points - visitor.points > h, 1, 0) ~ team.home + team.visitor + k(pmax(rH(x), 1 - rH(x))))In this case, the home team “won” if it scored more than h points more than the visiting team. Since rH(x) > 0.5 if x > h, then pmax() will assign the proper probability to the pseudo-winning team.
Finally, do note that using neutral() isn’t sufficient for adjusting for games played on neutral ground, because the adjustment is only taken into account in the logistic regression to produce probabilities, not the building of the transition matrix. Therefore, you’ll want to also account for neutral wins/losses in k() as well.
It’s also possible to compare teams’ skills using the Colley Matrix method, as outlined in Colley (2002). The coefficients to the Colley matrix formulation gives a ranking of teams. The differences in ranking are then fed into a logistic regession model to predict win status. Here ‘k’ denotes how convincing a win is; it represents the fraction of the win assigned to the winning team and the fraction of the loss assigned to the losing team. Setting ‘k’ = 1 emits the bias-free method presented by Colley. Any adjustments made using adjust() are also included in this logistic regression. You could also adjust the intercept for games played on neutral fields by using the neutral() function.
co <- elo.colley(score(points.Home, points.Visitor) ~ team.Home + team.Visitor, data = tournament,
subset = points.Home != points.Visitor)
summary(co)##
## An object of class 'elo.colley', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1565
## AUC: 0.8339
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 7 32
## (tie) 0 0
## FALSE 9 3rank.teams(co)## Athletic Armadillos Blundering Baboons Cunning Cats
## 1 7 4
## Defense-less Dogs Elegant Emus Fabulous Frogs
## 8 5 2
## Gallivanting Gorillas Helpless Hyenas
## 3 6predict(co, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE))## 1
## 0.9687583summary(elo.colley(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + neutral(neutral),
data = tournament, subset = points.Home != points.Visitor))##
## An object of class 'elo.colley', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.1565
## AUC: 0.8268
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 6 32
## (tie) 0 0
## FALSE 10 3These models can also be built “running”, where predictions for the next group of games are made based on past data. Consider using the skip= argument to skip the first few groups (otherwise the model might have trouble converging).
Note that predictions from this object use a model fit on all the data.
co <- elo.colley(score(points.Home, points.Visitor) ~ team.Home + team.Visitor + group(week), data = tournament,
subset = points.Home != points.Visitor, running = TRUE, skip = 5)
summary(co)##
## An object of class 'elo.colley', containing information on 8 teams and 51 matches.
##
## Mean Square Error: 0.2173
## AUC: 0.8339
## Favored Teams vs. Actual Wins:
## Actual
## Favored 0 1
## TRUE 4 19
## (tie) 6 13
## FALSE 6 3predict(co, newdata = data.frame(team.Home = "Athletic Armadillos", team.Visitor = "Blundering Baboons", stringsAsFactors = FALSE)) # the same thing## 1
## 0.9687583elo.glm(), elo.markovchain(), and elo.winpct() all allow for modeling of margins of victory instead of simple win/loss using the mov() function. Note that one must set the family="gaussian" argument to get linear regression instead of logistic regression.
summary(elo.glm(mov(points.Home, points.Visitor) ~ team.Home + team.Visitor, data = tournament,
family = "gaussian"))##
## Call:
## stats::glm(formula = wins.A ~ . - 1, family = family, data = dat,
## weights = wts, subset = NULL, na.action = stats::na.pass)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -10.6339 -2.8996 -0.0402 2.7879 12.9286
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## home.field 2.6964 0.6941 3.885 0.000313 ***
## `Athletic Armadillos` 3.1250 1.8363 1.702 0.095263 .
## `Blundering Baboons` -2.4375 1.8363 -1.327 0.190655
## `Cunning Cats` 0.3125 1.8363 0.170 0.865584
## `Defense-less Dogs` -3.5000 1.8363 -1.906 0.062646 .
## `Elegant Emus` -0.9375 1.8363 -0.511 0.612014
## `Fabulous Frogs` 0.6875 1.8363 0.374 0.709759
## `Gallivanting Gorillas` 0.2500 1.8363 0.136 0.892277
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 26.97582)
##
## Null deviance: 2161.0 on 56 degrees of freedom
## Residual deviance: 1294.8 on 48 degrees of freedom
## AIC: 352.81
##
## Number of Fisher Scoring iterations: 2
##
## Mean Square Error: 23.1221
## AUC: NA
## Favored Teams vs. Actual Wins:
## Actual
## Favored TRUE (tie) FALSE
## TRUE 31 3 10
## (tie) 0 0 0
## FALSE 4 2 6