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As I mentioned back in July, I haven’t had as much time (since summer of 2018) to write due to taking classes in pursuit of a degree from Georgia Tech’s Online Master of Science in Analytics (OMSA) program. On the other hand, the classes have given me some ideas for future content. And, in the case of the Bayesian Statistics class that I took this past fall, there’s content that translates well to a blog post directly. What follows is a lightly edited version of the report that I submitted at the end of the semster for this class.
Introduction
I model and predict English Premier League (EPL) game outcomes using Bayesian methods. Specifically, I estimated goals scored by each team in a given game as independent Poisson processes, taking the difference of the estimated points scored on each side to determine game winners. More broadly, one may call this a hierarchical Bayesian Poisson model.
Why model goals scored using a Poisson distribution? By definition, it “is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time with a known constant rate.” In the context of soccer, the fixed interval of time is the 90 minutes of a game (disregarding injury time and over time), and the known constant rate is the expected number of goals scored per minute. Importantly, I must make the assumption that the rate of scored goals is the same across all minutes of a game. ^{1} Additionally, when computing the difference between Poisson distributions, I must assume that the two distributions are independent of one another. ^{2}
Using Poisson distributions to model soccer scores is certainly not a novel concept. ^{3} ^{4} In particular, I would like to acknowledge the work of Rasmus Bååth’s, whose series of blog posts exemplifying the use of R and JAGS to model scores in La Liga games between the 200809 to 201213 season served as a guide for the analysis that I conduct here. ^{5} ^{6}
Data Collection
For this project I retrieved game scores and outcomes for the previous three seasons of EPL games (i.e. from the 201617 season through the 20182019 season).
Before impementing some models, I need to “wrangle” the raw data a bit in order to put it in a more workable format for OpenBUGs. (Specifically, I need to put it into a list.)
Modeling
My model is formally defined as follows.
\[
\begin{array}{c}
g_h \sim \mathcal{Pois}(\lambda_{h,i,j}) \\
g_a \sim \mathcal{Pois}(\lambda_{a,i,j}) \\
\log(\lambda_{h,i,j}) = \text{baseline}_h + (z_i – z_j) \\
\log(\lambda_{a,i,j}) = \text{baseline}_a + (z_j – z_i). \\
\end{array}
\]
This model estimates the goals scored by the home team, \(g_h\), and the goals scored by the away team, \(g_a\), in a given game between home team, \(\text{tm}_h\), and away team, \(\text{tm}_a\), as random variables coming from independent Poisson processes, \(\mathcal{Pois}(\lambda_{h,i,j})\) and \(\mathcal{Pois}(\lambda_{a,i,j})\). The log of the rate of goals scored by the home team, \(\lambda_{h,i,j}\), in a game between \(\text{tm}_i\) and \(\text{tm}_j\) is modeled as the sum of a “baseline” average of goals scored by any given team playing at home, \(\text{baseline}_h\), and the difference between the team “strength” \(z\) of teams \(i\) and \(j\) in a given game. I define the log of the goal rate by the away team, \(\lambda_{a,i,j}\), in a similar fashion. ^{7} It is important to distinguish the baseline levels for home and away so as to account for “home field advantage”. (One should expect to find that \(\text{baseline}_h > \text{baseline}_a\) in the posterior estimates.)
Since I am employing a Bayesian approach, I need to model priors as well. I define them as follows.
\[
\begin{array}{c}
\text{baseline}_h \sim \mathcal{N}(0, 2^2) \\
\text{baseline}_a \sim \mathcal{N}(0, 2^2) \\
z_{i} \sim \mathcal{N}(z_{\text{all}} , \sigma^2_{\text{all}}) \quad \text{tm}_i > 1 \\
z_{\text{all}} \sim \mathcal{N}(0, 2^2) \\
\sigma_{\text{all}} \sim \mathcal{U}(0, 2).
\end{array}
\]
There are a couple of things to note about these priors. First, I must “zero”anchor the strength estimate \(z\) of one team. (This is manifested by \(\text{tm}_i > 1\).) Here, I choose the first team alphabetically—Arsenal. Second, the priors are intentionally defined to be relatively vague (although not too vauge) so as to allow the posterior estimates to be heavily defined by the data rather than the priors. Note that the standard deviation of the overall team strength parameter \(z_{\text{all}}\), defined as \(2\) on a log scale, corresponds to an interval of \(\left[e^{2}, e^2\right] = \left[0.13, 7.40\right]\) on an unstransformed scale, i.e. goals scored per game.
I leverage the {R2OpenBUGs}
package to create this model with R on the “frontend” and generate the results using the OpenBUGS engine on the “backend”. Regarding the implementation itself, note that I run 100,000 simulations (n.iter
), minus 1,000 “burnin” runs (n.burn
).
The raw results are as follows. (As a quick “validation” of these results, note that \(\text{baseline}_h > \text{baseline}_a\), as hypothesized.)
Interpretation & Discussion
Next, I correspond the strength estimates \(z\) to teams. Notably, I must “readd” the zeroanchored team—Arsenal (whose \(z\) is assigned a dummy value of 1). To do this, I impute its credible set quantiles using the values of the overall strength term \(z_{\text{all}}\).
It’s not surprising to see that the strength (\(z\)) corresponding to all but three teams—Liverpool, Man City, and Tottenham—is negative. These three teams, followed closely by Arsenal have been regarded as the best teams for the past two or three EPL seasons. So, relative to Arsenal, all other teams (aside from the top three) are viewed as “worse” by the model.
Note that the \(z\) estimates above should not be interpreted as goals scored by the teams because they are relative to the strength of Arsenal. To facilitate such an interpretation, I need to translate \(z\) to goals scored per game. To do this, for each \(z\), I (1) subtract the average value of all \(z\)’s, (2) add the posterior mean of \(\text{baseline}_{h}\), and (3) exponentiate.
The plot below shows the results of this transformation.
Predictions
I can make predictions of game results for the historical data, given the model.
I make a couple of observations:
 The most common outcome is an actual win by the home team and a predicted win by the home team.
 The model never predicts a tie. (This may seem “unreasonable”, but Bååth also found this to be true for his final model.)
 The model predicts the outcome correctly in 447 + 216 = 663 of 1,140 games (i.e., 58%).
The next couple of visuals provide more details regarding the simulated outcomes.
From the above graph of the mode of goals scored by both sides, it’s apparent that a 21 scores in favor of the home side is the most common outcome.
The above histogram illustrating the mean (instead of the mode) of the simulated goals provides a bit more nuance to our understanding of modes shown before.
Finally, the above visual shows the predicted outcomes (inferred from the prior graph of predicted modes).
To better understand how the model works on a teamlevel basis, let’s look at how well it predicts for each team.
## # A tibble: 26 x 4
## tm_h is_correct n pct
##
## 1 Arsenal TRUE 46 81%
## 2 Man City TRUE 45 79%
## 3 Tottenham TRUE 43 75%
## 4 Liverpool TRUE 41 72%
## 5 Chelsea TRUE 39 68%
## 6 Cardiff TRUE 12 63%
## 7 Everton TRUE 36 63%
## 8 Fulham TRUE 12 63%
## 9 Man United TRUE 35 61%
## 10 Huddersfield TRUE 23 61%
## 11 Burnley TRUE 34 60%
## 12 Stoke TRUE 22 58%
## 13 Bournemouth TRUE 30 53%
## 14 Crystal Palace TRUE 30 53%
## 15 Sunderland TRUE 10 53%
## 16 Swansea TRUE 20 53%
## 17 West Ham TRUE 30 53%
## 18 Watford TRUE 29 51%
## 19 Newcastle TRUE 19 50%
## 20 Leicester TRUE 28 49%
## 21 Middlesbrough TRUE 9 47%
## 22 West Brom TRUE 18 47%
## 23 Wolves TRUE 9 47%
## 24 Brighton TRUE 15 39%
## 25 Southampton TRUE 22 39%
## 26 Hull TRUE 6 32%
In most cases, the model predicts the outcome correctly (see is_correct
) with greater than 50% accuracy, although there are also teams for which its accuracy is less than 50%.
Conclusion
In summary, I have created a hierarchical Poisson model to predict scores—and, consequently, game outcomes—for EPL games for the three seasons starting in 2016 and ending in 2018. The model has an training set prediction accuracy of 663. Bååth, whose work inspired mine, achieved an accuracy of 56% with his final model.
Future Work
My model can certainly be improved. One major flaw of the model is that it does not account for temporal effects, i.e. differences in team strength across seasons. ^{8} The consequences of this flaw are compounded by the fact that the pool of teams in each EPL season changes. At the end of each season, the three “worst” EPL teams (by winlosstie record) are “relegated” to a secondary league, and, in turn, three secondary league teams are “promoted” to the EPL in their place. ^{9} Consequently, one might say that the estimates of the teams that do not appear in all seasons are exaggerated.
Appendix
Code
See all relevant R code below.
library(tidyverse)
# Data Collection
seasons < 2016L:2018L
# Reference: https://github.com/jalapic/engsoccerdata/blob/master/R/england_current.R
scrape_epl_data < function(season = lubridate::year(Sys.Date())  1L) {
s1 < season %>% str_sub(3, 4) %>% as.integer()
s2 < s1 + 1L
path < sprintf('http://www.footballdata.co.uk/mmz4281/%2d%2d/E0.csv', s1, s2)
data_raw < path %>% read_csv()
data <
data_raw %>%
janitor::clean_names() %>%
mutate_at(vars(date), ~as.Date(., '%d/%m/%y')) %>%
select(
date,
tm_h = home_team,
tm_a = away_team,
g_h = fthg,
g_a = ftag
) %>%
mutate(
g_total = g_h + g_a,
g_diff = g_h  g_a,
result =
case_when(
g_h > g_a ~ 'h',
g_h < g_a ~ 'a',
TRUE ~ 't'
),
tm_winner =
case_when(
g_h > g_a ~ tm_h,
g_h < g_a ~ tm_a,
TRUE ~ NA_character_
)
) %>%
mutate_at(vars(matches('season^g_')), as.integer)
}
data <
tibble(season = seasons) %>%
mutate(data = map(season, scrape_epl_data)) %>%
unnest(data)
pull_distinctly < function (data, var = 1, ..., decreasing = FALSE) {
var < tidyselect::vars_pull(names(data), !!rlang::enquo(var))
sort(unique(data[[var]]), decreasing = decreasing, ...)
}
pull2 < function(data, ...) {
data %>%
pull(...) %>%
as.factor() %>%
as.integer()
}
tms < data %>% distinct(tm_h) %>% arrange(tm_h) %>% pull(tm_h)
n_tm < tms %>% length()
n_gm < data %>% nrow()
n_season < seasons %>% length()
data_list <
list(
g_h = data %>% pull(g_h),
g_a = data %>% pull(g_a),
tm_h = data %>% pull2(tm_h),
tm_a = data %>% pull2(tm_a),
season = data %>% pull2(season),
n_tm = n_tm,
n_gm = n_gm,
n_season = n_season
)
data_list
model_1 < glue::glue_collapse('model {
for(g in 1:n_gm) {
g_h[g] ~ dpois(lambda_h[tm_h[g], tm_a[g]])
g_a[g] ~ dpois(lambda_a[tm_h[g], tm_a[g]])
}
for(h in 1:n_tm) {
for(a in 1:n_tm) {
lambda_h[h, a] < exp(baseline_h + (z[h]  z[a]))
lambda_a[h, a] < exp(baseline_a + (z[a]  z[h]))
}
}
z[1] < 0
for(t in 2:n_tm) {
z[t] ~ dnorm(z_all, tau_all)
}
z_all ~ dnorm(0, 0.25)
tau_all < 1 / pow(sigma_all, 2)
sigma_all ~ dunif(0, 2)
baseline_h ~ dnorm(0, 0.25)
baseline_a ~ dnorm(0, 0.25)
}')
path_model_1 < 'model_1.txt'
write_lines(model_1, path_model_1)
if(eval_model_1) {
# inits_1 < list(n_tm = data_list$n_tm)
inits_1 < NULL
params_1 <
c(
paste0('baseline', c('_a', '_h')),
paste0('sigma_all'),
paste0('z', c('', '_all'))
)
res_sim_1 <
R2OpenBUGS::bugs(
# debug = TRUE,
data = data_list,
inits = inits_1,
model.file = path_model_1,
parameters.to.save = params_1,
DIC = FALSE,
n.chains = 1,
n.iter = 10000,
n.burnin = 1000
)
}
# Model 1 Interpretation
z_var_lvls < sprintf('z[%d]', 2:n_tm)
var_lvls < c(paste0('baseline', c('_a', '_h')), 'sigma_all', z_var_lvls, 'z_all')
res_sim_summ_1 <
res_sim_1$summary %>%
as_tibble(rownames = 'var') %>%
# Reorder these.
mutate_at(vars(var), ~factor(., levels = var_lvls)) %>%
arrange(var) %>%
# Then recoerce var back to its original data type.
mutate_at(vars(var), as.character)
res_sim_summ_1
tms_info <
tibble(tm = tms) %>%
mutate(tm_idx = row_number())
tms_info
# tms_colors <
# teamcolors::teamcolors %>%
# as_tibble() %>%
# filter(league == 'epl') %>%
# select(tm = name, color_1 = primary) %>%
# mutate_at(
# vars(tm),
# ~case_when(
# str_detect(., 'Bournemouth') ~ 'Bournemouth',
# str_detect(., 'Brighton') ~ 'Brighton',
# str_detect(., 'Leicester') ~ 'Leicester',
# str_detect(., 'Manchester C') ~ 'Man City',
# str_detect(., 'Manchester U') ~ 'Man United',
# str_detect(., 'Stoke') ~ 'Stoke',
# str_detect(., 'Swansea') ~ 'Swansea',
# str_detect(., 'Tottenham') ~ 'Tottenham',
# str_detect(., 'West Brom') ~ 'West Brom',
# str_detect(., 'West Ham') ~ 'West Ham',
# TRUE ~ .
# )
# )
# tms_colors
# tms_info %>% anti_join(tms_colors)
res_sim_summ_1_z <
bind_rows(
res_sim_summ_1 %>%
filter(var == 'z_all') %>%
mutate(var = 'z[1]') %>%
mutate(tm_idx = 1L) %>%
mutate_at(vars(matches('%$')), ~{.  mean}) %>%
mutate(mean = 0),
res_sim_summ_1 %>%
filter(var %>% str_detect('^z\\[')) %>%
mutate(
tm_idx =
var %>%
str_replace_all('(^z\\[)([09]+)(\\]$)', '\\2') %>%
as.integer()
)
) %>%
left_join(tms_info, by = 'tm_idx') %>%
select(tm_idx) %>%
select(tm, everything()) %>%
arrange(tm)
res_sim_summ_1_z
theme_custom < function(...) {
theme_light(base_size = 12) +
theme(
legend.position = 'bottom',
legend.title = element_blank(),
axis.title.x = element_text(hjust = 1),
axis.title.y = element_text(hjust = 1),
...
)
}
.lab_subtitle < 'Seasons 201617  201819'
.visualize_res_sim_summ < function(data, ...) {
data %>%
arrange(mean) %>%
mutate_at(vars(tm), ~forcats::fct_reorder(., mean)) %>%
ggplot() +
aes(x = tm) +
geom_pointrange(aes(y = mean, ymin = `2.5%`, ymax = `97.5%`)) +
theme_custom() +
coord_flip() +
labs(
subtitle = .lab_subtitle ,
caption = glue::glue(
'Posterior mean and 95% equitailed credible set depicted.
{tms[1]} used as "zero"anchor team.'
),
y = NULL
)
}
visualize_res_sim_summ_z < function(data, ...) {
data %>%
.visualize_res_sim_summ() +
geom_hline(aes(yintercept = 0)) +
labs(
title = glue::glue('Model\'s estimated strength (z)'),
x = 'Posterior mean of z'
)
}
visualize_res_sim_summ_z_adj < function(data, ...) {
data %>%
.visualize_res_sim_summ() +
labs(
title = glue::glue('Model\'s estimated goals scored per game'),
x = 'Posterior mean of z transformed to goals'
)
}
export_png <
function(x,
path,
...,
units = 'in',
width = 7,
height = 5) {
ggsave(
plot = x,
filename = path,
units = units,
width = width,
height = height,
...
)
}
n_sim < 1000
path_preds < file_path_out('preds_1.rds')
eval_preds < !fs::file_exists(path_preds)
if(eval_preds) {
.baseline_h < res_sim_summ_1 %>% filter(var == 'baseline_h') %>% pull(mean)
.baseline_a < res_sim_summ_1 %>% filter(var == 'baseline_a') %>% pull(mean)
.extract_tab_max < function(tab) {
tab[ which.max(tab)] %>% names() %>% as.integer()
}
do_predict < function(i) {
data_filt < data %>% slice(i)
.tm_h < data_filt %>% pull(tm_h)
.tm_a < data_filt %>% pull(tm_a)
z_h < res_sim_summ_1_z %>% filter(tm == .tm_h) %>% pull(mean)
z_a < res_sim_summ_1_z %>% filter(tm == .tm_a) %>% pull(mean)
g_h < rpois(n_sim, exp(.baseline_h + (z_h  z_a)))
g_a < rpois(n_sim, exp(.baseline_a + (z_a  z_h)))
tab_h < table(g_h)
tab_a < table(g_a)
result_sign < sign(g_h  g_a)
tab_result < table(result_sign)
result_mode < .extract_tab_max(tab_result)
tibble(
g_h_mode = .extract_tab_max(tab_h),
g_a_mode = .extract_tab_max(tab_a),
result_mode = result_mode,
g_h_mean = mean(g_h),
g_a_mean = mean(g_a)
)
}
set.seed(42)
preds <
tibble(idx = 1:n_gm) %>%
mutate(res = map(idx, do_predict)) %>%
unnest(res)
preds
}
preds_tidy <
preds %>%
gather(key = 'key', value = 'value', idx) %>%
select(idx, key, value)
preds_tidy
.key_lab_g_stem < 'Team Goals'
.key_lab_g_h_prefix < sprintf('Home %s, %%s', .key_lab_g_stem)
.key_lab_g_a_prefix < sprintf('Away %s, %%s', .key_lab_g_stem)
keys_info <
tribble(
~key, ~key_lab,
'g_h_mode', sprintf(.key_lab_g_h_prefix, 'Mode'),
'g_a_mode', sprintf(.key_lab_g_a_prefix, 'Mode'),
'g_h_mean', sprintf(.key_lab_g_h_prefix, 'Mean'),
'g_a_mean', sprintf(.key_lab_g_a_prefix, 'Mean'),
'result_mode', 'Result, Mode'
) %>%
mutate(idx = row_number()) %>%
mutate_at(vars(key_lab), ~forcats::fct_reorder(., idx)) %>%
select(idx)
keys_info
preds_tidy_aug <
preds_tidy %>%
inner_join(keys_info)
preds_tidy_aug
preds_aug <
preds %>%
inner_join(data %>% mutate(idx = row_number())) %>%
mutate_at(
vars(result_mode),
~case_when(
. == 1 ~ 'h',
. == 1 ~ 'a',
. == 0 ~ 't'
)
)
conf_mat_tidy <
preds_aug %>%
count(result_mode, result)
conf_mat_correct <
conf_mat_tidy %>%
group_by(is_correct = result_mode == result)
conf_mat_correct_summ <
conf_mat_correct %>%
summarise_at(vars(n), ~sum(.)) %>%
ungroup() %>%
mutate(frac = n / sum(n))
conf_mat_correct
conf_mat_correct_h < conf_mat_correct %>% filter(is_correct, result == 'h')
conf_mat_correct_t < conf_mat_correct %>% filter(is_correct, result == 't')
conf_mat_correct_a < conf_mat_correct %>% filter(is_correct, result == 'a')
conf_mat_correct_summ_yes < conf_mat_correct_summ %>% filter(is_correct)
viz_conf_mat <
conf_mat_tidy %>%
mutate(frac = n / sum(n)) %>%
mutate(n_lab = sprintf('%s (%s)', scales::comma(n), scales::percent(frac))) %>%
mutate_at(
vars(matches('result')),
~forcats::fct_relevel(., c('h', 't', 'a'))
) %>%
mutate_at(
vars(matches('result')),
~forcats::fct_recode(., Home = 'h', Away = 'a', Tie = 't')
) %>%
ggplot() +
aes(x = result_mode, y = result) +
geom_tile(aes(fill = n)) +
geom_text(aes(label = n_lab), size = 5, fontface = 'bold', color = 'black') +
# scale_fill_manual(limits = c(0, 0.5)) +
scale_fill_viridis_c(alpha = 0.5, begin = 0, end = 1, option = 'E') +
theme_custom() +
theme(
legend.position = 'none',
panel.grid.major = element_blank(),
panel.grid.minor = element_blank()
) +
labs(
title = 'Comparison of Predicted and Actual Game Outcomes',
subtitle = .lab_subtitle,
x = 'Predicted Result',
y = 'Actual Result'
)
viz_conf_mat
viz_g_mode <
preds_tidy_aug %>%
filter(key %>% str_detect('^g.*mode$')) %>%
ggplot() +
aes(x = value, fill = key_lab) +
scale_fill_brewer(palette = 'Set1') +
geom_bar(position = 'dodge', alpha = 0.8) +
theme_custom() +
theme(
panel.grid.major.x = element_blank()
) +
labs(
title = 'Simulated Mode of Goals Scored By Home and Away Teams',
subtitle = .lab_subtitle,
caption = .lab_caption_n_gm,
x = 'Goals Scored',
y = 'Count of Games'
)
viz_g_mode
viz_g_mean <
preds_tidy_aug %>%
# This is done so that the `aes()` can be defined before data is actually passed into the whole ggplot pipeline.
filter(row_number() == 0) %>%
ggplot() +
aes(x = value, fill = key_lab) +
scale_fill_brewer(palette = 'Set1') +
geom_histogram(data = filter(preds_tidy_aug, key == 'g_h_mean'), alpha = 1, binwidth = 0.2) +
geom_histogram(data = filter(preds_tidy_aug, key == 'g_a_mean'), alpha = 0.5, binwidth = 0.2) +
theme(
panel.grid.major.x = element_blank()
) +
theme_custom() +
labs(
title = 'Simulated Mean of Goals Scored By Home and Away Teams',
subtitle = .lab_subtitle ,
caption = .lab_caption_n_gm,
x = 'Goals Scored',
y = 'Count of Games'
)
viz_g_mean
preds_tidy_res <
preds_tidy %>%
filter(key == 'res_mode') %>%
mutate_at(
vars(value),
list(value_lab = ~case_when(
. == 1 ~ 'Home Team Wins',
. == 0 ~ 'Draw',
. == 1 ~ 'Away Team Wins'
))
) %>%
mutate_at(vars(value_lab), ~forcats::fct_reorder(., value))
preds_tidy_res %>% count(value_lab)
viz_result_mode <
preds_tidy %>%
filter(key == 'result_mode') %>%
mutate_at(
vars(value),
list(value_lab = ~case_when(
. == 1 ~ 'Home Team Wins',
. == 0 ~ 'Draw',
. == 1 ~ 'Away Team Wins'
))
) %>%
mutate_at(vars(value_lab), ~forcats::fct_reorder(., value)) %>%
ggplot() +
aes(x = value_lab) +
geom_bar(position = 'dodge') +
theme_custom() +
theme(
panel.grid.major.x = element_blank()
) +
labs(
title = 'Simulated Result of Games',
subtitle = .lab_subtitle ,
caption = .lab_caption_n_gm,
x = NULL,
y = 'Count of Games'
)
viz_result_mode
preds_by_tm <
preds_aug %>%
group_by(tm_h) %>%
count(is_correct = result_mode == result) %>%
mutate(frac = n / sum(n)) %>%
ungroup() %>%
filter(is_correct) %>%
arrange(frac) %>%
mutate(pct = scales::percent(frac)) %>%
select(frac)

This is arguably a “bad” assumption. Research has shown that goal rate per minute increases in the last 15 minutes of a game.↩

This may also be perceived to be a questionable assumption. One may argue that a matchup of “styles”—e.g. an aggressive team against another aggressive team—may distort the results from what would otherwise be expected.↩

This approach is arguably too “simplistic”, but it is certainly a valid approach.↩

See this Pinnacle blog post for a discussion of the topic. (Undoubtedly there are many more articles and papers that explore a similar notion.)↩

There are several notable differences with my work compared to that of Bååth: (1) I use the OpenBUGS software (and the
{R2OpenBUGS}
package) instead of JAGS; (2) I evaluate EPL teams instead of La Liga teams, and over a different time period; (3) I use a “tidy” approach (in terms of packages, plotting, coding style, etc.) instead of a more traditional “base R” approach; (4) I implement a modified version of the second of Bååth’s three proposed models (notably, using different priors).↩ 
Bååth’s work is licensed under the Creative Commons license, which allows for others to adapt the work of another.↩

There are certainly also changes in team strength within seasons, which are even more difficult to model.↩

There are certainly also changes in team strength within seasons, which are even more difficult to model.↩

This explains why there are more than 20 teams in thee data set even though there are only 20 teams in the EPL in a given season.↩
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