Data Science – NC233

Using R to build predictions for UEFA Euro 2020

Last friday, Euro 2020, one of the biggest events in International soccer, was kicked off by the inaugural match between Italy and Turkey (Italy won it 3-0). Euros (short for European Championships) are usually held every 4 years, but because of he-who-must-not-be-named, last year’s edition was postponed to this summer, while keeping the name “Euro 2020” (much like the Tokyo Olympics). 4 5 years ago, for Euro 2016, I basically wanted to try some cool methods based on splines on a real-world example. The model ended up doing fairly well, and most importantly it was a lot of fun, so we decided to do it again this year! The model, from training to the web app, is entirely built in R.

Click on this image to access the predictions!

How it works

The model is built on about 12.500 international football matches played since 2006. The main part actually consists in a combination of two models, one that predicts the number of goals scored by the reference team in a given game, the other one that predicts the number of goals they will allow. This basically models offense and defense for each team. A number of models for soccer have used similar ideas in the past, including FiveThirtyEight’s brilliant Soccer Power Index.

The features that are taken into account in the main models are:

Date
Indicator of home field advantage
Teams skills
Tournament importance (ex: Friendly Game rated as 1, Euro or World Cup Game rated as 8, etc.)
Confederation of opponent
Indicator showing if team is in list of “special playstyles” (compsition of the list is selected to maximize prediction accuracy)
Interaction term skill difference x tournament importance

Unfortunately the model does not take into account player-level data. For example, the absence of Milan AC’s superstar Zlatan Ibrahimovic is not taken into account in Sweden’s odds to win it all. Another great improvement could come from using expected goals instead of directly scored goals. The idea is that scored goals are a very small sample in soccer, so there’s much more signal in xG. Unfortunately, computing these expected goals requires play-by-play data, which I don’t have access to.

“I tried a lot of different methods, but none were able to beat my baseline set up with a simple linear regression”

As for the choice of the algorithm, although we had good success last time with splines, I also wanted to try various algorithms from the ML world in order to get the best possible results. In the end, I tried a lot of different methods, but none were able to beat my baseline set up with a simple polynomial regression. Perhaps unsurprisingly to many data science practitioners, linear regression wins again!

"The virgin non-parametric regression vs. The Chad linear regression" — “The virgin non-parametric regression vs. The Chad linear regression” – meme taken from Reddit

This first part of the prediction pipeline returns a double (see image below), which unfortunately we can’t directly use to create meaningful probabilities for the outcomes we’re interested in (group stage games, group rankings, teams final rankings).

Example of number goals predicted by the polynomial regression: results are floats, not integers

In order to get something useful, we have to resort to simulations. Simulations are really necessary in this case since a lot of non-linear factors can influence the final results: group composition, a somewhat convoluted qualifying system, etc. To me these “nonlinearities” are the fun part though! For this reason, we’ll update predictions after each day so that readers who are into this stuff can see how match results affect the likelihood of the global outcomes.

So, how do we transform a floating-point predicted number of goals into simulations? We start by using the predicted goals as parameter of a Poisson distribution. This generates all possible scores for each individual game and assigns a probability to each score. Then for each game we select at random one of the possible outcomes. This selection is done with unequal probabilities: the probability for each outcome to be selected corresponds to the probability indicated by the Poisson model (basically you don’t want to end with as many 8-3s as 1-0s in your simulations!). This is done through functions available in the excellent package sampling (see illustration below).

Screencap of the `draw_onesim` function, which uses sampling::UPsystematic to select an outcome at random with unequal probabilities — The main function used to select game outcomes at random according to probabilities generated by the Poisson process. It uses the method UPsystematic from the sampling package

And then… we repeat the process. 10.000 times, Monte-Carlo style. With a little help from the great package foreach for easy parallelization of the computations. And voila, by counting and averaging the simulated results, we get probabilities for each match, each team occupying each rank in their group, and for each team to reach each stage in the second round. 😊

A small statistical twist

The Poisson distribution is often used in simulations for sports predictions, and it is generally a great tool to use. However, the raw results present flaws that can come bite you if you don’t pay attention. For example, the Poisson tends to generate far fewer draws than expected (11% less in our case, to be precise). In order to account for this, we used one of my favorite tricks: weight calibration (implemented in the package Icarus). Basically, the idea is to modify the probabilities given by the Poisson distribution so that they match a certain pre-known result. There are infinite many ways to go about this, so weight calibration chooses the result that minimizes the distance to the initial probabilities. One of the other benefits of using this method is that we can calibrate on many probabilities at the same time, which we eventually did for:

Total number of goals scored per match and average
Share of draws
Among draws, share of 1-1s, 2-2s and draws with more than 6 goals
Share of draws in matches with “small” difference between teams skills
Share of draws in matches with “large” difference between teams skills

Output from the calibration process given by the Icarus package

Model selection and results

In order to select the best possible model, we ran a whole lot of tests and hyperparameter selection scripts. The main metrics for the selection process were prediction accuracy and the Brier score, which represents the quality of the estimated probabilities. Despite the simplicity of the model, the accuracy of the overall prediction is ~61% +/- 2% (both on the test set and in Cross-Validation), which is in line with most other soccer models. The 3-dimensional Brier score is 0.57 +/- 1%.

We get a clear trio of favorites: Belgium, France, and England. Although the first two are ranked 1st and 2nd in the FIFA rankings, England is probably favored because they’ll have home turf advantage for most of the second round. (But then as Dr Sean Elvidge writes, home turf advantage could be very relative this year ¯_(ツ)_/¯). It’s also interesting to see that leading soccer nations like Germany or Italy are not ranked favorably by our model, contrary to some others. I’m not totally sure I understand why btw, so feel free to let me know in the comments or on Twitter if you think you have an explanation!

Do you like data science and would be interested in building products that help entrepreneurs around the world start and grow their businesses? Shopify is hiring 2021 engineers and data scientists worldwide! Feel free to reach out if you’d like to know more 🙂

Featured image: Logo UEFA Euro 2020, © UEFA

Have you checked your features distributions lately?

tl;dr
Trying to debug a poorly performing machine learning model, I discovered that the distribution of one of the features varied from one date to another. I used a simple and neat affine rescaling. This simple quality improvement brought down the model’s prediction error by a factor 8

Data quality trumps any algorithm

I was recently working on a cool dataset that looked unusually friendly. It was tidy, neat, interesting… the kind of things that you rarely encounter in the wild! My goal was to build a super simple predictor for one of the features. However, I kept getting poor results and at first couldn’t figure out what was happening.

Data sets in tutorials are always cute, data sets in the wild are monstrous — Data sets in tutorials vs data sets in the wild. From Towards AI on Twitter

Here is the dataset I was working with. I tweaked the ids and dates, and muddied the variables a little to make sure I’m not leaking any critical info, but the gist of the problem is absolutely identical to what I was working with (i.e. it’s a real-world dataset^TM, not a toy example).

library(tidyverse)
## Load data
data_magictransfo <- readRDS("data/nc233_rescaling_df_04_2021.rds")

This dataset features 211 distinct units and 3 main variables (X₁, X₂, Y) describing them. We have values for these variables on 7 consecutive dates.

> length(unique(data_magictransfo$unit_id))
[1] 211
> table(data_magictransfo$date_num)

  1   2   3   4   5   6   7 
204 211 211 211 211 211 211

Y is the variable we’re interested in predicting, and we’ll use X₁ and X₂ as predictors. Since this is basically a problem of machine learning on time series, we’ll use the last date as test set, and train our predictor on the first 6 dates. Usual stuff. To measure the accuracy, we will use the ratio between the square root of the MSE and the mean of the true variable for the 7th date:

df_train <- data_magictransfo %>%  filter(date_num <= 6)
df_test <- data_magictransfo %>%
  filter(date_num > 6)

pred_error_ratio <- function(predict_Y, true_Y) {
 
  mse_rescale <- (predict_Y - true_Y)**2
  mse_rescale <- mean(mse_rescale)
  pred_error_ratio <- sqrt(mse_rescale) / mean(true_Y)
 
  return(pred_error_ratio)
}

The first thing I tried was a super simple linear regression:

simple_model <- lm(Y ~ X1 + X2, data=df_train)
df_test$Y_predict_lm <- predict(simple_model, newdata = df_test)

pred_error_ratio_lm <- pred_error_ratio(df_test$Y_predict_lm, df_test$Y)

> pred_error_ratio_lm
[1] 0.3352261

This extra simple model gives a 33% error rate. Not catastrophic, but not great. Literature on this dataset suggests that 5% should be easily obtained though. Maybe the poor performance comes from the simplicity of the model? Let’s fire up some neural networks and see:

library(nnet)

set.seed(1005192119)
nn <- nnet(Y ~ X1+X2, data=df_train %>% select(Y,X1,X2)
             , size=11, decay=1.0e-5, maxit=5000, linout=T)

df_test$Y_predict_nn <- predict(nn, newdata=df_test %>% select(Y,X1,X2), type="raw")
prediction_error_ratio_nn <- pred_error_ratio(df_test$Y_predict_nn, df_test$Y)

> prediction_error_ratio_nn
[1] 0.1481201

Stop cutting corners and start doing what should always be done first: explore the data

¯\_(ツ)_/¯ Definitely far from what I expected. At this point, I should probably stop cutting corners and start doing what should always be done first: explore the data. After a few summaries, I start suspecting that X1 hides a dirty little secret:

density_plot1 <- ggplot(df_train, aes(x=X1, color=date_num)) +  
  geom_density(fill="lightblue", alpha=0.25) +
  NULL
print(density_plot1)

Distribution (density plot) of X1 by date. We see that X1 is clearly sampled from a different distribution for dates 1 and 2 — Distribution of X1 by date. X₁ is clearly sampled from a different distribution for dates 1 and 2. All dates after 3 show identical distributions

It seems like the distribution of X₁ is different between dates 1, 2 and the rest of the dataset! 😱 Let’s zoom in on the first three dates and add vertical dashed lines for the average of X₁ by date:

# Add averages to viz
moments_df <- df_train %>%
  filter(date_num <= 3) %>%
  group_by(date_num) %>%
  summarize(mu = mean(X1), sd = sd(X1))
mu_vec <- moments_df$mu

density_plot3 <- ggplot(df_train %>% filter(date_num <= 3), aes(x=X1, color=date_num)) +
  geom_density(fill="lightblue", alpha=0.25) +
  geom_vline(xintercept=mu_vec, linetype="dashed", color="black", alpha=0.5) +
  NULL
print(density_plot3)

Distribution of X1 for dates 1, 2 and 3 — Distribution of X₁ for dates 1, 2 and 3. Shapes are different for all three dates but averages for dates 2 and 3 are identical

Clearly, X₁ for date 1 is a wildly different thing than other dates. At date 2, the distribution is closer to what it will be in the future: average is very close, but there is still a clear difference in the shape and the width of the tails.

As a data scientist, it’s your job to investigate the data-generating process 🕵️

This problem of features having different distributions by date (or by any other grouping than time, which could very well be possible as well) is common and can arise for multiple reasons. As a data scientist, it’s your job to investigate the data-generating process 🕵️ In this case, X₁ comes from a score a certain authority attributes to each unit. A quick investigation revealed that the scoring method used had changed between dates 1 and 2, and then another time between dates 2 and 3. The change only affected the score and not the rank of the units, which means a simple fix should do 🙌

Now that we’ve diagnosed the problem, let’s try and fix it! Let’s open the math toolbox and… No, don’t go away! I promise it will be simple! 😂 The simplest tool in the box will suffice at first. We’ll try a rescaling of the form:

\( Z = a X_1 + b \)

The expected value and variance of this new variable are easy to compute using well-know formulae:

\( E(Z) = a E(X_1) + b \\ Var(Z) = a^2 Var(X_1) \)

We want the rescaled variable Z to have the same average and variance as the distribution from date 3 onwards. Let’s denote them μ and σ². We thus have:

\( \mu = a \cdot E(X_1) + b \\ \sigma^2 = a^2 \cdot Var(X_1) \)

Since we know E(X₁) and Var(X₁) from the data, this is actually a very simple linear system with two variables (a and b) and two equations. The unique solution is:

\( a = \frac{\sigma}{Var(X_1)}\\ b = \mu ~ – ~ \frac{\sigma}{Var(X_1)} \cdot E(X_1) \)

which we can now apply on our data:

# Distribution parameters
library(zeallot) # Only used to improve code readability

moments2_df <- df_train %>% 
   mutate(date_select = case_when(
     date_num == "1" ~ "1",
     date_num == "2" ~ "2",
     date_num >= "3" ~ "3+",
     TRUE ~ "NA")
   ) %>% 
   select(date_select, X1) %>% 
   group_by(date_select) %>% 
   summarise(mu=mean(X1), sigma=sd(X1))

c(mu_1, mu_2, mu_Z) %<-% moments2_df$mu
c(sigma_1, sigma_2, sigma_Z) %<-% moments2_df$sigma
 
df_train <- df_train %>%
  mutate(X1_rescale = case_when(
    date_num == 1 ~ a1 * X1 + b1,
    date_num == 2 ~ a2 * X1 + b2,
    TRUE ~ X1
  ))

The new density plots look much better!

moments_df_rescale <- df_train %>%  filter(date_num <= 3) %>%
  group_by(date_num) %>%
  summarize(mu = mean(X1_rescale), sd = sd(X1_rescale))
mu_vec_rescale <- moments_df_rescale$mu

density_plot4 <- ggplot(df_train %>% filter(date_num <= 3), aes(x=X1_rescale, color=date_num)) +
  geom_density(fill="lightblue", alpha=0.25) +
  geom_vline(xintercept=mu_vec_rescale, linetype="dashed", color="black", alpha=0.5) +
  NULL
print(density_plot4)

Density plots of rescaled variables — Distribution (density) of rescaled X1 for dates 1, 2 and 3. The distributions are now centered on the same average and have same width

Now we can train our models on the rescaled data:

# New predictions
simple_model_rescale <- lm(Y ~ X1_rescale + X2, data=df_train)

df_test$X1_rescale <- df_test$X1
df_test$Y_predict_rescale <- predict(simple_model_rescale, newdata = df_test)
pred_error_ratio_rescale <- pred_error_ratio(df_test$Y_predict_rescale, df_test$Y)


set.seed(1005192119)
nn_check <- nnet(Y ~ X1_rescale+X2, data=df_train %>% select(Y,X1_rescale,X2)
           , size=13, decay=1.0e-5, maxit=5000, linout=T)

Y_predict_nn_check <- predict(nn_check, newdata=df_test %>% select(Y,X1_rescale,X2), type="raw")
prediction_error_ratio_nn_check <- pred_error_ratio(Y_predict_nn_check, df_test$Y)

The prediction errors go down by a lot! 🎉🎉🎉 8% for the linear model and 2.6% for the neural network:

> pred_error_ratio_rescale
[1] 0.08092733
> prediction_error_ratio_nn_check
[1] 0.02650212

As always, data quality trumps any algorithm, as powerful as it may be!
And lesson learned for me: never trust a cute-looking dataset. Data in the wild is always hard to tame.

To be continued…

Causal Inference cheat sheet for data scientists

Being able to make causal claims is a key business value for any data science team, no matter their size.
Quick analytics (in other words, descriptive statistics) are the bread and butter of any good data analyst working on quick cycles with their product team to understand their users. But sometimes some important questions arise that need more precise answers. Business value sometimes means distinguishing what is true insights from what is incidental noise. Insights that will hold up versus temporary marketing material. In other terms causation.

When answering these questions, absolute rigour is required. Failing to understand key mechanisms could mean missing out on important findings, rolling out the wrong version of a product, and eventually costing your business millions of dollars, or crucial opportunities.
Ron Kohavi, former director of the experimentation team at Microsoft, has a famous example: changing the place where credit card offers were displayed on amazon.com generated millions in revenue for the company.

The tech industry has picked up on this trend in the last 6 years, making Causal Inference a hot topic in data science. Netflix, Microsoft and Google all have entire teams built around some variations of causal methods. Causal analysis is also (finally!) gaining a lot of traction in pure AI fields. Having an idea of what causal inference methods can do for you and for your business is thus becoming more and more important.

The causal inference levels of evidence ladder

Hence the causal inference ladder cheat sheet! Beyond the value for data scientists themselves, I’ve also had success in the past showing this slide to internal clients to explain how we were processing the data and making conclusions.

The “ladder” classification explains the level of proof each method will give you. The higher, the easier it will be to make sure the results from your methods are true results and reproducible – the downside is that the set-up for the experiment will be more complex. For example, setting up an A/B test typically requires a dedicated framework and engineering resources.
Methods further down the ladder will require less effort on the set-up (think: observational data), but more effort on the rigour of the analysis. Making sure your analysis has true findings and is not just commenting some noise (or worse, is plain wrong) is a process called robustness checks. It’s arguably the most important part of any causal analysis method. The further down on the ladder your method is, the more robustness checks I’ll require if I’m your reviewer 🙂

I also want to stress that methods on lower rungs are not less valuable – it’s almost the contrary! They are brilliant methods that allow use of observational data to make conclusions, and I would not be surprised if people like Susan Athey and Guido Imbens, who have made significant contributions to these methods in the last 10 years, were awarded the Nobel prize one of these days!

causal_cheat_sheet — The causal inference levels of evidence ladder – click on the image to enlarge it

Rung 1 – Scientific experiments

On the first rung of the ladder sit typical scientific experiments. The kind you were probably taught in middle or even elementary school. To explain how a scientific experiment should be conducted, my biology teacher had us take seeds from a box, divide them into two groups and plant them in two jars. The teacher insisted that we made the conditions in the two jars completely identical: same number of seeds, same moistening of the ground, etc.
The goal was to measure the effect of light on plant growth, so we put one of our jars near a window and locked the other one in a closet. Two weeks later, all our jars close to the window had nice little buds, while the ones we left in the closet barely had grown at all.
The exposure to light being the only difference between the two jars, the teacher explained, we were allowed to conclude that light deprivation caused plants to not grow.

Sounds simple enough? Well, this is basically the most rigorous you can be when you want to attribute cause. The bad news is that this methodology only applies when you have a certain level of control on both your treatment group (the one who receives light) and your control group (the one in the cupboard). Enough control at least that all conditions are strictly identical but the one parameter you’re experimenting with (light in this case). Obviously, this doesn’t apply in social sciences nor in data science.

Then why do I include it in this article you might ask? Well, basically because this is the reference method. All causal inference methods are in a way hacks designed to reproduce this simple methodology in conditions where you shouldn’t be able to make conclusions if you followed strictly the rules explained by your middle school teacher.

Rung 2 – Statistical Experiments (aka A/B tests)

Probably the most well-known causal inference method in tech: A/B tests, a.k.a Randomized Controlled Trials for our Biostatistics friends. The idea behind statistical experiments is to rely on randomness and sample size to mitigate the inability to put your treatment and control groups in the exact same conditions. Fundamental statistical theorems like the law of large numbers, the Central Limit theorem or Bayesian inference gives guarantees that this will work and a way to deduce estimates and their precision from the data you collect.

Arguably, an Experiments platform should be one of the first projects any Data Science team should invest in (once all the foundational levels are in place, of course). The impact of setting up an experiments culture in tech companies has been very well documented and has earned companies like Google, Amazon, Microsoft, etc. billions of dollars.

Of course, despite being pretty reliable on paper, A/B tests come with their own sets of caveats. This white paper by Ron Kohavi and other founding members of the Experiments Platform at Microsoft is very useful.

Rung 3 – Quasi-Experiments

As awesome as A/B tests (or RCTs) can be, in some situations they just can’t be performed. This might happen because of lack of tooling (a common case in tech is when a specific framework lacks the proper tools to set up an experiment super quickly and the test becomes counter-productive), ethical concerns, or just simply because you want to study some data ex-post. Fortunately for you if you’re in one of those situations, some methods exist to still be able to get causal estimates of a factor. In rung 3 we talk about the fascinating world of quasi-experiments (also called natural experiments).

A quasi-experiment is the situation when your treatment and control group are divided by a natural process that is not truly random but can be considered close enough to compute estimates. In practice, this means that you will have different methods that will correspond to different assumptions about how “close” you are to the A/B test situation. Among famous examples of natural experiments: using the Vietnam war draft lottery to estimate the impact of being a veteran on your earnings, or the border between New Jersey and Pennsylvania to study the effect of minimum wages on the economy.

Now let me give you a fair warning: when you start looking for quasi-experiments, you can quickly become obsessed by it and start thinking about clever data collection in improbable places… Now you can’t say you haven’t been warned 😜 I have more than a few friends who were ~~lured into~~ attracted by a career in econometrics for the sheer love of natural experiments.

Most popular methods in the world of quasi-experiments are: differences-in-differences (the most common one, according to Scott Cunnigham, author of the Causal Inference Mixtape), Regression Discontinuity Design, Matching, or Instrumental variables (which is an absolutely brilliant construct, but rarely useful in practice). If you’re able to observe (i.e. gather data) on all factors that explain how treatment and control are separated, then a simple linear regression including all factors will give good results.

Rung 4 – The world of counterfactuals

Finally, you will sometimes want to try to detect causal factors from data that is purely observational. A classic example in tech is estimating the effect of a new feature when no A/B test was done and you don’t have any kind of group that isn’t receiving the feature that you could use as a control:

Slightly adapted from CausalImpact‘s documentation

Maybe right new you’re thinking: wait… are you saying we can simply look at the data before and after and be allowed to make conclusions? Well, the trick is that often it isn’t that simple to make a rigorous analysis or even compute an estimate. The idea here is to create a model that will allow to compute a counterfactual control group. Counterfactual means “what would have happened hadn’t this feature existed”. If you have a model of your number of users that you have enough confidence in to make some robust predictions, then you basically have everything

There is a catch though. When using counterfactual methods, the quality of your prediction is key. Without getting too much into the technical details, this means that your model not only has to be accurate enough, but also needs to “understand” what underlying factors are driving what you currently observe. If a confounding factor that is independent from your newest rollout varies (economic climate for example), you do not want to attribute this change to your feature. Your model needs to understand this as well if you want to be able to make causal claims.

This is why robustness checks are so important when using counterfactuals. Some cool Causal Inference libraries like Microsoft’s doWhy do these checks automagically for you 😲 Sensitivity methods like the one implemented in the R package tipr can be also very useful to check some assumptions. Finally, how could I write a full article on causal inference without mentioning DAGs? They are a widely used tool to state your assumptions, especially in the case of rung 4 methods.

(Quick side note: right now with the unprecedented Covid-19 crisis, it’s likely that most prediction models used in various applications are way off. Obviously, those cannot be used for counterfactual causal analysis)

Technically speaking, rung 4 methods look really much like methods from rung 3, with some small tweaks. For example, synthetic diff-in-diff is a combination of diff-in-diff and matching. For time series data, CausalImpact is a very cool and well-known R package. causalTree is another interesting approach worth looking at. More generally, models carefully crafted with domain expertise and rigorously tested are the best tools to do Causal Inference with only counterfactual control groups.

Hope this cheat sheet will help you find the right method for your causal analyses and be impactful for your business! Let us know about your best #causalwins on our Twitter, or in the comments!