A dumb introduction to league football and Principal Component Analysis (PCA)

Prologue

This blog post works best for those who know nothing about league football (soccer). In fact, the less you know about how a football league works, the fewer preconceived notions you’ll have about it, and the grander the outcomes of our statistical analysis would sound.

But this post is not just about football leagues. In the end, it is for those who (like me) are starting to analyze multidimensional datasets (or, more simply, datasets with many columns). Let’s get started by looking at our dataset.

The ISL 2024-25 dataset

The table below shows the outcomes of the 2024-25 season of the Indian Super League (ISL). The teams are arranged in alphabetical order of names.

Club	MP	W	D	L	GF	GA	GD
Bengaluru FC	24	11	5	8	40	31	9
Chennaiyin FC	24	7	6	11	34	39	-5
East Bengal FC	24	8	4	12	27	33	-6
FC Goa	24	14	6	4	43	27	16
Hyderabad FC	24	4	6	14	22	47	-25
Jamshedpur FC	24	12	2	10	37	43	-6
Kerala Blasters FC	24	8	5	11	33	37	-4
Mohammedan SC	24	2	7	15	12	43	-31
Mohun Bagan SG	24	17	5	2	47	16	31
Mumbai City FC	24	9	9	6	29	28	1
NorthEast United FC	24	10	8	6	46	29	17
Odisha FC	24	8	9	7	44	37	7
Punjab FC	24	8	4	12	34	38	-4

Abbreviation	Meaning
MP	Matches Played
W	Wins
D	Draws
L	Losses
GF	Goals For (Goals scored by the team)
GA	Goals Against (Goals conceded by the team)
GD	Goal Difference (GF minus GA)

The objective

Which team do you think won the ISL? The one with the most W? Or maybe, the one with the least GA? In the end, it all depends on how the league committee ranks the teams. It can also be something non-intuitive and bizarre (only to those who know league football), like the team with the fewest Ds would be crowned champions.

Our goal here is to figure out the ranking criteria that the league committee uses to decide the final standings.

Getting the dataset ready

There are 13 teams, and every team played 24 matches. Apart from the MP column (which is the same across all teams), we have six columns. Are all of them independent of each other?

Well, at least one is not. GD is a linear combination of GF and GA, and we’ll leave it for our further analysis. Thus, the (apparently) 5D independent dataset looks like:

Club	W	D	L	GF	GA
Bengaluru FC	11	5	8	40	31
Chennaiyin FC	7	6	11	34	39
East Bengal FC	8	4	12	27	33
FC Goa	14	6	4	43	27
Hyderabad FC	4	6	14	22	47
Jamshedpur FC	12	2	10	37	43
Kerala Blasters FC	8	5	11	33	37
Mohammedan SC	2	7	15	12	43
Mohun Bagan SG	17	5	2	47	16
Mumbai City FC	9	9	6	29	28
NorthEast United FC	10	8	6	46	29
Odisha FC	8	9	7	44	37
Punjab FC	8	4	12	34	38

Let’s look at the numbers in each column. Every column has a different range. For example, W can only have values between 0 and 24 (a team can win none or all of their matches). We can apply the same logic to L and D as well. However, there’s no theoretical lower or upper limit to GF and GA. Broadly speaking, they are higher than W, L, or D in our data. Throwing columns with very different value ranges into a variance-finder algorithm like PCA confuses it, as it tries to over-value the columns with higher magnitudes. We’ll be careful about this and scale the numbers in a way that we’re only dealing with “how” they are distributed, and not their actual magnitudes.

The league table now looks like the one below. A value near 0.000 means it is close to the league average.

Club	W	D	L	GF	GA
Bengaluru FC	0.510	-0.433	-0.286	0.567	-0.433
Chennaiyin FC	-0.551	0.079	0.510	-0.047	0.568
East Bengal FC	-0.286	-0.944	0.775	-0.764	-0.183
FC Goa	1.305	0.079	-1.346	0.875	-0.933
Hyderabad FC	-1.346	0.079	1.305	-1.276	1.568
Jamshedpur FC	0.775	-1.967	0.245	0.260	1.068
Kerala Blasters FC	-0.286	-0.433	0.510	-0.150	0.317
Mohammedan SC	-1.876	0.590	1.570	-2.301	1.068
Mohun Bagan SG	2.101	-0.433	-1.876	1.284	-2.309
Mumbai City FC	-0.020	1.613	-0.816	-0.559	-0.808
NorthEast United FC	0.245	1.102	-0.816	1.182	-0.683
Odisha FC	-0.286	1.613	-0.551	0.977	0.317
Punjab FC	-0.286	-0.944	0.775	-0.047	0.443

PCA-ing

Imagine a 5D space. It’s not easy to imagine one, but we’ll have to pretend that we can for a while. Each dimension of this 5D space is defined by an axis that corresponds to a column in our dataset, and each team is a data point. The idea is to draw lines through the data cloud along which the cloud is the widest, or, in other words, varies the most. These lines need to be orthogonal to each other so that they represent variations originating from completely independent factors. We call these lines Principal Components, or simply, PCs.

PC1 depicts the direction along which the cloud is the widest, PC2 depicts the direction along which the cloud is the second-widest, and so on. Can you guess the directions along which the first two PCs might be oriented in the image below?

This image is AI generated

Let’s first look at the amount of variation in the ISL dataset our PCs capture. You can think of the numbers on the y-axis as R-squared we commonly use in regression models.

The first two PCs capture a little more than 90% of the variation. In other words, most of the patterns in our dataset can be explained by just two lines, instead of five! The remaining variance (<10%) is likely statistical noise and doesn’t mean anything in real life, so we won’t worry about it much.

What are the PCs composed of? Seems like a weird question to ask: they are just lines depicting the direction of maximum variation, right? Why should they be “composed of” anything?

But every PC can also be thought of as a weighted linear combination of the five axes (or our five data columns). In fact, this is what PCA does: it calculates these weights automatically from our dataset. We call these weights “loadings”. We’ll now look at the recipe for the first two PCs.

PC1 has a strong positive dependence on W and GF, and a strong negative dependence on L and GA. This is the strongest pattern in the league: on average, teams that score more goals and concede fewer, win more and lose less (note that the number of matches is fixed). Makes sense, right? PC1 discovers this pattern and clumps all of these four columns into a single PC.

PC2 is heavily dominated by D, the only major leftover pattern in the dataset. In fact, PC2 – by nature – isn’t much different from the D column of our dataset.

Another cool geometric way to look at the (i) mutual relationship between the columns, as well as (ii) their relationship to the PCs, is the biplot. The smaller the angle between the lines, the more positively correlated they are. Interestingly, L and GA are even more positively correlated than W and GF. As the angle increases to 90°, the correlation drops to zero. For example, GF and D are at about 90° and aren’t correlated. As we go past 90°, the negative correlation goes up. Lines opposite to each other (or, in other words, at an angle of 180°) are perfectly negatively correlated. In our dataset, teams that win more lose less. Also note that the D arrow is aligned along PC2 and almost perpendicular to PC1; the others are aligned along PC1 but perpendicular to PC2.

Predicting the ISL champions

That’s what there is to PCA. It identifies correlations among the dataset’s columns (if any) and reduces the number of columns needed to describe the dataset’s major patterns. In our case, we have reduced the 5D ISL dataset to a more manageable 2D dataset. But do the first two PCs have any predictive ability? Can they predict who the champions were, and who finished at the bottom?

The short answer is: not necessarily. PCA doesn’t know how the league committee ranks the teams.

Nevertheless, let’s calculate each team’s PC scores and arrange them in descending order. More simply, let’s evaluate the following equations for each team, and arrange the numbers in descending order:

PC1 score = 0.507W + 0.036D - 0.526L + 0.480GF - 0.485GA

PC2 score = 0.301W - 0.925D + 0.178L + 0.077GF + 0.128GA

We get the following table. I’ve also included the actual ISL standings in the last column. Yes, Mohun Bagan SG were the champions of India in the 2024-25 season!

Standings based on PC1	Standings based on PC2	Real standings
Mohun Bagan SG	Jamshedpur FC	Mohun Bagan SG
FC Goa	Punjab FC	FC Goa
NorthEast United FC	East Bengal FC	NorthEast United FC
Bengaluru FC	Mohun Bagan SG	Bengaluru FC
Mumbai City FC	Bengaluru FC	Jamshedpur FC
Odisha FC	Kerala Blasters FC	Mumbai City FC
Jamshedpur FC	FC Goa	Odisha FC
Kerala Blasters FC	Chennaiyin FC	Kerala Blasters FC
Punjab FC	Hyderabad FC	Punjab FC
Chennaiyin FC	Mohammedan SC	East Bengal FC
East Bengal FC	NorthEast United FC	Chennaiyin FC
Hyderabad FC	Odisha FC	Hyderabad FC
Mohammedan SC	Mumbai City FC	Mohammedan SC

Wow, PC1 is very accurate at predicting the actual standings! Is this a mere coincidence? Yes and no. We were both careful and lucky.

Is coincidence just being careful and lucky?

The league calculates the points for every team using the following equation (three points for a win, one for a draw, and none for losses or goals scored/conceded):

point = 3W + 1D + 0L + 0GF + 0GA

It also uses tiebreakers when two or more teams have the same number of points. Goal difference (GF minus GA) is the first tiebreak. If the teams are still tied, GF is taken into account.

The PC1 score equation resembles this ranking criterion. This happens partly for two reasons:

(i) We’ve been careful while picking the columns on which we performed the PCA. If we decided to use dataset columns such as mean home stadium attendance or mean grass length on the training pitch, the PCs wouldn’t make much sense when it comes to predicting the final standings. We also didn’t leave out any necessary column.

(ii) We’re lucky that the league uses a linear combination of columns as the ranking criteria. If they used a system that’s something like the one below, we’d have run into trouble, as PCA can only linearly combine the columns.

point = 3W^3 + 1D^2 + …

But the resemblance is still uncanny. Why would the direction of maximum variance be such a good predictor of the league standings?

We’ve actually figured out the fundamentals of league football. To repeat our observations from the biplot: since the number of matches is fixed, a team winning essentially means a team loses. Also – on average – to win games, you need to have a higher GF than GA. Humans deliberately decided to set this pattern as the ranking criterion: teams that score more goals and concede fewer, win more and lose less – and are essentially considered better.

Epilogue

Sometimes, nature is kind and “sets” the direction(s) of large variance(s) in environmental datasets as good predictors of a target variable of our choice.

To put it simply, more often than not, variance in the real world isn’t just “spread” – it is a measure of the dominant physical forces/gradients. PCA – albeit blindly – finds that force. We can use this to search for linear relationships between smartly chosen environmental predictors and a target.