# The NFL Fractals

American football is certainly a complex sport. Given that complex systems are described many times through fractals (and we have also used fractlas in this blog to describe basketball players) can we get a fractal-based description of NFL teams? It turns out we can!

One of the benefits of using fractal theory to describe complex systems is that a non-integer dimensionality can be assigned to them. Therefore, comparissons between similar systems can be made possible in a way that is not when using  topological dimensions. For example, two NFL teams can be though of as having a topological dimensionality of 2 since they are spreading on a 2D space (in practice since the progress in the one dimesnion only matters, they can be thought of as being single dimensional).  However, it should also be clear that two teams do not necessarily march down the field the same way (e.g., some teams prefer to advance a few yards per snap by running the ball while others are more prone to deep passes).  This behavior can be captured by the so-called fractal dimension or correlation dimension.

In particular, let us consider a set of points $\mathcal{S}$. With $C(r)$ being the fraction of pairs of points from $\mathcal{S}$ that have distance smaller or equal to $r$, $\mathcal{S}$ behaves like a fractal with intrinsic fractal dimension $D_2$ in the range of scales $r_1$ to $r_2$ iff: $C(r) \propto r^{D_2},~r_1\le r\le r_2$. An infinitely complicated set $\mathcal{S}$ would exhibit this scaling over all possible ranges of $r$. However, real objects are finite and the equation holds only over a specific range of scales. For instance, a cloud of points uniformly distributed the unit square, has intrinsic dimension $D_2 = 2$, for the range of scales $[r_{min},1]$, where $r_{min}$ is the smallest distance among the pairs of $\mathcal{S}$.

In the case of NFL teams the point set  $\mathcal{S}$ can be considered to be the coordinates of the team at the beginning of every snap.  This essentially captures the way the teams progress towards the goal line.  Different teams progress differently.  Does this impact the probability of winning? Using data from the last 7 regular seasons I computed the fractal dimensionality of every NFL team during every game.  I then built a simple logistic regression model to examine how the fractal dimensionality of a team is correlated with its probability of winning.  The result is presented in the following figure.

As we can see there is a negative relation between the fractal dimensionality and the probability of winning.  This relation is also statistically significant (p-value = 0.002). What does this mean? Simply put the less uniform the advancement on the field the higher the chances of winning.  Of course, the notion of fractal dimensionality is very complicated and further examination is needed.  However, it is apparent that things are to be learnt by studying the teams through fractal theory!

Using the win-loss percentages of the last regular season I also calculated the spearman ranking correlation between the median fractal dimensionality of each NFL team and its final standing.

There is a significant negative correlation (-0.4, p-value= 0.02) as we can also see from the above scatter plot.  Given the value of the correlation as well as the logistic regression results, the fractal dimensionality itself cannot explain all of the winning-loss performance of a team.  However, it is a statistically significant explanatory variable that “adds” about 5% to the probability of winning per 0.1 decrease in the fractal dimensionality.

Drawing analogies between sports competitions and complex systems has the ability to provide a whole new way of viewing the teams and their performance that is yet to be explored!