Introducing the unicorn index: defining player uniqueness

Introduction

Each year, more and more “unicorns” enter the league. Many define unicorns to be unique big men, including Giannis, Jokic, or Porzingis. A unicorn big man will have some strong quality that’s uncommon among the typical big. For Giannis, it’s ball-handling and speed. For Jokic, it’s passing. For Porzingis, it’s a mix of shooting and mobility.

As more unicorn-like players enter the league, some lose their uniqueness. For example, a decade ago, a player like Porzingis would be unheard of. But, with the prevalence of stretch 5s today, he’s not as unique as we’d expect. To answer this question of how unique a player truly is, we’ll create the unicorn index.

The unicorn index measures the distance of a player’s stats from the average stats of the players in his position. This creates a metric of uniqueness for each player.

Methods

First, we collected 70 different statistics from both Basketball-Reference and NBA.com/Stats. These range from common counting and advanced stats to tracking stats such as touches and drives.

Adding the tracking stats from NBA.com helps us differentiate between players more. For example, only using PPG makes two bigs scoring 20 PPG seem similar. But, if one scores all his points off catch & shoot buckets and the other scores all his points off post plays, they’re distinct players.

The two tables below show the stats we collected.

The first table consists of stats collected from Basketball-Reference. The second table consists of stats collected from NBA.com/Stats. The general and specific touch stats are under “player tracking touches”. The specific shooting stats are under “player tracking shooting efficiency”. The defense stats are under “player tracking defense.”

After collecting the stats, we marked the players into positions. However, these positions were not the typical 5 positions. Instead, we separated players into guards, wings, and bigs. We also restricted the data to players who played at least 41 games and 10 MPG. Note that we used 2017-18 stats for Porzingis (injury) and Davis (trade saga).

To create the unicorn index, we will not calculate player-by-player distance among these raw stats. This would be somewhat useless, as many of the stats relate to each other. For example, VORP is a minutes-scaled stat of BPM, so we can predict it using BPM and MPG. Many of the stats are the sum of other stats (such as WS = OWS + DWS).

Having inter-related stats makes some stats useless. If we know some information, then knowing other related stats won’t give us more information about a player. So, we must first find a way to remove the relationships between these stats.

Principal component analysis

To make the stats independent, we’ll use something called principal component analysis (PCA). PCA transforms our data into uncorrelated components that still capture the variance of our initial data set. So, this lets us have fewer data points to consider while still encapsulating most of the data set.

Each component has no physical meaning in a basketball game. However, raw stats compose these components. So, we can see what stats contributed to each component the most. This will give us an initial idea of what differentiates players within a position.

With each extra component, we can explain more of the data’s variance. So, there are a couple different ways to pick the number of components (n_components). Some optimize n_components like marginal utility. They pick n_components based on benefit in explained variance vs. the previous n_components. However, we’re not concerned with having a very small n_components. So, we’ll say we want enough components to explain a certain percent of the variance. In this case, we’ll pick 90%. There is no specific reason for this; the analysis would work just as well if we explained 95% of the variance.

Because each position has different stats, we’ll do the PCA on each position. The graph below shows the explained variance ratio for each position with varying n_components.

For guards and bigs, the explained variance reaches 90% when n_components = 15. For wings, the explained variance reaches 90% when n_components = 13. This means it’s easier to differentiate between wings than guards and bigs, as it takes fewer components to capture the same amount of variance. Intuitively, we would expect this. There’s a lot more variety in wings than in guards or bigs. For example, most guards shoot, and most bigs can’t. Meanwhile, it’s mixed for wings, where some wings are league’s best shooters, while others don’t shoot.

So, we’ll proceed with n_components = 15 for guards and bigs, and n_components = 13 for wings.

Factor loadings

Each component has a factor loading, or how much our initial raw stats affected the component. This doesn’t matter for the sake of the unicorn index but it’s interesting to look at.

The factor loadings show us the composition of each component. So, the factor loadings for the first component are the first differentiating factor between players in the same position. For example, if these factors were 3P%, PTS, and EFG% in component 1 then shooting is the first differentiating factor. If component 2 had STL, BLK, and DBPM, then we know that after controlling for shooting, defense was the biggest differentiating factor. This follows for the rest of the components. Unfortunately, factor loadings won’t always group together like this. But, we will often see some trends.

Let’s look at the top 5 factor loadings for each component in the guards PCA. They are not in order of greatest to least impact on each component because the difference in effect is tiny. Scroll horizontally to view all factors.

We see that the first differentiating factor between guards is offensive production. After controlling for offensive production, shooting becomes the biggest differentiating factor. After controlling for both offensive production and shooting, ball handling becomes most important. The subsequent components have less of a clear connection between the factors. This is because we have so many touches-related stats and fewer defensive stats. So, we’d expect most groups to have some touch-related stats. This makes it unlikely to find a component composed of only defensive stats.

Next, let’s look at the top 5 factor loadings for each component in the wings PCA.

For wings, it seems that the first differentiating factor is offensive production, as it was for guards. Following offensive production, we see that defense and rebounding are important. Then, shooting is the next differentiating factor. After that, it becomes a bit less clear.

Finally, let’s look at the top 5 factor loadings for each component in the bigs PCA.

Like wings and guards, bigs differentiate themselves by their offensive production first. However, rebounding was also one of the most important factors in the first component. Following offensive production, it seems that shooting was the biggest differentiating factor. This seems surprising at first but it makes sense. Bigs should have the widest range of shooters to non-shooters because some players shoot a lot, while others don’t shoot at all. Following shooting, it seems that ball-handling/facilitation was the next most important factor. This follows the same reasoning as shooting; many bigs don’t pass at all or get touches, but some are among the best passers in the league and touch the ball often (Jokic, Giannis, etc.).

This gives us a general idea of the composition of the principal components.

Calculating the unicorn index

Calculating the unicorn index from the components has a couple steps. Before we jump in, we’ll want to describe the metrics we’re using.

Distance metrics composing the index

To calculate the unicorn index, we’ll use three different distance metrics. They are:

1. Euclidean distance. The Euclidean distance between two vectors (lists of values) equals the square root of the sum of their squared differences. Essentially, if we have two lists, p and q, of 3 elements, their Euclidean distance will be the square root of (p_1 – q_1)^2 + (p_2 – q_2)^2 + (p_3 – q_3)^2 where p_n and q_n are the nth elements the vector.
2. Manhattan distance (or city block/taxicab distance). The Manhattan distance between two vectors equals the sum of the absolute values of their differences. So, the only difference between this and Euclidean distance is that Euclidean distance squares these differences then takes the square root, giving us some different values. So, the Manhattan distance of two lists, p and q, of 3 elements will be |p_1 – q_1| + |p_2 – q_2| + |p_3 – q_3|
3. Chebyshev distance. The Chebyshev distance between two vectors equals the maximum difference between corresponding coordinates in the vectors. So, if we have two lists, p and q, of 3 elements and the difference between p_1 and q_1 (|p_1 – q_1|) is the greatest difference between elements, the Chebyshev distance will equal |p_1 – q_1|.

Calculation of distance

From the positional PCA data, we took the average of each component. This gave us a list of values that the “average” guard, wing, or big will have. Then, we calculated each player’s distance to these values. In each metric, a higher value indicates a higher distance from the positional average. A distance of 0 indicates that the player is perfectly average.

The graphs below show the Euclidean distance, Manhattan distance, and Chebyshev distance for guards.

The same 3 players ranked top 3 in each metric: James Harden, Russell Westbrook, and Ben Simmons. Westbrook and Simmons do have very unconventional stats for a guard.

However, we would not expect Harden to be “unique” for a guard. Because we’re measuring distance, someone could have a high distance by being amazing. So, even though Harden isn’t a “unicorn” by definition, his stats were so unique that he received a high score. We’ll notice this trend again later for other players.

Now, let’s look at these distances for wings. The three graphs below show the distances for wings.

Here, we see a pretty similar thing where the top 3 players (LeBron, Durant, George) all happen to be among the best wings. So, this contributes to them having a high “distance.” Still, they are all unique players. LeBron’s passing, Durant’s scoring, and George’s defense are all special for wings. Note that some of the more odd players here (like Svi Mykhailiuk) made it in because they are barely over the minutes and games played boundary. For example, Mykhailiuk played 42 games and 10.5 MPG. So, his stats are much worse than most players in the data set, making him technically unique.

Now, let’s look at the same results for bigs.

Here, we see that the common unicorn players do have the top distances. Intuitively, we’d expect the bigs to have the easiest to understand distances where the most distant players are both good and unique. This is because guards and wings are generally well-rounded. So, a high-distance guard or wing is either extremely unique (like Ben Simmons) or very good. Meanwhile, because a lot of bigs don’t shoot, pass, or dribble often, it’s easy for a player to differentiate themselves if they do one of these things well. Then, if a player does one of these things well as a big, they’re probably very good.

Now that we’ve seen how each distance metric ranks the players, we can create the final unicorn index.

Converting distances to the unicorn index

To convert these distances to the unicorn index, we’ll first normalize them between 0 and 1. So, the player with the highest distance in each metric for each position will receive a 1. The player with the lowest distance will receive a 0. For the rest of the players, the distribution remains as it was initially, but shifts between 0 and 1. This will let us compare the distances; we can’t do that now because they’re scaled differently. For example, notice that the Manhattan distance is always higher.

Scaling these distances will also give us a way to compare players across positions. It happens to be that in the raw distance metrics, guards had a wider range.

After scaling each distance, we can then take the average of the 3 distances to give us the unicorn index. The unicorn index is between 0 and 1. A player receiving a 1 means they had the highest distance from the average for their position in all 3 of our distance metrics. Therefore, they are the most unique player in that position.

The three graphs below show the unicorn index for guards, wings, and bigs.

Giannis was the only player to get a unicorn index of 1, meaning he is the most unique player in the NBA. Meanwhile, Tyler Johnson is the least unique player in the NBA.

The table below gives the unicorn index for every player who played at least 10 MPG and 41 games last year. The positional rank is how high the given player’s unicorn index ranks among players in their position. Next to the unicorn index, we have the normalized distance metrics. The unicorn index is the average of these normalized metrics. The table below is searchable and sortable.

Conclusion

The unicorn index spotted some conventional unicorns, while also bringing to light how unique some great players are. For example, Harden’s skill set isn’t unheard-of for a guard, but his production is very unique.

We can apply this same process to the league’s entire history to find the most unique player ever. We can also apply this to each player’s individual seasons relative to all player seasons in NBA history. This would give us the most unique season in NBA history. My bet for this would be some of Wilt’s seasons. If we restricted it to the 3-point era, maybe Curry’s unanimous MVP season would be the most unique.