“Who is the best forward in the NHL?” used to be a pretty simple question. As points used to be the main publicly available indicator of offensive production, it was pretty much a given the reigning Art Ross Trophy winner was also typically considered the best forward in the game.

As we all know, the number of different

statistics available to evaluate the performance of forwards has literally

exploded over the last 5 to 10 years.

While two-way forwards always had some value level, the backing of evidence of just how effective certain players can be in other areas than simply point production has changed the way we view NHL players significantly.

On corsica.hockey alone, we

can extract more than 130 (!) different statistics to evaluate a forward at

5v5.

Yet, it is simply impossible for a human being to properly consider 130

statistics at once.

Even if you have a Ph.D. in mathematics, when you try to

reconcile at once the information provided by more than 10 statistics, your

head pretty much begins to spin.

This is why, when evaluating the performance

of a forward, a typical approach is to select a few, maybe 3 or 4, key statistics

considered to be the most important and disregard most others unless something

out of the ordinary stands out. Such an approach is taken, for instance, when we

use a Vollman player usage chart to compare different forwards.

Some effort has

also been made to derive an ultimate “catch-all” statistic to discriminate the

performance of forwards from a single number. We recently attacked this question in the NHLNumbers “Stat of the Union” Roundtable, asking nine of hockey’s brightest minds about how they’d evaluate a player.

The search for an accurate

strategy to discriminate forward performance based on a few statistics is

understandable given the considerations mentioned above, but raises some very

important questions: can we accurately evaluate the performance of forwards

from 3-4 statistics?

Is it possible to develop a single statistic

discriminating the performance of multiple forwards? Or, more specifically:

what is the minimum number of statistics required to properly reflect the

performance of forwards? Luckily for us, we can get a pretty good answer using a

technique called principal component analysis.

## The

fabulous world of dimensionality reduction

Let’s say for a moment that

we evaluate the performance of a set of forwards using two statistics, called

S1 and S2. The forwards are distributed on a S2 *vs *S1 plot as follows:

Based on these two

statistics, the performance of the forward represented by the red dot was

similar to the forward with the blue dot, but very different from the forward

with the green dot. Let’s now make a simple linear regression between S1 and S2,

described by the following black line:

And then, let’s project each

forward on that line. Projecting means that, for each forward, we find the closest

point on the line, as represented by the orange arrows:

Given that each forward has

its own projection, we have essentially created a new statistic S3, which is an

implicit combination of S1 and S2. Considering that the projection has been obtained

from linear regression, our new statistic S3 represents the best statistic that

we can construct (linearly) from S1 and S2 to discriminate the performance of the

forwards from a single number. Here is a plot of each forward on our new S3

axis:

As you can see, S3 correctly

indicates again that the performance of the forward represented by the red dot

was similar to the forward with the blue dot, but very different from the forward

with the green dot. We have thus successfully created a new statistic S3, which

incorporates at once all the relevant information in S1 and S2 required to discriminate

the performance of the forwards from a single number. By considering S3 instead

of S1 and S2, our evaluation of forward performance has become a simpler one

dimensional problem.

Let’s now consider two other

statistics, that we will again call S1 and S2. This time, the forwards are

distributed on a S2 *vs *S1 plot as

follows:

In this scenario, the forwards

represented by the red, blue and green dots all performed very differently.

Let’s once again draw a line obtained by linear regression and project each forward

on that line to create a new statistic S3:

Finally, here is how this

time the forwards are distributed on the new S3 axis:

Our new statistic S3

correctly identifies the very different performance of the forward with the

blue dot with respect to those with the red and green dots, but incorrectly suggests that the performance of the forwards with the green and red dots was nearly identical. The poor accuracy of our new statistic (even though it is the best

one possible!) results from the lack of correlation between S1 and S2, meaning that

these two statistics provide different types of information. For instance, S1

could be a measure of offensive performance and S2 a measure of physicality. In

this case, our new statistic S3 may not successfully discriminate a strong offensive

but not very physical forward from a weak offensive but very physical one.

So, what is the moral of the

story? Sometimes, we can combine statistics to reduce the dimensionality of our

forward evaluation problem and still get an accurate picture of their

performance. Some other times, we simply can’t.

## Let’s

get multidimensional!

Now, back to our 137 (to be

exact) different statistics available on corsica.hockey to evaluate the

performance of forwards at 5v5.

I have extracted all the data for forwards at

5v5 over the last three seasons. I have removed forwards with less than 250 min

of 5v5 time on ice (roughly 20 games played) as well as statistics on FO% and

hits against, which were not available for all forwards, to finally obtain a

table of 1227 forwards x 135 statistics (a forward can appear up to three times

in the table if he has played more than 250 min in all of the previous three

seasons).

We first want to see if these 135 statistics can be combined into a single

new shiny statistic which still accurately reflects the performance of the forwards.

We can do that by projecting all forwards on a line, as in the examples above.

To quantify how well this single statistic discriminates the performance of the

forwards, we calculate the proportion of variance that it explains, which

corresponds to the ratio between the variance of each forward’s projection on

the line and the total variance in the 1227 x 135 table. This proportion of

variance is pretty similar to the *R2*

coefficient that we calculate to quantify the quality of a model’s fitting.

If

a single statistic is not sufficient, we then verify if a combination of two statistics

gives better results. To do so, we project every forwards on a 2-D plane

(instead of a line), which is where that principal component analysis technique

that I have mentioned comes into play. Without getting too much into the math

of it, the best possible combination of two statistics to discriminate all

forwards corresponds to the first two right-singular vectors of the 1227 x 135 table, which we can calculate by singular value decomposition

(*obviously*). We can again calculate

the proportion of variance explained by our two new statistics and if they

still do not contain enough information to accurately reflect the performance

of the forwards, we can project our forwards on a 3-D space and so on.

So, after performing all

these calculations, here is the proportion of variance explained according to

the number of statistics on which we project our forwards:

As you can see, if we

combine the 135 statistics into a single new shiny statistic, it can contain

at most 37% of the information in the 1227 x 135 table.

To keep at least half

of the information provided by the 135 statistics, the best we can do is to

combine them into 3 statistics. At the minimum, we need 10 statistics to

discriminate the performance of the forwards without losing more than 20% of

the information; it is pretty good, and certainly better than having to evaluate

135 statistics at once, but maybe not as good as one would have hoped.

## The

take-home message

If you evaluate the

performance of a forward using 2-3 statistics, it is all well and good, but

keep in mind that you are missing at least half of the story. Also, the

development of a single “catch-all” statistic may sadly be a problem without a

solution.

## Glossary

Term | Definition |

Principal component analysis | A mathematical technique used to combine variables (in this case hockey statistics) together in a way to retain the maximum possible amount of information. |

Dimensionality reduction | The process of reducing the number of variables (in this case hockey statistics) required to investigate a problem (in this case to evaluate the performance of forwards at 5v5). |

Proportion of variance explained | The cost of dimensionality reduction is that some information may be lost. The proportion of variance explained indicates the percentage of information that is retained by the variables (in this case hockey statistics) that we have obtained by dimensionality reduction. |