# Magnus 5: xG, Shooting, and Goalie-ing

## Estimating Shot Difficulty

##### This is the updated shooting and goaltending model, which is closely related to the model I introduced in the fall of 2020. Much of the exposition here is copied from that write-up.

Although shots are the basic currency of hockey, it is obvious that not all shots are equally difficult to create, nor equally difficult to score, nor equally dificult to save. The model described here estimates the goal likelihood of different patterns of shots, our main interest. Along the way we will also obtain estimates of shooting and goaltending impact.

## Vital Theoretical Preliminaries

My animating assumption is that all of the skaters (between six and eleven, in total, usually) are working together both to generate shots for their team and to suppress the generation of shots by the other team. In principle, I consider all of the skaters equally able to affect both processes, in the long run, even if a given skater (for either team) might be only minimally involved with a given shot. All of play in all three zones leading up to and including the decision by a given player to shoot the puck, I understand to be the product of the (combined) ability of all of the skaters, and I call the goal likelihood of shots generated, at the moment the decision to shoot is made, the "expected goal probability", or xG for short. Then, in addition to the xG of the pattern to which a given shot conforms, the shooter themself can, in principle, affect the goal likelihood of the shot, by shooting the puck skilfully, or perhaps (as I might) by woefully butchering whatever chances they and their teammates (and opponents) have conspired to generate. This ability I call "finishing impact" or "shooting talent". Finally, a goaltender can, in principle, affect the goal likelihood of a shot after it is taken, by sneakily interposing their body between the puck and the goal, or, (again as I might do) contriving to fail to do so.

This three-fold division&em;all of the skaters collectively produce the shot, the shooter shoots, and the goalie endeavours to save&em;is the animating idea behind the model I describe here. Even at this very basic level these choices have important consequences. For instance, a player who is skilled at receiving passes in (relatively) less-dangerous shooting positions and then skating untrammelled with the puck to more-dangerous locations will score more often for that reason, and that skill will appear in my accounting in their impact on xG (which will increase, since their team is taking more dangerous shots) and not in their finishing impact (which will presumably decrease, since they are shooting from locations where goals are easier to come by). Similarly, including goaltender effects only on shots already taken prevents us from making any estimate of goaltenders' impact on xG, conceded or generated, from, say, their tendency to handle the puck.

Throughout this article, when I say "shot" I will mean "unblocked shot", that is, goals, saves, and misses (including shots that hit the post or the crossbar). All shots in all strength situations that are taken against a team with a goaltender on the ice are considered.

#### Odds vs Probability

The machinery here uses the goal odds of a shot throughout, that is, that ratio of the probability of a shot going in the net to the probability of the shot not going in the net. A shot with a goal probability of 50% (that is, of 0.5) has odds of 1, a shot with a goal probability of 20% has odds of 0.20/0.80 = 0.25, and so on. In particular we do not use the word "odds" as if it were a synonym for probability, as if we were craven headline writers.

## Method

Once a shot is being taken by a given player from a certain spot against a specific goaltender, I estimate the probability that such a shot will be a goal. This process is modelled with a linear model, and fit with a generalized ridge logistic regression. For a detailed exposition about how such models can be fit, please see section five of Wessel van Wieringen's excellent lecture notes. Briefly: I use a design matrix $$X$$ for which every row encodes a shot with the following columns:

• Two indicators for the shooter; (in 2019-2020 there were 879 shooters)
• One for tip/deflections
• Another for all other shot types
• An indicator for the goaltender; (in 2019-2020 there were 86)
• A set of geometric terms and shot types, described below;
• An indicator for "rush shots", that is, in-zone shots for which the previous recorded play-by-play event is in a different zone and no more than four seconds prior;
• An indicator for "rebound shots", that is, shots for which the previous recorded play-by-play event is another shot taken by the same team no more than three seconds prior;
• An indicator for teams which are leading and another for teams which are trailing; to be interpreted as representing change in configurations surrounding shots compared to when teams are tied;
• Four indicators for different skater strength situations:
• SH for short-handed shots, that is, ones taken by a team with fewer skaters;
• PPv3 for shots taken by a team with more than three skaters against a team with exactly three skaters;
• PP for all other power-play shots, that is, ones taken by a team with more skaters;
• 3v3 for play where both teams have exactly three skaters (mostly in overtime)
All shots are assigned exactly one of the above indicators, which should all be understood as the change compared to a similar shot at even-strength, that is, all 4v4 and 5v5 shots gathered together.
• An interaction term for shots which are slapshots and also on the power-play of any kind, this term is meant to proxy for one-timers.

### The Geometric Terms

The various shot types (wrist, slap, backhand, tip, and wraparound) are all qualitatively different in execution and, we observe, arise in different situations. In particular, the pattern of shot locations is quite different for each shot type. Consequently, I choose to model the geometry of each shot type separately. To do so flexibly, consider a hexagonal grid of cells, centred on the split line, covering the offensive zone. To each of the four main shot types, I define a "fabric", that is, a subset of these cells which contain at least 95% of all of the shots of that type since 2007-2008.

• Wrist/Snap shots (388 hexes)
• Slap shots (682 hexes)
• Backhand shots (126 hexes)
• Tips and Deflections (126 hexes)

Each hex in each fabric corresponds to a model covariate, in addition to five indicator covariates for each of the five shot types. The purpose, of course, of making such fabrics is to account for how the goal likelihood of a given shot depends on where the shot is taken. For wraparounds, I found that shot locations clustered very tightly in the two obvious locations (near the goalposts) and that the spatial variation in goal likelihood was extremely small. Thus, I chose to use a simple indicator for wraparounds, without encoding additional geometric detail for those shots.

I especially appreciate Michael Schuckers, who brought the importance of attacking geometric variation separately by shot type to my attention.

Every shot is assigned a single location indicator. If a shot's location does not fall into a hex for the associated fabric, I mark it either as:

• "below the goal line",
• "out of zone" (for shots from beyond the blue line), or
• "perimeter", for in-zone shots above the goal line but far from the goal.

New in this version of the model is an additional distance term, only for slapshots. Distance from the centre of the net (0,89) is divided by 89ft, so that a distance of 1 is at the intersection of the split line with the blue line. I added this term because in every year, but especially 2019-2020, the variation due to location in goal odds for slapshots is larger than the location-variation for other shot types. Using an extra term distance term here permits me to keep the hex-to-hex fusion penalties (described below) consistent from fabric to fabric and also consistent through all three axes of the fabric.

## Fitting

This section is quite technical and explains the mathematical intricacies of how I fit the above model with NHL data to obtain covariate estimates. If you like you can skip to the results section below. If you read the above section with impatience, you should skip forward; if you read about all those more-than-a-thousand-different hex coordinates and thought "Micah you poor sap this is gonna be overfit to hell and back" you should read this section first.

The observation vector $$Y$$ is 1 for goals and 0 for saves or misses. The model itself is the a generalized linear one: $$Y \sim l\left(X\beta\right)$$ where $$\beta$$ is the vector of covariate values and $$X$$ is the design matrix described above, and $$l$$ is the logistic function $$x \mapsto l(x) = \frac{1}{1 + \exp(-x)}$$ (extended here to vectors pointwise) after which the regression is named.

Logistic regression coefficient values (the entries of $$\beta$$ which we are trying to find) can be difficult to interpret, but negative values always mean "less likely to become a goal" and positive values mean "more likely to become a goal". To compute the goal probability of a shot with a given description (encoded by a particular row $$X_i$$ of $$X$$): form the sum $$X_i\beta$$ of the model covariates to obtain a number, and then apply the logistic function to it to obtain the goal probability.

The logistic function is very convenient for modelling probabilities, since it monotonically takes the midpoint of the number line (that is, zero) to 50% while taking large negative numbers to positive numbers close to zero and very large positive numbers to positive numbers close to one.

The model is fit by maximizing the likelihood of the model, that is, for a given model, form the product of the predicted probabilities for all of the events that did happen (90% chance of a save here times 15% of that goal there, etc.). This product, with one factor for every shot in the data at hand, is called the likelihood, $$L$$. Large products are awkward, so instead of maximising $$L$$ directly we instead solve the equivalent problem of maximizing the logarithm of the likelihood, denoted by $$\mathcal L$$.

Before we compute the covariate values $$\beta$$ which minimize the log-likelihood $$\mathcal L$$, we bias the results with two so-called ridge penalty terms. These penalty encode our prior knowledge about the terms of the model before we consider the data at hand. The first ridge penalty encodes "static" information (things that we know about hockey generally) and the second one encodes "dynamic" prior information, that is, things learned about the covariates from previous years. By fitting the model with penalties, we obtain estimates which represent a compromise between the data at hand and our pre-existing understanding of what the covariates in the model mean.

#### Static penalties

The static penalties are encoded as a matrix $$K$$ and a term of the form $$\beta^TK\beta$$ is subtracted from the overall model likelihood. The entries of $$K$$ correspond to pairs of covariates, and we have at our disposal two different types of penalties. We can use diagonal entries $$(i,i)$$ of $$K$$ to penalize large a large covariate value in the $$i$$-th entry, and we can use so-called "fusion" penalties to penalize differences between two specified covariates. In our case we use a diagonal penalty of 100 for each goalie and shooter term, a diagonal penalty of 0.1 for each fabric hex, and a fusion penalty of 5 between each two adjacent hexes. The other terms (the strength indicators, the leading/trailing/rush/rebound indicators, the neutral zone/below goal line/shot type indicators) are unpenalized, so at these terms the model behaves like one fit with ordinary least squares.

It may be helpful to think of attaching elastic bands to each covariate. The diagonal penalties are very strong bands which attach each player value to zero, very weak bands that attach each hex value to zero, and medium-strength bands attaching each hex to its neighbours. Then, after the bands are prepared, the data is permitted to flow in, to push the covariate values while the tension in the bands constrains the resulting fit.

The substantial diagonal penalty for shooters and goalies encodes our prior understanding that all of the shooters and goalies are (by definition) NHL players, whose abilities cannot therefore are understood to be not-too-far from NHL average (that is, zero). The hex penalties work together to allow slow variation in the impact of geometry, encoding our prior belief that shots from nearby locations ought to have similar properties for that reason. In particular, fusing these many hex terms to one another in this way effectively lowers the number of covariates in the model, presumably helping to mitigate over-fitting.

#### Dynamic Penalties

In addition to the above penalties, I want to suitably accumulate specific knowledge learned from previous years when I form estimates of the impacts of the same factors in the future&em;in particular we imagine that our estimates for players describe athletic ability, which varies slowly. If you were truly interested only in a single season (or any length of games, thought of as a unity for whatever secret purposes of your own), these dynamic penalties would not be relevant. However, I am interested in fitting this model over each season from 2007-2008 until the present, and so I want to preserve the information gained from each season as I pass to the next. As we shall see, fitting this model produces both a point estimate and an uncertainty estimate for each covariate. The point estimates can be gathered into a vector $$\beta_0$$ of expected covariate values, and the uncertainties can be gathered into the diagonal of a matrix $$\Lambda$$, and then a penalty $$(\beta - \beta_0)^T \Lambda (\beta - \beta_0)$$ can be subtracted from the overall likelihood. For players for whom there is no prior (rookies, for instance, but also all players in 2007-2008, since I do not have any data for before that season), I use a prior of 0 (that is, average) with a very mild diagonal penalty of 0.001.

#### Computational delicacies

In the end, then, the task at hand to fit the model is to discover the covariate values $$\beta$$ which maximize the penalized log-likelihood, $$\mathcal L - \beta^T K \beta - (\beta - \beta_0)^T \Lambda (\beta - \beta_0)$$

Simple formulas for the $$\beta$$ which maximixes this penalized likelihood do not seem to exist, but we can still find it by iteratively. (following a very-slightly modified treatment of section 5.2 of these previously mentioned notes): Specfically, beginning with $$\beta_0$$ as obtained from prior data (or using the zero vector, where necessary), we will define a way to iteratively obtain a new estimate $$\beta_{n+1}$$ from a previous estimate $$\beta_n$$. Define $$W_n$$ to be the diagonal matrix whose $$i$$-th entry is $$l'(X_i \beta_n)$$, where $$X_i$$ is the $$i$$-th row of the design matrix $$X$$ and $$l'$$ is the derivative of the logistic function $$l$$ above. Similarly, define $$Y_n$$ to be the vector whose $$i$$-th entry is $$l(X_i \beta_n)$$. Then define $$\beta_{n+1} = ( X^TW_nX + \Lambda + K )^{-1} X^T ( W_n X \beta_n + Y - Y_n ) + \Lambda ( X^TW_nX + \Lambda + K )^{-1} \beta_n$$ Repeating this computation until convergence, we obtain estimates of shooter ability, goaltending ability, with suitable modifications for shot location and type, as well as the score and the skater strength.

Curious readers may wonder if such convergence is guaranteed; it suffices that $$K$$ and $$\Lambda$$ be positive-definite, which they are.

## Results

I have fitted this model to each regular season from 2007-2008 through to 2020-2021; using the estimates from each summer as the priors to feed into the following year's regression, as explained above. I explain the results of the 2020-2021 regression in detail here; historical results can be found via the following links:

### Geometry Results

First, the geometric terms, converted to probabilities for display. Each hex is shown with the constant term for that shot type added in already, so that the fabrics can be compared to one another.

 Wrist/Snap shots. Highest results are in tight as expected and drop off sharply outside the dots. Slap shots. Highest in the slot again as expected, but without the sharp drop off side to side and instead falling off roughly with total distance to the net. Backhand shots. High-quality areas are even more tightly concentrated than with wrist shots. Tips and Deflections. Here the sides of the net are quite high.

The base probabilities for:

• shots from below the goal line: 10.93%;
• shots from the neutral zone: 2.24%;
• wraparounds: 4.03%
• perimeter shots: 1.45%

#### Strength, Score, Rush, and Rebound Terms

Since precisely one of the geometric terms above applies to every shot, we can consider them as providing starting odds for us. Then, we can interpret the other covariates in terms of how they modify those odds, so we quote the remaining terms as odds ratios.
CovariateImpact on Goal Odds
SH 3%
PP 29%
PPv3 96%
PP-Slapshot 22%
Rush 10%
Rebound 178%
Leading 15%
Trailing 1%

Positive values indicate factors that improve the goal odds of a shot, relative to an unadorned even-strength shot. All of the strength terms are considerably positive. Unsurprisingly, we expect that power-play shots, and especially power-play shots facing only three skaters, should have plenty of room in which to fabricate those extra details which we can't measure directly at this time which we nevertheless know make goals more likely; like more time and space to aim and shoot without immediate pressure, more pre-shot movement, and so on. More interestingly, even short-handed shots (rare as they are) have better goal odds than similar even-strength shots.

Rushes and rebounds have associated odds ratios of around double, which accords with our intuition that they make goal odds substantially better. More interestingly, leading teams shoot better than tied teams, but trailing teams show negligible change for that reason.