Elo-based shot evaluation

Important ADDENDUM
The interest in this research was fueled by the presentation of Micah Blake McCurdy at SEAHAC2019, where he described evaluation of quality of goaltenders as the evaluation of the quality of the shots saved.

The author of this article is not a mathematician or a statistician, so possibly the below is an utter absurd, or heresy, however it passed a few sanity checks the author was able to establish, and therefore he likes it.

1. Why Elo?

Shot (or shot attempt, but like Micah, I will use the term shot for all three location-tracked events: GOAL, SHOT and MISS) is an essential zero-sum outcome: either the goal is scored (1) or it is not (0). That always invokes the question - can we use the Elo ratings to evaluate the expectation of a shot being stopped based on the difficulty of the shot, the shooter and the goaltender.

Thus, given the probability of the shot $P_s$ and the outcome of the shot $X (0|1)$, we can adjust the ratings of the shooter and the goalie:

$R_s=R_s_0 + (X-P_s)$ (1)

$R_g=R_s_0 + (P_s-X)$ (2)

For example, if a shot had a probability 0.06 of going in, and still the goal was scored, the rating of the shooter will increase by 0.94 and the rating of the goalie will decrease by 0.94.
But you may ask, how in the formulas above do ratings affect the probability of the shot going in?

2. The factors of the shot.

If we take the shots, for which coordinates are defined, i.e. from the 2010/11 season through 2018/19, we can observe that the average scoring rate is about 0.06% (remember, misses are included).

However as also described by Micah, not all shots are created equal. Therefore we dissect them by the following factors:

• Location on the rink
• Season (due to equipment changes)
• Stage (due to playing style changes)
• Shooter (Here comes Rs)
• Shooting team (Team attacking style)
• Goalie (Here comes Rg)
• Goalie team (Team defending style)
• Side (generalized code, see below)
• Shot type
• On-Ice strength
• Score differential
• Rebound length
• Is it a rush?
• Is it a giveaway in Def. zone?
• Is it a takeaway in Off. zone?

We will discuss the factors in depth below.

Location:

For the sake of simplicity we divide the rink into the following segments:

• All of the rink outside the offensive zone
• All of the rink below the goal line of the offensive zone
• The remainder of the offensive zone split into squares of three to eight feet. Some tests run provided that the optimal size of such a quadrant is five feet.

We realize that this dissection is imperfect, but for this first run it's simple enough. Later we may try to do the same process with more complicated sector shapes.

Season:

The annual changes in the rules of the NHL dictate that a given season may provide different average shot quality than other ones.

Stage:

The famous statement of playoffs being won through defense implies the chances to score are smaller in the postseason.

Shooter:

We use current $R_s$ from (1) for the shooter.

Shooting team:

The team's style of play may rely more on the quantity of shots or the quality, thus affecting the overall chances of a given shot going in.

Goalie:

We use current $R_g$ from (2) for the goalie

Goalie team:

The defending style and capability are not dissimilar to the ones of the shooting team.

Side:

We try to estimate the effect of the shooting side of the skater and the catching side of the goalie together with relative location of the shot:

• Shot location:
• C - center, in front of the goal
• R - to the right of the goal, attacker wise
• L - to the left of the goal, attacker wise
• Shooter's side - R|L
• Goalie's side - R|L

Altogether we have 12 values (LLL, ..., CRL, ..., CRR)

Shot type:

We use all available shot types:

• Slap
• Snap
• Wrist
• Backhand
• Wrap-around
• Tip-in
• Deflection

Naturally, all unassigned shots are discarded.

Strength:

On-ice strength, from 33 to 55. Penalty shots are discarded. A future development may introduce either special cases of goalies pulled, or incorporating those into different strengths, including the new 63-65 ones.

Therefore the shots that do not have on-ice count are discarded.

Score differential:

We treat the score score situations differently for all well-known reasons. We define the scores differences between -2 (shooting team 2 goals behind) to 2 separately, and the larger differences fall all under -3 and 3.

Rebound:

Rebound is defined when a shot attempt (including blocked shot) happened recently prior to this shot being analyzed. Not only we test for such an event in the 3 seconds before the shot, we also catalog them separately by the time passed between the previous attempt and the rebound (0, 1, 2, 3 seconds).

Rush:

We define rush similarly to others, as a shot attempt from offensive zone which is preceded by a non-faceoff event in non-offensive zone within five seconds.

Give:

We consider a defensive zone giveaway an event seriously tilting the chances of a score due to the goaltender being caught somewhat unaware. We define a qualifying giveaway event to have occurred in the defensive zone within 6 seconds of the shot.

Take:

We consider a offensive zone takeaway an event seriously tilting the chances of a score due to the goaltender being caught somewhat unaware. We define a qualifying takeaway event to have occurred in the offensive zone within 6 seconds of the shot.

Schedule aspects, such as back-to-back games (or prolonged breaks) or Home/Away are possible factors that were not considered. Maybe, next summer.

3. The application of the factors

Now that we managed to classify all the shots by the factors described above, we can try to appraise the effect of the factors.

We begin with setting some starting/default values. We obtain these values from two first seasons of our research period, i.e. from 2010/11 and 2011/12. Later we will incorporate the remainder of the data into it.

As we said earlier, let's assume an overall probability of the shot going in is about 0.0622 (from these two seasons). That corresponds to Elo rating difference of about 471 points in favor of the goaltender. Therefore we can say that an average shooter from an average location against an average goaltender is like a match between a 2029 and a 2500 rated players.

Now the factors from the previous chapter come in handy. For each of these factors (excluding the personal ratings for now) we compute the probability of success for a shot with each value of the factor. For example, for a binary factor like takeaway we get the following table:

Takeaway
0	0.0618
1	0.1014

which means that the Elo differential of the shot which is not preceded by takeaway is lower by 0.8 points, and but higher by 92.5(!) points if there was a takeaway.

For a non-binary factor, e.g. strength (no penalty shots):

Strength	p	Δelo
33	0.0500	-40.0
34	0.0968	83.5
35	0.1053	99.7
43	0.1099	108.1
44	0.0645	6.9
45	0.0656	10.1
53	0.1683	193.9
54	0.0852	59.1
55	0.0563	-18.3

Note that these numbers predate the 3on3 overtime. Also, it's tougher to score in full strength than on average.

Then to get the overall shot rating we add these differences to the original value of $R_{base}$==2029 (divided by 1.5 due to the behavior of the Elo sigmoid at low probabilities), and also add the difference between $R_s$ and 2029 (also assigned as the initial shooter rating), and subtract the difference between $R_g$ 2500 (also assigned as the initial goalie rating).

So we have

$$R_{shot} = R_{base} + 3/2∑Δ_f +(R_s - R_{base} + (2500-R_g)$$

or, effectively

$R_{shot} = R_s +3/2∑Δ_f + (2500 - R_g)$ (3)

Then we can estimate the chances of the goal going in by the Elo formula:

 $$P_s = 1 / ( 1 + 10 ^ (( 2500 - R_{shot} ) / 400))$$ 

 or, effectively

$P_s = 1 / ( 1 + 10 ^ ((R_g - R_s - 3/2∑Δ_f)/400))$ (4)

Note that we never do an explicit match of shooter vs goalie. We could do that instead of including $R_s$ and $R_g$ in the formula, but that way proved to be more complicated and provided less consistent results.

Looks straightforward? Unfortunately, it isn't.

4. Confounders

If the factors were completely independent, our job would be done. Alas, they are not, they are implicitly confounding each other, i.e. there might be more deflecting shots resulting in goal on a powerplay, or more rebounds in quadrants close to the goal, and so on. Therefore we try to mitigate these dependencies in the following way:

I. As a base line we calculate the probabilities of success for shots in each quadrant we defined.

II. Then we compute the probability of success for shots with each separate factor value in the given quadrant.

III. We calculate the ratio the freshly computed probability to the general probability of success in this quadrant.

IV. The resulting confounding effect is then calculated according to the following formula:

$C_f = 1 / (|log(ratio)| + 1)$

Thus, when the probabilities are the same in the quadrant with or without the factor set ($ratio == 1$), we get $C = 1$.

V. We multiply each $Δ$ factor$ by the corresponding $C$, thus formulas (3) and (4) become:

$R_{shot} = R_s + 3/2∑Δ_fC_f/1.5 + (2500 - R_g)$ (5)

and

$P_s = 1 / ( 1 + 10 ^ ((R_g - R_s - 3/2∑Δ_fC_f) /400))$ (6)

Obviously, for the quadrant factor $C_f == 1$.

To test the validity of the math above, we tested log loss of betting each shot not being a goal. By just using the base probability of 0.0622 the log loss was about 0.240. By using the probabilities computed through (5) and (6) the log loss was reduced to 0.210 with each factor contributing to the reduction.

5. The eXpected goal value and the save above expectation

Now by using (5) we can calculate the number of expected goals against each goalie in a game:

$xG = ∑↙{goalie}(P_s)$ in a given game.

We know how many goals were scored against the goalie and we can easily apply (2). The new rating of the goalie will be used in the calculations for the next game he participates in. Same applies for the shooters, only the sum is of the shots taken by the shooter. Empty Net shots are not accounted for.

We calculate the $xG$ and $G$ for games on each date starting with the 2012/13 season onward. After all games for a given date had been processed, we feed them back into the probabilities of the modifiers to keep them current, and in a way that gives the data from the current season double weight compared to the past data, whereas data from the earliest available date is tossed out.

Here is the sample of best and worst performances in $xG-G$ for goalies and skaters, single game, and season (playoffs excluded):

Goals saved above expectation (game)

Player	Date	Delta
ALEXANDAR GEORGIEV	20190210	6.034
EVGENI NABOKOV	20140323	5.787
LAURENT BROSSOIT	20150409	5.245
MIKE CONDON	20170119	5.178
RYAN MILLER	20160117	5.153

Goals saved below expectation (game)

Player	Date	Delta
AL MONTOYA	20161104	6.152
SERGEI BOBROVSKY	20181204	5.706
ROBIN LEHNER	20140227	5.313
SERGEI BOBROVSKY	20181013	5.279
JOEY MACDONALD	20130403	5.251

Goals scored above expectation (game)

Player	Date	Delta
PATRIK LAINE	20181124	4.436
CHRIS KUNITZ	20130203	3.646
ALEX OVECHKIN	20131210	3.582
AUSTON MATTHEWS	20161012	3.531
BRAD RICHARDSON	20190228	3.525

Goals scored below expectation (game)

Player	Date	Delta
NAZEM KADRI	20190210	2.355
BROCK NELSON	20141213	1.873
GABRIEL LANDESKOG	20190109	1.868
LOGAN COUTURE	20150217	1.827
RYAN O'REILLY	20190329	1.693

Goals saved above expectation (season)

Player	Date	Delta
SERGEI BOBROVSKY	2016	35.787
CAREY PRICE	2013	32.607
JOHN GIBSON	2016	26.296
CAREY PRICE	2014	24.424
THOMAS GREISS	2015	24.153

Goals saved below expectation (season)

Player	Season	Delta
JONATHAN QUICK	2018	42.428
CAREY PRICE	2017	33.573
CRAIG ANDERSON	2017	28.724
THOMAS GREISS	2017	25.168
SCOTT DARLING	2017	24.161

Goals scored above expectation (season)

Player	Season	Delta
LEON DRAISAITL	2018	23.047
PATRIK LAINE	2017	22.6
ALEX DEBRINCAT	2018	20.249
STEVEN STAMKOS	2018	19.833
ALEX OVECHKIN	2013	19.052

Goals scored below expectation (season)

Player	Season	Delta
ALEX CHIASSON	2013	12.174
MIKE RICHARDS	2013	11.016
ERIC STAAL	2015	10.476
TYLER TOFFOLI	2018	10.319
BRAYDEN SCHENN	2014	9.894

As another validity check, we checked for inflation of ratings over time. We found that the goalie ratings inflated by the total of just 318 points for 186 goaltenders, and, correspondingly, deflated by the same amount for 1683 skaters. These values are pretty admissible for inflation.

6. Predictive aspects

Shooter

If a shooter has the rating $R_s$ above base shot rating $R_{base}$, then he increases the probability of a goal (and vice versa). The difference should be computed for each separate case, but on average, given a nearly linear behavior of the Elo function at low probabilities, each extra 10 points would account for 0.0035 difference in the probability of the shot. We can do a more particular job by surveying which factors dominate the shots of the player, and what's their probability altogether, excluding the shooter and thus compute the difference more precisely.

Goalie

If a goalie has the rating $R_g$ above 2500 (base goaltender rating) then he decreases the probability of a goal (and vice versa) in exactly reverse way that the shooter does. However, the goaltenders face the shots from all possible factor values, therefore we must adjust the probability from the base probability (e.g. 0.0622). The only factor that possibly should be taken into account is the goalie's team.

Team

We can approach the $xG$ (or rather $pG$ (projected goals)) of a team by two ways: iterating over the projected or published roster, or by blanket-weighted-averaging the shots the team takes per game and their probabilities. The first way is more complex, but supposedly more precise.

Season

We do not see any particular implications of a season-wide projection at any level, team, skater or goaltender. For the first two we just multiple a single game projection by the number of games in a season. For the latter one, an estimate of the number of games would be necessary.

Playoffs

In the playoffs we can hone our predictions to the given shooter and goalie's team. Maybe, that when home/away factors will also become more prominent.

Here's current (EOS 2018/19) top 5 goalie and skater rankings:

Top 5 Goalies after 18/19

Player	Rating
ANTTI RAANTA	2535.5
BEN BISHOP	2534.8
JOHN GIBSON	2533.9
ROBIN LEHNER	2524.7
JUUSE SAROS	2523.1

Top 5 Colanders after 18/19

Player	Rating
KEITH KINKAID	2480.6
MAXIME LAGACE	2482.7
GARRET SPARKS	2485.4
CRAIG ANDERSON	2485.9
CHAD JOHNSON	2487.7

Top 5 Scorers after 18/19

Player	Rating
ALEX OVECHKIN	2104.4
STEVEN STAMKOS	2097.2
NIKITA KUCHEROV	2079.8
PATRICK KANE	2077.3
PATRIK LAINE	2076.0

Top 5 Whiffers after 18/19

Player	Rating
JORDAN STAAL	2002.8
MATT MOULSON	2002.9
JUSTIN ABDELKADER	2004.8
PATRIC HORNQVIST	2004.9
KYLE CLIFFORD	2005.1

Concluding, the author wants to underline once again, that he realizes the insufficient theoretical background for the task undertaken, and that many assumptions that are made smell of ad hoc approach. However, we hope that the model finds its usefulness among the hockey fans, and that this research attracts people of better qualification that would be interested to polish and improve it.

On Players Evaluation - Part VI (Skater's [and Goaltender non-SVP] Elo)

Part I

Part II

Part III

Part IV

Part V

The most important conclusion of the last chapter that dealt with goalies' Elos is that it is defined by actual performance of a goaltender versus the expected performance of the team he is facing. That is the approach we are going to inherit for evaluating skaters.

For the start we compute the average stats of a league for each season. We do that for most of the stats that are measured, from goals and assists to faceoffs taken, up to the time on ice for the goaltenders. This is a trivial calculation. Thus we obtain season stat averages S_avg.

Now we can begin to work with the skaters. We assign them a rating of 2000 in each stat. The first and the most difficult step is to coerce the actual performance of a skater in each stat to a chess-like result, on the scale from 0 to 1. This is a real problem, since the result distribution for the number of players looks something like one of these chi-squares:

Therefore we need to rebalance it somehow while preserving the following rules:

• They should be more or less distributive, i.e. scoring 1 goal thrice in a row in a game should produce approximately the same performance as scoring a hat trick in one game and going scoreless in the other two.
• They should still have the same shape as the original one.
• The average rating of the league in each stat should remain 2000 at the end of the season.

So first, we do not apply rating changes after a single game. We take a committing period, for example, five games, and average players' performance in every rated stat over that period. Second, we apply the following transformation to the performance:

P'_player = (P_player - S_avg) / S_avg

where S_avg is the season average on that stat. It could be more precise to compute against the averages against of the teams played (see the first paragraph), but we decided to go via a simpler route at this stage.

Then we scale the performance by the Adjustment Factor A:

P'_playeradj = P'_player / A

The adjustment factor sets the result between -0.5 and 0.5. More or less. There still are outliers, but they are very infrequently beyond 0.5 . The A factor depends on the rarity of the scoring in the stat and varies from 6 (Shot on Goal) to 90 (Shorthanded goal). The adjustment for goals, is, for example, 9. The adjustment for faceoffs won is 20. The latter one might look a bit surprising, but remember that many players do not ever take faceoffs, e.g. defensemen. Naturally, only skaters stats are computed for skaters, only goalie stats for goaltenders.

The final Result R_player is then:

R_player = P'_playeradj + 0.5

So for the rare events we have a lot of results in the 0.48-0.5 area and a few going to 1. For the frequent events (shots, blocks, hits), the distribution is more even.

Now that we got the player's "result" R, we can compute the elo change through the familiar formula:

ΔElo = K * (R - (1/(1+10^{(2000 - Elo_player)/400})))

where K is the volatility coefficient which we define as:

16 * √(A) * √(4 / (C + 1))

A is the aforementioned Adjustment Factor and C is the Career Year for the rookies (1) and the sophomores (2), and 3 for all other players.

'What is 2000', an attentive reader would ask? 2000 is the average rating of the league in each stat. We use, because the "result" of the player was "against" the league average. If we used team averages, we would put the average "Elo against" of the teams faced instead.

After we have the ΔElo, the new Elo' of a player in a specific stat becomes:

Elo' = Elo + ΔElo

And from that we can derive the expected average performance of a player in each stat, per game:

R_exp = 1/(1+10^{(2000-Elo')/400})

P_exp = (R_exp - 0.5) * A * S_avg + S_avg

which is an "unwinding" of the calculations that brought us from the actual performance to the new rating.

The calculation differs for the three following stats:

1. SVP - processed as described in Part V.
2. Win/Loss - processed as a chess game against a 2000 opponent, where the result is:

R_w = P_w/(P_w+P_l), R_l = P_l(P_w+P_l)

over the committing period.

The only subtlety here is that sometimes a hockey game may result in goalie win without a goalie loss.

3. PlusMinus -

R_+/- = 0.5 * (P_+/- - Savg_+/-) / 10 (10 skaters on ice on average)

Then, via the regular route we get the Elo' and the expected "result" R_exp, and the expected performance is:

P_exp+/- = (R_exp+/- - 0.5) * 10 + S_avg+/-

Please note that we do not compute "derived" stats, i.e. the number of points (or SHP, or PPP), or the GAA, given the GA and TOI, or GA, given SA and SV.

An example of the computed expected performances that lists expectations of top 30 Centers in Assists (Adjustment Factor 9) can be seen below:

#	Player	Pos	Team	Games	A	a/g	Avg. g.	Avg.a	E a/g	E a/fs
1	CONNOR MCDAVID	C	EDM	43	34	0.791	44.00	33.00	0.706	61.54
2	JOE THORNTON	C	SJS	41	24	0.585	74.11	52.00	0.665	51.27
3	NICKLAS BACKSTROM	C	WSH	40	24	0.600	69.20	50.10	0.663	51.85
4	EVGENI MALKIN	C	PIT	39	27	0.692	62.09	44.73	0.659	55.33
5	SIDNEY CROSBY	C	PIT	33	18	0.545	61.67	51.50	0.655	46.15
6	RYAN GETZLAF	C	ANA	36	25	0.694	68.58	45.42	0.648	50.26
7	EVGENY KUZNETSOV	C	WSH	40	22	0.550	54.75	27.75	0.605	47.43
8	ANZE KOPITAR	C	LAK	36	16	0.444	72.73	41.55	0.594	40.33
9	ALEXANDER WENNBERG	C	CBJ	40	28	0.700	59.00	25.67	0.583	52.50
10	CLAUDE GIROUX	C	PHI	43	25	0.581	61.70	37.60	0.579	47.56
11	TYLER SEGUIN	C	DAL	42	26	0.619	66.86	31.14	0.566	48.65
12	RYAN O'REILLY	C	BUF	30	16	0.533	66.00	26.38	0.553	39.23
13	DAVID KREJCI	C	BOS	44	18	0.409	60.64	32.36	0.528	38.05
14	RYAN JOHANSEN	C	NSH	41	22	0.537	65.33	27.00	0.523	43.43
15	JOE PAVELSKI	C	SJS	41	23	0.561	69.64	29.09	0.517	44.21
16	HENRIK SEDIN	C	VAN	43	17	0.395	75.56	47.81	0.517	37.17
17	DEREK STEPAN	C	NYR	42	22	0.524	68.00	30.86	0.508	42.31
18	VICTOR RASK	C	CAR	41	19	0.463	67.00	22.67	0.497	39.37
19	MARK SCHEIFELE	C	WPG	40	20	0.500	44.50	17.83	0.493	39.23
20	JASON SPEZZA	C	DAL	35	18	0.514	62.71	37.79	0.490	37.60
21	JOHN TAVARES	C	NYI	38	16	0.421	68.50	35.00	0.488	37.46
22	MITCHELL MARNER	C	TOR	39	21	0.538	39.00	21.00	0.484	41.82
23	STEVEN STAMKOS	C	TBL	17	11	0.647	65.11	29.00	0.474	29.97
24	ALEKSANDER BARKOV	C	FLA	36	18	0.500	56.75	21.00	0.463	36.51
25	MIKAEL GRANLUND	C	MIN	39	21	0.538	55.80	24.40	0.460	40.80
26	PAUL STASTNY	C	STL	40	13	0.325	65.09	34.55	0.457	31.74
27	JEFF CARTER	C	LAK	41	15	0.366	69.67	24.33	0.448	33.35
28	MIKE RIBEIRO	C	NSH	41	18	0.439	62.88	33.06	0.447	36.32
29	MIKKO KOIVU	C	MIN	39	16	0.410	66.83	34.25	0.445	35.14
30	ERIC STAAL	C	MIN	39	22	0.564	74.46	36.77	0.442	40.99

You can see more of such expectation evaluations on our website, http://morehockeystats.com/fantasy/evaluation .

Now, we ask ourselves, how can we use these stats evaluations to produce an overall evaluation of a player?

To be concluded...

On Players Evaluation - Part V (Goaltender's Elo)

Part I
Part II
Part III
Part IV

The goalkeeper is half of the whole team

Soviet proverb from Lev Yashin's times.

After a foray into the calmer lands of teams' evaluation using the Elo rating, it's time to turn our attention to the really juicy stuff - the evaluation of a single player. And we'll start with the most important one - the goaltender. DISCLAIMER: this evaluation concept is still a work in progress and one of several possible implementations of the idea.

By coincidence, it's also the simplest evaluation to make. While many stats describe the performance of a skater (goals, assists, shots, hits, blocks, faceoff wins, etc. - and even one that is accounted usually for goaltenders) only one stat truly describe the goalie's performance: the saves percentage. Usually, whole four stats are used to compare the goalies: wins (W), saves percentage (SVP), goals against average (GAA) and shutouts (SHO), but will show you first, why three of them are mostly unnecessary. Also, the name saves percentage is a bit of a misnomer, since the values of svp are usually not multiplied by 100 to look like real percent, but are shown more frequently between 0 and 1, and therefore would be more properly named as 'Saves Ratio', or 'Saves Share'.

Wins are truly results of team efforts. I always cringe when I read that a goaltender "outdueled" his opponent, when the both barely got see each other. The GAA is much more of an indication of how well the defense operates in front of the goalie. Shootouts are first, and foremost, a very rare thing, and secondly a 15-save shootout should not be the same as 40-save shootout, although for any of the four stats listed above they create two identical entry.

Therefore we feel ourselves on a firm ground evaluating goalie's performance through SVP only (with a slight input from shootouts, as described below) - and the Elo function, of course. For the start, each goaltender is assigned an Elo rating of 2000 for his first career appearance. We discard performances in which goalies faced less than four shots, because these usually are late relief appearances in the garbage time, not really an evidence of goaltending in a true hockey game. We only account for them to display the real SVP accrued in the season so far, and we consider dropping these appearances completely.

After the game we get the pure SVP from the real time stats. We adjust it in two ways:

1. If, in the very rare case, the performance is below 0.7, we set it to 0.7 .
2. If there was a shootout (not the shootout as defined by the NHL, but a performance where a goaltender was on the ice for at least 3420 seconds and did not let a single goal in during that time), we add a shootout bonus for the performance:

Bonus = (Saves - 10) / 200

If there were less than fifteen saves in the shootout, the bonus is assigned the minimum value of 0.025. We consider adding this bonus necessary, because the opposing team is usually gives an extra effort to avoid being shut out even during the garbage time.

Then, given the actual performance we can calculate the "Elo performance rating":

R_perf = 2000 + (SVP - SVP_vsopp) * 5000

Where SVP_vsopp is the SVP against the opponent the goalie is facing - effectively the shooting % of that team minus the shots resulting in empty-net goals, sort of "Expected SVP against that opponent". That means that for every thousandth of the SVP above the expectation, the performance is five points above 2000 (the absolute average).

Wait, there seems to be an inconsistency. Don't we need ratings of opponents for Elo changes calculation? Actually, no. Given an Elo performance of a player, we can calculate the rating change as a "draw" against a virtual opponent with that Elo performance, i.e.

ΔR = K * (0.5 - 1 / ( 1 + 10 ** (( R_perf - R_g)/ 400)) ) )

Where K is the volatility factor mentioned in the earlier posts. Right now we are using the volatility factor of 32, but that may change - including introducing a dependency of this factor on goaltender's experience.

And the new rating, is naturally,

R_g' = R_g + ΔR

Now we can calculate the expected remaining svp:

SVP_rem = SVPav_g + (R_g' - 2000) / 5000

Where SVP_avg is the league average SVP. It would be more correct to substitute that value with the weighted averages of the remaining teams to face (with accordance to the matches remaining), and we'll be switching to this index soon.

We can also calculate the SVP expected from the goalie at the start of the season:

SVP_exp = SVP_avg0 + (R_g0 - 2000) / 5000

where SVP_avg0is the average SVP of the league during the previous season and the R_g0 is the rating of the goalie at the conclusion of the previous season (including playoffs), or the initial rating of 2000.

We post a weekly update on our Elo ratings for goaltenders, and their actual and expected SVPs on our Twitter feed. You can also access our daily stats on our website page.

It looks like we're ready to try to take on the skaters' performances. But I'm not sure it's going to fit into one posting.

To be continued...

On Players Evaluation - Part IV (Teams Elo Projections)

On Players Evaluation - Part IV (Teams Elo Projections)

Part I
Part II
Part III

We left our reader at the point where we demonstrated how to produce Elo ratings for hockey teams over season (and over postseason too, if anyone wondered) and how to apply it to the up and coming next games of the rated teams.

However, in its main eparchy, chess, Elo is rarely used to produce single match outcome projections. It's much more popular when used to create a long-term projection, such as the whole tournament, which in chess lasts between five to thirteen rounds, usually.

Therefore, the question arises, shouldn't we try to use our newborn Elo ratings to long-term projections? And the answer is an unambiguous 'Yes!' We can and should create the projections for the team over longer spans such as a seven days ahead, thirty, or even through the end of the season!

How do we do it? Since we computed the Elo ratings for all teams, and we know the schedule ahead of all teams, we can run the Elo expectation on all matchups during the requested span and sum them. And since we assume that each team performs according the expectation, their Elo ratings do not change during the evaluation span.

E_team = Σ(E_match1, E_match2, ... , E_matchn)

All good? No. There is one more finesse to add. The produced expectations will all be calculated in 2-0 span per game, assuming only 2 points are in play in each matchup. However, due to the loser's point it's not so. Therefore on average there are 2 + N_OT/SO / N_total points are handed out during the season in every match (where N_OT/SO is the number of games that get decided in OT or SO). So we need to compute the N_OT/SO value, divide it by two (because there are two teams in each match) and multiply the expectation of each team by this factor. By doing so we receive the reliable Elo expectation, such as one in the table below, as of Jan 2nd, 2017. Spans of 7 days, 30 days and through the end of the season are displayed (games, expected points and total).

Elo ratings for season 2016

#	Team	Div	Elo	Pts	Gin7	Pin7	Tin7	Gin30	Pin30	Tin30	GinS	PinS	TinS
1	Columbus Blue Jackets	MET	2265.22	56	4	6	62	14	23	79	47	79	135
2	Pittsburgh Penguins	MET	2186.57	55	1	2	57	11	16	71	44	65	120
3	Minnesota Wild	CEN	2180.88	50	3	4	54	14	21	71	46	68	118
4	San Jose Sharks	PAC	2137.87	47	3	4	51	14	20	67	45	62	109
5	Washington Capitals	MET	2135.54	49	4	4	53	15	18	67	46	59	108
6	Montreal Canadiens	ATL	2117.99	50	4	5	55	14	18	68	45	58	108
7	New York Rangers	MET	2135.43	53	3	4	57	11	14	67	43	54	107
8	Chicago Blackhawks	CEN	2103.27	51	3	4	55	12	15	66	42	52	103
9	Anaheim Ducks	PAC	2105.41	46	3	4	50	13	18	64	43	55	101
10	Edmonton Oilers	PAC	2092.89	45	4	4	49	14	16	61	44	53	98
11	Ottawa Senators	ATL	2088.34	44	2	2	46	11	11	55	45	52	96
12	Toronto Maple Leafs	ATL	2097.27	41	3	4	45	12	14	55	46	54	95
13	St. Louis Blues	CEN	2066.58	43	2	2	45	12	12	55	44	51	94
14	Boston Bruins	ATL	2079.41	44	4	5	49	15	17	61	43	49	93
15	Carolina Hurricanes	MET	2093.06	39	4	5	44	13	13	52	46	53	92
16	Los Angeles Kings	PAC	2066.68	40	4	4	44	14	16	56	45	52	92
17	Philadelphia Flyers	MET	2079.35	45	3	3	48	12	13	58	43	46	91
18	Calgary Flames	PAC	2076.79	42	4	5	47	14	16	58	43	49	91
19	Tampa Bay Lightning	ATL	2068.90	42	4	4	46	13	14	56	44	48	90
20	New York Islanders	MET	2070.87	36	2	3	39	12	14	50	46	51	87
21	Florida Panthers	ATL	2059.66	40	4	5	45	13	14	54	44	46	86
22	Nashville Predators	CEN	2055.15	38	4	4	42	14	14	52	46	48	86
23	Dallas Stars	CEN	2052.77	39	3	3	42	13	13	52	44	46	85
24	Vancouver Canucks	PAC	2049.05	37	4	5	42	12	15	52	44	46	83
25	Detroit Red Wings	ATL	2033.62	37	3	3	40	13	12	49	45	43	80
26	Winnipeg Jets	CEN	2017.50	37	4	4	41	14	14	51	43	40	77
27	Buffalo Sabres	ATL	2009.45	34	3	3	37	13	12	46	46	41	75
28	New Jersey Devils	MET	1994.66	35	5	4	39	14	12	47	45	37	72
29	Arizona Coyotes	PAC	1921.41	27	3	2	29	12	8	35	45	30	57
30	Colorado Avalanche	CEN	1910.42	25	3	2	27	12	7	32	46	29	54

The N_OT/SO value right now is about 1.124 (i.e. about quarter of all games are decided past the regulation).

So you know what's good for the people?

But the people consists of men...

Iconic Soviet movie

The team projection leaves us wanting more. After all, don't we want to be able to evaluate individual players and factor it somehow in the projection to reflect the injuries and other reasons that force top players out of the lineups? Stay tuned.

To be continued...