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...