Simple NBA Elo Ratings¶

In this post, we will build a very simple example of the Elo rating system and apply it to rate NBA teams. We will look at the key assumptions and math behind Elo ratings, and show you how to implement the system in Python.

In [1]:

import numpy as np
import pandas as pd
pd.options.display.max_rows = 999
pd.options.display.max_columns = 999

In [2]:

%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(context='notebook', palette='colorblind')

In [3]:

import math
import operator
from collections import OrderedDict
from enum import Enum

We will use the pracnbastats package I developed for scraping stats.nba.com. You can install this package in your sports analytics Python environment by executing pip install pracnbastats in Terminal or at the Anaconda Prompt in Windows.

In [4]:

import pracnbastats as nba

In [5]:

from pathlib import Path

This code assumes the existence of a directory to hold scraped NBA data. You can create and name this directory however you want, and adjust the code in the cell below to suit your preferences.

In [6]:

PROJECT_DIR = Path.cwd().parent
DATA_DIR = PROJECT_DIR / 'data' / 'stats-nba-com'

In [7]:

filehandler = nba.FileHandler(DATA_DIR)

Scraping NBA 2017-18 Regular Season Match Ups¶

First, we will get a list of all the current NBA teams. We will need to use this as a lookup table to get the stats.nba.com team ID.

In [8]:

nba.currentteams.load(filehandler)
teams = nba.currentteams.data()
teams

Out[8]:

	id	abbr	code	conference	division	city	name	since
0	1610612738	BOS	celtics	East	Atlantic	Boston	Celtics	1946
1	1610612752	NYK	knicks	East	Atlantic	New York	Knicks	1946
2	1610612751	BKN	nets	East	Atlantic	Brooklyn	Nets	1976
3	1610612761	TOR	raptors	East	Atlantic	Toronto	Raptors	1995
4	1610612755	PHI	sixers	East	Atlantic	Philadelphia	76ers	1949
5	1610612749	MIL	bucks	East	Central	Milwaukee	Bucks	1968
6	1610612741	CHI	bulls	East	Central	Chicago	Bulls	1966
7	1610612739	CLE	cavaliers	East	Central	Cleveland	Cavaliers	1970
8	1610612754	IND	pacers	East	Central	Indiana	Pacers	1976
9	1610612765	DET	pistons	East	Central	Detroit	Pistons	1948
10	1610612737	ATL	hawks	East	Southeast	Atlanta	Hawks	1949
11	1610612748	MIA	heat	East	Southeast	Miami	Heat	1988
12	1610612766	CHA	hornets	East	Southeast	Charlotte	Hornets	1988
13	1610612753	ORL	magic	East	Southeast	Orlando	Magic	1989
14	1610612764	WAS	wizards	East	Southeast	Washington	Wizards	1961
15	1610612757	POR	blazers	West	Northwest	Portland	Trail Blazers	1970
16	1610612762	UTA	jazz	West	Northwest	Utah	Jazz	1974
17	1610612743	DEN	nuggets	West	Northwest	Denver	Nuggets	1976
18	1610612760	OKC	thunder	West	Northwest	Oklahoma City	Thunder	1967
19	1610612750	MIN	timberwolves	West	Northwest	Minnesota	Timberwolves	1989
20	1610612746	LAC	clippers	West	Pacific	LA	Clippers	1970
21	1610612758	SAC	kings	West	Pacific	Sacramento	Kings	1948
22	1610612747	LAL	lakers	West	Pacific	Los Angeles	Lakers	1948
23	1610612756	PHX	suns	West	Pacific	Phoenix	Suns	1968
24	1610612744	GSW	warriors	West	Pacific	Golden State	Warriors	1946
25	1610612763	MEM	grizzlies	West	Southwest	Memphis	Grizzlies	1995
26	1610612742	DAL	mavericks	West	Southwest	Dallas	Mavericks	1980
27	1610612740	NOP	pelicans	West	Southwest	New Orleans	Pelicans	2002
28	1610612745	HOU	rockets	West	Southwest	Houston	Rockets	1967
29	1610612759	SAS	spurs	West	Southwest	San Antonio	Spurs	1976

Now, let's get all of the team box scores for the 2017-18 regular season.

In [9]:

nba1718 = nba.team.BoxScores(filehandler=filehandler)
nba1718.data.shape

Out[9]:

(2460, 30)

In [10]:

nba1718.data.head()

Out[10]:

	season	season_type	team_id	team_abbr	game_id	date	opp_team_abbr	home_road	win_loss	min	fgm	fga	fg_pct	fg3m	fg3a	fg3_pct	ftm	fta	ft_pct	oreb	dreb	reb	ast	stl	blk	tov	pf	pts	mov	video
0	2017	reg	1610612757	POR	21701229	2018-04-11	UTA	H	W	240	41	89	0.461	9	24	0.375	11	16	0.688	7	39	46	19	9	9	10	22	102	9	Y
1	2017	reg	1610612762	UTA	21701229	2018-04-11	POR	R	L	240	33	89	0.371	8	23	0.348	19	24	0.792	14	39	53	18	5	7	15	19	93	-9	Y
2	2017	reg	1610612746	LAC	21701228	2018-04-11	LAL	H	L	240	41	83	0.494	6	18	0.333	12	28	0.429	8	33	41	27	8	1	17	17	100	-15	Y
3	2017	reg	1610612747	LAL	21701228	2018-04-11	LAC	R	W	240	42	88	0.477	17	39	0.436	14	16	0.875	10	39	49	25	8	6	17	23	115	15	Y
4	2017	reg	1610612758	SAC	21701230	2018-04-11	HOU	H	W	240	38	80	0.475	7	26	0.269	13	20	0.650	6	42	48	22	6	3	11	14	96	13	Y

To compute Elo ratings, what we really need is the match up information. This view of the data combines the home and road team information on one row per game.

In [11]:

nba1718.matchups.head()

Out[11]:

	game_id	season	season_type	date	team_id_h	team_abbr_h	pts_h	win_loss_h	team_id_r	team_abbr_r	pts_r	win_loss_r	hr_winner	winner	loser	mov
0	21701229	2017	reg	2018-04-11	1610612757	POR	102	W	1610612762	UTA	93	L	H	POR	UTA	9
1	21701228	2017	reg	2018-04-11	1610612746	LAC	100	L	1610612747	LAL	115	W	R	LAL	LAC	15
2	21701230	2017	reg	2018-04-11	1610612758	SAC	96	W	1610612745	HOU	83	L	H	SAC	HOU	13
3	21701221	2017	reg	2018-04-11	1610612748	MIA	116	W	1610612761	TOR	109	L	H	MIA	TOR	7
4	21701225	2017	reg	2018-04-11	1610612750	MIN	112	W	1610612743	DEN	106	L	H	MIN	DEN	6

In [12]:

nba1718.matchups.shape

Out[12]:

(1230, 16)

Here's a summary of all 30 NBA teams' regular season records, broken out by home and road games.

In [13]:

nba1718.team_records

Out[13]:

	team_abbr	season	season_type	team_id	games	wins	losses	home	road	home_wins	home_losses	road_wins	road_losses	win_pct	home_win_pct	road_win_pct
0	POR	2017	reg	1610612757	82	49	33	41	41	28	13	21	20	0.597561	0.682927	0.512195
1	LAC	2017	reg	1610612746	82	42	40	41	41	22	19	20	21	0.512195	0.536585	0.487805
2	SAC	2017	reg	1610612758	82	27	55	41	41	14	27	13	28	0.329268	0.341463	0.317073
3	MIA	2017	reg	1610612748	82	44	38	41	41	26	15	18	23	0.536585	0.634146	0.439024
4	MIN	2017	reg	1610612750	82	47	35	41	41	30	11	17	24	0.573171	0.731707	0.414634
5	ORL	2017	reg	1610612753	82	25	57	41	41	17	24	8	33	0.304878	0.414634	0.195122
6	PHI	2017	reg	1610612755	82	52	30	41	41	30	11	22	19	0.634146	0.731707	0.536585
7	OKC	2017	reg	1610612760	82	48	34	41	41	27	14	21	20	0.585366	0.658537	0.512195
8	NOP	2017	reg	1610612740	82	48	34	41	41	24	17	24	17	0.585366	0.585366	0.585366
9	CHI	2017	reg	1610612741	82	27	55	41	41	17	24	10	31	0.329268	0.414634	0.243902
10	CLE	2017	reg	1610612739	82	50	32	41	41	29	12	21	20	0.609756	0.707317	0.512195
11	BOS	2017	reg	1610612738	82	55	27	41	41	27	14	28	13	0.670732	0.658537	0.682927
12	DAL	2017	reg	1610612742	82	24	58	41	41	15	26	9	32	0.292683	0.365854	0.219512
13	IND	2017	reg	1610612754	82	48	34	41	41	27	14	21	20	0.585366	0.658537	0.512195
14	LAL	2017	reg	1610612747	82	35	47	41	41	20	21	15	26	0.426829	0.487805	0.365854
15	WAS	2017	reg	1610612764	82	43	39	41	41	23	18	20	21	0.524390	0.560976	0.487805
16	UTA	2017	reg	1610612762	82	48	34	41	41	28	13	20	21	0.585366	0.682927	0.487805
17	ATL	2017	reg	1610612737	82	24	58	41	41	16	25	8	33	0.292683	0.390244	0.195122
18	BKN	2017	reg	1610612751	82	28	54	41	41	15	26	13	28	0.341463	0.365854	0.317073
19	DET	2017	reg	1610612765	82	39	43	41	41	25	16	14	27	0.475610	0.609756	0.341463
20	NYK	2017	reg	1610612752	82	29	53	41	41	19	22	10	31	0.353659	0.463415	0.243902
21	DEN	2017	reg	1610612743	82	46	36	41	41	31	10	15	26	0.560976	0.756098	0.365854
22	SAS	2017	reg	1610612759	82	47	35	41	41	33	8	14	27	0.573171	0.804878	0.341463
23	MIL	2017	reg	1610612749	82	44	38	41	41	25	16	19	22	0.536585	0.609756	0.463415
24	PHX	2017	reg	1610612756	82	21	61	41	41	10	31	11	30	0.256098	0.243902	0.268293
25	TOR	2017	reg	1610612761	82	59	23	41	41	34	7	25	16	0.719512	0.829268	0.609756
26	MEM	2017	reg	1610612763	82	22	60	41	41	16	25	6	35	0.268293	0.390244	0.146341
27	CHA	2017	reg	1610612766	82	36	46	41	41	21	20	15	26	0.439024	0.512195	0.365854
28	GSW	2017	reg	1610612744	82	58	24	41	41	29	12	29	12	0.707317	0.707317	0.707317
29	HOU	2017	reg	1610612745	82	65	17	41	41	34	7	31	10	0.792683	0.829268	0.756098

Simple Elo Ratings¶

Now we are ready to implement the simple Elo system as described in this post.

The Elo home team win probability formula is:

$P(\textrm{Home team wins}) = \frac{1}{1 + 10^{-\frac{(H-R+A)}{400}}}$

The win probability for the road team is just $1 - P(\textrm{Home team wins})$ .

The function below computes the win probabilities for both the home and road teams. The function takes a parameter defining the value of home court, expressed in units of Elo rating points. As described in the main post, this function actually uses the more convenient Elo formula in terms of odds:

$P(\textrm{Home team wins}) = \frac{ah}{r + ah}$

and

$P(\textrm{Road team wins}) = \frac{r}{r + ah}$ where

$h = 10^{\frac{H}{400}} \\ r = 10^{\frac{R}{400}} \\ a = 10^{\frac{A}{400}}$

In [14]:

def win_probs(*, home_elo, road_elo, hca_elo):
    """Home and road team win probabilities implied by Elo ratings and home court adjustment."""
    h = math.pow(10, home_elo/400)
    r = math.pow(10, road_elo/400)
    a = math.pow(10, hca_elo/400)
    denom = r + a*h
    home_prob = a*h / denom
    road_prob = r / denom
    return home_prob, road_prob

We will also look at Elo ratings in terms of odds. This function computes the odds on the home team. The odds on the road team are just the reciprocal of the home team odds.

In [15]:

def home_odds_on(*, home_elo, road_elo, hca_elo):
    """Odds in favor of home team implied by Elo ratings and home court adjustment."""
    h = math.pow(10, home_elo/400)
    r = math.pow(10, road_elo/400)
    a = math.pow(10, hca_elo/400)
    return a*h/r

Let's test our our functions, and examine some basic properties of the Elo ratings system.

If two teams have the same Elo ratings, and there is no home court advantage, the win probabilit should be 50%.

In [16]:

win_probs(home_elo=1500, road_elo=1500, hca_elo=0)

Out[16]:

(0.5, 0.5)

This is the same as saying that the odds on a team winning are 1:1.

In [17]:

home_odds_on(home_elo=1500, road_elo=1500, hca_elo=0)

Out[17]:

1.0

Using the standard Elo scaling parameters, a 400 point ratings differential in favor of the home team should multiply the odds on the home team winning by 10.

In [18]:

home_odds_on(home_elo=1900, road_elo=1500, hca_elo=0)

Out[18]:

10.0

Expressing this shift in the odds in terms of probability, we see that the home team win probabilty is now almost 91%.

In [19]:

win_probs(home_elo=1900, road_elo=1500, hca_elo=0)

Out[19]:

(0.9090909090909091, 0.09090909090909091)

Only the difference in the ratings matter for the Elo system. For example, 2000 versus 1600 is the same as 1900 versus 1500.

In [20]:

home_odds_on(home_elo=2000, road_elo=1600, hca_elo=0)

Out[20]:

10.0

In [21]:

win_probs(home_elo=2000, road_elo=1600, hca_elo=0)

Out[21]:

(0.9090909090909091, 0.09090909090909091)

A 400 point ratings differential in the other direction (in favor of the road team) divides the odds on the home team winning by 10.

In [22]:

home_odds_on(home_elo=1100, road_elo=1500, hca_elo=0)

Out[22]:

0.09999999999999999

Now the road team has a probabilty of almost 91% of winning the game.

In [23]:

win_probs(home_elo=1100, road_elo=1500, hca_elo=0)

Out[23]:

(0.0909090909090909, 0.9090909090909091)

Plotting the Elo Ratings Logistic Function¶

The probability assumptions used in the Elo rating system mean that the win probabilities are computed using a version of the logistic function.

Let's see what this function looks like for the specific parameters used in Elo ratings. The plot below doesn't include any adjustment for home court advantage.

In [24]:

x = np.linspace(-1200, 1200, 240)
x.shape

Out[24]:

(240,)

In [25]:

vec_win_probs = np.vectorize(win_probs)

In [26]:

y = vec_win_probs(home_elo=x, road_elo=0, hca_elo=0)[0]
y.shape

Out[26]:

(240,)

In [27]:

fig, ax = plt.subplots(figsize=(10, 7))
ax.plot(x, y)
ax.set_xlabel('Elo Rating Difference')
ax.set_ylabel('Win Probability')
ax.set_title('Win Probability versus Elo Rating Difference')
ax.axhline(y=0.5, linestyle='dashed', color='grey')
plt.show()

Home Court Advantage¶

Now let's estimate the parameter to use to adjust for home court advantage. As described in this post, the average NBA home team win percentage since the 1996-97 season has been roughly 59.8%. We can calibrate our adjustment parameter to this historical value.

In [28]:

def hca_calibrate(*, home_win_prob):
    """Calibrate Elo home court adjustment to a given historical home team win percentage."""
    if home_win_prob <= 0 or home_win_prob >= 1:
        raise ValueError('invalid home win probability', home_win_prob)
    a = home_win_prob / (1 - home_win_prob)
    print(f'a = {a}')
    hca = 400 * math.log10(a)
    return hca

In [29]:

hca_elo = hca_calibrate(home_win_prob=0.598)
hca_elo

a = 1.4875621890547261

Out[29]:

68.99005236157628

We will use a home court adjustment of 69 simple Elo rating points to analyze NBA teams.

Here is a modifed plot of the Elo logistic function with and without home court advantage adjustment. Note how the adjustment shifts the curve so that an Elo rating differential of 69 points in favor of the road team is a 50-50 match up. Equivalently, for an Elo rating differential of zero, the home team win probabilty is 59.8%.

In [30]:

hca_y = vec_win_probs(home_elo=x, road_elo=0, hca_elo=hca_elo)[0]

In [31]:

fig, ax = plt.subplots(figsize=(10, 7))
ax.plot(x, y, label='No HCA')
ax.plot(x, hca_y, label='With HCA')
ax.set_xlabel('Elo Rating Difference')
ax.set_ylabel('Win Probability')
ax.set_title('Win Probability versus Elo Rating Difference')
ax.axhline(y=0.5, linestyle='dashed', color='grey')
ax.legend(loc='lower right')
plt.show()

Computing the Team Elo Ratings¶

Now we can compute Elo ratings for all 30 NBA teams over the course of the 2017-18 regular season. A detailed discussion of the assumptions and math behind Elo ratings is included in this post.

Elo Ratings Update¶

The method is very simple. First, we need a function to update the Elo ratings, based upon the prior ratings, the adjustment for home court advantage, and which team won the game. We need to specify the parameter $K$ , which controls how much to adjust the Elo ratings for each game. We will use the value $K = 20$ in this notebook.

Here are the formulas used to update Elo ratings:

$H_{new} = H_{prior} + K \times (W_H - p_H)$

$R_{new} = R_{prior} + K \times (W_R - p_R)$

The $W$ variables are 1 depending upon whether the home or road team won. Notice that $W_R = 1 - W_H$ since a win by one team is a loss by the other.

The $p$ variables are the Elo win probabilities for the home and road team, computed using the function above. Notice that $p_R = 1 - p_H$ since there are no ties in basketball.

In [32]:

def update(*, winner, home_elo, road_elo, hca_elo, k, probs=False):
    """Update Elo ratings for a given match up."""
    home_prob, road_prob = win_probs(home_elo=home_elo, road_elo=road_elo, hca_elo=hca_elo)
    if winner[0].upper() == 'H':
        home_win = 1
        road_win = 0
    elif winner[0].upper() in ['R', 'A', 'V']: # road, away or visitor are treated as synonyms
        home_win = 0
        road_win = 1
    else:
        raise ValueError('unrecognized winner string', winner)
    new_home_elo = home_elo + k*(home_win - home_prob)
    new_road_elo = road_elo + k*(road_win - road_prob)
    if probs:
        return new_home_elo, new_road_elo, home_prob, road_prob
    else:
        return new_home_elo, new_road_elo

Let's test out our Elo update function for various situations. If the home and road team are equally matched and there is no home court advantage, the winner's Elo rating goes up by 10 points, and the loser drops by 10 points. Elo ratings are zero-sum, since the loser gives up points to the winner.

In [33]:

update(winner='H', home_elo=1500, road_elo=1500, hca_elo=0, k=20)

Out[33]:

(1510.0, 1490.0)

In [34]:

update(winner='R', home_elo=1500, road_elo=1500, hca_elo=0, k=20)

Out[34]:

(1490.0, 1510.0)

Now, let's add in a home court advantage. This shifts the win probabilities.

In [35]:

update(winner='H', home_elo=1500, road_elo=1500, hca_elo=hca_elo, k=20)

Out[35]:

(1508.04, 1491.96)

In [36]:

update(winner='R', home_elo=1500, road_elo=1500, hca_elo=hca_elo, k=20)

Out[36]:

(1488.04, 1511.96)

The teams are still equally rated, but now we see that the home team is the slight favorite. When the road team wins, it's Elo rating afterwards is almost 1512. Compare that to the 1508 updated Elo rating the home team would have gotten if it had won.

Computing Elo Ratings for the Whole Season¶

In our simple Elo rating system, we will set all teams to an average rating of 1500 at the beginning of the season. In future posts, we will examine ways to extend the system to incorporate information across seasons. Doing this properly requires some thought around roster and coaching changes.

Setting all teams to an initial 1500 rating means that we start off knowing nothing about the teams. By the end of the regular season, hopefully our ratings system will have learned enough about team strength to make useful playoff predictions.

The below function computes the Elo ratings over an entire NBA regular season. The function is long, but not very complicated. Here is an outline of what it does:

Set all the initial team ratings to 1500
Get the match up results and sort them in chronological order
Loop through all game results:
- Get the Elo ratings from the home and road teams prior to the game
- Figure out which team won the game
- Get the new Elo ratings based on who won the game and home court advantage
- Store the before and after Elo ratings and the date of the game so we can plot time series charts later
- Store the new Elo ratings
At the end, check for data consistentcy
Return the end-of-season Elo ratings and a time series of Elo ratings for all the teams, along with the match up information used to compute the ratings

In [37]:

def simple_nba_elo(*, box_scores, teams, hca_elo, k):
    """Compute simple Elo ratings over the course of an NBA season."""
    latest_elos = {abbr: 1500 for abbr in teams['abbr']}
    matchups = box_scores.matchups.sort_values(by='date', ascending=True).copy()
    home_probs = []
    road_probs = []
    home_elos = []
    road_elos = []
    index_check = []
    elo_ts = []
    for game in matchups.itertuples(index=True):
        index = game.Index
        home_team = game.team_abbr_h
        road_team = game.team_abbr_r
        winner = game.hr_winner
        home_elo = latest_elos[home_team]
        road_elo = latest_elos[road_team]
        (new_home_elo, new_road_elo, home_prob, road_prob) = update(
            winner=winner,
            home_elo=home_elo,
            road_elo=road_elo,
            hca_elo=hca_elo,
            k=k,
            probs=True
        )
        home_info = OrderedDict({
            'date': game.date,
            'game_id': game.game_id,
            'abbr': home_team,
            'matchup_index': index,
            'opp_abbr': road_team,
            'home_road': 'H',
            'win_loss': 'W' if winner == 'H' else 'L',
            'win_prob': home_prob,
            'opp_prior_elo': latest_elos[road_team],
            'prior_elo': latest_elos[home_team],
            'new_elo': new_home_elo,
        })
        elo_ts.append(home_info)
        road_info = OrderedDict({
            'date': game.date,
            'game_id': game.game_id,
            'abbr': road_team,
            'matchup_index': index,
            'opp_abbr': home_team,
            'home_road': 'R',
            'win_loss': 'W' if winner == 'R' else 'L',
            'win_prob': road_prob,
            'opp_prior_elo': latest_elos[home_team],
            'prior_elo': latest_elos[road_team],
            'new_elo': new_road_elo,
        })
        elo_ts.append(road_info)
        latest_elos[home_team] = new_home_elo
        latest_elos[road_team] = new_road_elo
        home_probs.append(home_prob)
        road_probs.append(road_prob)
        home_elos.append(new_home_elo)
        road_elos.append(new_road_elo)
        index_check.append(index)
    matchups['home_prob'] = home_probs
    matchups['road_prob'] = road_probs
    matchups['home_elos'] = home_elos
    matchups['road_elos'] = road_elos
    matchups['index_check'] = index_check
    if not all(matchups['index_check'] == matchups.index):
        raise RuntimeError('indices do not match!')
    matchups = matchups.drop(columns=['index_check'])
    return matchups, pd.DataFrame(elo_ts), latest_elos

Let's run the function to get the 2017-18 Elo ratings for all 30 NBA teams. As discussed above, we are using a home court advantage adjustment of 69 Elo rating points and a $K$ value of 20.

In [38]:

matchups, elo_hist, curr_elos = simple_nba_elo(box_scores=nba1718, teams=teams, hca_elo=hca_elo, k=20)

The Results, and Some Plots¶

Here are the Elo ratings for all 30 NBA teams as of the end of the regular season, ranked best to worst.

In [39]:

sorted(curr_elos.items(), key=operator.itemgetter(1), reverse=True)

Out[39]:

[('HOU', 1718.7573739310872),
 ('TOR', 1642.509234000021),
 ('PHI', 1637.8605055619669),
 ('UTA', 1616.2033964377317),
 ('POR', 1595.065188211564),
 ('GSW', 1590.3946390420315),
 ('NOP', 1587.2519843938655),
 ('BOS', 1586.1753617623024),
 ('IND', 1584.6564274796217),
 ('OKC', 1580.234680473253),
 ('CLE', 1570.556557419121),
 ('DEN', 1568.7112357272322),
 ('SAS', 1541.9814371909993),
 ('MIN', 1540.460835899911),
 ('MIL', 1521.473057845756),
 ('LAC', 1512.9777428218233),
 ('MIA', 1512.1543637174311),
 ('WAS', 1484.998604454014),
 ('DET', 1467.7427511888648),
 ('LAL', 1461.962359042527),
 ('CHA', 1460.0431212254493),
 ('BKN', 1403.3122168209793),
 ('SAC', 1392.8896638705985),
 ('ATL', 1374.0186175508284),
 ('NYK', 1367.6683600631823),
 ('CHI', 1365.4361876958365),
 ('DAL', 1351.5809302667749),
 ('ORL', 1344.7360522769086),
 ('MEM', 1314.910648773433),
 ('PHX', 1303.2764648548855)]

Notice that since Elo ratings are zero-sum, the league average at the end of the season is the same as at the start of season.

In [40]:

sum(elo for elo in curr_elos.values()) / len(curr_elos)

Out[40]:

1499.9999999999998

(The reason the average isn't exactly 1500 is due to small numerical round-off errors during all these calculations.)

One of the results of the function is a time series of the Elo ratings by date and team.

In [41]:

elo_hist.tail(20)

Out[41]:

	date	game_id	abbr	matchup_index	opp_abbr	home_road	win_loss	win_prob	opp_prior_elo	prior_elo	new_elo
2440	2018-04-11	21701221	MIA	3	TOR	H	W	0.378393	1654.941373	1499.722224	1512.154364
2441	2018-04-11	21701221	TOR	3	MIA	R	L	0.621607	1499.722224	1654.941373	1642.509234
2442	2018-04-11	21701225	MIN	4	DEN	H	W	0.531625	1578.078736	1531.093335	1540.460836
2443	2018-04-11	21701225	DEN	4	MIN	R	L	0.468375	1531.093335	1578.078736	1568.711236
2444	2018-04-11	21701222	ORL	5	WAS	H	W	0.364324	1497.712134	1332.022523	1344.736052
2445	2018-04-11	21701222	WAS	5	ORL	R	L	0.635676	1332.022523	1497.712134	1484.998604
2446	2018-04-11	21701220	CLE	10	NYK	H	L	0.853405	1350.600259	1587.624658	1570.556557
2447	2018-04-11	21701220	NYK	10	CLE	R	W	0.146595	1587.624658	1350.600259	1367.668360
2448	2018-04-11	21701227	OKC	7	MEM	H	W	0.869252	1317.525610	1577.619719	1580.234680
2449	2018-04-11	21701227	MEM	7	OKC	R	L	0.130748	1577.619719	1317.525610	1314.910649
2450	2018-04-11	21701226	NOP	8	SAS	H	W	0.639873	1549.183969	1580.049452	1587.251984
2451	2018-04-11	21701226	SAS	8	NOP	R	L	0.360127	1580.049452	1549.183969	1541.981437
2452	2018-04-11	21701224	CHI	9	DET	H	L	0.479677	1458.149204	1375.029734	1365.436188
2453	2018-04-11	21701224	DET	9	CHI	R	W	0.520323	1375.029734	1458.149204	1467.742751
2454	2018-04-11	21701219	BOS	11	BKN	H	W	0.802880	1407.254609	1582.232970	1586.175362
2455	2018-04-11	21701219	BKN	11	BOS	R	L	0.197120	1582.232970	1407.254609	1403.312217
2456	2018-04-11	21701223	PHI	6	MIL	H	W	0.732127	1526.830528	1632.503036	1637.860506
2457	2018-04-11	21701223	MIL	6	PHI	R	L	0.267873	1632.503036	1526.830528	1521.473058
2458	2018-04-11	21701229	POR	0	UTA	H	W	0.542422	1625.354961	1585.913624	1595.065188
2459	2018-04-11	21701229	UTA	0	POR	R	L	0.457578	1585.913624	1625.354961	1616.203396

We can use this time series data to plot the path of teams' Elo ratings over the course of the season.

In [42]:

def plot_elo(*, data, team, ax):
    ts = data.loc[data['abbr'] == team, ['date', 'new_elo']].set_index(['date'])
    ts = ts.rename(columns={'new_elo': team})
    ts.plot(ax=ax, label=team)
    return ax

In [43]:

def plot_elo_for_teams(*, data, teams, title):
    fig, ax = plt.subplots(figsize=(10, 7))
    for team in teams:
        ax = plot_elo(data=elo_hist, team=team, ax=ax)
    ax.set_xlabel('Date')
    ax.set_ylabel('Simple Elo Rating')
    ax.set_title(title)
    plt.show()

Here is a plot of the Elo ratings for all six of the NBA division winners.

In [44]:

plot_elo_for_teams(
    data=elo_hist,
    teams=['TOR', 'MIA', 'CLE', 'HOU', 'GSW', 'POR',],
    title='Simple Elo Ratings for 2017-18 NBA Division Winners',
)

Here are the Elo ratings for the 4 teams in the Eastern Conference middle playoff brackets.

In [45]:

plot_elo_for_teams(
    data=elo_hist,
    teams=['CLE', 'IND', 'PHI', 'MIA',],
    title='Elo Ratings Eastern Conference Middle Playoff Brackets',
)

Here is a similar plot for the 4 teams in the Western Conference middle playoff brackets.

In [46]:

plot_elo_for_teams(
    data=elo_hist,
    teams=['POR', 'NOP', 'OKC', 'UTA',],
    title='Elo Ratings Western Conference Middle Playoff Brackets',
)

In [ ]: