5. Keener’s Method

Keener’s Method states that the strength \(s\) of a team should

• consider its interaction with opponents and these opponents’ strengths, as well as,

• be proportionally constant to its rating \(r\)

  • \(s = \lambda r\).

The strength statistics \(a_{ij}\) created by the i-th team when playing the j-th team can be derived from something meaning such as wins, yards, passes, etc… For example, \(a_{ij}\) can be the number of wins by the i-th team over the j-th team.

\(a_{ij} = w_{ij}\)

Or it can consider ties as well.

\(a_{ij} = w_{ij} - \dfrac{t_{ij}}{2}\)

Whatever strength statistics you use, we can denoted as \(S_{ij}\).

\(a_{ij} = S_{ij}\)

Typically, we want want to normalize \(a_{ij}\) as follows to avoid disproportionate effects on the ratings.

\(a_{ij} = \dfrac{S_{ij}}{S_{ij} + S_{ji}}\)

But, since there are some pairs of teams that might have not played, we can use Laplace’s rule of succession to normalize as follows.

\(a_{ij} = \dfrac{S_{ij} + 1}{S_{ij} + S_{ji} + 2}\)

Keener also suggests that we apply a skewing function to combat skew in the data. Skew could come from teams inflating their strength statistics to increase their ratings and thereby ranking. A particular skew function recommended by Keener is as follows.

\(h(x) = \dfrac{1}{2} + \dfrac{\mathrm{sgn} \{x - \frac{1}{2} \} \sqrt{|2x - 1|}}{2}\)

The strength \(s\) of each team is given as

\(s = Ar\),

where

• \(A\) is a \(n\) x \(n\) matrix with \(a_{ij}\) element representing the strength statistic of the i-th team when it plays the j-th team, and

• \(r\) is the rating.

Remember that we stated two things:

• \(s = \lambda r\), and

• \(s = Ar\).

Thus,

\(Ar = \lambda r\),

and we can interpret

• \(\lambda\) as an eigenvalue, and

• \(r\) as the corresponding eigenvector to the eigenvalue.

Using any numerical method we want, when can compute the eigenvectors and eigenvalues of \(A\). The eigenvector associated with the largest eigenvalue will have its elements correspond to the teams and be the ratings. When the ratings are sorted (the values of the eigenvector), then the ranking is produced.

5.1. NFL 2009

Let’s look at the whole NFL 2009 season. This data has scores, yards and turnovers. We will use scores to create the strength statistics.

import pandas as pd

def get_nfl_2009():
    return pd.read_csv('./nfl/2009.csv')

df = get_nfl_2009()
teams = sorted(list(set(df.t1) | set(df.t2)))

df

	t1	t2	s1	s2	yd1	yd2	to1	to2	home_team	week
0	Steelers	Titans	13	10	357	320	3	2.0	Steelers	1
1	Broncos	Bengals	12	7	302	307	0	2.0	Bengals	1
2	Ravens	Chiefs	38	24	501	188	1	0.0	Ravens	1
3	Colts	Jaguars	14	12	365	228	2	0.0	Colts	1
4	Cowboys	Buccaneers	34	21	462	450	0	0.0	Buccaneers	1
...	...	...	...	...	...	...	...	...	...	...
251	Cowboys	Eagles	24	0	474	228	1	1.0	Cowboys	17
252	Chiefs	Broncos	44	24	524	512	2	3.0	Broncos	17
253	Ravens	Raiders	21	13	330	325	0	2.0	Raiders	17
254	Titans	Seahawks	17	13	304	309	2	1.0	Seahawks	17
255	Jets	Bengals	37	0	320	72	0	3.0	Jets	17

256 rows × 10 columns

\(a_{ij}\) will actually be the total number of points scored by the i-th team on the j-th team during the whole season. These are the cummulative points.

def get_cummulative_score(df, t1, t2, f1='s1', f2='s2'):
    s1 = df[(df.t1 == t1) & (df.t2 == t2)][f1].sum()
    s2 = df[(df.t1 == t2) & (df.t2 == t1)][f2].sum()
    s = s1 + s2
    return s

rscore_df = pd.DataFrame([[get_cummulative_score(df, t1, t2) if t1 != t2 else 0 for t2 in teams]
              for t1 in teams], index=teams, columns=teams)
rscore_df

	49ers	Bears	Bengals	Bills	Broncos	Browns	Buccaneers	Cardinals	Chargers	Chiefs	...	Raiders	Rams	Ravens	Redskins	Saints	Seahawks	Steelers	Texans	Titans	Vikings
49ers	0	10	0	0	0	0	0	44	0	0	...	0	63	0	0	0	40	0	21	27	24
Bears	6	0	10	0	0	30	0	21	0	0	...	0	17	7	0	0	25	17	0	0	46
Bengals	0	45	0	0	7	39	0	0	24	17	...	17	0	34	0	0	0	41	17	0	10
Bills	0	0	0	0	0	3	33	0	0	16	...	0	0	0	0	7	0	0	10	17	0
Broncos	0	0	12	0	0	27	0	0	37	68	...	42	0	7	17	0	0	10	0	0	0
Browns	0	6	27	6	6	0	0	0	23	41	...	23	0	3	0	0	0	27	0	0	20
Buccaneers	0	0	0	20	0	0	0	0	0	0	...	0	0	0	13	27	24	0	0	0	0
Cardinals	25	41	0	0	0	0	0	0	0	0	...	0	52	0	0	0	58	0	28	17	30
Chargers	0	0	27	0	55	30	0	0	0	80	...	48	0	26	23	0	0	28	0	42	0
Chiefs	0	0	10	10	57	34	0	0	21	0	...	26	0	24	14	0	0	27	0	0	0
Colts	18	0	0	7	28	0	0	31	0	0	...	0	42	17	0	0	34	0	55	58	0
Cowboys	0	0	0	0	10	0	34	0	17	26	...	24	0	0	24	24	38	0	0	0	0
Dolphins	0	0	0	52	0	0	25	0	13	0	...	0	0	0	0	34	0	24	20	24	0
Eagles	27	24	0	0	30	0	33	0	23	34	...	9	0	0	54	22	0	0	0	0	0
Falcons	45	21	0	31	0	0	40	0	0	0	...	0	0	0	31	50	0	0	0	0	0
Giants	0	0	0	0	6	0	24	17	20	27	...	44	0	0	68	27	0	0	0	0	7
Jaguars	3	0	0	18	0	17	0	17	0	24	...	0	23	0	0	0	0	0	54	50	0
Jets	0	0	37	32	0	0	26	0	0	0	...	38	0	0	0	10	0	0	24	24	0
Lions	6	47	13	0	0	38	0	24	0	0	...	0	10	3	19	27	20	20	0	0	23
Packers	30	42	24	0	0	31	28	33	0	0	...	0	36	27	0	0	48	36	0	0	49
Panthers	0	0	0	9	0	0	44	34	0	0	...	0	0	0	20	43	0	0	0	0	26
Patriots	0	0	0	42	17	0	35	0	0	0	...	0	0	27	0	17	0	0	27	59	0
Raiders	0	0	20	0	23	9	0	0	36	23	...	0	0	13	13	0	0	27	6	0	0
Rams	6	9	0	0	0	0	0	23	0	0	...	0	0	0	7	23	17	0	13	7	10
Ravens	0	31	21	0	30	50	0	0	31	38	...	21	0	0	0	0	0	40	0	0	31
Redskins	0	0	0	0	27	0	16	0	20	6	...	34	9	0	0	30	0	0	0	0	0
Saints	0	0	0	27	0	0	55	0	0	0	...	0	28	0	33	0	0	0	0	0	0
Seahawks	30	19	0	0	0	0	7	23	0	0	...	0	55	0	0	0	0	0	7	13	9
Steelers	0	14	32	0	28	33	0	0	38	24	...	24	0	40	0	0	0	0	0	13	27
Texans	24	0	28	31	0	0	0	21	0	0	...	29	16	0	0	0	34	0	0	51	0
Titans	34	0	0	41	0	0	0	20	17	0	...	0	47	0	0	0	17	10	51	0	0
Vikings	27	66	30	0	0	34	0	17	0	0	...	0	38	33	0	0	35	17	0	0	0

32 rows × 32 columns

We then apply Laplace's rule of succession.

def get_strength(df, i, j):
    s_ij = df.iloc[i].iloc[j]
    s_ji = df.iloc[j].iloc[i]
    s = (s_ij + 1) / (s_ij + s_ji + 2)
    return s

sscore_df = pd.DataFrame([[get_strength(rscore_df, i, j) for j in range(len(teams))]
                          for i in range(len(teams))], index=teams, columns=teams)
sscore_df

	49ers	Bears	Bengals	Bills	Broncos	Browns	Buccaneers	Cardinals	Chargers	Chiefs	...	Raiders	Rams	Ravens	Redskins	Saints	Seahawks	Steelers	Texans	Titans	Vikings
49ers	0.500000	0.611111	0.500000	0.500000	0.500000	0.500000	0.500000	0.633803	0.500000	0.500000	...	0.500000	0.901408	0.500000	0.500000	0.500000	0.569444	0.500000	0.468085	0.444444	0.471698
Bears	0.388889	0.500000	0.192982	0.500000	0.500000	0.815789	0.500000	0.343750	0.500000	0.500000	...	0.500000	0.642857	0.200000	0.500000	0.500000	0.565217	0.545455	0.500000	0.500000	0.412281
Bengals	0.500000	0.807018	0.500000	0.500000	0.380952	0.588235	0.500000	0.500000	0.471698	0.620690	...	0.461538	0.500000	0.614035	0.500000	0.500000	0.500000	0.560000	0.382979	0.500000	0.261905
Bills	0.500000	0.500000	0.500000	0.500000	0.500000	0.363636	0.618182	0.500000	0.500000	0.607143	...	0.500000	0.500000	0.500000	0.500000	0.222222	0.500000	0.500000	0.255814	0.300000	0.500000
Broncos	0.500000	0.500000	0.619048	0.500000	0.500000	0.800000	0.500000	0.500000	0.404255	0.543307	...	0.641791	0.500000	0.205128	0.391304	0.500000	0.500000	0.275000	0.500000	0.500000	0.500000
Browns	0.500000	0.184211	0.411765	0.636364	0.200000	0.500000	0.500000	0.500000	0.436364	0.545455	...	0.705882	0.500000	0.072727	0.500000	0.500000	0.500000	0.451613	0.500000	0.500000	0.375000
Buccaneers	0.500000	0.500000	0.500000	0.381818	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.451613	0.333333	0.757576	0.500000	0.500000	0.500000	0.500000
Cardinals	0.366197	0.656250	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	...	0.500000	0.688312	0.500000	0.500000	0.500000	0.710843	0.500000	0.568627	0.461538	0.632653
Chargers	0.500000	0.500000	0.528302	0.500000	0.595745	0.563636	0.500000	0.500000	0.500000	0.786408	...	0.569767	0.500000	0.457627	0.533333	0.500000	0.500000	0.426471	0.500000	0.704918	0.500000
Chiefs	0.500000	0.500000	0.379310	0.392857	0.456693	0.454545	0.500000	0.500000	0.213592	0.500000	...	0.529412	0.500000	0.390625	0.681818	0.500000	0.500000	0.528302	0.500000	0.500000	0.500000
Colts	0.558824	0.500000	0.500000	0.205128	0.630435	0.500000	0.500000	0.744186	0.500000	0.500000	...	0.500000	0.860000	0.529412	0.500000	0.500000	0.660377	0.500000	0.554455	0.686047	0.500000
Cowboys	0.500000	0.500000	0.500000	0.500000	0.379310	0.500000	0.614035	0.500000	0.461538	0.562500	...	0.757576	0.500000	0.500000	0.781250	0.581395	0.684211	0.500000	0.500000	0.500000	0.500000
Dolphins	0.500000	0.500000	0.500000	0.557895	0.500000	0.500000	0.520000	0.500000	0.368421	0.500000	...	0.500000	0.500000	0.500000	0.500000	0.426829	0.500000	0.446429	0.428571	0.471698	0.500000
Eagles	0.666667	0.543478	0.500000	0.500000	0.525424	0.500000	0.693878	0.500000	0.428571	0.700000	...	0.416667	0.500000	0.500000	0.567010	0.319444	0.500000	0.500000	0.500000	0.500000	0.500000
Falcons	0.807018	0.594595	0.500000	0.888889	0.500000	0.500000	0.594203	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.640000	0.451327	0.500000	0.500000	0.500000	0.500000	0.500000
Giants	0.500000	0.500000	0.500000	0.500000	0.205882	0.500000	0.961538	0.418605	0.488372	0.622222	...	0.849057	0.500000	0.500000	0.696970	0.363636	0.500000	0.500000	0.500000	0.500000	0.150943
Jaguars	0.160000	0.500000	0.500000	0.542857	0.500000	0.428571	0.500000	0.360000	0.500000	0.531915	...	0.500000	0.533333	0.500000	0.500000	0.500000	0.023256	0.500000	0.561224	0.515152	0.500000
Jets	0.500000	0.500000	0.974359	0.523810	0.500000	0.500000	0.870968	0.500000	0.500000	0.500000	...	0.975000	0.500000	0.500000	0.500000	0.305556	0.500000	0.500000	0.757576	0.581395	0.500000
Lions	0.250000	0.358209	0.368421	0.500000	0.500000	0.506494	0.500000	0.438596	0.500000	0.500000	...	0.500000	0.379310	0.075472	0.571429	0.378378	0.388889	0.420000	0.500000	0.500000	0.303797
Packers	0.553571	0.589041	0.438596	0.500000	0.500000	0.888889	0.426471	0.809524	0.500000	0.500000	...	0.500000	0.672727	0.651163	0.500000	0.500000	0.816667	0.493333	0.500000	0.500000	0.420168
Panthers	0.500000	0.500000	0.500000	0.322581	0.500000	0.500000	0.616438	0.614035	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.538462	0.517647	0.500000	0.500000	0.500000	0.500000	0.771429
Patriots	0.500000	0.500000	0.500000	0.551282	0.461538	0.500000	0.818182	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.560000	0.500000	0.315789	0.500000	0.500000	0.444444	0.983607	0.500000
Raiders	0.500000	0.500000	0.538462	0.500000	0.358209	0.294118	0.500000	0.500000	0.430233	0.470588	...	0.500000	0.500000	0.388889	0.285714	0.500000	0.500000	0.528302	0.189189	0.500000	0.500000
Rams	0.098592	0.357143	0.500000	0.500000	0.500000	0.500000	0.500000	0.311688	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.444444	0.452830	0.243243	0.500000	0.451613	0.142857	0.220000
Ravens	0.500000	0.800000	0.385965	0.500000	0.794872	0.927273	0.500000	0.500000	0.542373	0.609375	...	0.611111	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.484848
Redskins	0.500000	0.500000	0.500000	0.500000	0.608696	0.500000	0.548387	0.500000	0.466667	0.318182	...	0.714286	0.555556	0.500000	0.500000	0.476923	0.500000	0.500000	0.500000	0.500000	0.500000
Saints	0.500000	0.500000	0.500000	0.777778	0.500000	0.500000	0.666667	0.500000	0.500000	0.500000	...	0.500000	0.547170	0.500000	0.523077	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000
Seahawks	0.430556	0.434783	0.500000	0.500000	0.500000	0.500000	0.242424	0.289157	0.500000	0.500000	...	0.500000	0.756757	0.500000	0.500000	0.500000	0.500000	0.500000	0.186047	0.437500	0.217391
Steelers	0.500000	0.454545	0.440000	0.500000	0.725000	0.548387	0.500000	0.500000	0.573529	0.471698	...	0.471698	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.560000	0.608696
Texans	0.531915	0.500000	0.617021	0.744186	0.500000	0.500000	0.500000	0.431373	0.500000	0.500000	...	0.810811	0.548387	0.500000	0.500000	0.500000	0.813953	0.500000	0.500000	0.500000	0.500000
Titans	0.555556	0.500000	0.500000	0.700000	0.500000	0.500000	0.500000	0.538462	0.295082	0.500000	...	0.500000	0.857143	0.500000	0.500000	0.500000	0.562500	0.440000	0.500000	0.500000	0.500000
Vikings	0.528302	0.587719	0.738095	0.500000	0.500000	0.625000	0.500000	0.367347	0.500000	0.500000	...	0.500000	0.780000	0.515152	0.500000	0.500000	0.782609	0.391304	0.500000	0.500000	0.500000

32 rows × 32 columns

Finally, we combat skew in the data.

import numpy as np

def h(x):
    h = 0.5 + ((np.sign(x - 0.5) * np.sqrt(np.abs(2 * x - 1))) / 2)
    return h

X = sscore_df.applymap(h)
X

	49ers	Bears	Bengals	Bills	Broncos	Browns	Buccaneers	Cardinals	Chargers	Chiefs	...	Raiders	Rams	Ravens	Redskins	Saints	Seahawks	Steelers	Texans	Titans	Vikings
49ers	0.500000	0.735702	0.500000	0.500000	0.500000	0.500000	0.500000	0.758653	0.500000	0.500000	...	0.500000	0.948000	0.500000	0.500000	0.500000	0.686339	0.500000	0.373677	0.333333	0.381042
Bears	0.264298	0.500000	0.108198	0.500000	0.500000	0.897360	0.500000	0.220492	0.500000	0.500000	...	0.500000	0.767261	0.112702	0.500000	0.500000	0.680579	0.650756	0.500000	0.500000	0.290573
Bengals	0.500000	0.891802	0.500000	0.500000	0.256025	0.710042	0.500000	0.500000	0.381042	0.745652	...	0.361325	0.500000	0.738783	0.500000	0.500000	0.500000	0.673205	0.258110	0.500000	0.154967
Bills	0.500000	0.500000	0.500000	0.500000	0.500000	0.238884	0.743086	0.500000	0.500000	0.731455	...	0.500000	0.500000	0.500000	0.500000	0.127322	0.500000	0.500000	0.150582	0.183772	0.500000
Broncos	0.500000	0.500000	0.743975	0.500000	0.500000	0.887298	0.500000	0.500000	0.281203	0.647151	...	0.766262	0.500000	0.116026	0.266874	0.500000	0.500000	0.164590	0.500000	0.500000	0.500000
Browns	0.500000	0.102640	0.289958	0.761116	0.112702	0.500000	0.500000	0.500000	0.321623	0.650756	...	0.820844	0.500000	0.037792	0.500000	0.500000	0.500000	0.344457	0.500000	0.500000	0.250000
Buccaneers	0.500000	0.500000	0.500000	0.256914	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.344457	0.211325	0.858870	0.500000	0.500000	0.500000	0.500000
Cardinals	0.241347	0.779508	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	...	0.500000	0.806848	0.500000	0.500000	0.500000	0.824687	0.500000	0.685240	0.361325	0.757539
Chargers	0.500000	0.500000	0.618958	0.500000	0.718797	0.678377	0.500000	0.500000	0.500000	0.878423	...	0.686772	0.500000	0.354444	0.629099	0.500000	0.500000	0.308259	0.500000	0.820092	0.500000
Chiefs	0.500000	0.500000	0.254348	0.268545	0.352849	0.349244	0.500000	0.500000	0.121577	0.500000	...	0.621268	0.500000	0.266146	0.801511	0.500000	0.500000	0.618958	0.500000	0.500000	0.500000
Colts	0.671499	0.500000	0.500000	0.116026	0.755377	0.500000	0.500000	0.849418	0.500000	0.500000	...	0.500000	0.924264	0.621268	0.500000	0.500000	0.783176	0.500000	0.665008	0.804997	0.500000
Cowboys	0.500000	0.500000	0.500000	0.500000	0.254348	0.500000	0.738783	0.500000	0.361325	0.676777	...	0.858870	0.500000	0.500000	0.875000	0.701737	0.803488	0.500000	0.500000	0.500000	0.500000
Dolphins	0.500000	0.500000	0.500000	0.670139	0.500000	0.500000	0.600000	0.500000	0.243505	0.500000	...	0.500000	0.500000	0.500000	0.500000	0.308727	0.500000	0.336337	0.311018	0.381042	0.500000
Eagles	0.788675	0.647442	0.500000	0.500000	0.612747	0.500000	0.811350	0.500000	0.311018	0.816228	...	0.295876	0.500000	0.500000	0.683044	0.199537	0.500000	0.500000	0.500000	0.500000	0.500000
Falcons	0.891802	0.717479	0.500000	0.940959	0.500000	0.500000	0.717029	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.764575	0.343999	0.500000	0.500000	0.500000	0.500000	0.500000
Giants	0.500000	0.500000	0.500000	0.500000	0.116518	0.500000	0.980384	0.298263	0.423751	0.747207	...	0.917766	0.500000	0.500000	0.813823	0.238884	0.500000	0.500000	0.500000	0.500000	0.082234
Jaguars	0.087689	0.500000	0.500000	0.646385	0.500000	0.311018	0.500000	0.235425	0.500000	0.626323	...	0.500000	0.629099	0.500000	0.500000	0.500000	0.011766	0.500000	0.674964	0.587039	0.500000
Jets	0.500000	0.500000	0.987011	0.609109	0.500000	0.500000	0.930678	0.500000	0.500000	0.500000	...	0.987340	0.500000	0.500000	0.500000	0.188195	0.500000	0.500000	0.858870	0.701737	0.500000
Lions	0.146447	0.233738	0.243505	0.500000	0.500000	0.556980	0.500000	0.324781	0.500000	0.500000	...	0.500000	0.254348	0.039279	0.688982	0.253402	0.264298	0.300000	0.500000	0.500000	0.186789
Packers	0.663663	0.710999	0.324781	0.500000	0.500000	0.940959	0.308259	0.893398	0.500000	0.500000	...	0.500000	0.793877	0.774921	0.500000	0.500000	0.897911	0.442265	0.500000	0.500000	0.300210
Panthers	0.500000	0.500000	0.500000	0.202158	0.500000	0.500000	0.741287	0.738783	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.638675	0.593934	0.500000	0.500000	0.500000	0.500000	0.868394
Patriots	0.500000	0.500000	0.500000	0.660128	0.361325	0.500000	0.898862	0.500000	0.500000	0.500000	...	0.500000	0.500000	0.673205	0.500000	0.196512	0.500000	0.500000	0.333333	0.991735	0.500000
Raiders	0.500000	0.500000	0.638675	0.500000	0.233738	0.179156	0.500000	0.500000	0.313228	0.378732	...	0.500000	0.500000	0.264298	0.172673	0.500000	0.500000	0.618958	0.105785	0.500000	0.500000
Rams	0.052000	0.232739	0.500000	0.500000	0.500000	0.500000	0.500000	0.193152	0.500000	0.500000	...	0.500000	0.500000	0.500000	0.333333	0.346426	0.141701	0.500000	0.344457	0.077423	0.125834
Ravens	0.500000	0.887298	0.261217	0.500000	0.883974	0.962208	0.500000	0.500000	0.645556	0.733854	...	0.735702	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.412961
Redskins	0.500000	0.500000	0.500000	0.500000	0.733126	0.500000	0.655543	0.500000	0.370901	0.198489	...	0.827327	0.666667	0.500000	0.500000	0.392583	0.500000	0.500000	0.500000	0.500000	0.500000
Saints	0.500000	0.500000	0.500000	0.872678	0.500000	0.500000	0.788675	0.500000	0.500000	0.500000	...	0.500000	0.653574	0.500000	0.607417	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000
Seahawks	0.313661	0.319421	0.500000	0.500000	0.500000	0.500000	0.141130	0.175313	0.500000	0.500000	...	0.500000	0.858299	0.500000	0.500000	0.500000	0.500000	0.500000	0.103797	0.323223	0.124095
Steelers	0.500000	0.349244	0.326795	0.500000	0.835410	0.655543	0.500000	0.500000	0.691741	0.381042	...	0.381042	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.500000	0.673205	0.733126
Texans	0.626323	0.500000	0.741890	0.849418	0.500000	0.500000	0.500000	0.314760	0.500000	0.500000	...	0.894215	0.655543	0.500000	0.500000	0.500000	0.896203	0.500000	0.500000	0.500000	0.500000
Titans	0.666667	0.500000	0.500000	0.816228	0.500000	0.500000	0.500000	0.638675	0.179908	0.500000	...	0.500000	0.922577	0.500000	0.500000	0.500000	0.676777	0.326795	0.500000	0.500000	0.500000
Vikings	0.618958	0.709427	0.845033	0.500000	0.500000	0.750000	0.500000	0.242461	0.500000	0.500000	...	0.500000	0.874166	0.587039	0.500000	0.500000	0.875905	0.266874	0.500000	0.500000	0.500000

32 rows × 32 columns

5.2. Numpy

We will use numpy to find the eigenvalues and eigenvectors.

e, E = np.linalg.eig(X)

e

array([1.58321633e+01+0.j        , 2.22768644e-03+1.85772142j,
       2.22768644e-03-1.85772142j, 3.39712442e-03+1.35674872j,
       3.39712442e-03-1.35674872j, 1.00041396e-03+1.26017375j,
       1.00041396e-03-1.26017375j, 1.04401936e-02+1.20532274j,
       1.04401936e-02-1.20532274j, 4.22054558e-03+1.07142414j,
       4.22054558e-03-1.07142414j, 7.62218544e-04+0.95584753j,
       7.62218544e-04-0.95584753j, 1.50585060e-02+0.7822698j ,
       1.50585060e-02-0.7822698j , 2.70582449e-03+0.72218668j,
       2.70582449e-03-0.72218668j, 2.01788728e-03+0.63803178j,
       2.01788728e-03-0.63803178j, 4.25439229e-03+0.57011899j,
       4.25439229e-03-0.57011899j, 1.11927138e-02+0.4517283j ,
       1.11927138e-02-0.4517283j , 7.65441825e-03+0.35245526j,
       7.65441825e-03-0.35245526j, 1.15967256e-02+0.24209874j,
       1.15967256e-02-0.24209874j, 3.16542655e-03+0.18988739j,
       3.16542655e-03-0.18988739j, 2.97621199e-03+0.07044387j,
       2.97621199e-03-0.07044387j, 2.49615168e-03+0.j        ])

The sorted eigenvector shows that the Saints are at the top with the Rams at the bottom. The Saints won the Super Bowl during the 2009-2010 season.

pd.Series(E[:,0], index=teams).sort_values(ascending=True)

Saints       -0.203355+0.000000j
Packers      -0.201014+0.000000j
Patriots     -0.197235+0.000000j
Chargers     -0.197097+0.000000j
Colts        -0.195922+0.000000j
Vikings      -0.195730+0.000000j
Cowboys      -0.195317+0.000000j
Jets         -0.195166+0.000000j
Eagles       -0.190664+0.000000j
Ravens       -0.190316+0.000000j
Steelers     -0.188673+0.000000j
Texans       -0.188029+0.000000j
Falcons      -0.183949+0.000000j
Cardinals    -0.182012+0.000000j
49ers        -0.179368+0.000000j
Broncos      -0.178877+0.000000j
Bengals      -0.177157+0.000000j
Panthers     -0.173233+0.000000j
Titans       -0.171839+0.000000j
Giants       -0.171517+0.000000j
Dolphins     -0.167714+0.000000j
Bears        -0.165493+0.000000j
Redskins     -0.163789+0.000000j
Bills        -0.163556+0.000000j
Jaguars      -0.162974+0.000000j
Chiefs       -0.157590+0.000000j
Browns       -0.157127+0.000000j
Seahawks     -0.153406+0.000000j
Raiders      -0.147552+0.000000j
Buccaneers   -0.147396+0.000000j
Lions        -0.144028+0.000000j
Rams         -0.140011+0.000000j
dtype: complex128

5.3. Scikit-Learn

Here, we use Scikit-Learn to retrieve the eigenvalues and eigenvectors.

from sklearn.decomposition import TruncatedSVD

svd = TruncatedSVD(n_components=2, n_iter=100, random_state=37)
svd.fit(X)

TruncatedSVD(n_iter=100, random_state=37)

svd.singular_values_

array([16.16520004,  1.85991954])

Again, we see the Saints at the top.

pd.Series(svd.components_[0,:], index=teams).sort_values(ascending=True)

Saints        0.148945
Packers       0.151001
Patriots      0.154887
Chargers      0.155021
Colts         0.156045
Vikings       0.156115
Cowboys       0.156881
Jets          0.156972
Eagles        0.161209
Ravens        0.161649
Steelers      0.163317
Texans        0.163902
Falcons       0.167846
Cardinals     0.169756
49ers         0.172460
Broncos       0.172764
Bengals       0.174560
Panthers      0.178314
Titans        0.179604
Giants        0.180136
Dolphins      0.183847
Bears         0.186135
Redskins      0.187734
Bills         0.188043
Jaguars       0.188691
Chiefs        0.193845
Browns        0.194392
Seahawks      0.198091
Buccaneers    0.203681
Raiders       0.203764
Lions         0.207208
Rams          0.211215
dtype: float64

Here are the ratings side-by-side.

pd.DataFrame({
    'numpy': pd.Series(E[:,0], index=teams),
    'scikit': pd.Series(svd.components_[0,:], index=teams)
})

	numpy	scikit
49ers	-0.179368+0.000000j	0.172460
Bears	-0.165493+0.000000j	0.186135
Bengals	-0.177157+0.000000j	0.174560
Bills	-0.163556+0.000000j	0.188043
Broncos	-0.178877+0.000000j	0.172764
Browns	-0.157127+0.000000j	0.194392
Buccaneers	-0.147396+0.000000j	0.203681
Cardinals	-0.182012+0.000000j	0.169756
Chargers	-0.197097+0.000000j	0.155021
Chiefs	-0.157590+0.000000j	0.193845
Colts	-0.195922+0.000000j	0.156045
Cowboys	-0.195317+0.000000j	0.156881
Dolphins	-0.167714+0.000000j	0.183847
Eagles	-0.190664+0.000000j	0.161209
Falcons	-0.183949+0.000000j	0.167846
Giants	-0.171517+0.000000j	0.180136
Jaguars	-0.162974+0.000000j	0.188691
Jets	-0.195166+0.000000j	0.156972
Lions	-0.144028+0.000000j	0.207208
Packers	-0.201014+0.000000j	0.151001
Panthers	-0.173233+0.000000j	0.178314
Patriots	-0.197235+0.000000j	0.154887
Raiders	-0.147552+0.000000j	0.203764
Rams	-0.140011+0.000000j	0.211215
Ravens	-0.190316+0.000000j	0.161649
Redskins	-0.163789+0.000000j	0.187734
Saints	-0.203355+0.000000j	0.148945
Seahawks	-0.153406+0.000000j	0.198091
Steelers	-0.188673+0.000000j	0.163317
Texans	-0.188029+0.000000j	0.163902
Titans	-0.171839+0.000000j	0.179604
Vikings	-0.195730+0.000000j	0.156115

Here’s the ranking side-by-side. The only difference is with the Buccaneers and Raiders.

ranking_df = pd.DataFrame({
    'numpy': pd.Series(E[:,0], index=teams).sort_values().index,
    'scikit': pd.Series(svd.components_[0,:], index=teams).sort_values().index
})
ranking_df['differs?'] = ranking_df.numpy != ranking_df.scikit
ranking_df

	numpy	scikit	differs?
0	Saints	Saints	False
1	Packers	Packers	False
2	Patriots	Patriots	False
3	Chargers	Chargers	False
4	Colts	Colts	False
5	Vikings	Vikings	False
6	Cowboys	Cowboys	False
7	Jets	Jets	False
8	Eagles	Eagles	False
9	Ravens	Ravens	False
10	Steelers	Steelers	False
11	Texans	Texans	False
12	Falcons	Falcons	False
13	Cardinals	Cardinals	False
14	49ers	49ers	False
15	Broncos	Broncos	False
16	Bengals	Bengals	False
17	Panthers	Panthers	False
18	Titans	Titans	False
19	Giants	Giants	False
20	Dolphins	Dolphins	False
21	Bears	Bears	False
22	Redskins	Redskins	False
23	Bills	Bills	False
24	Jaguars	Jaguars	False
25	Chiefs	Chiefs	False
26	Browns	Browns	False
27	Seahawks	Seahawks	False
28	Raiders	Buccaneers	True
29	Buccaneers	Raiders	True
30	Lions	Lions	False
31	Rams	Rams	False

5.4. Ranking by yards

Let’s use yardage gains instead.

rscore_df = pd.DataFrame([[get_cummulative_score(df, t1, t2, 'yd1', 'yd2') if t1 != t2 else 0 for t2 in teams]
              for t1 in teams], index=teams, columns=teams)
sscore_df = pd.DataFrame([[get_strength(rscore_df, i, j) for j in range(len(teams))]
                          for i in range(len(teams))], index=teams, columns=teams)
X = sscore_df.applymap(h)

e, E = np.linalg.eig(X)

svd = TruncatedSVD(n_components=2, n_iter=100, random_state=37)
svd.fit(X)

ranking_df = pd.DataFrame({
    'numpy': pd.Series(E[:,0], index=teams).sort_values().index,
    'scikit': pd.Series(svd.components_[0,:], index=teams).sort_values().index
})
ranking_df['differs?'] = ranking_df.numpy != ranking_df.scikit
ranking_df

	numpy	scikit	differs?
0	Browns	Packers	True
1	Buccaneers	Cowboys	True
2	Raiders	Texans	True
3	Lions	Jets	True
4	Rams	Steelers	True
5	Chiefs	Eagles	True
6	Bills	Patriots	True
7	49ers	Vikings	True
8	Seahawks	Saints	True
9	Falcons	Ravens	True
10	Bears	Giants	True
11	Dolphins	Chargers	True
12	Jaguars	Colts	True
13	Cardinals	Panthers	True
14	Titans	Broncos	True
15	Redskins	Bengals	True
16	Bengals	Redskins	True
17	Broncos	Titans	True
18	Panthers	Cardinals	True
19	Colts	Jaguars	True
20	Chargers	Dolphins	True
21	Giants	Bears	True
22	Ravens	Falcons	True
23	Saints	Seahawks	True
24	Vikings	49ers	True
25	Patriots	Bills	True
26	Eagles	Chiefs	True
27	Steelers	Rams	True
28	Jets	Lions	True
29	Texans	Raiders	True
30	Cowboys	Buccaneers	True
31	Packers	Browns	True

5.5. Turnovers

Let’s rank the teams by turnovers. It’s interesting that we can also rank teams to see who’s the worst.

rscore_df = pd.DataFrame([[get_cummulative_score(df, t1, t2, 'to1', 'to2') if t1 != t2 else 0 for t2 in teams]
              for t1 in teams], index=teams, columns=teams)
sscore_df = pd.DataFrame([[get_strength(rscore_df, i, j) for j in range(len(teams))]
                          for i in range(len(teams))], index=teams, columns=teams)
X = sscore_df.applymap(h)

e, E = np.linalg.eig(X)

svd = TruncatedSVD(n_components=2, n_iter=100, random_state=37)
svd.fit(X)

ranking_df = pd.DataFrame({
    'numpy': pd.Series(E[:,0], index=teams).sort_values().index,
    'scikit': pd.Series(svd.components_[0,:], index=teams).sort_values().index
})
ranking_df['differs?'] = ranking_df.numpy != ranking_df.scikit
ranking_df

	numpy	scikit	differs?
0	Lions	Lions	False
1	Raiders	Raiders	False
2	Rams	Rams	False
3	Browns	Browns	False
4	Giants	Giants	False
5	Seahawks	Seahawks	False
6	Dolphins	Dolphins	False
7	Steelers	Steelers	False
8	Buccaneers	Buccaneers	False
9	Titans	Titans	False
10	Redskins	Redskins	False
11	Jaguars	Jaguars	False
12	Bengals	Bengals	False
13	Bills	Bills	False
14	Bears	Bears	False
15	Texans	Texans	False
16	Chargers	Chargers	False
17	49ers	49ers	False
18	Cardinals	Cardinals	False
19	Colts	Colts	False
20	Saints	Saints	False
21	Cowboys	Cowboys	False
22	Falcons	Falcons	False
23	Jets	Jets	False
24	Broncos	Broncos	False
25	Panthers	Panthers	False
26	Vikings	Eagles	True
27	Eagles	Vikings	True
28	Ravens	Ravens	False
29	Chiefs	Chiefs	False
30	Patriots	Patriots	False
31	Packers	Packers	False