Rating, Ranking and Colley regression

Class 3

February 5th, 2020

Introduction

Summary of Massey: Atlantic coast conf.

Ratings are the solution $\mathbf{r}$ of $\overline{M}\mathbf{r} = \mathbf{p}$

The data that drives ratings is point difference.

Duke, Miami, U of North Carolina, U of Virginia, Virginia Tech (plus Georgia Tech and Pittsburgh now)

Critique

is point difference so clearly a reflection of the skills gap?
what if, in a short season, the final result is the only goal?

Colley’s Method

Idea: rating as winning

r_{i} = \frac{w_{i}}{t_{i}} = \frac{w_{i}}{w_{i} + l_{i}}

$r_i = \frac{w_i}{t_i} = \frac{w_i}{w_i + l_i}$

Simple
captures long-term trend (but not decay)
ignores strenght of the opponents
cold-start problem: $\frac{0}{0}\rightarrow \frac{0}{1} | \frac{1}{1}$ ???

Laplace correction

r_{i} = \frac{1 + w_{i}}{2 + t_{i}} = \frac{1 + w_{i}}{2 + w_{i} + l_{i}}

$r_i = \frac{1 + w_i}{2 + t_i} = \frac{1 + w_i}{2 + w_i + l_i}$

$\frac{1}{2}\rightarrow \frac{1}{3} | \frac{2}{3}$

$\frac{1}{2}\rightarrow \frac{1}{3} | \frac{2}{3} \rightarrow \frac{1}{4} | \frac{2}{4} | \frac{3}{4} \rightarrow \frac{1}{5} | \frac{2}{5} | \frac{3}{5} | \frac{4}{5}$

Ratings of 0 or 1 are not reachable.

Dependence on opponent’s strenght

Colley ratings can be rephrased to contain opponents’ strengt as one of the factors.

w_{i} = \frac{w_{i}}{2} + \frac{w_{i}}{2}

$w_i = \frac{w_i}{2} + \frac{w_i}{2}$

w_{i} = \frac{w_{i}}{2} + \frac{w_{i}}{2} + \frac{l_{i}}{2} - \frac{l_{i}}{2}

$w_i = \frac{w_i}{2} + \frac{w_i}{2} + \frac{l_i}{2} - \frac{l_i}{2}$

w_{i} = \frac{w_{i} - l_{i}}{2} + \frac{w_{i} + l_{i}}{2}

$w_i = \frac{w_i-l_i}{2} + \frac{w_i + l_i}{2}$

w_{i} = \frac{w_{i} - l_{i}}{2} + \frac{w_{i} + l_{i}}{2}

$w_i = \frac{w_i-l_i}{2} + \frac{w_i + l_i}{2}$

= \frac{w_{i} - l_{i}}{2} + \frac{t_{i}}{2}

$= \frac{w_i-l_i}{2} + \frac{t_i}{2}$

= \frac{w_{i} - l_{i}}{2} + \sum_{j = 1}^{t_{i}} \frac{1}{2}

$= \frac{w_i-l_i}{2} + \sum_{j=1}^{t_i}\frac{1}{2}$

since at the start, each $r_x = 1/2$ :

\sum_{j = 1}^{t_{i}} \frac{1}{2} = \sum_{j \in O_{i}} r_{j}

$\sum_{j=1}^{t_i}\frac{1}{2} = \sum_{j\in O_i}r_j$

As we progress

w_{i} \approx \frac{w_{i} - l_{i}}{2} + \sum_{j \in O_{i}} r_{j}

$w_i \approx \frac{w_i-l_i}{2} + \sum_{j\in O_i}r_j$

so we can interpret $w_i$ as winning balance plus sum of strenghts of the opponents

Colley in LA

Incorporate opponents’ strenght

r_{i} = \frac{1 + w_{i}}{2 + t_{i}}

$r_i = \frac{1 + w_i}{2 + t_i}$

r_{i} = \frac{1 + \frac{w_{i} - l_{i}}{2} + \sum_{j \in O_{i}} r_{i}}{2 + t_{i}}

$r_i = \frac{1 + \frac{w_i-l_i}{2} + \sum_{j\in O_i}r_i}{2 + t_i}$

Let’s compute all $r_i$ ’s at once.

Matrix form

C r = b

$C \mathbf{r} = \mathbf{b}$

C and b are defined to reflect Colley’s rating formula

$c_{ii} = 2 + t_i$

$c_{ij} = -n_{ij},$ i.e., the number of direct matches

(\begin{matrix} 6 & - 1 & - 1 & - 1 & - 1 \\ - 1 & 6 & - 1 & - 1 & - 1 \\ - 1 & - 1 & 6 & - 1 & - 1 \\ - 1 & - 1 & - 1 & 6 & - 1 \\ - 1 & - 1 & - 1 & - 1 & 6 \end{matrix})

$\begin{pmatrix} 6 & -1 & -1 & -1 & -1 \\ -1 & 6 & -1 & -1 & -1 \\ -1 & -1 & 6 & -1 & -1 \\ -1 & -1 & -1 & 6 & -1 \\ -1 & -1 & -1 & -1 & 6 \end{pmatrix}$

Essentially $C = 2I + M$

(\begin{matrix} 6 & - 1 & - 1 & - 1 & - 1 \\ - 1 & 6 & - 1 & - 1 & - 1 \\ - 1 & - 1 & 6 & - 1 & - 1 \\ - 1 & - 1 & - 1 & 6 & - 1 \\ - 1 & - 1 & - 1 & - 1 & 6 \end{matrix}) r = b

$\begin{pmatrix} 6 & -1 & -1 & -1 & -1 \\ -1 & 6 & -1 & -1 & -1 \\ -1 & -1 & 6 & -1 & -1 \\ -1 & -1 & -1 & 6 & -1 \\ -1 & -1 & -1 & -1 & 6 \end{pmatrix} \mathbf{r} = \mathbf{b}$

Now set $b_i = 1 + \frac{1}{2}(w_i - l_i)$

(\begin{matrix} 6 & - 1 & - 1 & - 1 & - 1 \\ - 1 & 6 & - 1 & - 1 & - 1 \\ - 1 & - 1 & 6 & - 1 & - 1 \\ - 1 & - 1 & - 1 & 6 & - 1 \\ - 1 & - 1 & - 1 & - 1 & 6 \end{matrix}) r = (\begin{matrix} - 1 \\ 3 \\ 1 \\ 0 \\ 2 \end{matrix})

$\begin{pmatrix} 6 & -1 & -1 & -1 & -1 \\ -1 & 6 & -1 & -1 & -1 \\ -1 & -1 & 6 & -1 & -1 \\ -1 & -1 & -1 & 6 & -1 \\ -1 & -1 & -1 & -1 & 6 \end{pmatrix} \mathbf{r} = \begin{pmatrix} -1 \\ 3 \\ 1 \\ 0 \\ 2 \end{pmatrix}$

Results and comparison with M.

Team	$r_{c}$	Colley	Massey
Miami	.79	1st	=
VT	.65	2nd	=
UNC	.50	3rd	4th
UVA	.36	4th	3rd
Duke	.21	5th	=

Colley: conclusions

Points to remember

Laplace correction
winning-only in fact includes the strenghts of the opponents
the total strenght of the league tends to remains constant

General Conclusions

Points to focus on

rating and rating is the fun side of Data Science!

latent variables represent non-measurable skills
they live in a feature space, possibly separated from the traditional data space

yet they may get a numeric estimate, and inform our predictions
M. and C. regress on the latent variable strenght

Further readings (Ch.3)

Colley can run with Massey’s points balances (and v. v.)

Both methods can be applied to collaborative filtering.

Rating, Ranking and Colley regression Class 3 February 5th, 2020