Sports ranking methods, 4

Yet Another Mathblog

This is the fourth of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the Elo rating.

This system was originally developed by Arpad Elo (Elo (1903-1992) was a physics professor at Marquette University in Milwaukee and a chess master, eight-time winner of the Wisconsin State Chess Championships.) Originally, it was developed for rating chess players in the 1950s and 1960s. Now it is used for table tennis, basketball, and other sports.

We use the following version of his rating system.

As above, assume all the $n$ teams play each other (ties allowed)
and let $latex r_i$ denote the rating of Team $latex i$, $latex i=1,2,dots,n$.

Let $latex A=(A_{ij})$ denote an $ntimes n$ matrix of score results:

$latex
A_{ij}=
left{
begin{array}{rr}
-1,& {rm if team } i {rm lost to team } j,
+1,& {rm if team } i {rm beat team } j,
0…

View original post 294 more words

Sports ranking methods, 3

Yet Another Mathblog

This is the third of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the random walker ranking.

We follow the presentation in the paper by Govan and Meyer (Ranking National Football League teams using Google’s PageRank). The table of “score differentials” based on the table in a previous post is:

$latex

begin{tabular}{c|cccccc}
verb+xy+ & Army & Bucknell & Holy Cross & Lafayette & Lehigh & Navy
hline
Army & 0 & 0 & 1 & 0 & 0 &…
Bucknell & 2 & 0 & 0 & 2 & 3 &…
Holy Cross & 0 & 3 & 0 & 4 & 14 &…
Lafayette & 10 & 0 & 0 & 0 & 0 &…
Lehigh & 2 & 0 & 0 & 11 & 0 &…
Navy & 11 & 14 & 8 & 22 & 6 &…
end{tabular}

$
This leads…

View original post 400 more words

Sports ranking methods, 2

Yet Another Mathblog

This is the second of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses Keener’s method (see J.P. Keener, The Perron-Frobenius theorem and the ranking of football, SIAM Review 35 (1993)80-93 for details).

See the first post in the series for a discussion of the data we’re using to explain this method. We recall the table of results:

XY	Army	Bucknell	Holy Cross	Lafayette	Lehigh	Navy
Army	x	14-16	14-13	14-24	10-12	8-19
Bucknell	16-14	x	27-30	18-16	23-20	10-22
Holy Cross	13-14	30-27	x	19-15	17-13	9-16
Lafayette	24-14	16-18	15-19	x	12-23	17-39
Lehigh	12-10	20-23	13-17	23-12	x	12-18
Navy	19-8	22-10	16-9	39-17	18-12	x

Win-loss digraph of the Patriot league mens baseball from 2015

Suppose T teams play each other. Let $latex A=(a_{ij})_{1leq i,jleq T}$ be a non-negative square matrix determined by the results of their games, called the preference matrix

View original post 326 more words

Sports ranking methods, 1

Source: Sports ranking methods, 1

Simple unsolved math problem, 8

Source: Simple unsolved math problem, 8

Simple unsolved math problem, 7

Source: Simple unsolved math problem, 7

Simple unsolved math problem, 6

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia: The formal definition is as follo…

Source: Simple unsolved math problem, 6

Simple unsolved math problem, 5

It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with 1’s down each side and the inside numbers determined…

Source: Simple unsolved math problem, 5

Simple unsolved math problem, 4

Problem: Optimally pack n unit circles into the smallest possible equilateral triangle. Let L(n) denote the length of the side of the smallest equilateral triangle in which n circles have been pack…

Source: Simple unsolved math problem, 4

Simple unsolved math problem, 3

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,  1 + 2 + 3 = 6 i…

Source: Simple unsolved math problem, 3

Simple unsolved math problem, 2

In 1911, Otto Toeplitz asked the following question. Inscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square? This question, also known as the square…

Source: Simple unsolved math problem, 2

Simple unsolved math problem, 1

In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows. First, we define the function on the set of positive integers: If the number $…

Source: Simple unsolved math problem, 1

Fibonacci times Euler

Recall the Fibonacci numbers given by 1,1,2,3,5,8,13,21… There is no need to define them. You all know. Now take the Euler numbers (OEIS) 1,1,1,2,5,16,61,272… Th…

Source: Fibonacci times Euler

The nature of infinity, 2

On Thursday, the first in a series of public discussions on scientific topics was put on by an organisation called Mass Interaction (the name comes from a statement by Richard Feynman that “a…

Source: The nature of infinity, 2

Singularly Valuable SVD

This blog aims to show that mathematics is beautiful, useful and fun. The singular value decomposition has all these qualities. It is very elegant, and has a wide range of useful applications

via Singularly Valuable SVD.