# Example without ties
d <- 10
n <- matrix( sample( x = 0:5, size = d * d, replace = TRUE ), d, d )
n <- n + t( n )
diag( n ) <- 0
g <- rowSums( n )
# Number of win matches for each team
w <- sapply( 1:d, FUN = function( i ) sample( x = 1:g[i], size = 1, replace = TRUE ) )
# Number of lost matches for ech team
l <- rowSums( n ) - w
r <- colley_rating( n, w, l )
print( r )## [[1]]
## [1] 10 6 2 5 3 4 8 7 1 9
##
## [[2]]
## [1] 2.838612 3.383796 3.576764 3.461508 3.537240 3.477859 3.027758 3.155513
## [9] 3.622417 2.918533A <- matrix( c( 0, 7, 21, 7, 0,
52, 0, 34, 25, 27,
24, 16, 0, 7, 3,
38, 17, 5, 0, 14,
45, 7, 30, 52, 0 ),
nrow = 5, ncol = 5 )
r <- od_rating( A )
print( r )## $r
## [,1]
## [1,] 20.09702
## [2,] 188.78533
## [3,] 41.79470
## [4,] 84.80018
## [5,] 279.80681
##
## $o
## [,1]
## [1,] 33.98927
## [2,] 151.57374
## [3,] 48.66316
## [4,] 82.04849
## [5,] 114.83557
##
## $d
## [,1]
## [1,] 1.6912590
## [2,] 0.8028894
## [3,] 1.1643379
## [4,] 0.9675509
## [5,] 0.4104102
##
## $iter
## [1] 27We take \(C\) the finite set of options of candidates, ranking matrix \(R \in C^{m\times n}\), number of votes for each case \(v \in \mathbb{R}^n\), \(u \in \mathbb{R}^m\) weights for each rank, \(N = |C|\) number of options or candidates, \(l : C \longrightarrow \{1, \ldots, N\}\) labels by option, \(W \in \mathbb{R}^{N \times n}\) the ranking matrix. For each \(i \in \{1, \ldots, N\}\) and \(j \in \{1, \ldots, n\}\) we compute \[ k = l( R_{i,j} ), \qquad W_{k,j} = u_i \]
\(\text{order}\) returns the order index of a vector. \[ w = W v,\qquad r = \text{order}( w ) \]
# Position of each participant
R <- matrix( c( 2, 3, 1, 3, 1, 2, 1, 2, 3 ), nrow = 3, ncol = 3 )
v <- c( 7, 5, 2 )
bc <- borda_count( R, v )
print( bc )## $rank
## 1 2 3
## 3 2 1
##
## $vrank
## 1 2 3
## 23 30 31
##
## $mranks
## [,1] [,2] [,3]
## 1 1 2 3
## 2 3 1 2
## 3 2 3 1
##
## $labels
## 1 2 3
## 1 2 3
##
## $mlabels
## [,1] [,2] [,3]
## [1,] 2 3 1
## [2,] 3 1 2
## [3,] 1 2 3R <- matrix( c( 'A', 'B', 'C', 'B', 'A', 'C', 'C', 'A', 'B' ), nrow = 3, ncol = 3 )
v <- c( 5, 4, 3 )
bc <- borda_count( R, v )
print( bc )## $rank
## A B C
## 1 2 3
##
## $vrank
## A B C
## 29 25 18
##
## $mranks
## [,1] [,2] [,3]
## A 3 2 2
## B 2 3 1
## C 1 1 3
##
## $labels
## A B C
## 1 2 3
##
## $mlabels
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 1 1
## [3,] 3 3 2R <- matrix( c( 'Steak', 'Chicken', 'Barbacoa', 'Carnitas',
'Carnitas', 'Barbacoa', 'Chicken', 'Steak',
'Chicken', 'Carnitas', 'Steak', 'Barbacoa',
'Steak', 'Barbacoa', 'Chicken', 'Carnitas' ), nrow = 4, ncol = 4 )
v <- c( 8, 7, 5, 3 )
bc <- borda_count( R, v )
print( bc )## $rank
## Barbacoa Carnitas Chicken Steak
## 4 3 1 2
##
## $vrank
## Barbacoa Carnitas Chicken Steak
## 51 54 64 61
##
## $mranks
## [,1] [,2] [,3] [,4]
## Barbacoa 2 3 1 3
## Carnitas 1 4 3 1
## Chicken 3 2 4 2
## Steak 4 1 2 4
##
## $labels
## Barbacoa Carnitas Chicken Steak
## 1 2 3 4
##
## $mlabels
## [,1] [,2] [,3] [,4]
## [1,] 4 2 3 4
## [2,] 3 1 2 1
## [3,] 1 3 4 3
## [4,] 2 4 1 2R <- matrix( c( 'A', 'B', 'C', 'B', 'A', 'C', 'C', 'A', 'B' ), nrow = 3, ncol = 3 )
v <- c( 5, 4, 3 )
u <- c( 6, 4, 2 )
bc <- borda_count( R, v, u )
print( bc )## $rank
## A B C
## 1 2 3
##
## $vrank
## A B C
## 58 50 36
##
## $mranks
## [,1] [,2] [,3]
## A 6 4 4
## B 4 6 2
## C 2 2 6
##
## $labels
## A B C
## 1 2 3
##
## $mlabels
## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 2 1 1
## [3,] 3 3 2