Combinatorics
TODO
- link to sets
Install required packages
wants <- c("e1071", "permute")
has <- wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])Combinations
Number of \(k\)-combinations
\({n \choose k}\)
myN <- 5
myK <- 4
choose(myN, myK)[1] 5factorial(myN) / (factorial(myK)*factorial(myN-myK))[1] 5Enumerate all combinations
combn(c("a", "b", "c", "d", "e"), myK) [,1] [,2] [,3] [,4] [,5]
[1,] "a" "a" "a" "a" "b"
[2,] "b" "b" "b" "c" "c"
[3,] "c" "c" "d" "d" "d"
[4,] "d" "e" "e" "e" "e" combn(c(1, 2, 3, 4), 3) [,1] [,2] [,3] [,4]
[1,] 1 1 1 2
[2,] 2 2 3 3
[3,] 3 4 4 4combn(c(1, 2, 3, 4), 3, sum)[1] 6 7 8 9combn(c(1, 2, 3, 4), 3, weighted.mean, w=c(0.5, 0.2, 0.3))[1] 1.8 2.1 2.3 2.8Permutations
Number of permutations
factorial(7)[1] 5040Random permutation
set.seed(123)
set <- LETTERS[1:10]
sample(set, length(set), replace=FALSE) [1] "C" "H" "D" "G" "F" "A" "J" "I" "B" "E"library(permute)
shuffle(length(set)) [1] 10 5 6 9 1 7 8 4 3 2Enumerate all permutations
All permutations at once
set <- LETTERS[1:3]
len <- length(set)
library(e1071)
(mat <- permutations(len)) [,1] [,2] [,3]
[1,] 1 2 3
[2,] 2 1 3
[3,] 2 3 1
[4,] 1 3 2
[5,] 3 1 2
[6,] 3 2 1apply(mat, 1, function(x) set[x]) [,1] [,2] [,3] [,4] [,5] [,6]
[1,] "A" "B" "B" "A" "C" "C"
[2,] "B" "A" "C" "C" "A" "B"
[3,] "C" "C" "A" "B" "B" "A" Each permutation individually
(grp <- rep(letters[1:3], each=3))[1] "a" "a" "a" "b" "b" "b" "c" "c" "c"N <- length(grp)
nPerms <- 100
library(permute)
pCtrl <- how(nperm=nPerms, complete=FALSE)
for(i in 1:5) {
perm <- permute(i, n=N, control=pCtrl)
print(grp[perm])
}[1] "c" "b" "b" "c" "b" "a" "a" "c" "a"
[1] "a" "c" "c" "b" "b" "a" "c" "a" "b"
[1] "a" "a" "a" "c" "b" "c" "c" "b" "b"
[1] "b" "a" "c" "a" "c" "b" "b" "a" "c"
[1] "a" "c" "c" "a" "b" "a" "c" "b" "b"Restricted permutations
Permute group membership in a two-way ANOVA design. Here: artificially low number of permutations.
Njk <- 4 ## cell size
P <- 2 ## levels factor A
Q <- 3 ## levels factor B
N <- Njk*P*Q
nPerms <- 10 ## number of permutations
id <- 1:(Njk*P*Q)
IV1 <- factor(rep(1:P, each=Njk*Q)) ## factor A
IV2 <- factor(rep(1:Q, times=Njk*P)) ## factor B
(myDf <- data.frame(id, IV1, IV2)) id IV1 IV2
1 1 1 1
2 2 1 2
3 3 1 3
4 4 1 1
5 5 1 2
6 6 1 3
7 7 1 1
8 8 1 2
9 9 1 3
10 10 1 1
11 11 1 2
12 12 1 3
13 13 2 1
14 14 2 2
15 15 2 3
16 16 2 1
17 17 2 2
18 18 2 3
19 19 2 1
20 20 2 2
21 21 2 3
22 22 2 1
23 23 2 2
24 24 2 3# choose permutation schemes for tests of factor A and B
library(permute) ## for how(), permute()
## only permute across A (within B)
pCtrlA <- how(plots=Plots(strata=IV2), complete=FALSE, nperm=nPerms)
## only permute across B (within A)
pCtrlB <- how(plots=Plots(strata=IV1), complete=FALSE, nperm=nPerms)Get permutations for test of factor A
for(i in 1:3) {
perm <- permute(i, n=N, control=pCtrlA)
print(myDf[perm, ])
# not shown
}Get permutations for test of factor B
for(i in 1:3) {
perm <- permute(i, n=N, control=pCtrlB)
print(myDf[perm, ])
# not shown
}Enumerate all \(n\)-tuples
All combinations of elements from \(n\) different sets (cartesian product)
IV1 <- c("control", "treatment")
IV2 <- c("f", "m")
IV3 <- c(1, 2)
expand.grid(IV1, IV2, IV3) Var1 Var2 Var3
1 control f 1
2 treatment f 1
3 control m 1
4 treatment m 1
5 control f 2
6 treatment f 2
7 control m 2
8 treatment m 2Apply a function to all pairs of elements from two sets
outer(1:5, 1:5, FUN="*") [,1] [,2] [,3] [,4] [,5]
[1,] 1 2 3 4 5
[2,] 2 4 6 8 10
[3,] 3 6 9 12 15
[4,] 4 8 12 16 20
[5,] 5 10 15 20 25Detach (automatically) loaded packages (if possible)
try(detach(package:e1071))
try(detach(package:permute))Get the article source from GitHub
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