问题
你想要生成平衡序列用于实验。
方案
函数 latinsquare() (在下方定义) 可以被用来生成拉丁方。
latinsquare(4)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 2 4 3
#> [2,] 2 1 3 4
#> [3,] 3 4 1 2
#> [4,] 4 3 2 1
# To generate 2 Latin squares of size 4 (in sequence)
latinsquare(4, reps=2)
#> [,1] [,2] [,3] [,4]
#> [1,] 3 4 1 2
#> [2,] 4 3 2 1
#> [3,] 1 2 4 3
#> [4,] 2 1 3 4
#> [5,] 4 2 1 3
#> [6,] 2 3 4 1
#> [7,] 1 4 3 2
#> [8,] 3 1 2 4
# It is better to put the random seed in the function call, to make it repeatable
# This will return the same sequence of two Latin squares every time
latinsquare(4, reps=2, seed=5873)
#> [,1] [,2] [,3] [,4]
#> [1,] 1 4 2 3
#> [2,] 4 1 3 2
#> [3,] 2 3 4 1
#> [4,] 3 2 1 4
#> [5,] 3 2 4 1
#> [6,] 1 4 2 3
#> [7,] 4 3 1 2
#> [8,] 2 1 3 4
存在大小为4的576个拉丁方。函数latinsquare会随机选择其中n个并以序列形式返回它们。这被称为重复拉丁方设计 。
一旦你生成了自己的拉丁方，你需要确保不存在许多重复的序列。这在小型拉丁方中非常普遍 (3x3 or 4x4)。
生成拉丁方的函数
这个函数用来生成拉丁方。它使用了略微暴力的算法来生成每个拉丁方，有时候在给定位置的数字用完了它可能会失败。这种情况下，它会尝试再试一试。可能存在一种更好的办法吧，但我并不清楚。
## - len is the size of the latin square
## - reps is the number of repetitions - how many Latin squares to generate
## - seed is a random seed that can be used to generate repeatable sequences
## - returnstrings tells it to return a vector of char strings for each square,
## instead of a big matrix. This option is only really used for checking the
## randomness of the squares.
latinsquare <- function(len, reps=1, seed=NA, returnstrings=FALSE) {
# Save the old random seed and use the new one, if present
if (!is.na(seed)) {
if (exists(".Random.seed")) { saved.seed <- .Random.seed }
else { saved.seed <- NA }
set.seed(seed)
}
# This matrix will contain all the individual squares
allsq <- matrix(nrow=reps*len, ncol=len)
# Store a string id of each square if requested
if (returnstrings) { squareid <- vector(mode = "character", length = reps) }
# Get a random element from a vector (the built-in sample function annoyingly
# has different behavior if there's only one element in x)
sample1 <- function(x) {
if (length(x)==1) { return(x) }
else { return(sample(x,1)) }
}
# Generate each of n individual squares
for (n in 1:reps) {
# Generate an empty square
sq <- matrix(nrow=len, ncol=len)
# If we fill the square sequentially from top left, some latin squares
# are more probable than others. So we have to do it random order,
# all over the square.
# The rough procedure is:
# - randomly select a cell that is currently NA (call it the target cell)
# - find all the NA cells sharing the same row or column as the target
# - fill the target cell
# - fill the other cells sharing the row/col
# - If it ever is impossible to fill a cell because all the numbers
# are already used, then quit and start over with a new square.
# In short, it picks a random empty cell, fills it, then fills in the
# other empty cells in the "cross" in random order. If we went totally randomly
# (without the cross), the failure rate is much higher.
while (any(is.na(sq))) {
# Pick a random cell which is currently NA
k <- sample1(which(is.na(sq)))
i <- (k-1) %% len +1 # Get the row num
j <- floor((k-1) / len) +1 # Get the col num
# Find the other NA cells in the "cross" centered at i,j
sqrow <- sq[i,]
sqcol <- sq[,j]
# A matrix of coordinates of all the NA cells in the cross
openCell <-rbind( cbind(which(is.na(sqcol)), j),
cbind(i, which(is.na(sqrow))))
# Randomize fill order
openCell <- openCell[sample(nrow(openCell)),]
# Put center cell at top of list, so that it gets filled first
openCell <- rbind(c(i,j), openCell)
# There will now be three entries for the center cell, so remove duplicated entries
# Need to make sure it's a matrix -- otherwise, if there's just
# one row, it turns into a vector, which causes problems
openCell <- matrix(openCell[!duplicated(openCell),], ncol=2)
# Fill in the center of the cross, then the other open spaces in the cross
for (c in 1:nrow(openCell)) {
# The current cell to fill
ci <- openCell[c,1]
cj <- openCell[c,2]
# Get the numbers that are unused in the "cross" centered on i,j
freeNum <- which(!(1:len %in% c(sq[ci,], sq[,cj])))
# Fill in this location on the square
if (length(freeNum)>0) { sq[ci,cj] <- sample1(freeNum) }
else {
# Failed attempt - no available numbers
# Re-generate empty square
sq <- matrix(nrow=len, ncol=len)
# Break out of loop
break;
}
}
}
# Store the individual square into the matrix containing all squares
allsqrows <- ((n-1)*len) + 1:len
allsq[allsqrows,] <- sq
# Store a string representation of the square if requested. Each unique
# square has a unique string.
if (returnstrings) { squareid[n] <- paste(sq, collapse="") }
}
# Restore the old random seed, if present
if (!is.na(seed) && !is.na(saved.seed)) { .Random.seed <- saved.seed }
if (returnstrings) { return(squareid) }
else { return(allsq) }
}
检查函数的随机性
一些生成拉丁方的算法并不是非常的随机。4x4的拉丁方有576种，它们每一种都应该有相等的概率被生成，但一些算法没有做到。可能我们没有必要检查上面的函数，但这里确实有办法可以做到这一点。前面我使用的算法并没有好的随机分布，我们运行下面的代码可以发现这一点。
这个代码创建10,000个4x4的拉丁方，然后计算这576个唯一拉丁方出现的频数。计数结果应该形成一个不是特别宽的正态分布；否则这个分布就不是很随机了。我相信期望的标准差是根号(10000/576)（假设随机生成拉丁方）。
# Set up the size and number of squares to generate
squaresize <- 4
numsquares <- 10000
# Get number of unique squares of a given size.
# There is not a general solution to finding the number of unique nxn squares
# so we just hard-code the values here. (From http://oeis.org/A002860)
uniquesquares <- c(1, 2, 12, 576, 161280, 812851200)[squaresize]
# Generate the squares
s <- latinsquare(squaresize, numsquares, seed=122, returnstrings=TRUE)
# Get the list of all squares and counts for each
slist <- rle(sort(s))
scounts <- slist[[1]]
hist(scounts, breaks=(min(scounts):(max(scounts)+1)-.5))
cat(sprintf("Expected and actual standard deviation: %.4f, %.4f\n",
sqrt(numsquares/uniquesquares), sd(scounts) ))
#> Expected and actual standard deviation: 4.1667, 4.0883
原文链接：http://www.cookbook-r.com/Tools_for_experiments/Generating_counterbalanced_orders/