Array operation: Outer product

Incidentally, I ran into outer product function outer today. It is extremely powerful function, in that it computes all combinations of product of two objects. One simplest and obvious demonstration is the multiplication table of numbers. 😄

We can show multiplication table of numbers 1 through 12 each multiplied 1 through 10.

outer(1:10, 1:12, "*")

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
##  [1,]    1    2    3    4    5    6    7    8    9    10    11    12
##  [2,]    2    4    6    8   10   12   14   16   18    20    22    24
##  [3,]    3    6    9   12   15   18   21   24   27    30    33    36
##  [4,]    4    8   12   16   20   24   28   32   36    40    44    48
##  [5,]    5   10   15   20   25   30   35   40   45    50    55    60
##  [6,]    6   12   18   24   30   36   42   48   54    60    66    72
##  [7,]    7   14   21   28   35   42   49   56   63    70    77    84
##  [8,]    8   16   24   32   40   48   56   64   72    80    88    96
##  [9,]    9   18   27   36   45   54   63   72   81    90    99   108
## [10,]   10   20   30   40   50   60   70   80   90   100   110   120

Another immediate use can be seen in design and combinatorics. For instance,

# outer product sum and subtraction function
outer(1:5, 1:10, "+")

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,]    2    3    4    5    6    7    8    9   10    11
## [2,]    3    4    5    6    7    8    9   10   11    12
## [3,]    4    5    6    7    8    9   10   11   12    13
## [4,]    5    6    7    8    9   10   11   12   13    14
## [5,]    6    7    8    9   10   11   12   13   14    15

outer(1:5, 1:10, "-")

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,]    0   -1   -2   -3   -4   -5   -6   -7   -8    -9
## [2,]    1    0   -1   -2   -3   -4   -5   -6   -7    -8
## [3,]    2    1    0   -1   -2   -3   -4   -5   -6    -7
## [4,]    3    2    1    0   -1   -2   -3   -4   -5    -6
## [5,]    4    3    2    1    0   -1   -2   -3   -4    -5

Interestingly, outer subtraction of same two numeric vector gives a symmetric matrix with negative elements in upper with respect to lower.

outer(1:5, 1:5, "-")

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    0   -1   -2   -3   -4
## [2,]    1    0   -1   -2   -3
## [3,]    2    1    0   -1   -2
## [4,]    3    2    1    0   -1
## [5,]    4    3    2    1    0

table(outer(1:5, 1:5, "-"))

## 
## -4 -3 -2 -1  0  1  2  3  4 
##  1  2  3  4  5  4  3  2  1

In genetics, outer product is useful finding genotype combinations from multiple loci. For two loci, combination of genotypes is obtained as:

outer(c("AA", "Aa", "aa"), c("BB", "Bb", "bb"), paste0)

##      [,1]   [,2]   [,3]  
## [1,] "AABB" "AABb" "AAbb"
## [2,] "AaBB" "AaBb" "Aabb"
## [3,] "aaBB" "aaBb" "aabb"

Likewise for gamete formation, we can see that when there are 5 alleles in a diploid organism, possible genotype combinations are:

outer(LETTERS[1:5], LETTERS[1:5], paste0)

##      [,1] [,2] [,3] [,4] [,5]
## [1,] "AA" "AB" "AC" "AD" "AE"
## [2,] "BA" "BB" "BC" "BD" "BE"
## [3,] "CA" "CB" "CC" "CD" "CE"
## [4,] "DA" "DB" "DC" "DD" "DE"
## [5,] "EA" "EB" "EC" "ED" "EE"