A matrix is a rectangular collection of the same data type. It can be viewed as a collection of column vectors all of the same length and the same type (i.e. numeric, character or logical) OR a collection of row vectors, again all of the same type and length. A data.frame is also a rectangular array. All of the columns must be the same length, but they may be of different types. The rows and columns of a matrix or data frame can be given names. However these are implemented differently in R; many operations will work for one but not both, often a source of confusion.
In this section, we will be working with matrices. The data.frame will be dealt with elsewhere.
There are many ways to create a matrix in R. One of the simplest is
to use the matrix() function. In the code below, we’ll
create a matrix from a vector from 1:16.
mat1 <- matrix(1:16,nrow=4)
mat1## [,1] [,2] [,3] [,4]
## [1,] 1 5 9 13
## [2,] 2 6 10 14
## [3,] 3 7 11 15
## [4,] 4 8 12 16The same is possible, but specifying that the matrix be “filled” by row.
mat1 <- matrix(1:16,nrow=4,byrow = TRUE)
mat1## [,1] [,2] [,3] [,4]
## [1,] 1 2 3 4
## [2,] 5 6 7 8
## [3,] 9 10 11 12
## [4,] 13 14 15 16Notice the subtle difference in the order that the numbers go into the matrix.
We can also build a matrix from parts by “binding” vectors together:
x <- 1:10
y <- rnorm(10)Each of the vectors above is of length 10 and both are “numeric”, so
we can make them into a matrix. Using rbind binds rows
(r) into a matrix.
mat <- rbind(x,y)
mat## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## x 1.0000000 2.00000000 3.0000000 4.0000000 5.0000000 6.000000 7.0000000
## y -0.8430163 0.03101184 -0.1286222 0.2374506 -0.8538436 1.121041 0.9539837
## [,8] [,9] [,10]
## x 8.000000 9.000000 10.0000000
## y 1.503723 -1.518493 -0.8256927The alternative to rbind is cbind that
binds columns.
mat <- cbind(x,y)
mat## x y
## [1,] 1 -0.84301634
## [2,] 2 0.03101184
## [3,] 3 -0.12862220
## [4,] 4 0.23745063
## [5,] 5 -0.85384364
## [6,] 6 1.12104093
## [7,] 7 0.95398374
## [8,] 8 1.50372298
## [9,] 9 -1.51849281
## [10,] 10 -0.82569271Inspecting the names associated with rows and columns is often useful, particularly if the names have human meaning.
rownames(mat)## NULLcolnames(mat)## [1] "x" "y"We can also change the names of the matrix by assigning valid names to the columns or rows.
colnames(mat) = c('apples','oranges')
colnames(mat)## [1] "apples" "oranges"mat## apples oranges
## [1,] 1 -0.84301634
## [2,] 2 0.03101184
## [3,] 3 -0.12862220
## [4,] 4 0.23745063
## [5,] 5 -0.85384364
## [6,] 6 1.12104093
## [7,] 7 0.95398374
## [8,] 8 1.50372298
## [9,] 9 -1.51849281
## [10,] 10 -0.82569271Matrices have dimensions.
dim(mat)## [1] 10 2nrow(mat)## [1] 10ncol(mat)## [1] 2Indexing for matrices works as for vectors except that we now need to
include both the row and column (in that order). We can access elements
of a matrix using the square bracket [ indexing method. Elements can be
accessed as var[r, c]. Here, r and
c are vectors describing the elements of the matrix to
select.
# The 2nd element of the 1st row of mat
mat[1,2]## oranges
## -0.8430163# The first ROW of mat
mat[1,]## apples oranges
## 1.0000000 -0.8430163# The first COLUMN of mat
mat[,1]## [1] 1 2 3 4 5 6 7 8 9 10# and all elements of mat that are > 4; note no comma
mat[mat>4]## [1] 5 6 7 8 9 10## [1] 5 6 7 8 9 10Note that in the last case, there is no “,”, so R treats the matrix as a long vector (length=20). This is convenient, sometimes, but it can also be a source of error, as some code may “work” but be doing something unexpected.
We can also use indexing to exclude a row or column by prefixing the
selection with a - sign.
mat[,-1] # remove first column## [1] -0.84301634 0.03101184 -0.12862220 0.23745063 -0.85384364 1.12104093
## [7] 0.95398374 1.50372298 -1.51849281 -0.82569271mat[-c(1:5),] # remove first five rows## apples oranges
## [1,] 6 1.1210409
## [2,] 7 0.9539837
## [3,] 8 1.5037230
## [4,] 9 -1.5184928
## [5,] 10 -0.8256927We can create a matrix filled with random values drawn from a normal distribution for our work below.
m = matrix(rnorm(20),nrow=10)
summary(m)## V1 V2
## Min. :-1.5849 Min. :-2.9330
## 1st Qu.:-0.8206 1st Qu.:-0.5576
## Median :-0.6572 Median :-0.1636
## Mean :-0.4372 Mean :-0.2955
## 3rd Qu.:-0.4814 3rd Qu.: 0.7165
## Max. : 2.1681 Max. : 1.2195Multiplication and division works similarly to vectors. When multiplying by a vector, for example, the values of the vector are reused. In the simplest case, let’s multiply the matrix by a constant (vector of length 1).
# multiply all values in the matrix by 20
m2 = m*20
summary(m2)## V1 V2
## Min. :-31.698 Min. :-58.659
## 1st Qu.:-16.411 1st Qu.:-11.151
## Median :-13.143 Median : -3.272
## Mean : -8.743 Mean : -5.910
## 3rd Qu.: -9.629 3rd Qu.: 14.331
## Max. : 43.361 Max. : 24.389By combining subsetting with assignment, we can make changes to just part of a matrix.
# and add 100 to the first column of m
m2[,1] = m2[,1] + 100
# summarize m
summary(m2)## V1 V2
## Min. : 68.30 Min. :-58.659
## 1st Qu.: 83.59 1st Qu.:-11.151
## Median : 86.86 Median : -3.272
## Mean : 91.26 Mean : -5.910
## 3rd Qu.: 90.37 3rd Qu.: 14.331
## Max. :143.36 Max. : 24.389A somewhat common transformation for a matrix is to transpose which changes rows to columns. One might need to do this if an assay output from a lab machine puts samples in rows and genes in columns, for example, while in Bioconductor/R, we often want the samples in columns and the genes in rows.
t(m2)## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 68.3024107 84.853621 74.30254 83.16710 87.48558 143.361431 87.08400
## [2,] -0.7274469 -1.932346 19.34996 -10.19411 24.38908 -4.610796 21.81934
## [,8] [,9] [,10]
## [1,] 86.62978 91.33294 106.05043
## [2,] -11.47012 -58.65922 -37.06505Again, we just need a matrix to play with. We’ll use
rnorm again, but with a slight twist.
m3 = matrix(rnorm(100,5,2),ncol=10) # what does the 5 mean here? And the 2?Since these data are from a normal distribution, we can look at a row (or column) to see what the mean and standard deviation are.
mean(m3[,1])## [1] 4.345614sd(m3[,1])## [1] 2.104432# or a row
mean(m3[1,])## [1] 4.263493sd(m3[1,])## [1] 1.490555There are some useful convenience functions for computing means and sums of data in all of the columns and rows of matrices.
colMeans(m3)## [1] 4.345614 4.628478 5.097733 5.711541 5.171218 4.974755 4.847533 4.931225
## [9] 5.162024 6.280296rowMeans(m3)## [1] 4.263493 5.320054 5.441344 5.287590 5.633510 4.941104 5.039043 5.013403
## [9] 5.063764 5.147110rowSums(m3)## [1] 42.63493 53.20054 54.41344 52.87590 56.33510 49.41104 50.39043 50.13403
## [9] 50.63764 51.47110colSums(m3)## [1] 43.45614 46.28478 50.97733 57.11541 51.71218 49.74755 48.47533 49.31225
## [9] 51.62024 62.80296We can look at the distribution of column means:
# save as a variable
cmeans = colMeans(m3)
summary(cmeans)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.346 4.868 5.036 5.115 5.169 6.280Note that this is centered pretty closely around the selected mean of
5 above.
How about the standard deviation? There is not a colSd
function, but it turns out that we can easily apply functions that take
vectors as input, like sd and “apply” them across either
the rows (the first dimension) or columns (the second) dimension.
csds = apply(m3, 2, sd)
summary(csds)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.186 2.038 2.144 2.116 2.316 2.903Again, take a look at the distribution which is centered quite close to the selected standard deviation when we created our matrix.