Activity 7

At the beginning of the chapter, I said that a matrix is just a vector but with two additional attributes: the number of rows and the number of columns. Here, we’ll take a closer look at the vector nature of matrices. Consider this example:

z <- matrix(10:17,nrow=4)
z

     [,1] [,2]
[1,]   10   14
[2,]   11   15
[3,]   12   16
[4,]   13   17

-> creates matrix with values 1 through 8 and 4 rows, therefore, with 2 columns

length(z)#As z is still a vector, we can query its length

[1] 8

-> length of vector z

class(z)

[1] "matrix" "array" 

dim(z)

[1] 4 2

-> z class is matrix, array, and the dimensions are 4 rows and two columns -> 2-dimensional

Avoiding Unintended Dimension Reduction In the world of statistics, dimension reduction is a good thing, with many statistical procedures aimed to do it well. If we are working with, say, 10 variables and can reduce that number to 3 that still capture the essence of our data, we’re happy. However, in R, something else might merit the name dimension reduction that we may sometimes wish to avoid. Say we have a four-row matrix and extract a row from it:

z

     [,1] [,2]
[1,]    1    5
[2,]    2    6
[3,]    3    7
[4,]    4    8

r <- z[2,]
r

[1] 2 6

-> returns second row of z and assigns it to r in vector format

This seems innocuous, but note the format in which R has displayed r. It’s a vector format, not a matrix format. In other words, r is a vector of length 2, rather than a 1-by-2 matrix. We can confirm this in a couple of ways:

attributes(z)

$dim
[1] 4 2

attributes(r)

NULL

str(z)

 int [1:4, 1:2] 1 2 3 4 5 6 7 8

str(r)

 int [1:2] 2 6

-> shows dimensions of z and NULL for r since r is a vector -> also the structural differences between a matrix and a regular vector can be seen

Fortunately, R has a way to suppress this dimension reduction: the drop argument. Here’s an example, using the matrix z from above:

r <- z[2,, drop=FALSE]
r

     [,1] [,2]
[1,]    2    6

-> using the drop=false, the second row can be selected while keeping the matrix format

dim(r)

[1] 1 2

class(r)

[1] "matrix" "array" 

-> confirms the matrix format

Naming Matrix Rows and Columns

The natural way to refer to rows and columns in a matrix is via the row and column numbers. However, you can also give names to these entities. Here’s an example:

z

     [,1] [,2]
[1,]    1    5
[2,]    2    6
[3,]    3    7
[4,]    4    8

colnames(z)

NULL

-> matrix has no column names

colnames(z) <- c("cat","dog")

-> assigning column names to z

z

     cat dog
[1,]  10  14
[2,]  11  15
[3,]  12  16
[4,]  13  17

colnames(z)

[1] "cat" "dog"

-> confirms that columns are renamed and can be accessed directly

z[,"cat"]

[1] 10 11 12 13

-> accesses column of z by column name

Higher-Dimensional Arrays

In a statistical context, a typical matrix in R has rows corresponding to observations, say on various people, and columns corresponding to variables, such as weight and blood pressure. The matrix is then a two-dimensional data structure. But suppose we also have data taken at different times, one data point per person per variable per time. Time then becomes the third dimension, in addition to rows and columns. In R, such data sets are called arrays. As a simple example, consider students and test scores. Say each test consists of two parts, so we record two scores for a student for each test. Now suppose that we have two tests, and to keep the example small, assume we have only three students. Here’s the data for the first test:

firsttest <- matrix(nrow=3,ncol=2)
firsttest[1,1] <- 468522
firsttest[2,1] <- 21230
firsttest[1,2] <- 30016523
firsttest[2,2] <- 252331
firsttest[3,1] <- 509851
firsttest[3,2] <- 500156

firsttest

       [,1]     [,2]
[1,] 468522 30016523
[2,]  21230   252331
[3,] 509851   500156

Student 1 had scores of 46 and 30 on the first test, student 2 scored 21 and 25, and so on. Here are the scores for the same students on the second test:

secondtest <- matrix(nrow=3,ncol=2)
secondtest[1,1] <- 4695
secondtest[2,1] <- 411458
secondtest[1,2] <- 4335847
secondtest[2,2] <- 35456
secondtest[3,1] <- 5078563
secondtest[3,2] <- 50512

secondtest

        [,1]    [,2]
[1,]    4695 4335847
[2,]  411458   35456
[3,] 5078563   50512

Now let’s put both tests into one data structure, which we’ll name tests. We’ll arrange it to have two “layers”—one layer per test—with three rows and two columns within each layer. We’ll store firsttest in the first layer and secondtest in the second. In layer 1, there will be three rows for the three students’ scores on the first test, with two columns per row for the two portions of a test. We use R’s array function to create the data structure:

tests <- array(data=c(firsttest,secondtest),dim=c(3,2,2))
tests

, , 1

       [,1]     [,2]
[1,] 468522 30016523
[2,]  21230   252331
[3,] 509851   500156

, , 2

        [,1]    [,2]
[1,]    4695 4335847
[2,]  411458   35456
[3,] 5078563   50512

-> adding firsttest and secondtest in 3-dimensional manner

In the argument dim=c(3,2,2), we are specifying two layers (this is the second 2), each consisting of three rows and two columns. This then becomes an attribute of the data structure:

attributes(tests)

$dim
[1] 3 2 2

-> shows attributes: 3 rows, two columns, 2 layers -> 3-dimensional

Each element of tests now has three subscripts, rather than two as in the matrix case. The first subscript corresponds to the first element in the $dim vector, the second subscript corresponds to the second element in the vector, and so on. For instance, the score on the second portion of test 1 for student 3 is retrieved as follows:

tests[3,2,1]

[1] 500156

-> accesses score of row 3, column 2, in layer 1

tests

, , 1

       [,1]     [,2]
[1,] 468522 30016523
[2,]  21230   252331
[3,] 509851   500156

, , 2

        [,1]    [,2]
[1,]    4695 4335847
[2,]  411458   35456
[3,] 5078563   50512

-> returns both layers: shows three rows with two columns in each layer

Just as we built our three-dimensional array by combining two matrices, we can build four-dimensional arrays by combining two or more three dimensional arrays, and so on.