Chapter 5 Arrays and matrices

5.1 Arrays

An array can be considered as a multiply subscripted collection of data entries, for example numeric. R allows simple facilities for creating and handling arrays, and in particular the special case of matrices.

A dimension vector is a vector of non-negative integers. If its length is k then the array is k-dimensional, e.g. a matrix is a 2-dimensional array. The dimensions are indexed from one up to the values given in the dimension vector.

A vector can be used by R as an array only if it has a dimension vector as its dim attribute. Suppose, for example, z is a vector of 1500 elements. The assignment

> dim(z) <- c(3,5,100)

gives it the dim attribute that allows it to be treated as a 3 by 5 by 100 array.

Other functions such as matrix() and array() are available for simpler and more natural looking assignments, as we shall see in The array() function.

The values in the data vector give the values in the array in the same order as they would occur in FORTRAN, that is “column major order,” with the first subscript moving fastest and the last subscript slowest.

For example if the dimension vector for an array, say a, is c(3,4,2) then there are 3 * 4 * 2 = 24 entries in a and the data vector holds them in the order a[1,1,1], a[2,1,1], …, a[2,4,2], a[3,4,2].

Arrays can be one-dimensional: such arrays are usually treated in the same way as vectors (including when printing), but the exceptions can cause confusion.

5.2 Array indexing. Subsections of an array

Individual elements of an array may be referenced by giving the name of the array followed by the subscripts in square brackets, separated by commas.

More generally, subsections of an array may be specified by giving a sequence of index vectors in place of subscripts; however if any index position is given an empty index vector, then the full range of that subscript is taken.

Continuing the previous example, a[2,,] is a 4 * 2 array with dimension vector c(4,2) and data vector containing the values

c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
  a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])

in that order. a[,,] stands for the entire array, which is the same as omitting the subscripts entirely and using a alone.

For any array, say Z, the dimension vector may be referenced explicitly as dim(Z) (on either side of an assignment).

Also, if an array name is given with just one subscript or index vector, then the corresponding values of the data vector only are used; in this case the dimension vector is ignored. This is not the case, however, if the single index is not a vector but itself an array, as we next discuss.

5.3 Index matrices

As well as an index vector in any subscript position, a matrix may be used with a single index matrix in order either to assign a vector of quantities to an irregular collection of elements in the array, or to extract an irregular collection as a vector.

A matrix example makes the process clear. In the case of a doubly indexed array, an index matrix may be given consisting of two columns and as many rows as desired. The entries in the index matrix are the row and column indices for the doubly indexed array. Suppose for example we have a 4 by 5 array X and we wish to do the following:

Extract elements X[1,3], X[2,2] and X[3,1] as a vector structure, and
Replace these entries in the array X by zeroes.

In this case we need a 3 by 2 subscript array, as in the following example.

> x <- array(1:20, dim=c(4,5))   # Generate a 4 by 5 array.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    9   13   17
[2,]    2    6   10   14   18
[3,]    3    7   11   15   19
[4,]    4    8   12   16   20
> i <- array(c(1:3,3:1), dim=c(3,2))
> i                             # i is a 3 by 2 index array.
     [,1] [,2]
[1,]    1    3
[2,]    2    2
[3,]    3    1
> x[i]                          # Extract those elements
[1] 9 6 3
> x[i] <- 0                     # Replace those elements by zeros.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    0   13   17
[2,]    2    0   10   14   18
[3,]    0    7   11   15   19
[4,]    4    8   12   16   20
>

Negative indices are not allowed in index matrices. NA and zero values are allowed: rows in the index matrix containing a zero are ignored, and rows containing an NA produce an NA in the result.

As a less trivial example, suppose we wish to generate an (unreduced) design matrix for a block design defined by factors blocks (b levels) and varieties (v levels). Further suppose there are n plots in the experiment. We could proceed as follows:

> Xb <- matrix(0, n, b)
> Xv <- matrix(0, n, v)
> ib <- cbind(1:n, blocks)
> iv <- cbind(1:n, varieties)
> Xb[ib] <- 1
> Xv[iv] <- 1
> X <- cbind(Xb, Xv)

To construct the incidence matrix, N say, we could use

> N <- crossprod(Xb, Xv)

However a simpler direct way of producing this matrix is to use table():

> N <- table(blocks, varieties)

Index matrices must be numerical: any other form of matrix (e.g. a logical or character matrix) supplied as a matrix is treated as an indexing vector.

5.4 The `array()` function

As well as giving a vector structure a dim attribute, arrays can be constructed from vectors by the array function, which has the form

> Z <- array(data_vector, dim_vector)

For example, if the vector h contains 24 or fewer, numbers then the command

> Z <- array(h, dim=c(3,4,2))

would use h to set up 3 by 4 by 2 array in Z. If the size of h is exactly 24 the result is the same as

> Z <- h ; dim(Z) <- c(3,4,2)

However if h is shorter than 24, its values are recycled from the beginning again to make it up to size 24 (see The recycling rule) but dim(h) <- c(3,4,2) would signal an error about mismatching length. As an extreme but common example

> Z <- array(0, c(3,4,2))

makes Z an array of all zeros.

At this point dim(Z) stands for the dimension vector c(3,4,2), and Z[1:24] stands for the data vector as it was in h, and Z[] with an empty subscript or Z with no subscript stands for the entire array as an array.

Arrays may be used in arithmetic expressions and the result is an array formed by element-by-element operations on the data vector. The dim attributes of operands generally need to be the same, and this becomes the dimension vector of the result. So if A, B and C are all similar arrays, then

> D <- 2*A*B + C + 1

makes D a similar array with its data vector being the result of the given element-by-element operations. However the precise rule concerning mixed array and vector calculations has to be considered a little more carefully.

5.4.1 Mixed vector and array arithmetic. The recycling rule

The precise rule affecting element by element mixed calculations with vectors and arrays is somewhat quirky and hard to find in the references. From experience we have found the following to be a reliable guide.

The expression is scanned from left to right.
Any short vector operands are extended by recycling their values until they match the size of any other operands.
As long as short vectors and arrays only are encountered, the arrays must all have the same dim attribute or an error results.
Any vector operand longer than a matrix or array operand generates an error.
If array structures are present and no error or coercion to vector has been precipitated, the result is an array structure with the common dim attribute of its array operands.

5.5 The outer product of two arrays

An important operation on arrays is the outer product. If a and b are two numeric arrays, their outer product is an array whose dimension vector is obtained by concatenating their two dimension vectors (order is important), and whose data vector is got by forming all possible products of elements of the data vector of a with those of b. The outer product is formed by the special operator %o%:

> ab <- a %o% b

An alternative is

> ab <- outer(a, b, "*")

The multiplication function can be replaced by an arbitrary function of two variables. For example if we wished to evaluate the function f(x; y) = cos(y)/(1 + x^2) over a regular grid of values with x- and y-coordinates defined by the R vectors x and y respectively, we could proceed as follows:

> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)

In particular the outer product of two ordinary vectors is a doubly subscripted array (that is a matrix, of rank at most 1). Notice that the outer product operator is of course non-commutative. Defining your own R functions will be considered further in Writing your own functions.

An example: Determinants of 2 by 2 single-digit matrices

As an artificial but cute example, consider the determinants of 2 by 2 matrices [a, b; c, d] where each entry is a non-negative integer in the range 0, 1, …, 9, that is a digit.

The problem is to find the determinants, ad - bc, of all possible matrices of this form and represent the frequency with which each value occurs as a high density plot. This amounts to finding the probability distribution of the determinant if each digit is chosen independently and uniformly at random.

A neat way of doing this uses the outer() function twice:

> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(fr, xlab="Determinant", ylab="Frequency")

Notice that plot() here uses a histogram like plot method, because it “sees” that fr is of class “table”. The “obvious” way of doing this problem with for loops, to be discussed in Loops and conditional execution, is so inefficient as to be impractical.

It is also perhaps surprising that about 1 in 20 such matrices is singular.

5.6 Generalized transpose of an array

The function aperm(a, perm) may be used to permute an array, a. The argument perm must be a permutation of the integers {1, …, k}, where k is the number of subscripts in a. The result of the function is an array of the same size as a but with old dimension given by perm[j] becoming the new j-th dimension. The easiest way to think of this operation is as a generalization of transposition for matrices. Indeed if A is a matrix, (that is, a doubly subscripted array) then B given by

> B <- aperm(A, c(2,1))

is just the transpose of A. For this special case a simpler function t() is available, so we could have used B <- t(A).

5.7 Matrix facilities

As noted above, a matrix is just an array with two subscripts. However it is such an important special case it needs a separate discussion. R contains many operators and functions that are available only for matrices. For example t(X) is the matrix transpose function, as noted above. The functions nrow(A) and ncol(A) give the number of rows and columns in the matrix A respectively.

5.7.1 Matrix multiplication

The operator %*% is used for matrix multiplication. An n by 1 or 1 by n matrix may of course be used as an n-vector if in the context such is appropriate. Conversely, vectors which occur in matrix multiplication expressions are automatically promoted either to row or column vectors, whichever is multiplicatively coherent, if possible, (although this is not always unambiguously possible, as we see later).

If, for example, A and B are square matrices of the same size, then

> A * B

is the matrix of element by element products and

> A %*% B

is the matrix product. If x is a vector, then

> x %*% A %*% x

is a quadratic form.¹⁶

The function crossprod() forms “crossproducts”, meaning that crossprod(X, y) is the same as t(X) %*% y but the operation is more efficient. If the second argument to crossprod() is omitted it is taken to be the same as the first.

The meaning of diag() depends on its argument. diag(v), where v is a vector, gives a diagonal matrix with elements of the vector as the diagonal entries. On the other hand diag(M), where M is a matrix, gives the vector of main diagonal entries of M. This is the same convention as that used for diag() in MATLAB. Also, somewhat confusingly, if k is a single numeric value then diag(k) is the k by k identity matrix!

5.7.2 Linear equations and inversion

Solving linear equations is the inverse of matrix multiplication. When after

> b <- A %*% x

only A and b are given, the vector x is the solution of that linear equation system. In R,

> solve(A,b)

solves the system, returning x (up to some accuracy loss). Note that in linear algebra, formally x = A^{-1} %*% b where A^{-1} denotes the inverse of A, which can be computed by

solve(A)

but rarely is needed. Numerically, it is both inefficient and potentially unstable to compute x <- solve(A) %*% b instead of solve(A,b).

The quadratic form x %% A^{-1} %% x which is used in multivariate computations, should be computed by something like¹⁷ x %*% solve(A,x), rather than computing the inverse of A.

5.7.3 Eigenvalues and eigenvectors

The function eigen(Sm) calculates the eigenvalues and eigenvectors of a symmetric matrix Sm. The result of this function is a list of two components named values and vectors. The assignment

> ev <- eigen(Sm)

will assign this list to ev. Then ev\(val</code> is the vector of eigenvalues of <code class="calibre2">Sm</code> and <code class="calibre2">ev\)vec is the matrix of corresponding eigenvectors. Had we only needed the eigenvalues we could have used the assignment:

> evals <- eigen(Sm)$values

evals now holds the vector of eigenvalues and the second component is discarded. If the expression

> eigen(Sm)

is used by itself as a command the two components are printed, with their names. For large matrices it is better to avoid computing the eigenvectors if they are not needed by using the expression

> evals <- eigen(Sm, only.values = TRUE)$values

5.7.4 Singular value decomposition and determinants

The function svd(M) takes an arbitrary matrix argument, M, and calculates the singular value decomposition of M. This consists of a matrix of orthonormal columns U with the same column space as M, a second matrix of orthonormal columns V whose column space is the row space of M and a diagonal matrix of positive entries D such that M = U %% D %% t(V). D is actually returned as a vector of the diagonal elements. The result of svd(M) is actually a list of three components named d, u and v, with evident meanings.

If M is in fact square, then, it is not hard to see that

> absdetM <- prod(svd(M)$d)

calculates the absolute value of the determinant of M. If this calculation were needed often with a variety of matrices it could be defined as an R function

> absdet <- function(M) prod(svd(M)$d)

after which we could use absdet() as just another R function. As a further trivial but potentially useful example, you might like to consider writing a function, say tr(), to calculate the trace of a square matrix. [Hint: You will not need to use an explicit loop. Look again at the diag() function.]

R has a builtin function det to calculate a determinant, including the sign, and another, determinant, to give the sign and modulus (optionally on log scale),

5.7.5 Least squares fitting and the QR decomposition

The function lsfit() returns a list giving results of a least squares fitting procedure. An assignment such as

> ans <- lsfit(X, y)

gives the results of a least squares fit where y is the vector of observations and X is the design matrix. See the help facility for more details, and also for the follow-up function ls.diag() for, among other things, regression diagnostics. Note that a grand mean term is automatically included and need not be included explicitly as a column of X. Further note that you almost always will prefer using lm(.) (see Linear models) to lsfit() for regression modelling.

Another closely related function is qr() and its allies. Consider the following assignments

> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)

These compute the orthogonal projection of y onto the range of X in fit, the projection onto the orthogonal complement in res and the coefficient vector for the projection in b, that is, b is essentially the result of the MATLAB ‘backslash’ operator.

It is not assumed that X has full column rank. Redundancies will be discovered and removed as they are found.

This alternative is the older, low-level way to perform least squares calculations. Although still useful in some contexts, it would now generally be replaced by the statistical models features, as will be discussed in Statistical models in R.

5.8 Forming partitioned matrices, `cbind()` and `rbind()`

As we have already seen informally, matrices can be built up from other vectors and matrices by the functions cbind() and rbind(). Roughly cbind() forms matrices by binding together matrices horizontally, or column-wise, and rbind() vertically, or row-wise.

In the assignment

> X <- cbind(arg_1, arg_2, arg_3, …)

the arguments to cbind() must be either vectors of any length, or matrices with the same column size, that is the same number of rows. The result is a matrix with the concatenated arguments arg_1, arg_2, … forming the columns.

If some of the arguments to cbind() are vectors they may be shorter than the column size of any matrices present, in which case they are cyclically extended to match the matrix column size (or the length of the longest vector if no matrices are given).

The function rbind() does the corresponding operation for rows. In this case any vector argument, possibly cyclically extended, are of course taken as row vectors.

Suppose X1 and X2 have the same number of rows. To combine these by columns into a matrix X, together with an initial column of 1s we can use

> X <- cbind(1, X1, X2)

The result of rbind() or cbind() always has matrix status. Hence cbind(x) and rbind(x) are possibly the simplest ways explicitly to allow the vector x to be treated as a column or row matrix respectively.

5.9 The concatenation function, `c()`, with arrays

It should be noted that whereas cbind() and rbind() are concatenation functions that respect dim attributes, the basic c() function does not, but rather clears numeric objects of all dim and dimnames attributes. This is occasionally useful in its own right.

The official way to coerce an array back to a simple vector object is to use as.vector()

> vec <- as.vector(X)

However a similar result can be achieved by using c() with just one argument, simply for this side-effect:

> vec <- c(X)

There are slight differences between the two, but ultimately the choice between them is largely a matter of style (with the former being preferable).

5.10 Frequency tables from factors

Recall that a factor defines a partition into groups. Similarly a pair of factors defines a two way cross classification, and so on. The function table() allows frequency tables to be calculated from equal length factors. If there are k factor arguments, the result is a k-way array of frequencies.

Suppose, for example, that statef is a factor giving the state code for each entry in a data vector. The assignment

> statefr <- table(statef)

gives in statefr a table of frequencies of each state in the sample. The frequencies are ordered and labelled by the levels attribute of the factor. This simple case is equivalent to, but more convenient than,

> statefr <- tapply(statef, statef, length)

Further suppose that incomef is a factor giving a suitably defined “income class” for each entry in the data vector, for example with the cut() function:

> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef

Then to calculate a two-way table of frequencies:

> table(incomef,statef)
         statef
incomef   act nsw nt qld sa tas vic wa
  (35,45]   1   1  0   1  0   0   1  0
  (45,55]   1   1  1   1  2   0   1  3
  (55,65]   0   3  1   3  2   2   2  1
  (65,75]   0   1  0   0  0   0   1  0

Extension to higher-way frequency tables is immediate.