Set Operations in R

R can perform different operations in sets, such as union, intersection, asymmetric difference of two sets, etc. Specifically, the following operations are available in R for set operations.

Operator	Usage	Definition
union	union(x, y)	Union of sets x and y
intersect	intersect(x, y)	Intersection of sets x and y
setdiff	setdiff(x, y)	Asymmetric difference between sets x and y (Elements in x but not in y)
setequal	setequal(x, y)	If sets x and y have the same elements
is.element	is.element(el, set)	If el is an element of set

Examples:

> x <- c(sort(sample(1:20, 9)),NA)

> y <- c(sort(sample(3:23, 7)),NA)

> x

[1]  1  3  5  8 11 17 18 19 20 NA

> y

[1]  7 11 15 16 17 19 22 NA

> union(x, y)

[1]  1  3  5  8 11 17 18 19 20 NA  7 15 16 22

> intersect(x, y)

[1] 11 17 19 NA
> setdiff(x, y)

[1]  1  3  5  8 18 20

> setdiff(y, x)

[1]  7 15 16 22

> setequal(x, y)

[1] FALSE

Note that each of union, intersect, setdiff and setequal will discard any duplicated values in the arguments. Look at the following example:

> x

[1]  1  3  5  8 11 17 18 19 20 NA

> x2 <- c(x, 1, 3, 5, 8)

> x2

[1]  1  3  5  8 11 17 18 19 20 NA  1  3  5  8

> setdiff(x, y)

[1]  1  3  5  8 18 20

> setdiff(x2, y)

[1]  1  3  5  8 18 20

> setequal(x, x2)

[1] TRUE

Although x and x2 have different length, they have the same UNIQUE elements so setequal(x, x2) returns a TRUE value.

is.element(x, y) is identical to x %in% y which is already discussed here. The return value of is.element is a vector of TRUE and FALSE with the same length as x, which indicates whether each element of x is an element of y or not.

> is.element(x, y)  # vector of length 10

[1] FALSE FALSE FALSE FALSE  TRUE  TRUE FALSE  TRUE FALSE  TRUE

> is.element(y, x)  # vector of length 8

[1] FALSE  TRUE FALSE FALSE  TRUE  TRUE FALSE  TRUE

Running time in R

In R, proc.time determines how much real and CPU time (in seconds) the currently running R process has already taken. proc.time returns five elements for backwards compatibility, but prints a named vector of length 3. The first two entries are the total user and system CPU times of the current R process and any child processes on which it has waited, and the third entry is the ‘real’ elapsed time since the process was started. system.time(expr) is used for timing a valid R expression which calls the function proc.time, evaluates expr, and then calls proc.time once more, returning the difference between the two proc.time calls. For example,

> ptm <- proc.time()

> for (i in 1:10000) x <- rnorm(1000)

> proc.time() - ptm

   user  system elapsed

   2.10    0.01    2.14
> system.time(for (i in 1:10000) x <- rnorm(1000))
   user  system elapsed
   2.01    0.00    2.06 

The definition of 'user' and 'system' times is from your OS. Typically it is something like

The 'user time' is the CPU time charged for the execution of user instructions of the calling process. The 'system time' is the CPU time charged for execution by the system on behalf of the calling process.

proc.time/system.time can be used to compare the running speed of different methods doing the same job. For example, we want to find out the maximum of a vector of 10000000 randomly generated Uniform[0,1] random variables.

> x<-runif(10000000)

> system.time(max(x))

   user  system elapsed

   0.05    0.00    0.05

> pc <- proc.time()

> cmax <- x[1]

> for (i in 2:10000000)

+ {

+   if(x[i] > cmax) cmax <- x[i]

+ }

> proc.time() - pc

   user  system elapsed

  16.88    0.11   18.21

We can see that there is a huge difference in running time between the two methods (0.05 seconds versus 18.21 seconds). Do not 'grow' data sets in loops or recursive function calls. Use R built-in functions whenever possible.

Complex numbers in R

We sometimes encounter the situations of using complex numbers in our computation. For example, the square root of -1 can be denoted as 1*i. Complex numbers are implemented in the "base" package, it’s very easy to work with them. To construct a complex number x + iy, you use complex and specify its real and imaginary components explicitly as follows:

> x <- 2

> y <- 3

> z1 <- complex(real = x, imaginary = y)

> z1

[1] 2+3i

You can convert other objects to class "complex" using as.complex and test if an object is complex with is.comple

> z2 <- as.complex(-5)
> z2
[1] -5+0i
> is.complex(z2)
[1] TRUE

There are five basic mathematical operations that works on complex numbers, Re, Im, Mod, Arg, and Conj. First, you may want to extract the real and imaginary components of a complex number. You can do this using Re and Im, respectively. You can also find the modulus and complex argument of a complex number with Mod and Arg. Finally, you can take the complex conjugate of a complex number with the help of Conj.

> z3 <- complex(real = 1.3, imaginary = 6) 

> z3

[1] 1.3+6i

> Re(z3)

[1] 1.3

> Im(z3)

[1] 6

> Mod(z3)

[1] 6.139218

> Arg(z3)

[1] 1.357428

> Conj(z3)

[1] 1.3-6i

Special symbols and math formulas in R

Sometimes one wants to put special symbols, such as Greek letters and superscripts on the plots. In R we can use function expression() to do this job.

> xlabel <- expression(paste(Delta, italic(s), sep = ""))

> ylabel <- expression(alpha[1] * " in (kg)"^2)

> plotname <- expression(sin * (beta))

> plot(rnorm(50), rnorm(50), xlab = xlabel, ylab = ylabel, main = plotname, xlim = c(-pi, pi), ylim = c(-3, 3), axes = FALSE)

> axis(1, at = c(-pi, -pi/2, 0, pi/2, pi), labels = expression(-pi, -pi/2, 0, pi/2, pi))

> axis(2)
> box()

> text(-pi/2, -2, expression(hat(beta) == (X^t * X)^{-1} * X^t * y))

> text(pi/2, 2, expression(paste(frac(1, sigma*sqrt(2*pi)), exp*(frac(-(x-mu)^2, 2*sigma^2)), sep = "")), cex = 1.5)

If you want to know more about expression and putting math symbols on the plot, run the following demos in R.

> demo(plotmath)