Data Vectors

The Plane

Pairs of numbers can be depicted as points on a plane.

The plane is normally denoted by $\mathbb{R}^2$.

Details

Pairs of numbers can be depicted as points on a plane.

Definition

A plane is a perfectly flat surface with no thickness and no end, it can extend forever in all directions. It has two-dimensions, length and width. We need two values to find a point on the plane.

Normally we talk about "the plane" (or "the $xy$-plane") as the collection of all pairs of numbers and denote it by

$\mathbb{R}^2 = \{ (x,y) : x,y \in \mathbb{R} \},$

giving coordinates to each point. The plane is also sometimes called The Cartesian coordinate system, named after its inventor, the French polymath René Descartes.

Examples

Example

Plotting the point $(2,4)$ in the $xy$-plane using R:

plot(2,4,xlim=c(0,6),ylim=c(0,6),xlab="x",ylab="y",cex=2)
text(2,4,"(2,4)",pos=4,cex=2)

Additional points can be added using the points() function:

points(3,5, cex = 0.5) ## a point at (3,5)

If you have two sets of coordinates on a plane you, can calculate the distance between the two points and graph the line connecting the points.

Example

What is the distance between the two points $(3,9)$ and $(5,1)$? What is the distance between the 2 points (3,9) and (5,1)?

We will use the Pythagorean theorem:

$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

We insert our values into the formula:

$d=\sqrt{(5-3)^{2}+(1-9)^{2}}$

When we combine inside the parentheses we get:

$d=\sqrt{(2)^{2}+(-8)^{2}}$

Squaring both terms:

$d=\sqrt{4+64}$

Then we take the square root:

$d=\sqrt{68}$

The result:

$d \approx 8.2462$

Simple Plots in R

Graphing functions in R

• plot() - plots a scatter plot (as a line plot)

• points() - adds points to a plot

• text() - adds text to a plot

• lines() - adds lines to a plot

Figure: Points on a plane, drawn with R.

Examples

Example

plot(2,3)

gives a single plot and

plot(2,3, xlim=c(0,5), ylim=c(0,5))

gives a single plot but forces both axes to range from 0 to 5.

Example

The following R commands can be used to generate a plot with two points:

plot(1,2,xlim=c(0,5),ylim=c(0,5),xlab="x",ylab="y")
points(3,1)
text(1,2,"(1,2)",pos=4, cex=2)
text(3,1,"(3,1)",pos=4, cex=2)

Example

In this example, we plot three points. The first two arguments of the plot function. The third plot was added with the points are by including vectors with a length of $2$ as the $x$ and $y$ arguments of the plot function. The third point was added with the points function. The second and third points were labeled using the text function and a line was drawn between them using the lines function.

plot(c(2,3),c(3,4),xlim=c(2,6),ylim=c(1,5),xlab="x",ylab="y")
points(4,2)
text(3,4,"(3,4)",pos=4, cex=2)
text(4,2,"(4,2)",pos=4, cex=2)
lines(c(3,4), c(4,2))

Note: Note that if you are unsure of what format the arguments of an R function needs to be, you can call a help file by typing ? before the function name (e.g. ?lines).

Data

Data are usually a sequence of numbers, typically called a vector.

Details

When we collect data these are one or more sequences of numbers, collected into data vectors. We commonly think of these data vectors as columns in a table.

Examples

Example

In R, if the command

x <- c(4,5,3,7)

is given, then x contains a vector of numbers.

Example

Create a function in R, give it a name Myfunction which takes the sum of x and y:

Myfunction <- function(x,y) { sum(x,y) }

If you input the vectors 1:3 and 4:7 into the function it will calculate the sum of x <- (1+2+3) and y <- (4+5+6+7) as follows:

> Myfunction(1:3,4:7)
[1] 28

Indices for a Data Vector

If data are in a vector x, then we use indices to refer to individual elements.

Details

If i is an integer then $x_i$ denotes the $i^{th}$ element of $x$.

Note: Although we do not distinguish (much) between row- and column vectors, usually a vector is thought of as a column. If we need to specify the type of vector, row or column, then for vector $x$, the column vector would be referred to as $x'$ and the row vector as $x^T$.

(the transpose of the original).

Examples

Example

If $x=(4,5,3,7)$ then $x_1=4$ and $x_4=7$

Example

How to remove all indices below a certain value in R?

> x <- c(1,5,8,9,4,16,12,7,11)

> x
[1]  1  5  8  9  4 16 12  7 11

> y <- x[x>10]

> y
[1] 16 12 11

Example

Consider a function that takes to vectors

$a \in \mathbb{R}^n, b \in \mathbb{N}^m$

as arguments with:

$n \ge m$

and:

$1 \le b_1,\dots,b_m \le n.$

The function returns the sum:

$\sum_{i = 1}^m {a_b}_i$

Long version:

> fn <- function(a,b) {
+ result <- sum(a[b])
+ return(result)
+ }

Short version:

fN <- function(a,b) sum(a[b])

Summation

We use the symbol $\Sigma$ to denote sums.

In R, the sum function adds numbers.

Examples

Example

If $x=(4,5,3,7)$

then

$\sum_{i=1}^{4} x_i = x_1+x_2+x_3+x_4 = 4+5+3+7 = 19$

and

$\sum_{i=2}^{4} x_i = x_2+x_3+x_4 = 5+3+7 = 15.$

In R one can give the corresponding commands:

> x <- c(4,5,3,7)

> x
[1] 4 5 3 7

> sum(x)
[1] 19

> sum(x[2:4])
[1] 15