• Matrices
• Arrays
• Lists
• Dataframes
• Structures of structures

Many data structures in R are made by adding bells and whistles to vectors, so “vector structures”

A matrix in R is a collections of homogeneous elements arranged in 2 dimensions

matrix(1:15, nrow = 4)

     [,1] [,2] [,3] [,4]
[1,]    1    5    9   13
[2,]    2    6   10   14
[3,]    3    7   11   15
[4,]    4    8   12    1

• A matrix is a vector with a dim attribute, i.e. an integer vector giving the number or rows and columns
  
• To create matrices us matrix()
  
• The functions dim(), nrow() and ncol() provide the attributes of the matrix
  
• Rows and columns can have names, dimnames(), rownames(), colnames()
  

Factory makes cars and trucks, using labor and steel

• a car takes 40 hours of labor and 1 ton of steel
• a truck takes 60 hours and 3 tons of steel
• resources: 1600 hours of labor and 70 tons of steel each week

factory <- matrix(c(40,1,60,3),nrow=2)
is.array(factory)

[1] TRUE

is.matrix(factory)

[1] TRUE

could also specify ncol, and/or byrow=TRUE to fill by rows.

Element-wise operations with the usual arithmetic and comparison operators (e.g., factory/3)

Compare whole matrices with identical() or all.equal()

Gets a special operator

six.sevens <- matrix(rep(7,6),ncol=3)
six.sevens

     [,1] [,2] [,3]
[1,]    7    7    7
[2,]    7    7    7

factory %*% six.sevens # [2x2] * [2x3]

     [,1] [,2] [,3]
[1,]  700  700  700
[2,]   28   28   28

What happens if you try six.sevens %*% factory?

Transpose:

t(factory)

     [,1] [,2]
[1,]   40    1
[2,]   60    3

Determinant:

det(factory)

[1] 60

The diag() function can extract the diagonal entries of a matrix:

diag(factory)

[1] 40  3

It can also change the diagonal:

diag(factory) <- c(35,4)
factory

     [,1] [,2]
[1,]   35   60
[2,]    1    4

Re-set it for later:

diag(factory) <- c(40,3)

diag(c(3,4))

     [,1] [,2]
[1,]    3    0
[2,]    0    4

diag(2)

     [,1] [,2]
[1,]    1    0
[2,]    0    1

We can name either rows or columns or both, with rownames() and colnames()

These are just character vectors, and we use the same function to get and to set their values

Names help us understand what we're working with

Names can be used to coordinate different objects

Take the mean: rowMeans(), colMeans(): input is matrix, output is vector. Also rowSums(), etc.

summary(): vector-style summary of column

colMeans(factory)

  cars trucks 
  20.5   31.5 

summary(factory)

      cars           trucks     
 Min.   : 1.00   Min.   : 3.00  
 1st Qu.:10.75   1st Qu.:17.25  
 Median :20.50   Median :31.50  
 Mean   :20.50   Mean   :31.50  
 3rd Qu.:30.25   3rd Qu.:45.75  
 Max.   :40.00   Max.   :60.00  

apply(), takes 3 arguments: the array or matrix, then 1 for rows and 2 for columns, then name of the function to apply to each

rowMeans(factory)

labor steel 
   50     2 

apply(factory,1,mean)

labor steel 
   50     2 

What would apply(factory,1,sd) do?

arrays are basically matrices in higher dimensions

x <- c(7, 8, 10, 45 , 70, 80 , 100, 250)
x.arr <- array(x,dim=c(2,2,2))
x.arr

, , 1

     [,1] [,2]
[1,]    7   10
[2,]    8   45

, , 2

     [,1] [,2]
[1,]   70  100
[2,]   80  250

dim says how many rows and columns; filled by columns

Can have \( 3, 4, \ldots n \) dimensional arrays; dim is a length-\( n \) vector

Can access a 2-D array either by pairs of indices or by the underlying vector:

x <- c(7, 8, 10, 45)
x.arr <- array(x,dim=c(2,2))
x.arr

     [,1] [,2]
[1,]    7   10
[2,]    8   45

x.arr[1,2]

[1] 10

x.arr[3]

[1] 10

Using a vector-style function on a vector structure will go down to the underlying vector, unless the function is set up to handle arrays specially:

which(x.arr > 9)

[1] 3 4

Sequence of values, not necessarily all of the same type

my.distribution <- list("exponential",7,FALSE)
my.distribution

[[1]]
[1] "exponential"

[[2]]
[1] 7

[[3]]
[1] FALSE

Most of what you can do with vectors you can also do with lists

Add to lists with c() (also works with vectors):

my.distribution <- c(my.distribution,7)
my.distribution

[[1]]
[1] "exponential"

[[2]]
[1] 7

[[3]]
[1] FALSE

[[4]]
[1] 7

Can use [ ] as with vectors
or use [[ ]], but only with a single index
[[ ]] drops names and structures, [ ] does not

is.character(my.distribution)

[1] FALSE

is.character(my.distribution[[1]])

[1] TRUE

my.distribution[[2]]^2

[1] 49

What happens if you try my.distribution[2]^2? What happens if you try [[ ]] on a vector?

Dataframe = the classic data table, \( n \) rows for cases, \( p \) columns for variables

Not just a matrix because columns can have different types

Many matrix functions also work for dataframes (rowSums(), summary(), apply())

but no matrix multiplication of dataframes, even if all columns are numeric

• 2D tables of data
• Each case/unit is a row
• Each variable is a column
• Variables can be of any type (numbers, text, Booleans, …)
• Both rows and columns can get names

library(datasets)
states <- data.frame(state.x77, abb=state.abb, region=state.region, division=state.division)

data.frame() is combining here a pre-existing matrix (state.x77), a vector of characters (state.abb), and two vectors of qualitative categorical variables (factors; state.region, state.division)

Column names are preserved or guessed if not explicitly set

• By row and column index

states[49,3]

[1] 0.7

• By row and column names

states["Wisconsin","Illiteracy"]

[1] 0.7

• All of a row:

states["Wisconsin",]

          Population Income Illiteracy Life.Exp Murder HS.Grad Frost  Area
Wisconsin       4589   4468        0.7    72.48      3    54.5   149 54464
          abb        region           division
Wisconsin  WI North Central East North Central

Exercise: what class is states["Wisconsin",]?

• All of a column:

head(states[,3])

[1] 2.1 1.5 1.8 1.9 1.1 0.7

head(states[,"Illiteracy"])

[1] 2.1 1.5 1.8 1.9 1.1 0.7

head(states$Illiteracy)

[1] 2.1 1.5 1.8 1.9 1.1 0.7

• Rows matching a condition:

states[states$division=="New England", "Illiteracy"]

[1] 1.1 0.7 1.1 0.7 1.3 0.6

states[states$region=="South", "Illiteracy"]

 [1] 2.1 1.9 0.9 1.3 2.0 1.6 2.8 0.9 2.4 1.8 1.1 2.3 1.7 2.2 1.4 1.4

Parts or all of the dataframe can be assigned to:

summary(states$HS.Grad)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  37.80   48.05   53.25   53.11   59.15   67.30 

states$HS.Grad <- states$HS.Grad/100
summary(states$HS.Grad)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.3780  0.4805  0.5325  0.5311  0.5915  0.6730 

states$HS.Grad <- 100*states$HS.Grad

We can add rows or columns to an array or data-frame with rbind() and cbind(), but be careful about forced type conversions

Error in rbind(a.data.frame, list(v1 = -3, v2 = -5, logicals = TRUE)) : 
  object 'a.data.frame' not found