More Data Structures

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

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

Most useful: arrays

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

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

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.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

Census data for California and Pennsylvania on housing prices, by Census “tract”

calif_penn <- read.csv("http://www.stat.cmu.edu/~cshalizi/uADA/13/hw/01/calif_penn_2011.csv")
penn <- calif_penn[calif_penn[,"STATEFP"]==42,]
coefficients(lm(Median_house_value ~ Median_household_income, data=penn))

            (Intercept) Median_household_income 
             -26206.564                   3.651 

Fit a simple linear model, predicting median house price from median household income

penn.coefs <- coefficients(lm(Median_house_value ~ Median_household_income, data=penn))
penn.coefs

            (Intercept) Median_household_income 
             -26206.564                   3.651 

allegheny.rows <- 24:425
allegheny.medinc <- penn[allegheny.rows,"Median_household_income"]
allegheny.values <- penn[allegheny.rows,"Median_house_value"]
allegheny.fitted <- penn.coefs["(Intercept)"]+penn.coefs["Median_household_income"]*allegheny.medinc

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

In R, a matrix is a specialization of a 2D array

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?

Numeric vectors can act like proper vectors:

output <- c(10,20)
factory %*% output

     [,1]
[1,] 1600
[2,]   70

output %*% factory

     [,1] [,2]
[1,]  420  660

R silently casts the vector as either a row or a column matrix

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

solve(factory)

         [,1]    [,2]
[1,]  0.05000 -1.0000
[2,] -0.01667  0.6667

factory %*% solve(factory)

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

Solving the linear system \( \mathbf{A}\vec{x} = \vec{b} \) for \( \vec{x} \):

available <- c(1600,70)
solve(factory,available)

[1] 10 20

factory %*% solve(factory,available)

     [,1]
[1,] 1600
[2,]   70

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.   : 1.0   Min.   : 3.0  
 1st Qu.:10.8   1st Qu.:17.2  
 Median :20.5   Median :31.5  
 Mean   :20.5   Mean   :31.5  
 3rd Qu.:30.2   3rd Qu.:45.8  
 Max.   :40.0   Max.   :60.0  

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

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?

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

We can name some or all of the elements of a list

names(my.distribution) <- c("family","mean","is.symmetric")
my.distribution

$family
[1] "exponential"

$mean
[1] 7

$is.symmetric
[1] FALSE

my.distribution[["family"]]

[1] "exponential"

my.distribution["family"]

$family
[1] "exponential"

Creating a list with names:

another.distribution <- list(family="gaussian",mean=7,sd=1,is.symmetric=TRUE)

Adding named elements:

my.distribution$was.estimated <- FALSE
my.distribution[["last.updated"]] <- "2011-08-30"

Removing a named list element, by assigning it the value NULL:

my.distribution$was.estimated <- NULL

Lists give us a way to store and look up data by name, rather than by position

A really useful programming concept with many names: key-value pairs, dictionaries, associative arrays, hashes

If all our distributions have components named family, we can look that up by name, without caring where it is in the list

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

Lots of the really-statistical parts of R presume data frames penn from last time was really a dataframe

Not just a matrix because columns can have different types

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

but no matrix multiplying dataframes, even if all columns are numeric

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

rbind(a.data.frame,list(v1=-3,v2=-5,logicals=TRUE))

  v1 v2 logicals
1 35 10     TRUE
2  8  4    FALSE
3 -3 -5     TRUE

rbind(a.data.frame,c(3,4,6))

  v1 v2 logicals
1 35 10        1
2  8  4        0
3  3  4        6

So far, every list element has been a single data value

List elements can be other data structures, e.g., vectors and matrices:

plan <- list(factory=factory, available=available, output=output)
plan$output

trucks   cars 
    20     10 

Internally, a dataframe is basically a list of vectors

List elements can even be other lists
which may contain other data structures
including other lists
which may contain other data structures…

This recursion lets us build arbitrarily complicated data structures from the basic ones

Most complicated objects are (usually) lists of data structures

eigen() finds eigenvalues and eigenvectors of a matrix
Returns a list of a vector (the eigenvalues) and a matrix (the eigenvectors)

eigen(factory)

$values
[1] 41.556  1.444

$vectors
        [,1]    [,2]
[1,] 0.99966 -0.8413
[2,] 0.02593  0.5406

class(eigen(factory))

[1] "list"

• Arrays add multi-dimensional structure to vectors
• Matrices act like you'd hope they would
• Lists let us combine different types of data
• Dataframes are hybrids of matrices and lists, for classic tabular data
• Recursion lets us build complicated data structures out of the simpler ones