## Agenda

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

## Arrays

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

Some properties of the array:

dim(x.arr)
## [1] 2 2
is.vector(x.arr)
## [1] FALSE
is.array(x.arr)
## [1] TRUE
typeof(x.arr)
## [1] "double"
str(x.arr)
##  num [1:2, 1:2] 7 8 10 45
attributes(x.arr)
## $dim ## [1] 2 2 typeof() returns the type of the elements str() gives the structure: here, a numeric array, with two dimensions, both indexed 1–2, and then the actual numbers Exercise: try all these with x ## Accessing and operating on arrays 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 Omitting an index means "all of it": x.arr[c(1:2),2] ## [1] 10 45 x.arr[,2] ## [1] 10 45 ## Functions on arrays 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 Many functions do preserve array structure: y <- -x y.arr <- array(y,dim=c(2,2)) y.arr + x.arr ## [,1] [,2] ## [1,] 0 0 ## [2,] 0 0 Others specifically act on each row or column of the array separately: rowSums(x.arr) ## [1] 17 53 We will see a lot more of this idea ## Example: Price of houses in PA 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 ## -26206.564325 3.651256 Fit a simple linear model, predicting median house price from median household income Census tracts 24–425 are Allegheny county Tract 24 has a median income of$14,719; actual median house value is $34,100 — is that above or below what's? 34100 < -26206.564 + 3.651*14719 ## [1] FALSE Tract 25 has income$48,102 and house price $155,900 155900 < -26206.564 + 3.651*48102 ## [1] FALSE What about tract 26? We could just keep plugging in numbers like this, but that's • boring and repetitive • error-prone (what if I forget to change the median income, or drop a minus sign from the intercept?) • obscure if we come back to our work later (what are these numbers?) ## Use variables and names penn.coefs <- coefficients(lm(Median_house_value ~ Median_household_income, data=penn)) penn.coefs ## (Intercept) Median_household_income ## -26206.564325 3.651256 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 plot(x=allegheny.fitted, y=allegheny.values, xlab="Model-predicted median house values", ylab="Actual median house values", xlim=c(0,5e5),ylim=c(0,5e5)) abline(a=0,b=1,col="grey") ## Running example: resource allocation ("mathematical programming") 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 ## Matrices 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() ## Matrix multiplication 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? ## Multiplying matrices and vectors 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 ## Matrix operators Transpose: t(factory) ## [,1] [,2] ## [1,] 40 1 ## [2,] 60 3 Determinant: det(factory) ## [1] 60 ## The diagonal 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) ## Creating a diagonal or identity matrix diag(c(3,4)) ## [,1] [,2] ## [1,] 3 0 ## [2,] 0 4 diag(2) ## [,1] [,2] ## [1,] 1 0 ## [2,] 0 1 ## Inverting a matrix solve(factory) ## [,1] [,2] ## [1,] 0.05000000 -1.0000000 ## [2,] -0.01666667 0.6666667 factory %*% solve(factory) ## [,1] [,2] ## [1,] 1 0 ## [2,] 0 1 ## Why's it called "solve"" anyway? 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 ## Names in matrices 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 rownames(factory) <- c("labor","steel") colnames(factory) <- c("cars","trucks") factory ## cars trucks ## labor 40 60 ## steel 1 3 available <- c(1600,70) names(available) <- c("labor","steel") output <- c(20,10) names(output) <- c("trucks","cars") factory %*% output # But we've got cars and trucks mixed up! ## [,1] ## labor 1400 ## steel 50 factory %*% output[colnames(factory)] ## [,1] ## labor 1600 ## steel 70 all(factory %*% output[colnames(factory)] <= available[rownames(factory)]) ## [1] TRUE Notice: Last lines don't have to change if we add motorcycles as output or rubber and glass as inputs (abstraction again) ## Doing the same thing to each row or column 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? ## Lists 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 ## Accessing pieces of 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? ## Expanding and contracting 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 Chop off the end of a list by setting the length to something smaller (also works with vectors): length(my.distribution) ## [1] 4 length(my.distribution) <- 3 my.distribution ## [[1]] ## [1] "exponential" ## ## [[2]] ## [1] 7 ## ## [[3]] ## [1] FALSE ## Naming list elements 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" Lists have a special short-cut way of using names, $ (which removes names and structures):

my.distribution[["family"]]
## [1] "exponential"
my.distribution$family ## [1] "exponential" ## Names in lists Creating a list with names: another.distribution <- list(family="gaussian",mean=7,sd=1,is.symmetric=TRUE) Adding named elements: my.distribution$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 ## Key-Value pairs 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 ## Dataframes 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 a.matrix <- matrix(c(35,8,10,4),nrow=2) colnames(a.matrix) <- c("v1","v2") a.matrix ## v1 v2 ## [1,] 35 10 ## [2,] 8 4 a.matrix[,"v1"] # Try a.matrix$v1 and see what happens
## [1] 35  8
a.data.frame <- data.frame(a.matrix,logicals=c(TRUE,FALSE))
a.data.frame
##   v1 v2 logicals
## 1 35 10     TRUE
## 2  8  4    FALSE
a.data.frame$v1 ## [1] 35 8 a.data.frame[,"v1"] ## [1] 35 8 a.data.frame[1,] ## v1 v2 logicals ## 1 35 10 TRUE colMeans(a.data.frame) ## v1 v2 logicals ## 21.5 7.0 0.5 ## Adding rows and columns 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 ## Structures of Structures 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

## Structures of Structures

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

## Example: Eigenstuff

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

eigen(factory)
## eigen() decomposition
## $values ## [1] 41.556171 1.443829 ## ##$vectors
##            [,1]       [,2]
## [1,] 0.99966383 -0.8412758
## [2,] 0.02592747  0.5406062
class(eigen(factory))
## [1] "eigen"

With complicated objects, you can access parts of parts (of parts…)

factory %*% eigen(factory)$vectors[,2] ## [,1] ## labor -1.2146583 ## steel 0.7805429 eigen(factory)$values[2] * eigen(factory)$vectors[,2] ## [1] -1.2146583 0.7805429 eigen(factory)$values[2]
## [1] 1.443829
eigen(factory)[[1]][[2]] # NOT [[1,2]]
## [1] 1.443829

## Creating an example dataframe

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

colnames(states)
##  [1] "Population" "Income"     "Illiteracy" "Life.Exp"   "Murder"
##  [6] "HS.Grad"    "Frost"      "Area"       "abb"        "region"
## [11] "division"
states[1,]
##         Population Income Illiteracy Life.Exp Murder HS.Grad Frost  Area
## Alabama       3615   3624        2.1    69.05   15.1    41.3    20 50708
##         abb region           division
## Alabama  AL  South East South Central

## Dataframe access

• By row and column index
states[49,3]
## [1] 0.7
• By row and column names
states["Wisconsin","Illiteracy"]
## [1] 0.7

## Dataframe access

• 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",]?

## Dataframe access

• 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$Illiteracy) ## [1] 2.1 1.5 1.8 1.9 1.1 0.7 ## Dataframe access • Rows matching a condition: states[states$division=="New England", "Illiteracy"]
## [1] 1.1 0.7 1.1 0.7 1.3 0.6
states[states$region=="South", "Illiteracy"] ## [1] 2.1 1.9 0.9 1.3 2.0 1.6 2.8 0.9 2.4 1.8 1.1 2.3 1.7 2.2 1.4 1.4 ## Dataframe access Parts or all of the dataframe can be assigned to: summary(states$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$HS.Grad ## with() What percentage of literate adults graduated HS? head(100*(states$HS.Grad/(100-states\$Illiteracy)))
## [1] 42.18590 67.71574 59.16497 40.67278 63.29626 64.35045

with() takes a data frame and evaluates an expression "inside" it:

with(states, head(100*(HS.Grad/(100-Illiteracy))))
## [1] 42.18590 67.71574 59.16497 40.67278 63.29626 64.35045

## Data arguments

Lots of functions take data arguments, and look variables up in that data frame:

plot(Illiteracy~Frost, data=states)

$$R^2 =0.45$$, $$p \approx {10}^{-7}$$

## Summary

• 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