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More Data Structures
===
author: John S
date: 10 Sept 2015
font-family: 'Garamond'

Agenda
===
- Arrays
- Matrices
- Lists
- Dataframes
- Structures of structures

Vector structures, starting with arrays
===

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

Most useful: **arrays**
```{r}
x <- c(7, 8, 10, 45)
x.arr <- array(x,dim=c(2,2))
x.arr
```
`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:
```{r}
dim(x.arr)
is.vector(x.arr)
is.array(x.arr)
```

===
```{r}
typeof(x.arr)
str(x.arr)
attributes(x.arr)
```
`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:
```{r}
x.arr[1,2]
x.arr[3]
```

===
Omitting an index means "all of it":
```{r}
x.arr[c(1:2),2]
x.arr[,2]
```

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:
```{r}
which(x.arr > 9)
```

===
Many functions _do_ preserve array structure:
```{r}
y <- -x
y.arr <- array(y,dim=c(2,2))
y.arr + x.arr
```

Others specifically act on each row or column of the array separately:
```{r}
rowSums(x.arr)
```

<small>We will see a lot more of this idea</small>


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

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

```{r}
six.sevens <- matrix(rep(7,6),ncol=3)
six.sevens
factory %*% six.sevens # [2x2] * [2x3]
```

<small>What happens if you try `six.sevens %*% factory`?</small>

Multiplying matrices and vectors
===
Numeric vectors can act like proper vectors:
```{r}
output <- c(10,20)
factory %*% output
output %*% factory
```
<small>R silently casts the vector as either a row or a column matrix</small>

Matrix operators
===

Transpose:
```{r}
t(factory)
```

Determinant:
```{r}
det(factory)
```

The diagonal
===
The `diag()` function can extract the diagonal entries of a matrix:
```{r}
diag(factory)
```
It can also _change_ the diagonal:
```{r}
diag(factory) <- c(35,4)
factory
```
Re-set it for later:
```{r}
diag(factory) <- c(40,3)
```

Creating a diagonal or identity matrix
===
```{r}
diag(c(3,4))
diag(2)
```

Inverting a matrix
===
```{r}
solve(factory)
factory %*% solve(factory)
```

Why's it called "solve"" anyway?
===
Solving the linear system $\mathbf{A}\vec{x} = \vec{b}$ for $\vec{x}$:

```{r}
available <- c(1600,70)
solve(factory,available)
```

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

===
```{r}
rownames(factory) <- c("labor","steel")
colnames(factory) <- c("cars","trucks")
factory
available <- c(1600,70)
names(available) <- c("labor","steel")
```

===
```{r}
output <- c(20,10)
names(output) <- c("trucks","cars")
factory %*% output # But we've got cars and trucks mixed up!
factory %*% output[colnames(factory)]
all(factory %*% output[colnames(factory)] <= available[rownames(factory)])
```
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

```{r}
colMeans(factory)
summary(factory)
```

===
`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

```{r}
rowMeans(factory)
apply(factory,1,mean)
```

<small>What would `apply(factory,1,sd)` do?</small>

Lists
====
Sequence of values, _not_ necessarily all of the same type

```{r}
my.distribution <- list("exponential",7,FALSE)
my.distribution
```

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
```{r}
is.character(my.distribution)
is.character(my.distribution[[1]])
my.distribution[[2]]^2
```
<small>What happens if you try `my.distribution[2]^2`?</small>
<small>What happens if you try `[[ ]]` on a vector?</small>

Expanding and contracting lists
===
Add to lists with `c()` (also works with vectors):
```{r}
my.distribution <- c(my.distribution,7)
my.distribution
```

===
Chop off the end of a list by setting the length to something smaller (also works with vectors):
```{r}
length(my.distribution)
length(my.distribution) <- 3
my.distribution
```

Naming list elements
===
We can name some or all of the elements of a list
```{r}
names(my.distribution) <- c("family","mean","is.symmetric")
my.distribution
my.distribution[["family"]]
my.distribution["family"]
```


===

Lists have a special short-cut way of using names, `$` (which removes names and structures):
```{r}
my.distribution[["family"]]
my.distribution$family
```

Names in lists (cont'd.)
===
Creating a list with names:
```{r}
another.distribution <- list(family="gaussian",mean=7,sd=1,is.symmetric=TRUE)
```

Adding named elements:
```{r}
my.distribution$was.estimated <- FALSE
my.distribution[["last.updated"]] <- "2011-08-30"
```

Removing a named list element, by assigning it the value `NULL`:
```{r}
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
<small>`penn` from last time was really a dataframe</small>

Not just a matrix because *columns can have different types*

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

<small>but no matrix multiplying dataframes, even if all columns are numeric</small>

===
```{r}
a.matrix <- matrix(c(35,8,10,4),nrow=2)
colnames(a.matrix) <- c("v1","v2")
a.matrix
a.matrix[,"v1"]  # Try a.matrix$v1 and see what happens
```

===
```{r}
a.data.frame <- data.frame(a.matrix,logicals=c(TRUE,FALSE))
a.data.frame
a.data.frame$v1
a.data.frame[,"v1"]
a.data.frame[1,]
colMeans(a.data.frame)
```

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
```{r}
rbind(a.data.frame,list(v1=-3,v2=-5,logicals=TRUE))
rbind(a.data.frame,c(3,4,6))
```
<small>A dataframe is actually a list.</small>

Example: Triathlon Data
===
Results from a triathlon 
```{r}
tri.data <- read.csv("http://www.math.umass.edu/~jstauden/IMdata.csv")
US.only <- tri.data[tri.data$Country=="USA",]
head(US.only)
coefficients(lm(Finish ~ Run, data=US.only))
```

=======
```{r}
plot(US.only$Run,US.only$Finish,xlab="Run (Hours)",
     ylab="Finish (Hours)")
```
<small>Common error: Read in data. You think a column (x) is a number, but R thinks it's a character. (solution: as.numeric(x) or as.numeric(as.character(x)) )</small>


Structures of Structures
===
So far, every list element has been same length.

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

```{r}
eigen(factory)
class(eigen(factory))
```

===
With complicated objects, you can access parts of parts (of parts...)
```{r}
factory %*% eigen(factory)$vectors[,2]
eigen(factory)$values[2] * eigen(factory)$vectors[,2]
eigen(factory)$values[2]
eigen(factory)[[1]][[2]] # NOT [[1,2]]
```

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