# Vectors

```{r setup, include = FALSE}
library(purrr)
library(dplyr)
```

So far this book has focussed on data frames and packages that work with them. But as you start to write your own functions, and dig deeper into R, you need to learn about vectors, the objects that underpin data frames. If you've learned R in a more traditional way, you're probably already familiar with vectors, as most R resources start with vectors and work their way up to data frames. I think it's better to start with data frames because they're immediately useful, and then work your way down to the underlying components.

Vectors are particularly important as its to learn to write functions that work with vectors, rather than data frames. The technology that lets ggplot2, tidyr, dplyr etc work with data frames is considerably more complex and not currently standardised. While I'm currently working on a new standard that will make life much easier, it's unlikely to be ready in time for this book.

## Vector overview

There are two types of vectors:

1. __Atomic__ vectors, which are further broken down into six types:
  __logical__, __integer__, __double__,  __character__, __complex__, and 
  __raw__. Integer and double vectors are collectively known as
  __numeric__ vectors. 

1. __Lists__, are sometimes called recursive vectors, because lists can 
  contain other lists. This is the chief difference between atomic vectors
  and lists: atomic vectors are homogeneous, lists can be heterogeneous.

There's a somewhat related object: `NULL`. It's often used to represent the absence of a vector (as opposed to `NA` which is used to represent the absence of a value in a vector). `NULL` typically behaves like a vector of length 0.

The structure of the vector types is summarised in the following diagram:

```{r, echo = FALSE}
knitr::include_graphics("diagrams/data-structures-overview.png")
```

Every vector has two key properties: 

1.  Its __type__, which you can determine with `typeof()`.

    ```{r}
    typeof(letters)
    typeof(1:10)
    ```

1. Its __length__, which you can determine with `length()`.

    ```{r}
    x <- list("a", "b", 1:10)
    length(x)
    ```

Vectors can also contain arbitrary additional metadata in the form of attributes. These attributes are used to create __augmented vectors__ which build on additional behaviour. There are four important types of augmented vector:

* Factors and dates are built on top of integer vectors.
* Date times (POSIXct) are built on of double vectors.
* Data frames and tibbles are built on top of lists.

This chapter will introduce you to these important vectors from simplest to most complicated. You'll start with atomic vectors, then build up to lists, and finally learn about augmented vectors.

## Important types of atomic vector

The four most important types of atomic vector are logical, integer, double, and character. Raw and complex are rarely used during a data analysis, so I won't discuss them here.

Each type of atomic vector has its own missing value:

```{r}
NA            # logical
NA_integer_   # integer
NA_real_      # double
NA_character_ # character
```

Normally, you don't need to know about these different types because you can always use `NA` it will be converted to the correct type. However, there are some functions that are strict about their inputs, so it's useful to have this knowledge sitting in your back pocket so you can be specific when needed.

### Logical

Logical vectors are the simplest type of atomic vector because they can take only three possible values: `FALSE`, `TRUE`, and `NA`. Logical vectors are usually constructed with comparison operators, as described in [comparisons]. You can also create them by hand with `c()`:

```{r}
c(TRUE, TRUE, FALSE, NA)
```

### Numeric

Integer and double vectors are known collectively as numeric vectors. In R, numbers are doubles by default. To make an integer, place a `L` after the number:

```{r}
typeof(1)
typeof(1L)
1.5L
```

Most of the time the distinction between integers and doubles is not important. However, there are two important differences that you need to be aware of:

1. Doubles are approximations, 

1. Integers have one special value: `NA_integer_`, while doubles have four:
   `NA_real_`, `NaN`, `Inf` and `-Inf`

Doubles represent floating point numbers that can not always be precisely represented with a fixed amount of memory. This means that you should consider all doubles to be approximations. For example, what is square of the square root of two?

```{r}
x <- sqrt(2) ^ 2
x
```

It certainly looks like we get what we expect: 2. But things are not exactly as they seem:

```{r}
x == 2
x - 2
```

This behaviour is common when working with floating point numbers: most calculations include some approximation error. Instead of comparing floating point numbers using `==`, you should use `dplyr::near()` which allows for some numerical tolerance.

```{r, eval = packageVersion("dplyr") >= "0.4.3.9000"}
dplyr::near(x, 2)
```

Doubles also have three special values in addition to `NA`:

```{r}
c(NA, -1, 0, 1) / 0
```

Avoid using `==` to check for these other special values. Instead use the helper functions `is.finite()`, `is.infinite()`, and `is.nan()`:

|                  |  0  | Inf | NA  | NaN |
|------------------|-----|-----|-----|-----|
| `is.finite()`    |  x  |     |     |     |
| `is.infinite()`  |     |  x  |     |     |
| `is.na()`        |     |     |  x  |  x  |
| `is.nan()`       |     |     |     |  x  |

Note that `is.finite(x)` is not the same as `!is.infinite(x)`.

### Character

Character vectors are the most complex type of atomic vector, because each element of a character vector is a string, and a string can contain an arbitrary amount of data. Strings are such an important data type, they have their own chapter: [strings].

Here I wanted to mention one important feature of the underlying string implementation: R uses a global string pool. This means that each unique string is only stored in memory once, and every use of the string points to that representation. This reduces the amount of memory needed by duplicated strings. You can see this behaviour in practice with `pryr::object_size()`:

```{r}
x <- "This is a reasonably long string."
pryr::object_size(x)

 y <- rep(x, 1000)
pryr::object_size(y)
```

`y` doesn't take up 1,000x as much memory as `x`, because each element of `y` is just a pointer to that same string. A pointer is 8 bytes, so 1000 pointers to a 136 B string is 8 * 1000 + 136 = 8.13 kB.

### Exercises

1.  Read the source code for `dplyr::near()`. How does it work?

1.  A logical vector can take 3 possible values. How many possible
    values can an integer vector take?

1.  Brainstorm at least four functions that allow you to convert a double to an
    integer. How do they differ? Be precise.
    
1.  What functions from the readr package allow you to turn a string
    into a logical, integer, or double vector?

## Using atomic vectors

Now that you understand the different types of atomic vector, it's useful to review some of the important tools for working with them. These include:

1.  The implicit coercion rules which govern what happen when, for example,
    you use a logical vector in a numeric context.

1.  Tools to test if an function input is a specific type of vector.

1.  R's recycling rules which govern what happens when you attempt to work
    with vectors of different lengths.

1.  Naming the elements of a vector.

1.  Subsetting a vector to pull out elements of interest.

### Coercion

There are two ways to convert, or coerce, one type of vector to another:

1.  Explicit coercion happesn when you call a function like `as.logical()`,
    `as.integer()`, `as.double()`, or `as.character()`. Whenever you find
    yourself using explicit coercion, you should always check whether you can
    make the fix upstream, so that the vector never had the wrong type in 
    the first place. For example, you may need to tweak you readr 
    `col_types` specification.

1.  Implicit coercion happens when you use a vector in a specific context
    that expects a certain type of vector. For example, when you use a logical
    vector with a numeric summary function, or when you use a double vector
    where an integer vector is expected.
    
Because explicit coercion is used relatively rarely (and is largely easy to understand), it's more important to understand implicit coercion. 

The most important type of implicit coercion is using a logical vector in a numeric context. In this case `TRUE` is converted to `1` and `FALSE` converted to 0. That means the sum of a logical vector is the number of trues, and the mean of a logical vector is the proportion of trues:

```{r}
x <- sample(20, 100, replace = TRUE)
y <- x > 10
sum(y)  # how many are greater than 10?
mean(y) # what proportion are greater than 10?
```

You may see some code (typically older) that relies on the implicit coercion in the opposite direction, from integer to logical:

```{r, eval = FALSE}
if (length(x)) {
  # do something
}
```

In this case, 0 is converted to `FALSE` and everything else is converted to `TRUE`. I think this makes it harder to understand your code, and I don't recommend it.

It's also important to understand what happens when you try and create a vector containing multiple types with `c()`: the most complex type always wins.

```{r}
str(c(TRUE, 1L))
str(c(1L, 1.5))
str(c(1.5, "a"))
```

An atomic vector can not have a mix of different types because the type is a property of the complete vector, not of the individual elements. If you need to mix multiple types in the same vector, you should use a list, which you'll learn about shortly.

### Test functions

Sometimes you want to do different things based on the type of vector you get. One option is to use `typeof()`. Another is to use a test function which returns a `TRUE` or `FALSE` (broadly, functions that return a single logical value are often called __predicate__ functions). 

Base R provides many functions like `is.vector()` and `is.atomic()`, but they are often surprising. Instead, it's safer to use the `is_*` functions provided by purrr, which are summarised in the table below.

|                  | lgl | int | dbl | chr | list |
|------------------|-----|-----|-----|-----|------|
| `is_logical()`   |  x  |     |     |     |      |
| `is_integer()`   |     |  x  |     |     |      |
| `is_double()`    |     |     |  x  |     |      |
| `is_numeric()`   |     |  x  |  x  |     |      |
| `is_character()` |     |     |     |  x  |      |
| `is_atomic()`    |  x  |  x  |  x  |  x  |      |
| `is_list()`      |     |     |     |     |  x   |
| `is_vector()`    |  x  |  x  |  x  |  x  |  x   |

Each predicate also comes with a "scalar" version, which checks that the length is 1. This is useful if you want to check (for example) that the inputs to your function are as you expect.

### Scalars and recycling rules

As well as implicitly coercion the types of vectors to be compatible, R will also implicit coerce the length of vectors. This is called vector "recycling", because the shorter vector is repeated, or __recycled__, to be the same length as the longer vector. 

This is generally most useful when you are mixing vectors and "scalars". But note that R does not actually have scalars. In R, a single number is a vector of length 1. Because there are no scalars, most built-in functions are __vectorised__, meaning that they will operate on a vector of numbers. That's why, for example, this code works:

```{r}
sample(10) + 100
runif(10) > 0.5
```

In R, basic mathematical operations work with vectors, not scalars like in most programming languages. This means that you should never need to perform explicit iteration (either with a loop or a map function) performing simple mathematical computations.

It's intuitive what should happen if you add two vectors of the same length, or a vector and a "scalar", but what happens if you add two vectors of different lengths?

```{r}
1:10 + 1:2
```

Here, R will expand the shortest vector to the same length as the longest, so called __recycling__. This is silent except in the case where the length of the longer is not an integer multiple of the length of the longer:

```{r}
1:10 + 1:3
```

While vector recycling can be used to create very succinct, clever code, it can also silently conceal problems. For this reason, the vectorised functions in dplyr, purrr, etc will throw errors when you recycle anything other than a scalar.

```{r, error = TRUE}
data.frame(x = 1:4, y = 1:2)
dplyr::data_frame(x = 1:4, y = 1:2)
purrr::map2(1:4, 1:2, `+`)
```

### Naming vectors

All types of vectors can be named. You can either name them during creation with `c()`:

```{r}
c(x = 1, y = 2, z = 4)
```

Or after the fact with `purrr::set_names()`:

```{r}
1:3 %>% set_names(c("a", "b", "c"))
```

Named vectors are most useful for subsetting, described next.

### Subsetting

So far we've used `dplyr::filter()` to filter the rows in a data frame. `filter()`, however, does not work with vectors, so we need to learn a new tool: `[`. `[` is the subsetting function, and is called like `x[a]`. We're not going to cover 2d and higher data structures here, but the idea generalises in a straightforward way: `x[a, b]` for 2d, `x[a, b, c]` for 3d, and so on. 

There are four types of thing that you can subset a vector with:

1.  A numeric vector containing only integers. The integers must either be all 
    positive, all negative, or zero.
    
    Subsetting with positive integers keeps the elements at those positions:
    
    ```{r}
    x <- c("one", "two", "three", "four", "five")
    x[c(3, 2, 5)]
    ```
    
    By repeating a position, you can actually make an longer output than 
    input:
    
    ```{r}
    x[c(1, 1, 5, 5, 5, 2)]
    ```
    
    Negative values drop the elements at the specified positions:
    
    ```{r}
    x[c(-1, -3, -5)]
    ```
    
    It's an error to mix positive and negative values:
    
    ```{r, error = TRUE}
    x[c(1, -1)]
    ```

    The error message mentions subsetting with zero, which returns no values:
    
    ```{r}
    x[0]
    ```
    
    This is not generally useful, but can be helpful if you want to create 
    unusual data structures with which to test your functions.
  
1.  Subsetting with a logical vector keeps all values corresponding to a
    `TRUE` value. This is most often useful in conjunction with a function
    that creates a logical vector.
    
    ```{r, eval = FALSE}
    # All non-missing values of x
    x[!is.na(x)]
    
    # All even values of x
    x[x %% 2 == 0]
    ```

1.  If you have a named vector, you can subset it with a character vector. 
    
    ```{r}
    x <- c(abc = 1, def = 2, xyz = 5)
    x[c("xyz", "def")]
    ```
    
    Like with positive integers, you can also use a character vector to 
    duplicate individual entries.

1.  The simplest type of subsetting is nothing, `x[]`, which returns the 
    complete `x`. This is not useful for subsetting vectors, but it is useful
    when subsetting matrices (and other high dimensional structures) because
    it lets you select all the rows or all the columns, by leaving that
    index blank. For example, if `x` is 2d, `x[1, ]` selects the first row and 
    all the columns, and `x[, -1]` selects all rows and all columns except
    the first.
    
I'd recommend reading <http://adv-r.had.co.nz/Subsetting.html#applications> to learn more about how you can use subsetting to achieve various goals. If you are working with data frames, you can typically use a dplyr function to achieve these goals, but the techniques are useful to know about when you are writing your own functions.

There is an important variation of `[` called `[[`. `[[` only ever extracts a single element, and always drops names. It's a good idea to use it whenever you want to make it clear that you're extracting one thing, as in a for loop. The distinction between `[` and `[[` is most important for lists, as we'll see shortly.

### Exercises

1.  Carefully read the documentation of `is.vector()`. What does it actually
    test for? Why does `is.atomic()` not agree with the definition of 
    atomic vectors above?

1.  Create functions that take a vector as input and returns:
    
    1. The last value.  Should you use `[` or `[[`?

    1. The elements at even numbered positions.
    
    1. Every element except the last value.

1.  Why is `x[-which(x > 0)]` not the same as `x[x <= 0]`? 

1.  What happens when you subset with a positive integer that's bigger
    than the length of the vector? What happens when you subset with a 
    name that doesn't exist?

## Recursive vectors (lists)

Lists are a step up in complexity from atomic vectors, because lists can contain other lists. This makes them suitable for representing hierarchical or tree-like structures. You create a list with `list()`:

```{r}
x <- list(1, 2, 3)
str(x)

x_named <- list(a = 1, b = 2, c = 3)
str(x_named)
```

Unlike atomic vectors, `lists()` can contain a mix of objects:

```{r}
y <- list("a", 1L, 1.5, TRUE)
str(y)
```

Lists can even contain other lists!

```{r}
z <- list(list(1, 2), list(3, 4))
str(z)
```

`str()` is very helpful when looking at lists because it focusses on the **str**ucture, not the contents.

### Visualising lists

To explain more complicated list manipulation functions, it's helpful to have a visual representation of lists. For example, take these three lists:

```{r}
x1 <- list(c(1, 2), c(3, 4))
x2 <- list(list(1, 2), list(3, 4))
x3 <- list(1, list(2, list(3)))
```

I'll draw them as follows:

```{r, echo = FALSE, out.width = "75%"}
knitr::include_graphics("diagrams/lists-structure.png")
```

* Lists are rounded rectangles that contain their children.
  
* I draw each child a little darker than its parent to make it easier to see 
  the hierarchy.
  
* The orientation of the children (i.e. rows or columns) isn't important, 
  so I'll pick a row or column orientation to either save space or illustrate 
  an important property in the example.

### Subsetting

There are three ways to subset a list, which I'll illustrate with `a`:

```{r}
a <- list(a = 1:3, b = "a string", c = pi, d = list(-1, -5))
```

*   `[` extracts a sub-list. The result will always be a list.

    ```{r}
    str(a[1:2])
    str(a[4])
    ```
    
    Like with vectors, you can subset with a logical, integer, or character
    vector.
    
*   `[[` extracts a single component from a list. It removes a level of 
    hierarchy from the list.

    ```{r}
    str(y[[1]])
    str(y[[4]])
    ```

*   `$` is a shorthand for extracting named elements of a list. It works
    similarly to `[[` except that you don't need to use quotes.
    
    ```{r}
    a$a
    a[["b"]]
    ```

The distinction between `[` and `[[` is really important for lists, because `[[` drills down into the list while `[` returns a new, smaller list. Compare the code and output above with the visual representation below.

```{r, echo = FALSE, out.width = "75%"}
knitr::include_graphics("diagrams/lists-subsetting.png")
```

### Lists of condiments

It's easy to get confused between `[` and `[[`, but it's important to understand the difference. A few months ago I stayed at a hotel with a pretty interesting pepper shaker that I hope will help you remember these differences:

```{r, echo = FALSE, out.width = "25%"} 
knitr::include_graphics("images/pepper.jpg")
```

If this pepper shaker is your list `x`, then, `x[1]` is a pepper shaker containing a single pepper packet:

```{r, echo = FALSE, out.width = "25%"} 
knitr::include_graphics("images/pepper-1.jpg")
```

`x[2]` would look the same, but would contain the second packet. `x[1:2]` would be a pepper shaker containing two pepper packets. 

`x[[1]]` is:

```{r, echo = FALSE, out.width = "25%"} 
knitr::include_graphics("images/pepper-2.jpg")
```

If you wanted to get the content of the pepper package, you'd need `x[[1]][[1]]`:

```{r, echo = FALSE, out.width = "25%"} 
knitr::include_graphics("images/pepper-3.jpg")
```

### Exercises

1.  Draw the following lists as nested sets:

    1.  `list(a, b, list(c, d), list(e, f))`
    1   `list(list(list(list(list(list(a))))))`

1.  What happens if you subset a data frame as if you're subsetting a list?
    What are the key differences between a list and a data frame?

## Augmented vectors

Atomic vectors and lists are the building blocks for four other important vector types: factors, dates, date times, and data frames. I call these __augmented vectors__, because they are vectors with additional __attributes__. 

Attributes are a way of adding arbitrary additional metadata to a vector. You can think of attributes as named list of vectors that can be attached to any object. You can get and set individual attribute values with `attr()` or see them all at once with `attributes()`.

```{r}
x <- 1:10
attr(x, "greeting")
attr(x, "greeting") <- "Hi!"
attr(x, "farewell") <- "Bye!"
attributes(x)
```

There are three very important attributes that are used to implement fundamental parts of R:

* "names" are used to name the elements of a vector. 
* "dims" make a vector behave like a matrix or array.
* "class" is used to implemenet the S3 object oriented system.

### S3

Class is particularly important because it changes what __generic functions__ do with the object. Generic functions are key to object oriented programming in R, and are what make augmented vectors behave differently to the vector they are built on top of. A detailed discussion of the S3 object oriented system is beyond the scope of this book, but you can read more about it at <http://adv-r.had.co.nz/OO-essentials.html#s3>.

Here's what a typical generic function looks like:

```{r}
as.Date
```

The call to "UseMethod" means that this is a generic function, and it will call a specific __method__, a function, based on the class of the first argument. (All methods are functions; not all functions are methods). You can list all the methods for a generic with `methods()`:

```{r}
methods("as.Date")
```

And you can see the specific implementation of a method with `getS3method()`:

```{r}
getS3method("as.Date", "default")
getS3method("as.Date", "numeric")
```

The most important S3 generic is `print()`: it controls how the object is printed when you type its name on the console. Other important generics are the subsetting functions `[`, `[[`, and `$`. 

### Factors

Factors are designed to represent categorical data that can take a fixed set of possible values. Factors are built on top of integers, and have a levels attribute:

```{r}
x <- factor(c("ab", "cd", "ab"), levels = c("ab", "cd", "ef"))
typeof(x)
attributes(x)
```

Historically, factors were much easier to work with than characters so many functions in base R automatically convert characters to factors (controlled by the dread `stringsAsFactors` argument). To get more historical context, you might want to read [stringsAsFactors: An unauthorized biography](http://simplystatistics.org/2015/07/24/stringsasfactors-an-unauthorized-biography/) by Roger Peng or [stringsAsFactors = \<sigh\>](http://notstatschat.tumblr.com/post/124987394001/stringsasfactors-sigh) by Thomas Lumley.  The motivation for factors is modelling. If you're going to fit a model to categorical data, you need to know in advance all the possible values. There's no way to make a prediction for "green" if all you've ever seen is "red", "blue", and "yellow".

The packages in this book keep characters as is, but you will need to deal with them if you are working with base R or many other packages. When you encounter a factor, you should first check to see if you can avoid creating it in the first. Often there will be `stringsAsFactors` argument that you can set to `FALSE`. Otherwise, you can apply `as.character()` to the column to explicitly turn back into a character vector.

```{r}
x <- factor(letters[1:5])
is.factor(x)
as.factor(letters[1:5])
```

### Dates and date times

Dates in R are numeric vectors (sometimes integers, sometimes doubles) that represent the number of days since 1 January 1970.

```{r}
x <- as.Date("1971-01-01")
unclass(x)

typeof(x)
attributes(x)
```

Date times are numeric vectors (sometimes integers, sometimes doubles) that represent the number of seconds since 1 January 1970:

```{r}
x <- lubridate::ymd_hm("1970-01-01 01:00")
unclass(x)

typeof(x)
attributes(x)
```

The `tzone` is optional. It controls how the time is printed, not what absolute time it refers to.

```{r}
attr(x, "tzone") <- "US/Pacific"
x
attr(x, "tzone") <- "US/Eastern"
x
log(-1)
1
```

There is another type of datetimes called POSIXlt. These are built on top of named lists:

```{r}
y <- as.POSIXlt(x)
typeof(y)
attributes(y)
```

If you use the packages outlined in this book, you should never encounter a POSIXlt. They do crop up in base R, because they are used extract specific components of a date (like the year or month). However, lubridate provides helpers for you to do this instead. Otherwise POSIXct's are always easier to work with, so if you find you have a POSIXlt, you should always convert it to a POSIXct with `as.POSIXct()`.

### Data frames and tibbles

Data frames are augmented lists: they have class "data.frame", and `names` (column) and `row.names` attributes:

```{r}
df1 <- data.frame(x = 1:5, y = 5:1)
typeof(df1)
attributes(df1)
```

The difference between a data frame and a list is that all the elements of a data frame must be the same length. All functions that work with data frames enforce this constraint.

In this book, we use tibbles, rather than data frames. Tibbles are identical to data frames, except that they have two additional components in the class:

```{r}
df2 <- dplyr::data_frame(x = 1:5, y = 5:1)
typeof(df2)
attributes(df2)
```

These extra components give tibbles the helpful behaviours defined in [tibbles].
