# Logical vectors {#logicals}

```{r, results = "asis", echo = FALSE}
status("drafting")
```

## Introduction

In this chapter, you'll learn useful tools for working with logical vectors.
Logical vectors are the simplest type of vector because each element can only be one of three possible values: `TRUE`, `FALSE`, and `NA`.
You'll find logical vectors directly in data relatively rarely, but despite that they're extremely powerful because you'll frequently create them during data analysis.

We'll begin with the most common way of creating logical vectors: numeric comparisons.
Then we'll talk about using Boolean algebra to combine different logical vectors, and some useful summaries for logical vectors.
We'll finish off with some other tool for making conditional changes.
Along the way, you'll also learn a little more about working with missing values, `NA`.

### Prerequisites

Most of the functions you'll learn about this package are provided by base R; I'll label any new functions that don't come from base R with `dplyr::`.
You don't need the tidyverse to use base R functions, but we'll still load it so we can use `mutate()`, `filter()`, and friends.
use plenty of functions .
We'll also continue to draw inspiration from the nyclights13 dataset.

```{r setup, message = FALSE}
library(tidyverse)
library(nycflights13)
```

However, as we start to discuss more tools, there won't always be a perfect real example.
So we'll also start to use more abstract examples where we create some dummy data with `c()`.
This makes it easiesr to explain the general point at the cost to making it harder to see how it might apply to your data problems.
Just remember that any manipulate we do to a free-floating vector, you can do to a variable inside data frame with `mutate()` and friends.

```{r}
x <- c(1, 2, 3, 5, 7, 11, 13)
x * 2

# Equivalent to:
df <- tibble(x)
df |> 
  mutate(y = x *  2)
```

## Comparisons

A very common way to create a logical vector is via a numeric comparison with `<`, `<=`, `>`, `>=`, `!=`, and `==`.
You'll learn other ways to create them in later chapters dealing with strings and dates.
So far, we've mostly created logical variables implicitly within `filter()` --- they are computed, used, and then throw away.
For example, the following filter finds all day time departures that leave roughly on time:

```{r}
flights |> 
  filter(dep_time > 600 & dep_time < 2000 & abs(arr_delay) < 20)
```

But it's useful to know that this is a shortcut and you can explicitly create the underlying logical variables with `mutate()`:

```{r}
flights |> 
  mutate(
    daytime = dep_time > 600 & dep_time < 2000,
    approx_ontime = abs(arr_delay) < 20,
    .keep = "used"
  )
```

This is useful because it allows you to name components, which can made the code easier to read, and it allows you to double-check the intermediate steps.
This is a particularly useful technique when you're doing more complicated Boolean algebra, as you'll learn about in the next section.

So the initial filter could also be written as:

```{r, results = FALSE}
flights |> 
  mutate(
    daytime = dep_time > 600 & dep_time < 2000,
    approx_ontime = abs(arr_delay) < 20,
  ) |> 
  filter(daytime & approx_ontime)
```

### Floating point comparison

Beware when using `==` with numbers as results might surprise you!
It looks like this vector contains the numbers 1 and 2:

```{r}
x <- c(1 / 49 * 49, sqrt(2) ^ 2)
x
```

But if you test them for equality, you surprisingly get `FALSE`:

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

That's because computers use finite precision arithmetic (they obviously can't store an infinite number of digits!) so in most cases, the number number you see is an actually approximation.
R usually rounds these numbers to avoid displaying a bunch of usually unimportant digits.

To see the details you can call `print()` with the the `digits`[^logicals-1] argument.
R normally calls print automatically for you (i.e. `x` is a shortcut for `print(x)`), but calling it explicitly is useful if you want to provide other arguments:

[^logicals-1]: A floating point number can hold roughly 16 decimal digits; the precise number is surprisingly complicated and depends on the number.

```{r}
print(x, digits = 16)
```

Now that you've seen why `==` is failing, what can you do about it?
One option is to use `round()` to round to any number of digits, or instead of `==`, use `dplyr::near()`, which does the comparison with a small amount of tolerance:

```{r}
near(x, c(1, 2))
```

### Missing values {#na-comparison}

Missing values represent the unknown so they missing values are "contagious": almost any operation involving an unknown value will also be unknown:

```{r}
NA > 5
10 == NA
```

The most confusing result is this one:

```{r}
NA == NA
```

It's easiest to understand why this is true with a bit more context:

```{r}
# Let x be Mary's age. We don't know how old she is.
x <- NA

# Let y be John's age. We don't know how old he is.
y <- NA

# Are John and Mary the same age?
x == y
# We don't know!
```

So if you want to find all flights with `dep_time` is missing, the following code won't work because `dep_time == NA` will yield a `NA` for every single row, and `filter()` automatically drops missing values:

```{r}
flights |> 
  filter(dep_time == NA)
```

Instead we'll need a new tool: `is.na()`.

### `is.na()`

There's one other very useful way to create logical vectors: `is.na()`.
This takes any type of vector and returns `TRUE` is the value is `NA`, and `FALSE` otherwise:

```{r}
is.na(c(TRUE, NA, FALSE))
is.na(c(1, NA, 3))
is.na(c("a", NA, "b"))
```

We can use `is.na()` to find all the rows with a missing `dep_time`:

```{r}
flights |> 
  filter(is.na(dep_time))
```

It can also be useful in `arrange()`, because by default, `arrange()` puts all the missing values at the end.
You can override this default by first sorting by `is.na()`:

```{r}
flights |> 
  arrange(arr_delay)

flights |> 
  arrange(desc(is.na(arr_delay)), arr_delay)
```

### Exercises

1.  How does `dplyr::near()` work? Read the source code to find out.
2.  Use `mutate()`, `is.na()`, and `count()` together to describe how the missing values in `dep_time`, `sched_dep_time` and `dep_delay` are connected.

## Boolean algebra

Once you have multiple logical vectors, you can combine them together using Boolean algebra.
In R, `&` is "and", `|` is "or", and `!` is "not", and `xor()` is exclusive or[^logicals-2].
Figure \@ref(fig:bool-ops) shows the complete set of Boolean operations and how they work.

[^logicals-2]: That is, `xor(x, y)` is true if x is true, or y is true, but not both.
    This is how we usually use "or" In English.
    Both is not usually an acceptable answer to the question "would you like ice cream or cake?".

```{r bool-ops}
#| echo: false
#| out.width: NULL
#| fig.cap: > 
#|    Complete set of boolean operations. `x` is the left-hand
#|    circle, `y` is the right-hand circle, and the shaded region show 
#|    which parts each operator selects."
#| fig.alt: >
#|    Six Venn diagrams, each explaining a given logical operator. The
#|    circles (sets) in each of the Venn diagrams represent x and y. 1. y &
#|    !x is y but none of x, x & y is the intersection of x and y, x & !y is
#|    x but none of y, x is all of x none of y, xor(x, y) is everything
#|    except the intersection of x and y, y is all of y none of x, and 
#|    x | y is everything.
knitr::include_graphics("diagrams/transform.png", dpi = 270)
```

As well as `&` and `|`, R also has `&&` and `||`.
Don't use them in dplyr functions!
These are called short-circuiting operators and only ever return a single `TRUE` or `FALSE`.
They're important for programming so you'll learn more about them in Section \@ref(conditional-execution).

The following code finds all flights that departed in November or December:

```{r, eval = FALSE}
flights |> 
   filter(month == 11 | month == 12)
```

Note that the order of operations doesn't work like English.
You can't think "find all flights that departed in November or December" and write `flights |> filter(month == 11 | 12)`.
This code will not error, but it will do something rather confusing.
First R evaluates `11 | 12` which is equivalent to `TRUE | TRUE`, which returns `TRUE`.
Then it evaluates `month == TRUE`.
Since month is numeric, this is equivalent to `month == 1`, so `flights |> filter(month == 11 | 12)` returns all flights in January!

### `%in%`

An easy way to avoid this issue is to use `%in%`.
`x %in% y` returns a logical vector the same length as `x` that is `TRUE` whenever a value in `x` is anywhere in `y` .

```{r}
letters[1:10] %in% c("a", "e", "i", "o", "u")
```

So we could instead write:

```{r, eval = FALSE}
flights |> 
  filter(month %in% c(11, 12))
```

Note that `%in%` obeys different rules for `NA` to `==`.

```{r}
c(1, 2, NA) == NA
c(1, 2, NA) %in% NA
```

This can make for a useful shortcut:

```{r}
flights |> 
  filter(dep_time %in% c(NA, 0800))
```

### Missing values {#na-boolean}

The rules for missing values in Boolean algebra are a little tricky to explain because they seem inconsistent at first glance:

```{r}
df <- tibble(x = c(TRUE, FALSE, NA))

df |> 
  mutate(
    and = x & NA,
    or = x | NA
  )
```

To understand what's going on, think about `NA | TRUE`.
A missing value means that the value could either be `TRUE` or `FALSE`.
`TRUE | TRUE` and `FALSE | TRUE` are both `TRUE`, so `NA | TRUE` must also be `TRUE`.
Similar reasoning applies with `NA & FALSE`.

### Exercises

1.  Find all flights where `arr_delay` is missing but `dep_delay` is not. Find all flights where neither `arr_time` nor `sched_arr_time` are missing, but `arr_delay` is.
2.  How many flights have a missing `dep_time`? What other variables are missing in these rows? What might these rows represent?
3.  Look at the number of cancelled flights per day. Is there a pattern? Is the proportion of cancelled flights related to the average delay?

## Summaries {#logical-summaries}

While, you can summarize logical variables directly with functions that work only with logicals, there are two other important summaries.
Numeric summaries like `sum()` and `mean()`, and using summaries as inline filters.

### Logical summaries

There are two important logical summaries: `any()` and `all()`.
`any(x)` is the equivalent of `|`; it'll return `TRUE` if there are any `TRUE`'s in `x`.
`all(x)` is equivalent of `&`; it'll return `TRUE` only if all values of `x` are `TRUE`'s.
Like all summary functions, they'll return `NA` if there are any missing values present, and like usual you can make the missing values go away with `na.rm = TRUE`.

For example, we could use `all()` to find out if there were days where every flight was delayed:

```{r}
not_cancelled <- flights |> 
  filter(!is.na(dep_delay), !is.na(arr_delay))

not_cancelled |> 
  group_by(year, month, day) |> 
  summarise(
    all_delayed = all(arr_delay >= 0),
    any_delayed = any(arr_delay >= 0),
    .groups = "drop"
  )
```

In most cases, however, `any()` and `all()` are a little too crude, and it would be nice to be able to get a little more detail about how many values are `TRUE` or `FALSE`.
That leads us to the numeric summaries.

### Numeric summaries

When you use a logical vector in a numeric context, `TRUE` becomes 1 and `FALSE` becomes 0.
This makes `sum()` and `mean()` are particularly useful with logical vectors because `sum(x)` will give the number of `TRUE`s and `mean(x)` gives the proportion of `TRUE`s.
That lets us see the distribution of delays across the days of the year:

```{r}
not_cancelled |> 
  group_by(year, month, day) |> 
  summarise(
    prop_delayed = mean(arr_delay > 0),
    .groups = "drop"
  ) |> 
  ggplot(aes(prop_delayed)) + 
  geom_histogram(binwidth = 0.05)
```

Or we could ask how many flights left before 5am, which usually are flights that were delayed from the previous day:

```{r}
not_cancelled |> 
  group_by(year, month, day) |> 
  summarise(
    n_early = sum(dep_time < 500),
    .groups = "drop"
  ) |> 
  arrange(desc(n_early))
```

### Logical subsetting

There's one final use for logical vectors in summaries: you can use a logical vector to filter a single variable to a subset of interest.
This makes use of the base `[` (pronounced subset) operator, which you'll learn more about this in Section \@ref(vector-subsetting).

Imagine we wanted to look at the average delay just for flights that were actually delayed.
One way to do so would be to first filter the flights:

```{r}
not_cancelled |> 
  filter(arr_delay > 0) |> 
  group_by(year, month, day) |> 
  summarise(
    ahead = mean(arr_delay),
    n = n(),
    .groups = "drop"
  )
```

This works, but what if we wanted to also compute the average delay for flights that left early?
We'd need to perform a separate filter step, and then figure out how to combine the two data frames together (which we'll cover in Chapter \@ref(relational-data)).
Instead you could use `[` to perform an inline filtering: `arr_delay[arr_delay > 0]` will yield only the positive arrival delays.

This leads to:

```{r}
not_cancelled |> 
  group_by(year, month, day) |> 
  summarise(
    ahead = mean(arr_delay[arr_delay > 0]),
    behind = mean(arr_delay[arr_delay < 0]),
    n = n(),
    .groups = "drop"
  )
```

Also note the difference in the group size: in the first chunk `n` gives the number of delayed flights per day; in the second, `n` gives the total number of flights.

### Exercises

1.  What will `sum(is.na(x))` tell you? How about `mean(is.na(x))`?
2.  What does `prod()` return when applied to a logical vector? What logical summary function is it equivalent to? What does `min()` return applied to a logical vector? What logical summary function is it equivalent to? Read the documentation and perform a few experiments.

## Conditional transformations

One of the most powerful features of logical vectors are their use for conditional transformations, i.e. returning one value for true values, and a different value for false values.
We'll see a couple of different ways to do this, and the

### `if_else()`

If you want to use one value when a condition is true and another value when it's `FALSE`, you can use `if_else()`[^logicals-3].

[^logicals-3]: This is equivalent to the base R function `ifelse`.
    There are two main advantages of `if_else()`over `ifelse()`: you can choose what should happen to missing values, and `if_else()` is much more likely to give you a meaningful error message if you use the wrong type of variable.

```{r}
df <- tibble(
  date = as.Date("2020-01-01") + 0:6,
  balance = c(100, 50, 25, -25, -50, 30, 120)
)
df |> 
  mutate(
    status = if_else(balance < 0, "overdraft", "ok")
  )
```

If you need to create more complex conditions, you can string together multiple `if_elses()`s, but this quickly gets hard to read.

```{r}
df |> 
  mutate(
    status = if_else(balance == 0, "zero", 
      if_else(balance < 0, "overdraft", "ok"))
  )
```

Instead, you can switch to `case_when()` instead.

### `case_when()`

`case_when()` has a special syntax that unfortunately looks like nothing else you'll use in the tidyverse.
it takes pairs that look like `condition ~ output`.
`condition` must make a logical a logical vector; when it's `TRUE`, `output` will be used.

```{r}
flights |> 
  mutate(
    status = case_when(
      is.na(arr_delay)      ~ "cancelled",
      arr_delay > 60        ~ "very late",
      arr_delay > 15        ~ "late",
      abs(arr_delay) <= 15  ~ "on time",
      arr_delay < -15       ~ "early",
      arr_delay < -30       ~ "very early",
    ),
    .keep = "used"
  )
```

(Note that I usually add spaces to make the outputs line up so it's easier to scan)

To explain how `case_when()` works, lets pull it out of the mutate and create some simple dummy data.

```{r}
x <- 1:10
case_when(
  x <  5 ~ "small",
  x >= 5 ~ "big"
)
```

-   If none of the cases match, the output will be missing:

    ```{r}
    case_when(
      x %% 2 == 0 ~ "even",
    )
    ```

-   You can create a catch all value by using `TRUE` as the condition:

    ```{r}
    case_when(
      x %% 2 == 0 ~ "even",
      TRUE        ~ "odd"
    )
    ```

-   If multiple conditions are `TRUE`, the first is used:

    ```{r}
    case_when(
      x < 5 ~ "< 5",
      x < 3 ~ "< 3",
      TRUE ~ "big"
    )
    ```

The simple examples I've shown you here all use just a single variable, but the logical conditions can use any number of variables.
And you can use variables on the right hand side.

## Cumulative tricks

Before we move on to the next chapter, I want to show you a grab bag of tricks that make use of cumulative functions (i.e. functions that depending on every previous value of a vector).
These all feel a bit magical, and I'm torn on whether or not they should be included in this book.
But in the end, some of them are just so useful I think it's important to mention them --- they're not particularly easy to understand and don't help with that many problems, but when they do, they provide a substantial advantage.

<!-- TODO: illustration of accumulating function -->

Another useful pair of functions are cumulative any, `dplyr::cumany()`, and cumulative all, `dplyr::cumall()`.
`cumany()` will be `TRUE` after it encounters the first `TRUE`, and `cumall()` will be `FALSE` after it encounters its first `FALSE`.

```{r}
cumany(c(FALSE, FALSE, TRUE, TRUE, FALSE, TRUE))
cumall(c(TRUE, FALSE, TRUE, TRUE, FALSE, TRUE))
```

These are particularly useful in conjunction with `filter()` because they allow you to select rows:

-   Before the first `FALSE` with `cumall(x)`.
-   Before the first `TRUE` with `cumall(!x)`.
-   After the first `TRUE` with `cumany(x)`.
-   After the first `FALSE` with `cumany(!x)`.

If you imagine some data about a bank balance, then these functions allow you t

```{r}
df <- tibble(
  date = as.Date("2020-01-01") + 0:6,
  balance = c(100, 50, 25, -25, -50, 30, 120)
)
# all rows after first overdraft
df |> filter(cumany(balance < 0))
# all rows until first overdraft
df |> filter(cumall(!(balance < 0)))
```

`cumsum()` as way of defining groups:

```{r}
df |> 
  mutate(
    negative = balance < 0,
    flip = negative != lag(negative),
    group = cumsum(coalesce(flip, FALSE))
  )
```

### Exercises

1.  For each plane, count the number of flights before the first delay of greater than 1 hour.
