joins.qmd

# Joins {#sec-joins}

```{r}
#| echo: false

source("_common.R")
```

## Introduction

It's rare that a data analysis involves only a single data frame.
Typically you have many data frames, and you must **join** them together to answer the questions that you're interested in.
This chapter will introduce you to two important types of joins:

-   Mutating joins, which add new variables to one data frame from matching observations in another.
-   Filtering joins, which filter observations from one data frame based on whether or not they match an observation in another.

We'll begin by discussing keys, the variables used to connect a pair of data frames in a join.
We cement the theory with an examination of the keys in the datasets from the nycflights13 package, then use that knowledge to start joining data frames together.
Next we'll discuss how joins work, focusing on their action on the rows.
We'll finish up with a discussion of non-equi joins, a family of joins that provide a more flexible way of matching keys than the default equality relationship.

### Prerequisites

In this chapter, we'll explore the five related datasets from nycflights13 using the join functions from dplyr.

```{r}
#| label: setup
#| message: false

library(tidyverse)
library(nycflights13)
```

## Keys

To understand joins, you need to first understand how two tables can be connected through a pair of keys, within each table.
In this section, you'll learn about the two types of key and see examples of both in the datasets of the nycflights13 package.
You'll also learn how to check that your keys are valid, and what to do if your table lacks a key.

### Primary and foreign keys

Every join involves a pair of keys: a primary key and a foreign key.
A **primary key** is a variable or set of variables that uniquely identifies each observation.
When more than one variable is needed, the key is called a **compound key.** For example, in nycflights13:

-   `airlines` records two pieces of data about each airline: its carrier code and its full name.
    You can identify an airline with its two letter carrier code, making `carrier` the primary key.

    ```{r}
    airlines
    ```

-   `airports` records data about each airport.
    You can identify each airport by its three letter airport code, making `faa` the primary key.

    ```{r}
    #| R.options:
    #|   width: 67
    airports
    ```

-   `planes` records data about each plane.
    You can identify a plane by its tail number, making `tailnum` the primary key.

    ```{r}
    #| R.options:
    #|   width: 67
    planes
    ```

-   `weather` records data about the weather at the origin airports.
    You can identify each observation by the combination of location and time, making `origin` and `time_hour` the compound primary key.

    ```{r}
    #| R.options:
    #|   width: 67
    weather
    ```

A **foreign key** is a variable (or set of variables) that corresponds to a primary key in another table.
For example:

-   `flights$tailnum` is a foreign key that corresponds to the primary key `planes$tailnum`.
-   `flights$carrier` is a foreign key that corresponds to the primary key `airlines$carrier`.
-   `flights$origin` is a foreign key that corresponds to the primary key `airports$faa`.
-   `flights$dest` is a foreign key that corresponds to the primary key `airports$faa`.
-   `flights$origin`-`flights$time_hour` is a compound foreign key that corresponds to the compound primary key `weather$origin`-`weather$time_hour`.

These relationships are summarized visually in @fig-flights-relationships.

```{r}
#| label: fig-flights-relationships
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   Connections between all five data frames in the nycflights13 package.
#|   Variables making up a primary key are colored grey, and are connected
#|   to their corresponding foreign keys with arrows.
#| fig-alt: |
#|   The relationships between airports, planes, flights, weather, and
#|   airlines datasets from the nycflights13 package. airports$faa
#|   connected to the flights$origin and flights$dest. planes$tailnum
#|   is connected to the flights$tailnum. weather$time_hour and
#|   weather$origin are jointly connected to flights$time_hour and 
#|   flights$origin. airlines$carrier is connected to flights$carrier.
#|   There are no direct connections between airports, planes, airlines, 
#|   and weather data frames.
knitr::include_graphics("diagrams/relational.png", dpi = 270)
```

You'll notice a nice feature in the design of these keys: the primary and foreign keys almost always have the same names, which, as you'll see shortly, will make your joining life much easier.
It's also worth noting the opposite relationship: almost every variable name used in multiple tables has the same meaning in each place.
There's only one exception: `year` means year of departure in `flights` and year of manufacturer in `planes`.
This will become important when we start actually joining tables together.

### Checking primary keys

Now that that we've identified the primary keys in each table, it's good practice to verify that they do indeed uniquely identify each observation.
One way to do that is to `count()` the primary keys and look for entries where `n` is greater than one.
This reveals that `planes` and `weather` both look good:

```{r}
planes |> 
  count(tailnum) |> 
  filter(n > 1)

weather |> 
  count(time_hour, origin) |> 
  filter(n > 1)
```

You should also check for missing values in your primary keys --- if a value is missing then it can't identify an observation!

```{r}
planes |> 
  filter(is.na(tailnum))

weather |> 
  filter(is.na(time_hour) | is.na(origin))
```

### Surrogate keys

So far we haven't talked about the primary key for `flights`.
It's not super important here, because there are no data frames that use it as a foreign key, but it's still useful to consider because it's easier to work with observations if we have some way to describe them to others.

After a little thinking and experimentation, we determined that there are three variables that together uniquely identify each flight:

```{r}
flights |> 
  count(time_hour, carrier, flight) |> 
  filter(n > 1)
```

Does the absence of duplicates automatically make `time_hour`-`carrier`-`flight` a primary key?
It's certainly a good start, but it doesn't guarantee it.
For example, are altitude and latitude a good primary key for `airports`?

```{r}
airports |>
  count(alt, lat) |> 
  filter(n > 1)
```

Identifying an airport by its altitude and latitude is clearly a bad idea, and in general it's not possible to know from the data alone whether or not a combination of variables makes a good a primary key.
But for flights, the combination of `time_hour`, `carrier`, and `flight` seems reasonable because it would be really confusing for an airline and its customers if there were multiple flights with the same flight number in the air at the same time.

That said, we might be better off introducing a simple numeric surrogate key using the row number:

```{r}
flights2 <- flights |> 
  mutate(id = row_number(), .before = 1)
flights2
```

Surrogate keys can be particularly useful when communicating to other humans: it's much easier to tell someone to take a look at flight 2001 than to say look at UA430 which departed 9am 2013-01-03.

### Exercises

1.  We forgot to draw the relationship between `weather` and `airports` in @fig-flights-relationships.
    What is the relationship and how should it appear in the diagram?

2.  `weather` only contains information for the three origin airports in NYC.
    If it contained weather records for all airports in the USA, what additional connection would it make to `flights`?

3.  The `year`, `month`, `day`, `hour`, and `origin` variables almost form a compound key for `weather`, but there's one hour that has duplicate observations.
    Can you figure out what's special about that hour?

4.  We know that some days of the year are special and fewer people than usual fly on them (e.g., Christmas eve and Christmas day).
    How might you represent that data as a data frame?
    What would be the primary key?
    How would it connect to the existing data frames?

5.  Draw a diagram illustrating the connections between the `Batting`, `People`, and `Salaries` data frames in the Lahman package.
    Draw another diagram that shows the relationship between `People`, `Managers`, `AwardsManagers`.
    How would you characterize the relationship between the `Batting`, `Pitching`, and `Fielding` data frames?

## Basic joins {#sec-mutating-joins}

Now that you understand how data frames are connected via keys, we can start using joins to better understand the `flights` dataset.
dplyr provides six join functions: `left_join()`, `inner_join()`, `right_join()`, `full_join()`, `semi_join()`, and `anti_join().` They all have the same interface: they take a pair of data frames (`x` and `y`) and return a data frame.
The order of the rows and columns in the output is primarily determined by `x`.

In this section, you'll learn how to use one mutating join, `left_join()`, and two filtering joins, `semi_join()` and `anti_join()`.
In the next section, you'll learn exactly how these functions work, and about the remaining `inner_join()`, `right_join()` and `full_join()`.

### Mutating joins

A **mutating join** allows you to combine variables from two data frames: it first matches observations by their keys, then copies across variables from one data frame to the other.
Like `mutate()`, the join functions add variables to the right, so if your dataset has many variables, you won't see the new ones.
For these examples, we'll make it easier to see what's going on by creating a narrower dataset with just six variables[^joins-1]:

[^joins-1]: Remember that in RStudio you can also use `View()` to avoid this problem.

```{r}
flights2 <- flights |> 
  select(year, time_hour, origin, dest, tailnum, carrier)
flights2
```

There are four types of mutating join, but there's one that you'll use almost all of the time: `left_join()`.
It's special because the output will always have the same rows as `x`, the data frame you're joining to[^joins-2].
The primary use of `left_join()` is to add in additional metadata.
For example, we can use `left_join()` to add the full airline name to the `flights2` data:

[^joins-2]: That's not 100% true, but you'll get a warning whenever it isn't.

```{r}
flights2 |>
  left_join(airlines)
```

Or we could find out the temperature and wind speed when each plane departed:

```{r}
flights2 |> 
  left_join(weather |> select(origin, time_hour, temp, wind_speed))
```

Or what size of plane was flying:

```{r}
flights2 |> 
  left_join(planes |> select(tailnum, type, engines, seats))
```

When `left_join()` fails to find a match for a row in `x`, it fills in the new variables with missing values.
For example, there's no information about the plane with tail number `N3ALAA` so the `type`, `engines`, and `seats` will be missing:

```{r}
flights2 |> 
  filter(tailnum == "N3ALAA") |> 
  left_join(planes |> select(tailnum, type, engines, seats))
```

We'll come back to this problem a few times in the rest of the chapter.

### Specifying join keys

By default, `left_join()` will use all variables that appear in both data frames as the join key, the so called **natural** join.
This is a useful heuristic, but it doesn't always work.
For example, what happens if we try to join `flights2` with the complete `planes` dataset?

```{r}
flights2 |> 
  left_join(planes)
```

We get a lot of missing matches because our join is trying to use `tailnum` and `year` as a compound key.
Both `flights` and `planes` have a `year` column but they mean different things: `flights$year` is the year the flight occurred and `planes$year` is the year the plane was built.
We only want to join on `tailnum` so we need to provide an explicit specification with `join_by()`:

```{r}
flights2 |> 
  left_join(planes, join_by(tailnum))
```

Note that the `year` variables are disambiguated in the output with a suffix (`year.x` and `year.y`), which tells you whether the variable came from the `x` or `y` argument.
You can override the default suffixes with the `suffix` argument.

`join_by(tailnum)` is short for `join_by(tailnum == tailnum)`.
It's important to know about this fuller form for two reasons.
Firstly, it describes the relationship between the two tables: the keys must be equal.
That's why this type of join is often called an **equi join**.
You'll learn about non-equi joins in @sec-non-equi-joins.

Secondly, it's how you specify different join keys in each table.
For example, there are two ways to join the `flight2` and `airports` table: either by `dest` or `origin`:

```{r}
flights2 |> 
  left_join(airports, join_by(dest == faa))

flights2 |> 
  left_join(airports, join_by(origin == faa))
```

In older code you might see a different way of specifying the join keys, using a character vector:

-   `by = "x"` corresponds to `join_by(x)`.
-   `by = c("a" = "x")` corresponds to `join_by(a == x)`.

Now that it exists, we prefer `join_by()` since it provides a clearer and more flexible specification.

`inner_join()`, `right_join()`, `full_join()` have the same interface as `left_join()`.
The difference is which rows they keep: left join keeps all the rows in `x`, the right join keeps all rows in `y`, the full join keeps all rows in either `x` or `y`, and the inner join only keeps rows that occur in both `x` and `y`.
We'll come back to these in more detail later.

### Filtering joins

As you might guess the primary action of a **filtering join** is to filter the rows.
There are two types: semi-joins and anti-joins.
**Semi-joins** keep all rows in `x` that have a match in `y`.
For example, we could use a semi-join to filter the `airports` dataset to show just the origin airports:

```{r}
airports |> 
  semi_join(flights2, join_by(faa == origin))
```

Or just the destinations:

```{r}
airports |> 
  semi_join(flights2, join_by(faa == dest))
```

**Anti-joins** are the opposite: they return all rows in `x` that don't have a match in `y`.
They're useful for finding missing values that are **implicit** in the data, the topic of @sec-missing-implicit.
Implicitly missing values don't show up as `NA`s but instead only exist as an absence.
For example, we can find rows that are missing from `airports` by looking for flights that don't have a matching destination airport:

```{r}
flights2 |> 
  anti_join(airports, join_by(dest == faa)) |> 
  distinct(dest)
```

Or we can find which `tailnum`s are missing from `planes`:

```{r}
flights2 |>
  anti_join(planes, join_by(tailnum)) |> 
  distinct(tailnum)
```

### Exercises

1.  Find the 48 hours (over the course of the whole year) that have the worst delays.
    Cross-reference it with the `weather` data.
    Can you see any patterns?

2.  Imagine you've found the top 10 most popular destinations using this code:

    ```{r}
    top_dest <- flights2 |>
      count(dest, sort = TRUE) |>
      head(10)
    ```

    How can you find all flights to those destinations?

3.  Does every departing flight have corresponding weather data for that hour?

4.  What do the tail numbers that don't have a matching record in `planes` have in common?
    (Hint: one variable explains \~90% of the problems.)

5.  Add a column to `planes` that lists every `carrier` that has flown that plane.
    You might expect that there's an implicit relationship between plane and airline, because each plane is flown by a single airline.
    Confirm or reject this hypothesis using the tools you've learned in previous chapters.

6.  Add the latitude and the longitude of the origin *and* destination airport to `flights`.
    Is it easier to rename the columns before or after the join?

7.  Compute the average delay by destination, then join on the `airports` data frame so you can show the spatial distribution of delays.
    Here's an easy way to draw a map of the United States:

    ```{r}
    #| eval: false

    airports |>
      semi_join(flights, join_by(faa == dest)) |>
      ggplot(aes(x = lon, y = lat)) +
        borders("state") +
        geom_point() +
        coord_quickmap()
    ```

    You might want to use the `size` or `color` of the points to display the average delay for each airport.

8.  What happened on June 13 2013?
    Draw a map of the delays, and then use Google to cross-reference with the weather.

    ```{r}
    #| eval: false
    #| include: false

    worst <- filter(flights, !is.na(dep_time), month == 6, day == 13)
    worst |>
      group_by(dest) |>
      summarize(delay = mean(arr_delay), n = n()) |>
      filter(n > 5) |>
      inner_join(airports, join_by(dest == faa)) |>
      ggplot(aes(x = lon, y = lat)) +
        borders("state") +
        geom_point(aes(size = n, color = delay)) +
        coord_quickmap()
    ```

## How do joins work?

Now that you've used joins a few times it's time to learn more about how they work, focusing on how each row in `x` matches rows in `y`.
We'll begin by introducing a visual representation of joins, using the simple tibbles defined below and shown in @fig-join-setup.
In these examples we'll use a single key called `key` and a single value column (`val_x` and `val_y`), but the ideas all generalize to multiple keys and multiple values.

```{r}
x <- tribble(
  ~key, ~val_x,
     1, "x1",
     2, "x2",
     3, "x3"
)
y <- tribble(
  ~key, ~val_y,
     1, "y1",
     2, "y2",
     4, "y3"
)
```

```{r}
#| label: fig-join-setup
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   Graphical representation of two simple tables. The colored `key`
#|   columns map background color to key value. The grey columns represent
#|   the "value" columns that are carried along for the ride. 
#| fig-alt: |
#|   x and y are two data frames with 2 columns and 3 rows, with contents
#|   as described in the text. The values of the keys are colored:
#|   1 is green, 2 is purple, 3 is orange, and 4 is yellow.

knitr::include_graphics("diagrams/join/setup.png", dpi = 270)
```

@fig-join-setup2 introduces the foundation for our visual representation.
It shows all potential matches between `x` and `y` as the intersection between lines drawn from each row of `x` and each row of `y`.
The rows and columns in the output are primarily determined by `x`, so the `x` table is horizontal and lines up with the output.

```{r}
#| label: fig-join-setup2
#| echo: false
#| out-width: ~
#| fig-cap: | 
#|   To understand how joins work, it's useful to think of every possible
#|   match. Here we show that with a grid of connecting lines.
#| fig-alt: |
#|   x and y are placed at right-angles, with horizonal lines extending 
#|   from x and vertical lines extending from y. There are 3 rows in x and 
#|   3 rows in y, which leads to nine intersections representing nine
#|   potential matches.

knitr::include_graphics("diagrams/join/setup2.png", dpi = 270)
```

To describe a specific type of join, we indicate matches with dots.
The matches determine the rows in the output, a new data frame that contains the key, the x values, and the y values.
For example, @fig-join-inner shows an inner join, where rows are retained if and only if the keys are equal.

```{r}
#| label: fig-join-inner
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   An inner join matches each row in `x` to the row in `y` that has the
#|   same value of `key`. Each match becomes a row in the output.
#| fig-alt: |
#|   x and y are placed at right-angles with lines forming a grid of
#|   potential matches. Keys 1 and 2 appear in both x and y, so we
#|   get a match, indicated by a dot. Each dot corresponds to a row
#|   in the output, so the resulting joined data frame has two rows.

knitr::include_graphics("diagrams/join/inner.png", dpi = 270)
```

We can apply the same principles to explain the **outer joins**, which keep observations that appear in at least one of the data frames.
These joins work by adding an additional "virtual" observation to each data frame.
This observation has a key that matches if no other key matches, and values filled with `NA`.
There are three types of outer joins:

-   A **left join** keeps all observations in `x`, @fig-join-left.
    Every row of `x` is preserved in the output because it can fall back to matching a row of `NA`s in `y`.

    ```{r}
    #| label: fig-join-left
    #| echo: false
    #| out-width: ~
    #| fig-cap: | 
    #|   A visual representation of the left join where every row in `x`
    #|   appears in the output.
    #| fig-alt: |
    #|   Compared to the previous diagram showing an inner join, the y table
    #|   gets a new virtual row containin NA that will match any row in x
    #|   that didn't otherwise match. This means that the output now has
    #|   three rows. For key = 3, which matches this virtual row, val_y takes
    #|   value NA.

    knitr::include_graphics("diagrams/join/left.png", dpi = 270)
    ```

-   A **right join** keeps all observations in `y`, @fig-join-right.
    Every row of `y` is preserved in the output because it can fall back to matching a row of `NA`s in `x`.
    The output still matches `x` as much as possible; any extra rows from `y` are added to the end.

    ```{r}
    #| label: fig-join-right
    #| echo: false
    #| out-width: ~
    #| fig-cap: | 
    #|   A visual representation of the right join where every row of `y` 
    #|   appears in the output. 
    #| fig-alt: |
    #|   Compared to the previous diagram showing an left join, the x table
    #|   now gains a virtual row so that every row in y gets a match in x.
    #|   val_x contains NA for the row in y that didn't match x.

    knitr::include_graphics("diagrams/join/right.png", dpi = 270)
    ```

-   A **full join** keeps all observations that appear in `x` or `y`, @fig-join-full.
    Every row of `x` and `y` is included in the output because both `x` and `y` have a fall back row of `NA`s.
    Again, the output starts with all rows from `x`, followed by the remaining unmatched `y` rows.

    ```{r}
    #| label: fig-join-full
    #| echo: false
    #| out-width: ~
    #| fig-cap: | 
    #|   A visual representation of the full join where every row in `x`
    #|   and `y` appears in the output.
    #| fig-alt: |
    #|   Now both x and y have a virtual row that always matches.
    #|   The result has 4 rows: keys 1, 2, 3, and 4 with all values 
    #|   from val_x and val_y, however key 2, val_y and key 4, val_x are NAs
    #|   since those keys don't have a match in the other data frames.

    knitr::include_graphics("diagrams/join/full.png", dpi = 270)
    ```

Another way to show how the types of outer join differ is with a Venn diagram, as in @fig-join-venn.
However, this is not a great representation because while it might jog your memory about which rows are preserved, it fails to illustrate what's happening with the columns.

```{r}
#| label: fig-join-venn
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   Venn diagrams showing the difference between inner, left, right, and
#|   full joins.
#| fig-alt: |
#|   Venn diagrams for inner, full, left, and right joins. Each join
#|   represented with two intersecting circles representing data frames x
#|   and y, with x on the right and y on the left. Shading indicates the
#|   result of the join. 
#|
#|   Inner join: the intersection is shaded. 
#|   Full join: Everything is shaded. 
#|   Left join: All of x is shaded.
#|   Right join: All of y is shaded.

knitr::include_graphics("diagrams/join/venn.png", dpi = 270)
```

The joins shown here are the so-called **equi** **joins**, where rows match if the keys are equal.
Equi joins are the most common type of join, so we'll typically omit the equi prefix, and just say "inner join" rather than "equi inner join".
We'll come back to non-equi joins in @sec-non-equi-joins.

### Row matching

So far we've explored what happens if a row in `x` matches zero or one row in `y`.
What happens if it matches more than one row?
To understand what's going let's first narrow our focus to the `inner_join()` and then draw a picture, @fig-join-match-types.

```{r}
#| label: fig-join-match-types
#| echo: false
#| out-width: ~
#| fig-cap: | 
#|   The three ways a row in `x` can match. `x1` matches
#|   one row in `y`, `x2` matches two rows in `y`, `x3` matches
#|   zero rows in y. Note that while there are three rows in
#|   `x` and three rows in the output, there isn't a direct
#|   correspondence between the rows.
#| fig-alt: |
#|   A join diagram where x has key values 1, 2, and 3, and y has
#|   key values 1, 2, 2. The output has three rows because key 1 matches
#|   one row, key 2 matches two rows, and key 3 matches zero rows.

knitr::include_graphics("diagrams/join/match-types.png", dpi = 270)
```

There are three possible outcomes for a row in `x`:

-   If it doesn't match anything, it's dropped.
-   If it matches 1 row in `y`, it's preserved.
-   If it matches more than 1 row in `y`, it's duplicated once for each match.

In principle, this means that there's no guaranteed correspondence between the rows in the output and the rows in `x`, but in practice, this rarely causes problems.
There is, however, one particularly dangerous case which can cause a combinatorial explosion of rows.
Imagine joining the following two tables:

```{r}
df1 <- tibble(key = c(1, 2, 2), val_x = c("x1", "x2", "x3"))
df2 <- tibble(key = c(1, 2, 2), val_y = c("y1", "y2", "y3"))
```

While the first row in `df1` only matches one row in `df2`, the second and third rows both match two rows.
This is sometimes called a `many-to-many` join, and will cause dplyr to emit a warning:

```{r}
df1 |> 
  inner_join(df2, join_by(key))
```

If you are doing this deliberately, you can set `relationship = "many-to-many"`, as the warning suggests.

### Filtering joins

The number of matches also determines the behavior of the filtering joins.
The semi-join keeps rows in `x` that have one or more matches in `y`, as in @fig-join-semi.
The anti-join keeps rows in `x` that match zero rows in `y`, as in @fig-join-anti.
In both cases, only the existence of a match is important; it doesn't matter how many times it matches.
This means that filtering joins never duplicate rows like mutating joins do.

```{r}
#| label: fig-join-semi
#| echo: false
#| out-width: null
#| fig-cap: |
#|   In a semi-join it only matters that there is a match; otherwise
#|   values in `y` don't affect the output.
#| fig-alt: |
#|   A join diagram with old friends x and y. In a semi join, only the 
#|   presence of a match matters so the output contains the same columns
#|   as x.

knitr::include_graphics("diagrams/join/semi.png", dpi = 270)
```

```{r}
#| label: fig-join-anti
#| echo: false
#| out-width: null
#| fig-cap: |
#|   An anti-join is the inverse of a semi-join, dropping rows from `x`
#|   that have a match in `y`.
#| fig-alt: |
#|   An anti-join is the inverse of a semi-join so matches are drawn with
#|   red lines indicating that they will be dropped from the output.

knitr::include_graphics("diagrams/join/anti.png", dpi = 270)
```

## Non-equi joins {#sec-non-equi-joins}

So far you've only seen equi joins, joins where the rows match if the `x` key equals the `y` key.
Now we're going to relax that restriction and discuss other ways of determining if a pair of rows match.

But before we can do that, we need to revisit a simplification we made above.
In equi joins the `x` keys and `y` are always equal, so we only need to show one in the output.
We can request that dplyr keep both keys with `keep = TRUE`, leading to the code below and the re-drawn `inner_join()` in @fig-inner-both.

```{r}
x |> inner_join(y, join_by(key == key), keep = TRUE)
```

```{r}
#| label: fig-inner-both
#| fig-cap: |
#|   An inner join showing both `x` and `y` keys in the output.
#| fig-alt: |
#|   A join diagram showing an inner join betwen x and y. The result
#|   now includes four columns: key.x, val_x, key.y, and val_y. The
#|   values of key.x and key.y are identical, which is why we usually
#|   only show one.
#| echo: false
#| out-width: ~

knitr::include_graphics("diagrams/join/inner-both.png", dpi = 270)
```

When we move away from equi joins we'll always show the keys, because the key values will often be different.
For example, instead of matching only when the `x$key` and `y$key` are equal, we could match whenever the `x$key` is greater than or equal to the `y$key`, leading to @fig-join-gte.
dplyr's join functions understand this distinction equi and non-equi joins so will always show both keys when you perform a non-equi join.

```{r}
#| label: fig-join-gte
#| echo: false
#| fig-cap: |
#|   A non-equi join where the `x` key must be greater than or equal to 
#|   the `y` key. Many rows generate multiple matches.
#| fig-alt: |
#|   A join diagram illustrating join_by(key >= key). The first row
#|   of x matches one row of y and the second and thirds rows each match
#|   two rows. This means the output has five rows containing each of the 
#|   following (key.x, key.y) pairs: (1, 1), (2, 1), (2, 2), (3, 1),
#|   (3, 2).
knitr::include_graphics("diagrams/join/gte.png", dpi = 270)
```

Non-equi join isn't a particularly useful term because it only tells you what the join is not, not what it is. dplyr helps by identifying four particularly useful types of non-equi join:

-   **Cross joins** match every pair of rows.
-   **Inequality joins** use `<`, `<=`, `>`, and `>=` instead of `==`.
-   **Rolling joins** are similar to inequality joins but only find the closest match.
-   **Overlap joins** are a special type of inequality join designed to work with ranges.

Each of these is described in more detail in the following sections.

### Cross joins

A cross join matches everything, as in @fig-join-cross, generating the Cartesian product of rows.
This means the output will have `nrow(x) * nrow(y)` rows.

```{r}
#| label: fig-join-cross
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   A cross join matches each row in `x` with every row in `y`.
#| fig-alt: |
#|   A join diagram showing a dot for every combination of x and y.
knitr::include_graphics("diagrams/join/cross.png", dpi = 270)
```

Cross joins are useful when generating permutations.
For example, the code below generates every possible pair of names.
Since we're joining `df` to itself, this is sometimes called a **self-join**.
Cross joins use a different join function because there's no distinction between inner/left/right/full when you're matching every row.

```{r}
df <- tibble(name = c("John", "Simon", "Tracy", "Max"))
df |> cross_join(df)
```

### Inequality joins

Inequality joins use `<`, `<=`, `>=`, or `>` to restrict the set of possible matches, as in @fig-join-gte and @fig-join-lt.

```{r}
#| label: fig-join-lt
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   An inequality join where `x` is joined to `y` on rows where the key 
#|   of `x` is less than the key of `y`. This makes a triangular
#|   shape in the top-left corner.
#| fig-alt: |
#|   A diagram depicting an inequality join where a data frame x is joined by 
#|   a data frame y where the key of x is less than the key of y, resulting 
#|   in a triangular shape in the top-left corner.

knitr::include_graphics("diagrams/join/lt.png", dpi = 270)
```

Inequality joins are extremely general, so general that it's hard to come up with meaningful specific use cases.
One small useful technique is to use them to restrict the cross join so that instead of generating all permutations, we generate all combinations:

```{r}
df <- tibble(id = 1:4, name = c("John", "Simon", "Tracy", "Max"))

df |> inner_join(df, join_by(id < id))
```

### Rolling joins

Rolling joins are a special type of inequality join where instead of getting *every* row that satisfies the inequality, you get just the closest row, as in @fig-join-closest.
You can turn any inequality join into a rolling join by adding `closest()`.
For example `join_by(closest(x <= y))` matches the smallest `y` that's greater than or equal to x, and `join_by(closest(x > y))` matches the biggest `y` that's less than `x`.

```{r}
#| label: fig-join-closest
#| echo: false
#| out-width: ~
#| fig-cap: |
#|   A rolling join is similar to a greater-than-or-equal inequality join
#|   but only matches the first value.
#| fig-alt: |
#|   A rolling join is a subset of an inequality join so some matches are
#|   grayed out indicating that they're not used because they're not the 
#|   "closest".
knitr::include_graphics("diagrams/join/closest.png", dpi = 270)
```

Rolling joins are particularly useful when you have two tables of dates that don't perfectly line up and you want to find (e.g.) the closest date in table 1 that comes before (or after) some date in table 2.

For example, imagine that you're in charge of the party planning commission for your office.
Your company is rather cheap so instead of having individual parties, you only have a party once each quarter.
The rules for determining when a party will be held are a little complex: parties are always on a Monday, you skip the first week of January since a lot of people are on holiday, and the first Monday of Q3 2022 is July 4, so that has to be pushed back a week.
That leads to the following party days:

```{r}
parties <- tibble(
  q = 1:4,
  party = ymd(c("2022-01-10", "2022-04-04", "2022-07-11", "2022-10-03"))
)
```

Now imagine that you have a table of employee birthdays:

```{r}
set.seed(123)
employees <- tibble(
  name = sample(babynames::babynames$name, 100),
  birthday = ymd("2022-01-01") + (sample(365, 100, replace = TRUE) - 1)
)
employees
```

And for each employee we want to find the first party date that comes after (or on) their birthday.
We can express that with a rolling join:

```{r}
employees |> 
  left_join(parties, join_by(closest(birthday >= party)))
```

There is, however, one problem with this approach: the folks with birthdays before January 10 don't get a party:

```{r}
employees |> 
  anti_join(parties, join_by(closest(birthday >= party)))
```

To resolve that issue we'll need to tackle the problem a different way, with overlap joins.

### Overlap joins

Overlap joins provide three helpers that use inequality joins to make it easier to work with intervals:

-   `between(x, y_lower, y_upper)` is short for `x >= y_lower, x <= y_upper`.
-   `within(x_lower, x_upper, y_lower, y_upper)` is short for `x_lower >= y_lower, x_upper <= y_upper`.
-   `overlaps(x_lower, x_upper, y_lower, y_upper)` is short for `x_lower <= y_upper, x_upper >= y_lower`.

Let's continue the birthday example to see how you might use them.
There's one problem with the strategy we used above: there's no party preceding the birthdays Jan 1-9.
So it might be better to be explicit about the date ranges that each party spans, and make a special case for those early birthdays:

```{r}
parties <- tibble(
  q = 1:4,
  party = ymd(c("2022-01-10", "2022-04-04", "2022-07-11", "2022-10-03")),
  start = ymd(c("2022-01-01", "2022-04-04", "2022-07-11", "2022-10-03")),
  end = ymd(c("2022-04-03", "2022-07-11", "2022-10-02", "2022-12-31"))
)
parties
```

Hadley is hopelessly bad at data entry so he also wanted to check that the party periods don't overlap.
One way to do this is by using a self-join to check if any start-end interval overlap with another:

```{r}
parties |> 
  inner_join(parties, join_by(overlaps(start, end, start, end), q < q)) |> 
  select(start.x, end.x, start.y, end.y)
```

Ooops, there is an overlap, so let's fix that problem and continue:

```{r}
parties <- tibble(
  q = 1:4,
  party = ymd(c("2022-01-10", "2022-04-04", "2022-07-11", "2022-10-03")),
  start = ymd(c("2022-01-01", "2022-04-04", "2022-07-11", "2022-10-03")),
  end = ymd(c("2022-04-03", "2022-07-10", "2022-10-02", "2022-12-31"))
)
```

Now we can match each employee to their party.
This is a good place to use `unmatched = "error"` because we want to quickly find out if any employees didn't get assigned a party.

```{r}
employees |> 
  inner_join(parties, join_by(between(birthday, start, end)), unmatched = "error")
```

### Exercises

1.  Can you explain what's happening with the keys in this equi join?
    Why are they different?

    ```{r}
    x |> full_join(y, join_by(key == key))

    x |> full_join(y, join_by(key == key), keep = TRUE)
    ```

2.  When finding if any party period overlapped with another party period we used `q < q` in the `join_by()`?
    Why?
    What happens if you remove this inequality?

## Summary

In this chapter, you've learned how to use mutating and filtering joins to combine data from a pair of data frames.
Along the way you learned how to identify keys, and the difference between primary and foreign keys.
You also understand how joins work and how to figure out how many rows the output will have.
Finally, you've gained a glimpse into the power of non-equi joins and seen a few interesting use cases.

This chapter concludes the "Transform" part of the book where the focus was on the tools you could use with individual columns and tibbles.
You learned about dplyr and base functions for working with logical vectors, numbers, and complete tables, stringr functions for working with strings, lubridate functions for working with date-times, and forcats functions for working with factors.

In the next part of the book, you'll learn more about getting various types of data into R in a tidy form.