r4ds/joins.qmd

# Joins {#sec-relational-data}

```{r}
#| results: "asis"
#| echo: false
source("_common.R")
status("restructuring")
```

## 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.
All the verbs in this chapter use a pair of data frames.
Fortunately this is enough, since you can solve any more complex problem a pair at a time.

You'll learn about important types of joins in this chapter:

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

If you're familiar with SQL, you should find the ideas in this chapter familiar, as their realization in dplyr is very similar.
We'll point out any important differences as we go.
Don't worry if you're not familiar with SQL as you'll learn more about it in @sec-import-databases.

### Prerequisites

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

The connection between two tables is defined by a pair of keys.
In this section, you'll learn what those terms mean, and how they apply to the datasets in 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

To understand joins, you need to first understand how two tables might be connected.
which come in pairs of primary and foreign key.
A **primary key** is a variable (or group of variables) that uniquely identifies an observation.
A **foreign key** is the value of a primary key in another table and is used to connect two tables.
Let's make those terms concrete by looking at four other data frames in nycfights13:

-   `airlines` lets you look up the full carrier name from its abbreviated code.
    Its primary key is the two letter `carrier` code.

    ```{r}
    airlines
    ```

-   `airports` gives information about each airport.
    Its primary key is the three `faa` airport code.

    ```{r}
    airports
    ```

-   `planes` gives information about each plane.
    It's primary key is the `tailnum`.

    ```{r}
    planes
    ```

-   `weather` gives the weather at each NYC airport for each hour.
    It has a compound primary key; to uniquely identify each observation you need to know both `origin` (the location) and `time_hour` (the time).

    ```{r}
    weather
    ```

These datasets are all connected via the `flights` data frame because the `tailnum`, `carrier`, `origin`, `dest`, and `time_hour` variables are all primary keys in other datasets making them foreign keys.

-   `flights$tailnum` connects to primary key `planes$tailnum`.

-   `flights$carrier` connecet to primary key `airlines$carrer`.

-   `flights$origin` connects to primary key `airports$faa`.

-   `flights$dest` connects to primary key `airports$faa` .

-   `flights$origin`-`flights$time_hour` connects to primary key `weather$origin`-`weather$time_hour`.

We can also draw these relationships, as in @fig-flights-relationships.
This diagram is a little overwhelming, but it's simple compared to some you'll see in the wild!
The key to understanding diagrams like this is that you'll solve real problems by working with pairs of data frames.
You don't need to understand the whole thing; you just need to understand the chain of connections between the two data frames that you're interested in.

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

### Checking primary keys

That that we've identified the primary keys, 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 think about because it makes it easier to work with observations if have some way to uniquely identify them.

There's clearly no one variable or even a pair of variables that uniquely identifies a flight, but we can find three together that work:

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

Does that 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 longitude a primary key for `airports`?

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

Identifying an airport by it's altitude and latitude is clearly a bad idea, and in general it's not possible to know from the data itself whether or not a combination of variables that uniquely identifies an observation is a primary key.
For flights, the combination of `time_hour`, `carrier`, and `flight` seems like a reasonable primary key because it would be really confusing for the airline if there were multiple flights with the same 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 particular 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 the 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 origin (NYC) airports.
    If it contained weather records for all airports in the USA, what additional relation would it define with `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 this time?

4.  We know that some days of the year are "special" and fewer people than usual fly on them.
    How might you represent that data as a data frame?
    What would be the primary keys of that data frame?
    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 characterise 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 to using them to better understand the `flights` dataset.
We'll first show you the mutating joins, so called because their primary role[^joins-1] is to add additional column to the `x` data frame, just like `mutate()`. You'll learn learn about join keys, and finish up with a discussion of the filtering joins, which work like a `filter()` rather than a `mutate()`.

[^joins-1]: They also affect the number of rows; we'll come back to that shortly.

### 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 you have a lot of variables already, you won't see the new variables.
For these examples, we'll make it easier to see what's going on in the examples by creating a narrower dataset:

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

(Remember that in RStudio you can also use `View()` to avoid this problem.)

As you'll learn shortly, there are four types of mutating join, but the one that should be your default is `left_join()`.
It preserves the rows in `x` even when there's no match in `y`, filling in the new variables with missing values.

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:

```{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 sort of plane was flying:

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

Note that in each of these cases the number of rows has stayed the same, but we've added new columns to the right.

### 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.
What happens if we try to join `flights` with the complete `planes`?

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

We get a lot of missing matches because both `flights` and `planes` have a `year` column but they mean different things: the year the flight occurred and the year the plane was built.
We only want to join on the `tailnum` column so we need switch to an explicit specification:

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

Note that the `year` variables are disambiguated in the output with a suffix.
You can control this with the `suffix` argument.

`join_by(tailnum)` is short for `join_by(tailnum == tailnum)`.
This fuller form is important because 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()` as it's a more flexible specification that supports more types of join, as you'll learn in @sec-non-equi-joins.

### 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` are useful for matching filtered summary data frames back to the original rows.
For example, we could use 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 figuring out what's missing.
For example, we can figure out which flights are missing information about the destination airport:

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

Or which flights lack metadata about their plane:

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

### Exercises

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

2.  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?

3.  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 that destination?

4.  What does it mean for a flight to have a missing `tailnum`?
    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.  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 above.

6.  Add the location of the origin *and* destination (i.e. the `lat` and `lon`) 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(lon, lat)) +
        borders("state") +
        geom_point() +
        coord_quickmap()
    ```

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

8.  What happened on June 13 2013?
    Display the spatial pattern of 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) |>
      summarise(delay = mean(arr_delay), n = n()) |>
      filter(n > 5) |>
      inner_join(airports, by = c("dest" = "faa")) |>
      ggplot(aes(lon, lat)) +
        borders("state") +
        geom_point(aes(size = n, colour = delay)) +
        coord_quickmap()
    ```

## How do joins work?

Now that you've used a few joins it's time to learn more about how they work, focusing especially on how each row in `x` matches with rows in `y`.
We'll begin by using @fig-join-setup to introduce a visual representation of the two simple tibbles defined below.
The column with colored cells represents the keys of the two data frames, here literally called `key`.
The grey columns represents the "value" columns that is carried along for the ride.
In these examples we'll use a single key variable, but the idea generalizes 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.
#| fig-alt: >
#|   x and y are two data frames with 2 columns and 3 rows each. The first
#|   column in each is the key and the second is the value. The contents of
#|   these data frames are given in the previous code chunk.

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

@fig-join-setup2 shows all potential matches between `x` and `y` as an intersection of a pair of lines.
For this example, the rows in the output will be primarily determined by `x`, so the `x` table is horizontal and will line 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 by drawing 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 leading to 9 intersections that represent nine potential
#|   matches.

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

In an actual join, matches will be indicated with dots, as in @fig-join-inner.
The number of dots equals the number of matches, which in turn equals the number of rows in the output, a new data frame that contains the key, the x values, and the y values.
The join shown here is a so-called **inner join**, where rows match if the keys are equal, so that the output contains only the rows with keys that appear in both `x` and `y`.

```{r}
#| label: fig-join-inner
#| echo: false
#| out-width: ~
#| fig-cap: >
#|   An inner join matches rows in `x` to rows in `y` that have the
#|   same value of `key`. Each match becomes a row in the output.
#| fig-alt: >
#|   Keys 1 and 2 appear in both x and y, so there values are equal and
#|   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)
```

An **outer join** keeps 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 row in `x` appears
    #|   in the output.
    #| fig-alt: >
    #|   Compared to the inner join, the `y` table gets a new virtual row
    #|   that will match any row in `x` that doesn't otherwise have a match.
    #|   This means that the output now has three rows. For key = 3, which
    #|   matches this virtual row, the value of val_y is 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`.
    Note the output will consist of all `x` rows that match a row in `y`, then all the rows of `y` that didn't match in `x`.

    ```{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: >
    #|   Keys 1 and 2 from x are matched to those in y, key 4 is 
    #|   also carried along to the joined result since it's on the right data
    #|   frame,  but key 3 from x is not carried along since it's on the left
    #|   but not on the right. The result is a data frame with 3 rows: keys 
    #|   1, 2, and 4, all values from val_y, and the corresponding values
    #|   from val_x for keys 1 and 2 with an NA for key 4, val_x. 

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

-   A **full join** keeps all observations in `x` and `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.
    Note the output will consist of all `x` rows followed by the remaining `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: >
    #|   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 outer joins differ is with a Venn diagram, @fig-join-venn.
This, however, 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: Only intersection is shaded. 
#|   Full join: Everything is shaded. 
#|   Left join: All of x is shaded.
#|   Right: All of y is shaded.

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

### Row matching

So far we've explored what happens if there's either zero or one matches.
What happens if there's more than one match?
To understand what's going let's first narrow our focus to the `inner_join()` and then consider the three possible options for each row in `x`:

-   If it doesn't match anything, it's dropped.
-   If it matches 1 row, it's kept as is.
-   If it matches more than 1 row, it's duplicated once for each match.

These three options are illustrated in @fig-join-match-type.

```{r}
#| label: fig-join-match-types
#| echo: false
#| out-width: ~
#| fig-cap: > 
#|   The three key 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)
```

In principle, this means that there are no guarantees about the number of rows in the output of an `inner_join()`:

-   There might be fewer rows if some rows in `x` don't match any rows in `y`.
-   There might be more rows if some rows in `x` match multiple rows in `y`.
-   There might be the same number of rows if every row in `x` matches one row in `y`.
-   There might be the same number of rows if the number of multiple matches precisely balances out with the rows that don't match.

Row expansion is a fundamental property of joins, but it feels dangerous to us so dplyr will warn whenever there are multiple matches:

```{r}
df1 <- tibble(key = c(1, 2, 3), val_x = c("x1", "x2", "x3"))
df2 <- tibble(key = c(1, 2, 2), val_y = c("y1", "y2", "y3"))

df1 |> 
  inner_join(df2, join_by(key))
```

This is another reason we recommend the `left_join()` --- every row in `x` is guaranteed to match a "virtual" row in `y` so it'll never drop rows, and you'll always get a warning when it duplicates rows.

You can further control over row matching with two arguments:

-   `unmatched` controls what happens when in `x` fails to match any rows in `y`. It defaults to `"drop"` which will silently drop any unmatched rows.
-   `multiple` controls what happens when a row in `x` matches more than one row in `y`. For equi-joins, it defaults to `"warn"` which emits a warning message if there are any multiple matches.

There are two common cases in which you might want to override the default: enforcing a one-to-one mapping or allowing multiple joins.

### One-to-one mapping

Both `unmatched` and `multiple` can take value `"error"` which means that the join will fail unless each row in `x` matches exactly one row in `y`:

```{r}
#| error: true
df1 |> 
  inner_join(df2, join_by(key), unmatched = "error", multiple = "error")
```

Note that `unmatched = "error"` is not useful with `left_join()` because, as described above, every row in `x` has a fallback match to a virtual row in `y` filled with missing values.

### Allow multiple rows

Sometimes it's useful to deliberately expand the number of rows in the output.
A natural way that this comes about is when you flip the direction of the question you're asking.
For example, as we've seen above, it's natural to supplement the `flights` data with information about the plane that flew each flight:

```{r}
#| results: false
flights2 |> 
  left_join(planes, by = "tailnum")
```

But it's also reasonable to ask what flights did each plane fly?

```{r}
plane_flights <- planes |> 
  left_join(flights2, by = "tailnum")
```

Since this duplicate rows in `x` (the planes), we need to explicitly say we're ok with the multiple matches by setting `multiple = "all"`:

```{r}
plane_flights <- planes |> 
  left_join(flights2, by = "tailnum", multiple = "all")

plane_flights
```

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

The number of matches is also closely related to 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 don't have a match in `y`, as in @fig-join-anti.
In both cases, only the existence of a match is important; it doesn't matter which observation is matched.
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: >
#|   Diagram of a semi join. Data frame x is on the left and has two columns 
#|   (key and val_x) with keys 1, 2, and 3. Diagram y is on the right and also 
#|   has two columns (key and val_y) with keys 1, 2, and 4. Semi joining these 
#|   two results in a data frame with two rows and two columns (key and val_x), 
#|   with keys 1 and 2 (the only keys that match between the two data frames).

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: >
#|   Diagram of an anti join. Data frame x is on the left and has two columns 
#|   (key and val_x) with keys 1, 2, and 3. Diagram y is on the right and also 
#|   has two columns (key and val_y) with keys 1, 2, and 4. Anti joining these 
#|   two results in a data frame with one row and two columns (key and val_x), 
#|   with keys 3 only (the only key in x that is not in y).

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

## Non-equi joins

So far you've only seen **equi-joins**, joins where the two rows match if the keys in equal the keys in y.
Now we're going to relax that restriction and discuss other ways of determining if a pair of rows match.

But before you learn about equi-joins we need to revisit a simplification we made above: because the x keys and y are equal, 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 |> left_join(y, by = "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
#|   omit one.
#| echo: false
#| out-width: ~

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

This distinction between the keys becomes much more important as we move away from equi-joins because the key values are much more likely to be different.
For example, instead matching when the `x$key` and `y$key` are equal, we could match whenever the `x$key` is greater than or equal the `y$key`, leading to @fig-join-gte.

```{r}
#| label: fig-join-gte
#| echo: false
#| fig-cap: >
#|   A non-equi join where the `x` key must greater than or equal to 
#|   than 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 particularly useful as term because it only tells you what the join is not, not what it is. dplyr helps a bit by identifying four particularly useful types of non-equi-join:

-   **Cross-joins** match every pair of rows.
-   **Inequality-joins** use `<`, `<=`, `>`, `>=` 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-cross-join, 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 you want to generate permutations.
For example, the code below generates every possible pair of names.
This is sometimes called a **self-join** because we're joining a table to itself.

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

### 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-cross-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`.
knitr::include_graphics("diagrams/join/cross-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 filter 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 |> left_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))` finds the smallest `y` that's greater than or equal to x, and `join_by(closest(x > y))` finds the biggest `y` that's less than x.

```{r}
#| label: fig-join-closest
#| echo: false
#| out-width: ~
#| fig-cap: >
#|   A following join is similar to a greater-than-or-equal inequality join
#|   but only matches the first value.
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 office birthdays.
Your company is rather cheap so instead of having individual parties, you only have a party once each quarter.
Parties are always on a Monday, and 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 = lubridate::ymd(c("2022-01-10", "2022-04-04", "2022-07-11", "2022-10-03"))
)
```

Now imagine that we have a table of employee birthdays:

```{r}
employees <- tibble(
  name = wakefield::name(100),
  birthday = lubridate::ymd("2022-01-01") + (sample(365, 100, replace = TRUE) - 1)
)
employees
```

For each employee we want to find the first party date that comes after (or on) their birthday:

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

```{r}
#| echo: false
employees |> 
  left_join(parties, join_by(preceding(birthday, party)))
```

### 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 used above: there's no party preceding the birthdays Jan 1-9.
So it might be better to to be explicit about the date ranges that each party spans, and make a special case for those early bithdays:

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

I'm hopelessly bad at data entry so I also want to check that my party periods don't overlap.
I can perform an self-join and check to see if any start-end interval overlaps with any other:

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

Let's fix that problem and continue:

```{r}
parties <- tibble(
  q = 1:4,
  party = lubridate::ymd(c("2022-01-10", "2022-04-04", "2022-07-11", "2022-10-03")),
  start = lubridate::ymd(c("2022-01-01", "2022-04-04", "2022-07-11", "2022-10-03")),
  end = lubridate::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 I want to find out if any employees didn't get assigned a birthday.

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

### Exercises

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

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

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

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