# Numeric vectors {#numbers}

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

## Introduction

In this chapter, you'll learn useful tools for creating and manipulating with numeric vectors.
We'll start by doing into a little more detail of `count()` before diving into various numeric transformations.
You'll then learn about more general transformations that are often used with numeric vectors, but also work with other types.
Then you'll learn about a few more useful summaries before we finish up with a comparison of function variants that have similar names and similar actions, but are each designed for a specific use case.

### Prerequisites

This chapter mostly uses functions from base R, which are available without loading any packages.
But we still need the tidyverse because we'll use these base R functions inside of tidyverse functions like `mutate()` and `filter()`.
Like in the last chapter, we'll again use real examples from nycflights13, as well as toy examples made inline with `c()` and `tribble()`.

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

### Counts

It's surprising how much data science you can do with just counts and a little basic arithmetic.
There are two ways to compute a count in dplyr.
The simplest is to use `count()`, which is great for quick exploration and checks during analysis:

```{r}
flights |> count(dest)
```

(Despite the advice in Chapter \@ref(code-style), I usually put `count()` on a single line because I'm usually using it at the console for a quick check that my calculation is working as expected.)

Alternatively, you can count "by hand" which allows you to compute other summaries at the same time:

```{r}
flights |> 
  group_by(dest) |> 
  summarise(
    n = n(),
    delay = mean(arr_delay, na.rm = TRUE)
  )
```

`n()` is a special a summary function because it doesn't take any arguments and instead reads information from the current group.
This means you can't use it outside of dplyr verbs:

```{r, error = TRUE}
n()
```

There are a couple of related counts that you might find useful:

-   `n_distinct(x)` counts the number of distinct (unique) values of one or more variables.
    For example, we could figure out which destinations are served by the most carriers?

    ```{r}
    flights |> 
      group_by(dest) |> 
      summarise(
        carriers = n_distinct(carrier)
      ) |> 
      arrange(desc(carriers))
    ```

-   A weighted count is just a sum.
    For example you could "count" the number of miles each plane flew:

    ```{r}
    flights |> 
      group_by(tailnum) |> 
      summarise(miles = sum(distance))
    ```

    This comes up enough that `count()` has a `wt` argument that does this for you:

    ```{r}
    flights |> count(tailnum, wt = distance)
    ```

-   `sum()` and `is.na()` is also a powerful combination, allowing you to count the number of missing values:

    ```{r}
    flights |> 
      group_by(dest) |> 
      summarise(n_cancelled = sum(is.na(dep_time))) 
    ```

### Exercises

1.  How can you use `count()` to count the number rows with a missing value for a given variable?
2.  Expand the following calls to `count()` to use the core verbs of dplyr:
    1.  `flights |> count(dest, sort = TRUE)`

    2.  `flights |> count(tailnum, wt = distance)`

## Numeric transformations

Base R provides many useful transformation functions that you can use with `mutate()`.
We'll come back to this distinction later in Section \@ref(variants), but the key property that they all possess is that the output is the same length as the input.

There's no way to list every possible function that you might use, so this section will aim give a selection of the most useful.
One category that I've deliberately omit is the trigonometric functions; R provides all the trig functions that you might expect, but they're rarely needed for data science.

### Arithmetic and recycling rules

We introduced the basics of arithmetic (`+`, `-`, `*`, `/`, `^`) in Chapter \@ref(workflow-basics) and have used them a bunch since.
They don't need a huge amount of explanation, because they do what you learned in grade school.
But we need to to briefly talk about the **recycling rules** which determine what happens when the left and right hand sides have different lengths.
This is important for operations like `air_time / 60` because there are 336,776 numbers on the left hand side, and 1 number on the right hand side.

R handles this by repeating, or **recycling**, the short vector.
We can see this in operation more easily if we create some vectors outside of a data frame:

```{r}
x <- c(1, 2, 10, 20)
x / 5
# is shorthand for
x / c(5, 5, 5, 5)
```

Generally, you want to recycle vectors of length 1, but R supports a rather more general rule where it will recycle any shorter length vector, usually (but not always) warning if the longer vector isn't a multiple of the shorter:

```{r}
x * c(1, 2)
x * c(1, 2, 3)
```

This recycling can lead to a surprising result if you accidentally use `==` instead of `%in%` and the data frame has an unfortunate number of rows.
For example, take this code which attempts to find all flights in January and February:

```{r}
flights |> 
  filter(month == c(1, 2))
```

The code runs without error, but it doesn't return what you want.
Because of the recycling rules it returns January flights that are in odd numbered rows and February flights that are in even numbered rows.
There's no warning because `nycflights` has an even number of rows.

To protect you from this silent failure, most tidyverse functions uses stricter recycling that only recycles single values.
Unfortunately that doesn't help here, or many other cases, because the key computation is performed by the base R function `==`, not `filter()`.

### Minimum and maximum

The arithmetic functions work with pairs of variables.
Two closely related functions are `pmin()` and `pmax()`, which when given two or more variables will return the smallest or largest value in each row:

```{r}
df <- tribble(
  ~x, ~y,
  1,  3,
  5,  2,
  7, NA,
)

df |> 
  mutate(
    min = pmin(x, y),
    max = pmax(x, y)
  )
```

These are different to the summary functions `min()` and `max()` which take multiple observations and return a single value.
We'll come back to those in Section \@ref(min-max-summary).

### Modular arithmetic

Modular arithmetic is the technical name for the type of math you did before you learned about real numbers, i.e. when you did division that yield a whole number and a remainder.
In R, these are provided by `%/%` which does integer division, and `%%` which computes the remainder:

```{r}
1:10 %/% 3
1:10 %% 3
```

Modular arithmetic is handy for the flights dataset, because we can use it to unpack the `sched_dep_time` variable into and `hour` and `minute`:

```{r}
flights |> 
  mutate(
    hour = sched_dep_time %/% 100,
    minute = sched_dep_time %% 100,
    .keep = "used"
  )
```

We can combine that with the `mean(is.na(x))` trick from Section \@ref(logical-summaries) to see how the proportion of delayed flights varies over the course of the day.
The results are shown in Figure \@ref(fig:prop-cancelled).

```{r prop-cancelled}
#| fig.cap: > 
#|   A line plot with scheduled departure hour on the x-axis, and proportion
#|   of cancelled flights on the y-axis. Cancellations seem to accumulate
#|   over the course of the day until 8pm, very late flights are much
#|   less likely to be cancelled.
#| fig.alt: >
#|   A line plot showing how proportion of cancelled flights changes over
#|   the course of the day. The proportion starts low at around 0.5% at
#|   6am, then steadily increases over the course of the day until peaking
#|   at 4% at 7pm. The proportion of cancelled flights then drops rapidly
#|   getting down to around 1% by midnight.
flights |> 
  group_by(hour = sched_dep_time %/% 100) |> 
  summarise(prop_cancelled = mean(is.na(dep_time)), n = n()) |> 
  filter(hour > 1) |> 
  ggplot(aes(hour, prop_cancelled)) +
  geom_line(colour = "grey50") + 
  geom_point(aes(size = n))
```

### Logarithms

Logarithms are an incredibly useful transformation for dealing with data that ranges across multiple orders of magnitude.
They also convert multiplicative relationships to additive.
For example, take compounding interest --- the amount of money you have at `year + 1` is the amount of money you had at `year` multiplied by the interest rate.
That gives a formula like `money = starting * interest ^ year`:

```{r}
starting <- 100
interest <- 1.05

money <- tibble(
  year = 2000 + 1:50,
  money = starting * interest^(1:50)
)
```

If you plot this data, you'll get a curve:

```{r}
ggplot(money, aes(year, money)) +
  geom_line()
```

Log transforming the y-axis gives a straight line:

```{r}
ggplot(money, aes(year, money)) +
  geom_line() + 
  scale_y_log10()
```

We get a straight line because (after a little algebra) we get `log(money) = log(starting) + n * log(interest)`, which matches the pattern for a straight line, `y = m * x + b`.
This is a useful pattern: if you see a (roughly) straight line after log-transforming the y-axis, you know that there's an underlying multiplicative relationship.

If you're log-transforming your data with dplyr, instead of relying on ggplot2 to do it for you, you have a choice of three logarithms: `log()` (the natural log, base e), `log2()` (base 2), and `log10()` (base 10).
I recommend using `log2()` or `log10()`.
`log2()` is easy to interpret because difference of 1 on the log scale corresponds to doubling on the original scale and a difference of -1 corresponds to halving; whereas `log10()` is easy to back-transform because (e.g) 3 is 10\^3 = 1000.

The inverse of `log()` is `exp()`; to compute the inverse of `log2()` or `log10()` you'll need to use `2^` or `10^`.

### Rounding

Use `round(x)` to round a number to the nearest integer:

```{r}
round(123.456)
```

You can control the precision of the rounding with the second argument, `digits`.
`round(x, digits)` rounds to the nearest `10^-n` so `digits = 2` will give you.
This definition is cool because it implies `round(x, -3)` will round to the nearest thousand:

```{r}
round(123.456, 2)  # two digits
round(123.456, 1)  # one digit
round(123.456, -1) # round to nearest ten
round(123.456, -2) # round to nearest hundred
```

There's one weirdness with `round()` that seems surprising at first glance:

```{r}
round(c(1.5, 2.5))
```

`round()` uses what's known as "round half to even" or Banker's rounding.
If a number is half way between two integers, it will be rounded to the **even** integer.
This is the right general strategy because it keeps the rounding unbiased: half the 0.5s are rounded up, and half are rounded down.

`round()` is paired with `floor()` to round down and `ceiling()` to round up:

```{r}
x <- 123.456

floor(x)
ceiling(x)
```

These functions don't have a digits argument, but instead, you can scale down, round, and then scale back up:

```{r}
# Round down to nearest two digits
floor(x / 0.01) * 0.01
# Round up to nearest two digits
ceiling(x / 0.01) * 0.01
```

You can use the same technique if you want to `round()` to a multiple of some other number:

```{r}
# Round to nearest multiple of 4
round(x / 4) * 4

# Round to nearest 0.25
round(x / 0.25) * 0.25
```

### Cumulative and rolling aggregates

Base R provides `cumsum()`, `cumprod()`, `cummin()`, `cummax()` for running, or cumulative, sums, products, mins and maxes, and dplyr provides `cummean()` for cumulative means.

```{r}
x <- 1:10
cumsum(x)
cummean(x)
```

If you need more complex rolling or sliding aggregates, try the [slider](https://davisvaughan.github.io/slider/) package by Davis Vaughan.
The example below illustrates some of its features.

```{r}
library(slider)

# Same as a cumulative sum
slide_vec(x, sum, .before = Inf)
# Sum the current element and the one before it
slide_vec(x, sum, .before = 1)
# Sum the current element and the two before and after it
slide_vec(x, sum, .before = 2, .after = 2)
# Only compute if the window is complete
slide_vec(x, sum, .before = 2, .after = 2, .complete = TRUE)
```

### Exercises

1.  Explain in words what each line of the code used to generate Figure \@ref(fig:prop-cancelled) does.

## General transformations

These are often used with numbers, but can be applied to most other column types.

### Ranks

dplyr provides a number of ranking functions, but you should start with `dplyr::min_rank()`.
It does the most usual way of dealing with ties (e.g. 1st, 2nd, 2nd, 4th).
The default gives smallest values the small ranks; use `desc(x)` to give the largest values the smallest ranks.

```{r}
y <- c(1, 2, 2, NA, 3, 4)
min_rank(y)
min_rank(desc(y))
```

If `min_rank()` doesn't do what you need, look at the variants `dplyr::row_number()`, `dplyr::dense_rank()`, `dplyr::percent_rank()`, `dplyr::cume_dist()`, `dplyr::ntile()`, as well as base R's `rank()`.

`row_number()` can also be used without a variable within `mutate()`.
When combined with `%%` and `%/%` this can be a useful tool for dividing data into similarly sized groups:

```{r}
flights |> 
  mutate(
    row = row_number(),
    group_3 = row %/% (n() / 3),
    group_3 = row %% 3,
    .keep = "none"
  )
```

### Offset

`dplyr::lead()` and `dplyr::lag()` allow you to refer to leading or lagging values.
They return a vector of the same length but padded with NAs at the start or end

```{r}
x <- c(2, 5, 11, 19, 35)
lag(x)
lag(x, 2)
lead(x)
```

-   `x - lag(x)` gives you the difference between the current and previous value.
-   `x == lag(x)` tells you when the current value changes. See Section XXX for use with cumulative tricks.

If the rows are not already ordered, you can provide the `order_by` argument.

### Positions

If your rows have a meaningful order, you can use base R's `[`, or dplyr's `first(x)`, `nth(x, 2)`, or `last(x)` to extract values at a certain position.
For example, we can find the first and last departure for each day:

```{r}
flights |> 
  group_by(year, month, day) |> 
  summarise(
    first_dep = first(dep_time), 
    last_dep = last(dep_time)
  )
```

The chief advantage of `first()` and `nth()` over `[` is that you can set a default value if that position does not exist (i.e. you're trying to get the 3rd element from a group that only has two elements).
The chief advantage of `last()` over `[`, is writing `last(x)` rather than `x[length(x)]`.

Additionally, if the rows aren't ordered, but there's a variable that defines the order, you can use `order_by` argument.
You can do this with `[` + `order_by()` but it requires a little thought.

Computing positions is complementary to filtering on ranks.
Filtering gives you all variables, with each observation in a separate row:

```{r}
flights |> 
  group_by(year, month, day) |> 
  mutate(r = min_rank(desc(sched_dep_time))) |> 
  filter(r %in% c(1, max(r)))
```

### Exercises

1.  Find the 10 most delayed flights using a ranking function.
    How do you want to handle ties?
    Carefully read the documentation for `min_rank()`.

2.  Which plane (`tailnum`) has the worst on-time record?

3.  What time of day should you fly if you want to avoid delays as much as possible?

4.  For each destination, compute the total minutes of delay.
    For each flight, compute the proportion of the total delay for its destination.

5.  Delays are typically temporally correlated: even once the problem that caused the initial delay has been resolved, later flights are delayed to allow earlier flights to leave.
    Using `lag()`, explore how the delay of a flight is related to the delay of the immediately preceding flight.

6.  Look at each destination.
    Can you find flights that are suspiciously fast?
    (i.e. flights that represent a potential data entry error).
    Compute the air time of a flight relative to the shortest flight to that destination.
    Which flights were most delayed in the air?

7.  Find all destinations that are flown by at least two carriers.
    Use that information to rank the carriers.

## Summaries

Just using means, counts, and sum can get you a long way, but R provides many other useful summary functions.

### Center

We've used `mean(x)`, but `median(x)` is also useful.
The mean is the sum divided by the length; the median is a value where 50% of `x` is above it, and 50% is below it.

```{r}
flights |>
  group_by(month) |>
  summarise(
    med_arr_delay = median(arr_delay, na.rm = TRUE),
    med_dep_delay = median(dep_delay, na.rm = TRUE)
  )
```

Don't forget what you learned in Section \@ref(sample-size): whenever creating numerical summaries, it's a good idea to include the number of observations in each group.

### Minimum, maximum, and quantiles {#min-max-summary}

Quantiles are a generalization of the median.
For example, `quantile(x, 0.25)` will find a value of `x` that is greater than 25% of the values, and less than the remaining 75%.
`min()` and `max()` are like the 0% and 100% quantiles: they're the smallest and biggest numbers.

```{r}
# When do the first and last flights leave each day?
flights |> 
  group_by(year, month, day) |> 
  summarise(
    first = min(dep_time, na.rm = TRUE),
    last = max(dep_time, na.rm = TRUE)
  )
```

Using the median and 95% quantile is coming in performance monitoring.
`median()` shows you what the (bare) majority of people experience, and 95% shows you the worst case, excluding 5% of outliers.

### Spread

The root mean squared deviation, or standard deviation `sd(x)`, is the standard measure of spread.

```{r}
# Why is distance to some destinations more variable than to others?
flights |> 
  group_by(origin, dest) |> 
  summarise(distance_sd = sd(distance), n = n()) |> 
  filter(distance_sd > 0)

# Did it move?
flights |> 
  filter(dest == "EGE") |> 
  select(time_hour, dest, distance, origin) |> 
  ggplot(aes(time_hour, distance, colour = origin)) +
  geom_point()
```

<https://en.wikipedia.org/wiki/Eagle_County_Regional_Airport> --- seasonal airport.
Nothing in wikipedia suggests a move in 2013.

The interquartile range `IQR(x)` and median absolute deviation `mad(x)` are robust equivalents that may be more useful if you have outliers.
IQR is `quantile(x, 0.75) - quantile(x, 0.25)`.
`mad()` is derivied similarly to `sd()`, but inside being the average of the squared distances from the mean, it's the median of the absolute differences from the median.

### With `mutate()`

As the names suggest, the summary functions are typically paired with `summarise()`, but they can also be usefully paired with `mutate()`, particularly when you want do some sort of group standardization.

-   `x / sum(x)` calculates the proportion of a total.
-   `(x - mean(x)) / sd(x)` computes a Z-score (standardised to mean 0 and sd 1).
-   `x / x[1]` computes an index based on the first observation.

### Exercises

1.  Currently `dep_time` and `sched_dep_time` are convenient to look at, but hard to compute with because they're not really continuous numbers.
    Convert them to a more convenient representation of number of minutes since midnight.

2.  What trigonometric functions does R provide?

3.  Brainstorm at least 5 different ways to assess the typical delay characteristics of a group of flights.
    Consider the following scenarios:

    -   A flight is 15 minutes early 50% of the time, and 15 minutes late 50% of the time.

    -   A flight is always 10 minutes late.

    -   A flight is 30 minutes early 50% of the time, and 30 minutes late 50% of the time.

    -   99% of the time a flight is on time.
        1% of the time it's 2 hours late.

    Which is more important: arrival delay or departure delay?

## Variants

We've seen a few variants of different functions

| Summary | Cumulative | Paired |
|---------|------------|--------|
| `sum`   | `cumsum`   | `+`    |
| `prod`  | `cumprod`  | `*`    |
| `all`   | `cumall`   | `&`    |
| `any`   | `cumany`   | `|`    |
| `min`   | `cummin`   | `pmin` |
| `max`   | `cummax`   | `pmax` |

-   Summary functions take a vector and always return a length 1 vector. Typically used with `summarise()`
-   Cumulative functions take a vector and return the same length. Used with `mutate()`.
-   Paired functions take a pair of functions and return a vector the same length (using the recycling rules if the vectors aren't the same length). Used with `mutate()`

```{r}
x <- c(1, 2, 3, 5)
sum(x)
cumsum(x)
x + 10
```
