# 5. Vector indexing¶

This open-access textbook is, and will remain, freely available for everyone’s enjoyment (also in PDF; a paper copy can also be ordered). It is a non-profit project. Although available online, it is a whole course, and should be read from the beginning to the end. Refer to the Preface for general introductory remarks. Any bug/typo reports/fixes are appreciated. Make sure to check outMinimalist Data Wrangling with Python[27] too.

We now know plenty of ways to process vectors *in their entirety*,
but how to extract and replace their specific *parts*? We will be
collectively referring to such activities as *indexing*.
This is because they are often performed through the
*index operator*, `**[**`.

## 5.1. **head** and **tail**¶

Let’s begin with something more lightweight, though.
The **head** function fetches
a few elements from the beginning of a vector.

```
x <- 1:10
head(x) # head(x, 6)
## [1] 1 2 3 4 5 6
head(x, 3) # get the first three
## [1] 1 2 3
head(x, -3) # skip the last three
## [1] 1 2 3 4 5 6 7
```

Similarly, **tail** extracts a couple of items from the end of a
sequence.

```
tail(x) # tail(x, 6)
## [1] 5 6 7 8 9 10
tail(x, 3) # get the last three
## [1] 8 9 10
tail(x, -3) # skip the first three
## [1] 4 5 6 7 8 9 10
```

Both functions work on lists too[1]. They are useful for
previewing the contents of *really big* objects. Also,
they never complain about our trying to fetch too many elements:

```
head(x, 100) # no more than the first 100 elements
## [1] 1 2 3 4 5 6 7 8 9 10
```

## 5.2. Subsetting and extracting from vectors¶

Given a vector `x`

, `x[i]`

returns its subset
comprised of elements indicated by the indexer `i`

, which
can be a *single* vector of:

nonnegative integers (gives the positions of elements to retrieve),

negative integers (gives the positions to omit),

logical values (states which items should be fetched or skipped),

character strings (locates the elements with specific names).

### 5.2.1. Nonnegative indexes¶

Consider example vectors:

```
(x <- seq(10, 100, 10))
## [1] 10 20 30 40 50 60 70 80 90 100
(y <- list(1, 11:12, 21:23))
## [[1]]
## [1] 1
##
## [[2]]
## [1] 11 12
##
## [[3]]
## [1] 21 22 23
```

The first element in a vector is at index 1. Hence:

```
x[1] # the first element
## [1] 10
x[length(x)] # the last element
## [1] 100
```

Important

We might have wondered why “`[1]`

” is displayed each time
we print out an atomic vector on the console:

```
print((1:51)*10)
## [1] 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
## [18] 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340
## [35] 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510
```

It is merely a visual hint indicating which vector element we output at the beginning in each line.

Vectorisation is a universal feature of R. It comes as no surprise that the indexer can also be of length greater than one.

```
x[c(1, length(x))] # the first and the last
## [1] 10 100
x[1:3] # the first three
## [1] 10 20 30
```

Take note of the boundary cases:

```
x[c(1, 2, 1, 0, 3, NA_real_, 1, 11)] # repeated, 0, missing, out of bound
## [1] 10 20 10 30 NA 10 NA
x[c()] # indexing by an empty vector
## numeric(0)
```

When applied on lists, the index operator always returns a list as well, even if we ask for a single element:

```
y[2] # a list that includes the second element
## [[1]]
## [1] 11 12
y[c(1, 3)] # not the same as x[1, 3] (a different story)
## [[1]]
## [1] 1
##
## [[2]]
## [1] 21 22 23
```

If we want to *extract* a component, i.e., to dig into what is inside
a list at a specific location, we can refer to `**[[**`:

```
y[[2]] # extract the second element
## [1] 11 12
```

This is exactly why R displays “`[[1]]`

”, “`[[2]]`

”, etc.
when lists are printed.

On a side note, calling `x[[i]]`

on an *atomic* vector, where `i`

is a single
value, has almost[2] the same effect as `x[i]`

.
However, `**[[**` generates an error if the subscript
is out of bounds.

Important

Let’s reflect on the operators’ behaviour in the case of nonexistent items:

```
c(1, 2, 3)[4]
## [1] NA
list(1, 2, 3)[4]
## [[1]]
## NULL
c(1, 2, 3)[[4]]
## Error in c(1, 2, 3)[[4]]: subscript out of bounds
list(1, 2, 3)[[4]]
## Error in list(1, 2, 3)[[4]]: subscript out of bounds
```

Note

(*)
`**[[**` also supports multiple indexers.

```
y[[c(1, 3)]]
## Error in y[[c(1, 3)]]: subscript out of bounds
```

Its meaning is different from `y[`

**c**`(1, 3)]`

, though;
we are about to extract a single value, remember?
Here, indexing is applied *recursively*.
Namely, the above is equivalent to `y[[1]][[3]]`

. We got an error
because `y[[1]]`

is of a length smaller than three.

More examples:

```
y[[c(3, 1)]] # y[[3]][[1]]
## [1] 21
list(list(7))[[c(1, 1)]] # 7, not list(7)
## [1] 7
```

### 5.2.2. Negative indexes¶

The indexer can also be a vector of negative integers.
This way, we can *exclude* the elements at given positions:

```
y[-1] # all but the first
## [[1]]
## [1] 11 12
##
## [[2]]
## [1] 21 22 23
x[-(1:3)] # all but the first three
## [1] 40 50 60 70 80 90 100
x[-c(1, 0, 2, 1, 1, 8:100)] # 0, repeated, out of bound indexes
## [1] 30 40 50 60 70
```

Note

Negative and positive indexes cannot be mixed.

```
x[-1:3] # recall that -1:3 == (-1):3
## Error in x[-1:3]: only 0's may be mixed with negative subscripts
```

Also, `NA`

indexes cannot be mixed with negative ones.

### 5.2.3. Logical indexer¶

A vector can also be subsetted by means of a logical vector.
If they both are of identical lengths,
the consecutive elements in the latter indicate whether
the corresponding elements of the indexed vector are supposed to be
selected (`TRUE`

) or omitted (`FALSE`

).

```
# 1*** 2 3 4 5*** 6*** 7 8*** 9? 10***
x[c(TRUE, FALSE, FALSE, FALSE, TRUE, TRUE, FALSE, TRUE, NA, TRUE)]
## [1] 10 50 60 80 NA 100
```

In other words, `x[l]`

, where `l`

is a logical vector,
returns all `x[i]`

with `i`

such that `l[i]`

is `TRUE`

.
We thus extracted the elements at indexes 1, 5, 6, 8, and 10.

Important

Be careful: if the element selector is `NA`

,
we will get a missing value (for atomic vectors) or `NULL`

(for lists).

```
c("one", "two", "three")[c(NA, TRUE, FALSE)]
## [1] NA "two"
list("one", "two", "three")[c(NA, TRUE, FALSE)]
## [[1]]
## NULL
##
## [[2]]
## [1] "two"
```

This, lamentably, comes with no warning, which might be
problematic when indexers are generated programmatically.
As a remedy, we sometimes pass the logical indexer to the **which**
function first. It returns the indexes of the elements equal to `TRUE`

,
ignoring the missing ones.

```
which(c(NA, TRUE, FALSE, TRUE, FALSE, NA, TRUE))
## [1] 2 4 7
c("one", "two", "three")[which(c(NA, TRUE, FALSE))]
## [1] "two"
```

Recall that in Chapter 3, we discussed
ample vectorised operations that generate logical vectors.
Anything that yields a logical vector of the same length as `x`

can be passed as an indexer.

```
x > 60 # yes, it is a perfect indexer candidate
## [1] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE
x[x > 60] # select elements in `x` that are greater than 60
## [1] 70 80 90 100
x[x < 30 | 70 < x] # elements not between 30 and 70
## [1] 10 20 80 90 100
x[x < mean(x)] # elements smaller than the mean
## [1] 10 20 30 40 50
x[x^2 > 7777 | log10(x) <= 1.6] # indexing via a transformed version of `x`
## [1] 10 20 30 90 100
(z <- round(runif(length(x)), 2)) # ten pseudorandom numbers
## [1] 0.29 0.79 0.41 0.88 0.94 0.05 0.53 0.89 0.55 0.46
x[z <= 0.5] # indexing based on `z`, not `x`: no problem
## [1] 10 30 60 100
```

The indexer is always evaluated first and then passed to the subsetting operation. The index operator does not care how an indexer is generated.

Furthermore, the recycling rule is applied when necessary:

```
x[c(FALSE, TRUE)] # every second element
## [1] 20 40 60 80 100
y[c(TRUE, FALSE)] # interestingly, there is no warning here
## [[1]]
## [1] 1
##
## [[2]]
## [1] 21 22 23
```

Consider a simple database about six people, their favourite dishes, and birth years.

```
name <- c("Graham", "John", "Terry", "Eric", "Michael", "Terry")
food <- c("bacon", "spam", "spam", "eggs", "spam", "beans")
year <- c( 1941, 1939, 1942, 1943, 1943, 1940 )
```

The consecutive elements in different vectors correspond to each other, e.g., Graham was born in 1941, and his go-to food was bacon.

List the names of people born in 1941 or 1942.

List the names of those who like spam.

List the names of those who like spam and were born after 1940.

Compute the average birth year of the lovers of spam.

Give the average age, in 1969, of those who didn’t find spam utmostly delicious.

The answers must be provided programmatically,
i.e., do not just write `"Eric"`

and `"Graham"`

.
Make the code generic enough so that it works
in the case of any other database of this kind, no matter
its size.

Remove missing values from a given vector without referring
to **na.omit**.

### 5.2.4. Character indexer¶

Let’s consider a vector equipped with the `names`

attribute:

```
x <- structure(x, names=letters[1:10]) # add names
print(x)
## a b c d e f g h i j
## 10 20 30 40 50 60 70 80 90 100
```

These labels can be referred to when extracting the elements. To do this, we use an indexer that is a character vector:

```
x[c("a", "f", "a", "g", "z")]
## a f a g <NA>
## 10 60 10 70 NA
```

Important

We have said that special object attributes add *extra* functionality
on top of the existing ones. Therefore, indexing by means
of positive, negative, and logical vectors is still available:

```
x[1:3]
## a b c
## 10 20 30
x[-(1:5)]
## f g h i j
## 60 70 80 90 100
x[x > 70]
## h i j
## 80 90 100
```

Lists can also be subsetted this way.

```
(y <- structure(y, names=c("first", "second", "third")))
## $first
## [1] 1
##
## $second
## [1] 11 12
##
## $third
## [1] 21 22 23
y[c("first", "second")]
## $first
## [1] 1
##
## $second
## [1] 11 12
y["third"] # result is a list
## $third
## [1] 21 22 23
y[["third"]] # result is the specific element unwrapped
## [1] 21 22 23
```

Important

Labels do not have to be unique. When we have repeated names, the first matching element is extracted:

```
structure(c(1, 2, 3), names=c("a", "b", "a"))["a"]
## a
## 1
```

There is no direct way to select all *but* given names,
just like with negative integer indexers. For a workaround,
see Section 5.4.1.

Rewrite the solution to Exercise 5.1 assuming that we now have three features wrapped inside a list.

```
(people <- list(
Name=c("Graham", "John", "Terry", "Eric", "Michael", "Terry", "Steve"),
Food=c("bacon", "spam", "spam", "eggs", "spam", "beans", "spam"),
Year=c( 1941, 1939, 1942, 1943, 1943, 1940, NA_real_)
))
## $Name
## [1] "Graham" "John" "Terry" "Eric" "Michael" "Terry" "Steve"
##
## $Food
## [1] "bacon" "spam" "spam" "eggs" "spam" "beans" "spam"
##
## $Year
## [1] 1941 1939 1942 1943 1943 1940 NA
```

Do not refer to `name`

, `food`

, and `year`

directly.
Instead, use the full `people[["Name"]]`

etc. accessors.
There is no need to pout: it is just a tiny bit of extra work.
Also, notice that Steve has joined the group; hello, Steve.

## 5.3. Replacing elements¶

### 5.3.1. Modifying atomic vectors¶

There are also *replacement* versions of the aforementioned indexing schemes.
They allow us to substitute some new content for the old one.

```
(x <- 1:12)
## [1] 1 2 3 4 5 6 7 8 9 10 11 12
x[length(x)] <- 42 # modify the last element
print(x)
## [1] 1 2 3 4 5 6 7 8 9 10 11 42
```

The principles of vectorisation, recycling rule, and implicit coercion are all in place:

```
x[c(TRUE, FALSE)] <- c("a", "b", "c")
print(x)
## [1] "a" "2" "b" "4" "c" "6" "a" "8" "b" "10" "c" "42"
```

Long story long: first, to ensure that the new content
can be poured into the old wineskin, R coerced the numeric
vector to a character one.
Then, every second element therein, a total of six items,
was replaced by a recycled version of the replacement sequence
of length three. Finally, the name `x`

was rebound to such
a brought-forth object and the previous one became forgotten.

Note

For more details on replacement functions in general, see Section 9.3.6. Such operations alter the state of the object they are called on (quite rare a behaviour in functional languages).

Replace missing values in a given numeric vector with the arithmetic mean of its well-defined observations.

### 5.3.2. Modifying lists¶

List contents can be altered as well. For modifying individual elements,
the safest option is to use the replacement version
of the `**[[**` operator:

```
y <- list(a=1, b=1:2, c=1:3)
y[[1]] <- 100:110
y[["c"]] <- -y[["c"]]
print(y)
## $a
## [1] 100 101 102 103 104 105 106 107 108 109 110
##
## $b
## [1] 1 2
##
## $c
## [1] -1 -2 -3
```

The replacement version
of `**[**` modifies a whole sub-list:

```
y[1:3] <- list(1, c("a", "b", "c"), c(TRUE, FALSE))
print(y)
## $a
## [1] 1
##
## $b
## [1] "a" "b" "c"
##
## $c
## [1] TRUE FALSE
```

Moreover:

```
y[1] <- list(1:10) # replace one element with one object
y[-1] <- 10:11 # replace two elements with two singletons
print(y)
## $a
## [1] 1 2 3 4 5 6 7 8 9 10
##
## $b
## [1] 10
##
## $c
## [1] 11
```

Note

Let `i`

be a vector of positive indexes of elements to be modified.
Overall, calling “`y[i] <- z`

” behaves as if we wrote:

`y[[ i[1] ]] <- z[[1]]`

,`y[[ i[2] ]] <- z[[2]]`

,`y[[ i[3] ]] <- z[[3]]`

,

and so forth.

Furthermore, `z`

(but not `i`

) will be recycled when necessary.
In other words, we retrieve `z[[j %% length(z)]]`

for consecutive `j`

s
from 1 to the length of `i`

.

Reflect on the results of the following expressions:

`y[1] <-`

**c**`("a", "b", "c")`

,`y[[1]] <-`

**c**`("a", "b", "c")`

,`y[[1]] <-`

**list**`(c("a", "b", "c"))`

,`y[1:3] <-`

**c**`("a", "b", "c")`

,`y[1:3] <-`

**list**`(c("a", "b", "c"))`

,`y[1:3] <- "a"`

,`y[1:3] <-`

**list**`("a")`

,`y[`

**c**`(1, 2, 1)] <-`

**c**`("a", "b", "c")`

.

Important

Setting a list item to `NULL`

removes it from the list completely.

```
y <- list(1, 2, 3, 4)
y[1] <- NULL # removes the first element (i.e., 1)
y[[1]] <- NULL # removes the first element (i.e., now 2)
y[1] <- list(NULL) # sets the first element (i.e., now 3) to NULL
print(y)
## [[1]]
## NULL
##
## [[2]]
## [1] 4
```

The same notation convention is used for dropping object attributes; see Section 9.3.6.

### 5.3.3. Inserting new elements¶

New elements can be pushed at the end of the vector easily[3].

```
(x <- 1:5)
## [1] 1 2 3 4 5
x[length(x)+1] <- 6 # insert at the end
print(x)
## [1] 1 2 3 4 5 6
x[10] <- 10 # insert at the end but add more items
print(x)
## [1] 1 2 3 4 5 6 NA NA NA 10
```

The elements to be inserted can be named as well:

```
x["a"] <- 11 # still inserts at the end
x["z"] <- 12
x["c"] <- 13
x["z"] <- 14 # z is already there; replace
print(x)
## a z c
## 1 2 3 4 5 6 NA NA NA 10 11 14 13
```

Note that `x`

was not equipped with the `names`

attribute before.
The unlabelled elements were assigned blank labels (empty strings).

Note

It is not possible to
insert new elements at the beginning or in the middle of
a sequence, at least not with the index operator.
By writing “`x[3:4] <- 1:5`

” we do not replace two elements
in the middle with five other ones.
However, we can always use the **c** function to slice
parts of the vector and intertwine them with some new content:

```
x <- seq(10, 100, 10)
x <- c(x[1:2], 1:5, x[5:7])
print(x)
## [1] 10 20 1 2 3 4 5 50 60 70
```

## 5.5. Preserving and losing attributes¶

Attributes are conceived of as *extra* data. It is thus up to a function’s
authors what they will decide to do with them.
Generally, it is safe to assume that much thought has been put into
the design of base R functions. Oftentimes, they behave fairly reasonably.
This is why we are now going to spend some time now exploring
their approaches to the handling of attributes.

Namely, for functions and operators that aim at transforming vectors passed as their inputs, the assumed strategy may be to:

ignore the input attributes completely,

equip the output object with the same set of attributes, or

take care only of a few special attributes, such as

`names`

, if that makes sense.

Below we explore some common patterns; see also Section 1.3 of [68].

### 5.5.1. **c**¶

First, **c** drops[5] all
attributes except `names`

:

```
(x <- structure(1:4, names=c("a", "b", "c", "d"), attrib1="<3"))
## a b c d
## 1 2 3 4
## attr(,"attrib1")
## [1] "<3"
c(x) # only `names` are preserved
## a b c d
## 1 2 3 4
```

We can therefore end up calling this function chiefly for this nice
side effect. Also, recall that **unname** drops the labels.

```
unname(x)
## [1] 1 2 3 4
## attr(,"attrib1")
## [1] "<3"
```

### 5.5.2. **as.something**¶

**as.vector**, **as.numeric**, and similar drop all attributes
in the case where the output is an atomic vector,
but it might not necessarily do so in other cases
(because they are S3 generics; see Chapter 10).

```
as.vector(x) # drops all attributes if x is atomic
## [1] 1 2 3 4
```

### 5.5.3. Subsetting¶

Subsetting with `**[**` (except where the indexer is not given)
drops all attributes but `names`

(as well as `dim`

and `dimnames`

;
see Chapter 11), which is adjusted accordingly:

```
x[1] # subset of labels
## a
## 1
x[[1]] # this always drops the labels (makes sense, right?)
## [1] 1
```

The replacement version of the index operator modifies the values in an existing vector whilst preserving all the attributes. In particular, skipping the indexer replaces all the elements:

```
y <- x
y[] <- c("u", "v") # note that c("u", "v") has no attributes
print(y)
## a b c d
## "u" "v" "u" "v"
## attr(,"attrib1")
## [1] "<3"
```

### 5.5.4. Vectorised functions¶

Vectorised unary functions tend to copy all the attributes.

```
round(x)
## a b c d
## 1 2 3 4
## attr(,"attrib1")
## [1] "<3"
```

Binary operations are expected to get the attributes from the longer input. If they are of equal sizes, the first argument is preferred to the second.

```
y <- structure(c(1, 10), names=c("f", "g"), attrib1=":|", attrib2=":O")
y * x # x is longer
## a b c d
## 1 20 3 40
## attr(,"attrib1")
## [1] "<3"
y[c("h", "i")] <- c(100, 1000) # add two new elements at the end
y * x
## f g h i
## 1 20 300 4000
## attr(,"attrib1")
## [1] ":|"
## attr(,"attrib2")
## [1] ":O"
x * y
## a b c d
## 1 20 300 4000
## attr(,"attrib1")
## [1] "<3"
## attr(,"attrib2")
## [1] ":O"
```

Also, Section 9.3.6 mentions a way to copy all attributes from one object to another.

Important

Even in base R, the foregoing rules are not enforced strictly. We consider them inconsistencies that should be, for the time being, treated as features (with which we need to learn to live as they have not been fixed for years, but hope springs eternal).

As far as third-party extension packages are concerned, suffice it to say that a lot of R programmers do not know what attributes are whatsoever. It is always best to refer to the documentation, perform a few experiments, and/or manually ensure the preservation of the data we care about.

Check what attributes are preserved by **ifelse**.

## 5.6. Exercises¶

Answer the following questions (contemplate first, then use R to find the answer).

What is the result of

`x[`

**c**`()]`

? Is it the same as`x[]`

?Is

`x[`

**c**`(1, 1, 1)]`

equivalent to`x[1]`

?Is

`x[1]`

equivalent to`x["1"]`

?Is

`x[`

**c**`(-1, -1, -1)]`

equivalent to`x[-1]`

?What does

`x[`

**c**`(0, 1, 2, NA)]`

do?What does

`x[0]`

return?What does

`x[1, 2, 3]`

do?What about

`x[`

**c**`(0, -1, -2)]`

and`x[`

**c**`(-1, -2, NA)]`

?Why

`x[NA]`

is so significantly different from`x[`

**c**`(1, NA)]`

?What is

`x[`

**c**`(FALSE, TRUE, 2)]`

?What will we obtain by calling

`x[x<`

**min**`(x)]`

?What about

`x[`

**length**`(x)+1]`

?Why

`x[`

**min**`(y)]`

is most probably a mistake? What could it mean? How can it be fixed?Why cannot we mix indexes of different types and write

`x[`

**c**`(1, "b", "c", 4)]`

? Or can we?Why would we call

**as.vector**`(`

**na.omit**`(x))`

instead of just**na.omit**`(x)`

?What is the difference between

**sort**and**order**?What is the type and the length of the object returned by a call to

**split**`(a, u)`

? What about**split**`(a,`

**c**`(u, v))`

?How to get rid of the seventh element from a list of ten elements?

How to get rid of the seventh, eight, and ninth elements from a list with ten elements?

How to get rid of the seventh element from an atomic vector of ten elements?

If

`y`

is a list, by how many elements “`y[`

**c**`(`

**length**`(y)+1,`

**length(y)**`+1,`

**length**`(y)+1)] <-`

**list**`(1, 2, 3)`

” will extend it?What is the difference between

`x[x>0]`

and`x[`

**which**`(x>0)]`

?

If `x`

is an atomic vector of length `n`

, `x[5:n]`

obviously
extracts everything from the fifth element to the end.
Does it, though? Check what happens when `x`

is of length less than five,
including 0. List different ways to correct this expression
so that it makes (some) sense in the case of shorter vectors.

Similarly, `x[`

**length**`(x) + 1 - 5:1]`

is supposed to return the last five elements in `x`

. Propose a few
alternatives that are correct also for short `x`

s.

Given a numeric vector, fetch its five largest elements. Ensure the code works for vectors of length less than five.

We can compute a *trimmed* mean of some `x`

by setting the `trim`

argument to the **mean** function.
Compute a similar robust estimator of location – the \(p\)-winsorised mean,
\(p\in[0, 0.5]\) defined as the arithmetic mean of all elements
in `x`

clipped to the \([Q_p, Q_{1-p}]\) interval,
where \(Q_p\) is the vector’s \(p\)-quantile;
see **quantile**.
For example, if `x`

is (8, 5, 2, 9, 7, 4, 6, 1, 3),
we have \(Q_{0.25}=3\) and \(Q_{0.75}=7\) and hence the \(0.25\)-winsorised mean
will be equal to the arithmetic mean of (7, 5, 3, 7, 7, 4, 6, 3, 3).

Let `x`

and `y`

be two vectors of the same length, \(n\), and no ties.
Implement the formula for the Spearman rank correlation coefficient:

where \(d_i=r_i-s_i\), \(i=1,\dots,n\),
and \(r_i\) and \(s_i\) denote the rank of \(x_i\) and \(y_i\), respectively;
see also **cor**.

(*)
Given two vectors `x`

and `y`

both of length \(n\),
a call to **approx**`(x, y, ...)`

can be used to
interpolate linearly between
the points \((x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\).
We can use it to generate new \(y\)s for previously
unobserved \(x\)s (somewhere “in-between” the data we already have).
Moreover, **spline**`(x, y, ...)`

can perform a cubic
spline interpolation, which is smoother; see Figure 5.2.

```
x <- c(1, 3, 5, 7, 10)
y <- c(1, 15, 25, 6, 0)
x_new <- seq(1, 10, by=0.25)
y_new1 <- approx(x, y, xout=x_new)[["y"]]
y_new2 <- spline(x, y, xout=x_new)[["y"]]
plot(x, y, ylim=c(-10, 30)) # the points to interpolate between
lines(x_new, y_new1, col="black", lty="solid") # linear interpolation
lines(x_new, y_new2, col="darkred", lty="dashed") # cubic interpolation
legend("topright", legend=c("linear", "cubic"),
lty=c("solid", "dashed"), col=c("black", "darkred"), bg="white")
```

Using a call to one of the above, impute missing data in
`euraud-20200101-20200630.csv`

,
e.g., the blanks in `(0.60, 0.62, NA, 0.64, NA, NA, 0.58)`

should
be filled so as to obtain `(0.60, 0.62, 0.63, 0.64, 0.62, 0.60, 0.58)`

.

Given some `1`

≤ `from`

≤ `to`

≤ `n`

, use **findInterval**
to generate a logical vector of length `n`

with `TRUE`

elements only at indexes between `from`

and `to`

, inclusive.

Implement expressions that give rise to the same results
as calls to **which**, **which.min**,
**which.max**, and **rev** functions.
What is the difference between `x[x>y]`

and
`x[`

**which**`(x>y)]`

?
What about **which.min**`(x)`

vs
**which**`(x == `

**min**`(x))`

?

Given two equal-length vectors `x`

and `y`

,
fetch the value from the former that corresponds
to the smallest value in the latter.
Write three versions of such an expression, each dealing with potential
ties in `y`

differently. For example:

```
x <- c("a", "b", "c", "d", "e", "f")
y <- c( 3, 1, 2, 1, 1, 4)
```

It should choose the first (`"b"`

), last (`"e"`

), or random
element from `x`

fulfilling the above property
(`"b"`

, `"d"`

, or `"e"`

with equal probability).
Make sure your code works for `x`

being of the type `character`

or `numeric`

as well as an empty vector.

Implement an expression that yields the same result as
**duplicated**`(x)`

for a numeric vector `x`

, but using **diff** and **order**.

Based on **match** and **unique**,
implement your versions of **union**`(x, y)`

,
**intersect**`(x, y)`

, **setdiff**`(x, y)`

,
**is.element**`(x, y)`

, and **setequal**`(x, y)`

for `x`

and `y`

being nonempty numeric vectors.