2. Numeric Vectors

The open-access textbook Deep R Programming by Marek Gagolewski is, and will remain, freely available for everyone’s enjoyment (also in PDF). It is a non-profit project. This book is still a work in progress. Beta versions of Chapters 1–12 are already complete, but there will be more. In the meantime, any bug/typos reports/fixes are appreciated. Although available online, it is a whole course; it should be read from the beginning to the end. Refer to the Preface for general introductory remarks. Also, check out my other book, Minimalist Data Wrangling with Python [20].

In this chapter, we discuss the uttermost common operations on numeric vectors. They are so fundamental that we will also find them in other scientific computing environments, including Python with NumPy or TensorFlow, Julia, MATLAB, GNU Octave, or Scilab.

At first blush, the number of functions we are going to explore may seem quite large. Still, the reader is kindly asked to place some trust (a rare thing these days) in yours truly. It is because our selection is comprised only of the most representative and educational amongst the plethora of possible choices. More complex building blocks can either be reduced to a creative combination of the former or be easily found – should the need arise – in a number additional packages or libraries (e.g., the GNU GSL [23]).

A solid understanding of base R programming is necessary for the effective dealing with the popular packages (such as data.table, dplyr, or caret). Most importantly, base R’s API is stable, hence the code we write today will most likely work the same way in 10 years. This is often not the case when we rely on third-party add-ons.

In the sequel, we will be advocating a minimalist, keep-it-simple approach to the art of programming of data processing pipelines, one that is a good balance between “doing it all by oneself”, “minimising the information overload”, “being lazy”, and “standing on the shoulders of giants”.

Note

The exercises that we suggest below are all self-contained, unless explicitly stated otherwise. The use of language constructs that are yet to be formally introduced (in particular, if, for, and while which we will explain in Chapter 8) is not only unnecessary, but discouraged. Moreover, we recommend against taking shortcuts by looking up partial solutions on the internet. Rather, to get the most out of this course, the reader should be seeking relevant information within the current and preceding chapters as well as the R help system.

2.1. Creating Numeric Vectors

2.1.1. Numeric Constants

The simplest numeric vectors are those of length one:

-3.14
## [1] -3.14
1.23e-4
## [1] 0.000123

The latter is in what we call the scientific notation which is convenient means of entering numbers of very large or small order of magnitude. Here, “e” stands for “… times 10 to the power of…”. Therefore, 1.23e-4 is equal to \(1.23\times 10^{-4}=0.000123\). In other words, given \(1.23\), we move the decimal separator by 4 digits towards the left.

In real life, some information items may be inherently or temporarily missing, unknown, or Not Available. R is data processing-oriented, hence it is equipped with a special indicator:

NA_real_  # numeric NA (missing value)
## [1] NA

This is similar to the Null marker in database query languages such as SQL. Note that NA_real_ is displayed simply as “NA”, chiefly for readability.

Moreover, Inf denotes the infinity (\(\infty\); a value that is larger than the largest representable double precision – 64 bit – floating point number) and NaN stands for not-a-number (it is returned as the result of some illegal operations, e.g., \(0/0\) or \(\infty-\infty\)).

2.1.2. Concatenating Vectors with c

Let us provide some ways to create numeric vectors with possibly more than 1 element.

First, the c function we introduced in the previous chapter, can be used to combine (concatenate) many numeric vectors, each of any length, so as to form a single object:

c(1, 2, 3)  # 3 vectors of length 1  ->  1 vector of length 3
## [1] 1 2 3
c(1, c(2, NA_real_, 4), 5, c(6, c(7, Inf)))
## [1]   1   2  NA   4   5   6   7 Inf

Note

Running help("c"), we will see that its usage is like “c(...)”. In the current context, this means that the c function takes an arbitrary number of arguments. In Section 9.5.6 we will study the dot-dot-dot (ellipsis) parameter in more detail.

2.1.3. Repeating Entries with rep

Second, rep replicates the elements in a given vector a given number of times.

rep(1, 5)
## [1] 1 1 1 1 1
rep(c(1, 2, 3), 4)
##  [1] 1 2 3 1 2 3 1 2 3 1 2 3

In the second case, the whole vector (1, 2, 3) has been recycled (tiled) four times. Interestingly, if the second argument was a vector of the same length as the first one, the behaviour would be quite different:

rep(c(1, 2, 3), c(2, 1, 4))
## [1] 1 1 2 3 3 3 3
rep(c(1, 2, 3), c(4, 4, 4))
##  [1] 1 1 1 1 2 2 2 2 3 3 3 3

Here, each element is repeated the corresponding number of times.

If we call help("rep"), we will come across the notion like “rep(x, ...)” in the Usage section. Unfortunately, it is rather peculiar, but reading further we discover the dot-dot-dot stands for one of the following further parameters (see the Arguments section):

  • times,

  • length.out,

  • each.

So far, we have been playing with times, which is listed second in the parameter list (after x – the vector whose elements are to be repeated).

Important

It turns out that the following function calls are all equivalent:

rep(c(1, 2, 3), 4)   # positional matching of arguments: `x`, then `times`
rep(c(1, 2, 3), times=4)    # `times` is the second argument
rep(x=c(1, 2, 3), times=4)  # keyword arguments of the form name=value
rep(times=4, x=c(1, 2, 3))  # keyword arguments can be given in any order
rep(times=4, c(1, 2, 3))    # mixed positional and keyword arguments

We can also pass each or length.out (a dot has no special meaning in R; see Section 2.2), but their names should be mentioned explicitly:

rep(c(1, 2, 3), length.out=7)
## [1] 1 2 3 1 2 3 1
rep(c(1, 2, 3), each=3)
## [1] 1 1 1 2 2 2 3 3 3
rep(c(1, 2, 3), length.out=7, each=3)
## [1] 1 1 1 2 2 2 3

Note

Whether it was a good programming practice to actually implement a range of varied behaviours inside a single function is a matter of taste. On the one hand, in all of the examples above, we do repeat the input elements somehow, so remembering just one function name is really convenient. Nevertheless, a drastic change in the repetition pattern depending, e.g., on the length of the times argument can be bug-prone. Anyway, we have been warned[1].

Zero-length vectors are also possible:

rep(c(1, 2, 3), 0)
## numeric(0)

Even though their handling might be a little tricky (compare Chapter 9), we will see later that they are useful in contexts like “create an empty data frame with a specific column structure”.

2.1.4. Generating Arithmetic Progressions with seq and `:`

Third, we can call the seq function to create a sequence of equally-spaced numbers (on a linear scale, i.e., an arithmetic progression).

seq(1, 15, 2)
## [1]  1  3  5  7  9 11 13 15

Reading the function’s help page, we note that it has the following parameters: from, to, by, length.out, amongst others.

Thus, the above call is equivalent to:

seq(from=1, to=15, by=2)
## [1]  1  3  5  7  9 11 13 15

Note that to actually means “up to”:

seq(from=1, to=16, by=2)
## [1]  1  3  5  7  9 11 13 15

We can also pass length.out instead of by. In such a case, the increments or decrements will be computed via the formula ((to - from)/(length.out - 1)); this default value is reported in the Usage section in help("seq").

seq(1, 0, length.out=5)
## [1] 1.00 0.75 0.50 0.25 0.00

Also, this:

seq(length.out=5)  # default `from` is 1
## [1] 1 2 3 4 5

Arithmetic progressions with step equal to 1 or -1 can also be generated via the `:` operator.

1:10    # seq(1, 10) or seq(1, 10, 1)
##  [1]  1  2  3  4  5  6  7  8  9 10
-1:10   # seq(-1, 10) or seq(-1, 10, 1)
##  [1] -1  0  1  2  3  4  5  6  7  8  9 10
-1:-10  # seq(-1, -10) or seq(-1, -10, -1)
##  [1]  -1  -2  -3  -4  -5  -6  -7  -8  -9 -10

Note the order of precedence of this operator: “-1:10” means “(-1):10” and not “-(1:10)”; compare Section 2.4.3.

Exercise 2.1

Take a look at the manual page of seq_along and seq_len and determine whether they can easily be done without, having seq[2] at hand.

2.1.5. Generating Pseudorandom Numbers

We can also generate sequences drawn independently from a range of univariate probability distributions.

runif(7)  # uniform U(0, 1)
## [1] 0.287578 0.788305 0.408977 0.883017 0.940467 0.045556 0.528105
rnorm(7)  # normal N(0, 1)
## [1]  1.23950 -0.10897 -0.11724  0.18308  1.28055 -1.72727  1.69018

These correspond to seven pseudorandom deviates following the uniform distribution on the unit interval (i.e., (0, 1)) and the standard normal distribution (i.e., with expectation 0 and standard deviation 1), respectively; compare Figure 2.3.

For more named distribution classes (frequently occurring in various real-world statistical modelling exercises), see Section 2.3.4.

Another useful function samples a number of values from a given vector, either with or without replacement:

sample(1:10, 20, replace=TRUE)  # 20 with replacement (allow repetitions)
##  [1]  3  3 10  2  6  5  4  6  9 10  5  3  9  9  9  3  8 10  7 10
sample(1:10, 5, replace=FALSE)  # 5 without replacement (do not repeat)
## [1] 9 3 4 6 1

The distribution of the sampled values does not need to be uniform; the prob argument may be fed with a vector of the corresponding probabilities. For example, here are 20 independent realisations of the random variable \(X\) such that \(\Pr(X=0)=0.9\) (the probability that we obtain 0 is equal to 90%) and \(\Pr(X=1)=0.1\):

sample(0:1, 20, replace=TRUE, prob=c(0.9, 0.1))
##  [1] 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

Note

If n is a single number (a numeric vector of length 1), then sample(n, ...) is equivalent to sample(1:n, ...). Similarly, seq(n) is a synonym for seq(1, n) or seq(1, length(n)), depending on the length of n. This is a dangerous behaviour which can occasionally backfire and lead to bugs (check what happens when n is, e.g., 0). Nonetheless, we have been warned and from now on are going to be extra careful (but are we really?). Read more at help("sample") and help("seq").

Let us stress that the numbers we obtain are merely pseudorandom, because they are generated algorithmically. R uses the Mersenne-Twister MT19937 method [37] by default; see help("RNG") and [16, 24, 34]. By setting the seed of the random number generator, i.e., re-setting its state to a given one, we can obtain results that are reproducible.

set.seed(12345)  # seeds are specified with integers
sample(1:10, 5, replace=TRUE) # a,b,c,d,e
## [1]  3 10  8 10  8
sample(1:10, 5, replace=TRUE) # f,g,h,i,j
## [1]  2  6  6  7 10

Setting the seed to the one used previously gives:

set.seed(12345)
sample(1:10, 5, replace=TRUE) # a,b,c,d,e
## [1]  3 10  8 10  8

We did not(?) expect that! And now for something completely different:

set.seed(12345)
sample(1:10, 10, replace=TRUE) # a,b,c,d,e,f,g,h,i,j
##  [1]  3 10  8 10  8  2  6  6  7 10

Reproducibility is a crucial feature of each truly scientific experiment. The same initial condition (here: the same seed), leads to exactly the same outcomes.

Note

Some claim that the only unsuspicious seed is 42, but each programmer can have their own picks. Yours truly, for example, uses 123, 1234, and 12345 as well. When performing many runs of Monte Carlo experiments, it may be a good idea to call set.seed(i) in the i-th iteration of a simulation we are trying to program.

Anyhow, we should make sure that our seed settings are applied consistently across all our scripts. Otherwise, we might be accused of tampering with evidence. For instance, here is the ultimate proof that we are very lucky today:

set.seed(1679619)  # totally unsuspicious, right?
sample(0:1, 20, replace=TRUE)  # so random
##  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

This is exactly why reproducible scripts and auxiliary data should be published alongside all research reports or papers. Only open, transparent science can be fully trustworthy.

If set.seed is not called explicitly, and the random state is not restored from the previously saved R session (see Chapter 16), then the random generator is initialised based on the current wall time and the identifier of the running R instance (PID). This may give the impression that the numbers we generate are surprising.

In order to understand the “pseudo” part of the said randomness better, in Section 8.3, we will build a very simple random generator ourselves.

2.1.6. Reading Data with scan

The example text file named euraud-20200101-20200630.csv gives the EUR to AUD exchange rates (how many Australian Dollars can one buy for 1 Euro) from 1 January to 30 June 2020 (remember COVID-19?). Let us preview the first couple of lines:

# EUR/AUD Exchange Rates
# Source: Statistical Data Warehouse of the European Central Bank System
# https://www.ecb.europa.eu/stats/policy_and_exchange_rates/
# (provided free of charge)
NA
1.6006
1.6031
NA

The four first lines that begin with “#” merely serve as comments for us, humans; they should be ignored by the interpreter. The first “real” value, NA corresponds to 1 January (Wednesday; New Years Day; Forex markets were closed, hence a missing observation).

The scan function can be used to read all the inputs and convert them to a single numeric vector:

scan(paste0("https://github.com/gagolews/teaching-data/raw/",
    "master/marek/euraud-20200101-20200630.csv"), comment.char="#")
##  [1]     NA 1.6006 1.6031     NA     NA 1.6119 1.6251 1.6195 1.6193 1.6132
## [11]     NA     NA 1.6117 1.6110 1.6188 1.6115 1.6122     NA     NA 1.6154
## [21] 1.6177 1.6184 1.6149 1.6127     NA     NA 1.6291 1.6290 1.6299 1.6412
## [31] 1.6494     NA     NA 1.6521 1.6439 1.6299 1.6282 1.6417     NA     NA
## [41] 1.6373 1.6260 1.6175 1.6138 1.6151     NA     NA 1.6129 1.6195 1.6142
## [51] 1.6294 1.6363     NA     NA 1.6384 1.6442 1.6565 1.6672 1.6875     NA
## [61]     NA 1.6998 1.6911 1.6794 1.6917 1.7103     NA     NA 1.7330 1.7377
## [71] 1.7389 1.7674 1.7684     NA     NA 1.8198 1.8287 1.8568 1.8635 1.8226
## [81]     NA     NA 1.8586 1.8315 1.7993 1.8162 1.8209     NA     NA 1.8021
## [91] 1.7967 1.8053 1.7970 1.8004     NA     NA 1.7790 1.7578 1.7596
##  [ reached getOption("max.print") -- omitted 83 entries ]

We used the paste0 function to concatenate two long (too long to fit a single line of code) strings to form a single URL; see Section 6.1.3.

We can also read the files located on our computer, for example:

scan("~/teaching-data/marek/euraud-20200101-20200630.csv",
    comment.char="#")

uses an absolute file path that starts at the user’s home directory, denoted “~”: yours truly’s case is /home/gagolews/.

Note

For portability reasons, we should use slashes, “/”, as path separators (but see help("file.path") and help(".Platform")). These are not only recognised by all Unix-like boxes but also other popular operating systems. Note that URLs (such as https://www.r-project.org/) feature slashes too.

Paths can also be relative to the current working directory, denoted “.”. It can be read via a call to getwd. Usually, it is the directory from where the R session has been started.

For instance, if the current working directory was /home/gagolews/teaching-data/marek, we could have written the file path equivalently as ./euraud-20200101-20200630.csv or even euraud-20200101-20200630.csv.

On as side note, ../ would denote the parent directory of the current working directory. For instance, ../r/iris.csv would be equivalent to /home/gagolews/teaching-data/r/iris.csv.

Exercise 2.2

Read the help page about scan. Take note of the following formal arguments and their meaning: dec, sep, what, comment.char, and na.strings.

Later we will discuss the read.table and read.csv, which are wrappers around scan that can be used to read tabular data. Note that write can be used to export an atomic vector’s contents to a text file.

Example 2.3

Figure 2.1 shows the graph of the aforementioned exchange rates, which was generated by calling:

plot(scan(paste0("https://github.com/gagolews/teaching-data/raw/",
    "master/marek/euraud-20200101-20200630.csv"), comment.char="#"),
    xlab="Day", ylab="EUR/AUD")
../_images/euraud-plot-1.png

Figure 2.1 EUR/AUD exchange rates from 2020-01-01 (day 1) to 2020-06-30 (day 182)

Somewhat misleadingly (and for the reasons that will become apparent later), the documentation of plot can be accessed by calling help("plot.default"). Read about, and experiment with, different values of the main, xlab, ylab, type, col, pch, cex, lty, and lwd arguments. More plotting routines will be discussed in Chapter 13.

2.2. Creating Named Objects

Often, the objects we bring forth will need to be memorised so that they can be referred to in further computations. The assignment operator, `<-`, can be used for this very purpose:

x <- 1:3  # creates a numeric vector and binds the name `x` to it

The now-named object can be recalled and dealt with as we please:

print(x)  # or just `x` in the R console
## [1] 1 2 3
sum(x)    # example operation: compute the sum of all elements in `x`
## [1] 6

Important

In R, all names are case-sensitive. Hence, x and X can coexist peacefully: when set, they refer to two different objects. Also, if we tried to call Print(x) above, we would get an error.

Typically, we will be using what we refer to as syntactic names (see Section 9.4.1 for an exception though). In the R help system (see help("make.names") and also help("Quotes")), we read: A syntactically valid name consists of letters, numbers and the dot or underline characters and starts with a letter or the dot not followed by a number. Names such as .2way are not valid, and neither are the reserved words. For the list of the latter, see help("Reserved").

A good name is self-explanatory and thus reader-friendly: patients, mean, and average_scores are way better (if they really are what they claim they are) than xyz123, crap, or spam. Also, it might not be such a bad idea to get used to denoting:

  • vectors with x, y, z,

  • matrices (and matrix-like objects) with A, B, …, X, Y, Z,

  • integer indexes with letters i, j, k, l,

  • object sizes with n, m, d, p or nx, ny, etc.,

especially when they are only of temporary nature (for storing some auxiliary results, iterating over collections of objects, etc.).

There are numerous naming conventions that we can adopt, but most often they are a matter of taste; snake_case, lowerCamelCase, UpperCamelCase, flatcase, or dot.case are equally good as long as they are used coherently (for instance, some use snake_case for vectors and UpperCamelCase for functions). It may even be the case that we have little choice but to adhere to the naming conventions agreed upon in the project we are about to contribute to.

Note

Let us stress that a dot, “.”, has no special meaning (however, see Chapter 10 and Chapter 16 for some asterisks); na.omit is as good a name as na_omit, naOmit, NAOMIT, naomit, and NaOmit. Users coming from some other (C, C++, Java, Python, etc.) programming languages will need to habituate themselves to this convention.

R, as a dynamic language, allows for introducing new variables at any time. Moreover, existing names can be re-bound to new values. For instance:

(y <- c(1, 10, 100))  # bracketed expression - printing not suppressed
## [1]   1  10 100
x <- y
print(x)
## [1]   1  10 100

Now x refers to a verbatim copy of y.

Note

Objects are automatically destroyed when there are no more names bound with them. In particular, by now the garbage collector should have got rid of the 1:3 vector begotten above (to which the name x was bound previously). See Section 14.4 for more details on memory management.

2.3. Vectorised Mathematical Functions

Mathematically, we will be denoting a given vector \(\boldsymbol{x}\) of length n as \((x_1, x_2, \dots, x_n)\). In other words, its i-th element is equal to \(x_i\).

Let us review some ubiquitous operations in numerical computing.

2.3.1. abs and sqrt

R implements vectorised versions of the most popular mathematical functions, e.g., abs (absolute value, \(|x|\)) and sqrt (square root, \(\sqrt{x}\)).

abs(c(2, -1, 0, -3, NA_real_))
## [1]  2  1  0  3 NA

Here, vectorised means that instead of being defined to act on a single numeric value, the function of interest is applied on each element in a vector. The i-th resulting item is a transformed version of the i-th input. If an input is a missing value, the corresponding output will be marked as “don’t know” as well.

Another example:

x <- c(4, 2, -1)
(y <- sqrt(x))
## Warning in sqrt(x): NaNs produced
## [1] 2.0000 1.4142    NaN

To attract our attention to the fact that computing the square root of a negative value yields a not-a-number, R generated an informative warning. A warning is not an error though: the result is being reckoned as usual.

Also the fact that the irrational \(\sqrt{2}\) is displayed[3] as 1.4142 does not mean that it is such a crude approximation to \(1.41421356237309504880168872420969807856967187537694\dots\); it is only rounded when printing, for aesthetic reasons. In fact, in Section 3.2.3 we will point out that the computer’s floating-point arithmetic allows for roughly 16 decimal digits precision (but we shall see that the devil is in the detail).

print(y, digits=16)  # display more significant figures
## [1] 2.000000000000000 1.414213562373095               NaN

2.3.2. Rounding

The following functions get rid of all or portions of fractional parts of numbers:

  • floor(x) (rounds down to the nearest integer, denoted \(\lfloor x\rfloor\)),

  • ceiling(x) (rounds up, denoted \(\lceil x\rceil\)),

  • trunc(x) (rounds towards zero), and

  • round(x, digits=0) (rounds to the nearest number with digits decimal digits).

For instance:

x <- c(7.0001, 6.9999, -4.3149, -5.19999, 123.4567, -765.4321, 0.5, 1.5, 2.5)
floor(x)
## [1]    7    6   -5   -6  123 -766    0    1    2
ceiling(x)
## [1]    8    7   -4   -5  124 -765    1    2    3
trunc(x)
## [1]    7    6   -4   -5  123 -765    0    1    2

Note

If we call help("round"), we will read that its usage is like round(x, digits=0), which means that the digits parameter is equipped with the default value of 0. In other words, if rounding to 0 decimal digits is what we need, the second argument can be omitted.

round(x)  # the same as round(x, 0)
## [1]    7    7   -4   -5  123 -765    0    2    2
round(x, 1)
## [1]    7.0    7.0   -4.3   -5.2  123.5 -765.4    0.5    1.5    2.5
round(x, -2)
## [1]    0    0    0    0  100 -800    0    0    0

2.3.3. Natural Exponential Function and Logarithm

Moreover:

  • exp(x) outputs the natural exponential function, \(e^x\), where the Euler’s number \(e\simeq 2.718\),

  • log(x, base=exp(1)) computes, by default, the natural logarithm of \(x\), \(\log_e x\) (which is most often denoted simply as \(\log x\)).

Recall that if \(x=e^y\), then \(\log_e x = y\), i.e., one is the inverse of the other.

log(c(0, 1, 2.7183, 7.3891, 20.0855))  # grows slowly
## [1] -Inf    0    1    2    3
exp(c(0, 1, 2, 3))                     # grows fast
## [1]  1.0000  2.7183  7.3891 20.0855

Note

These functions enjoy a number of very useful identities and inequalities, including:

  • \(\log(x\cdot y)=\log x + \log y\),

  • \(\log(x^y) = y\log x\),

  • \(e^{x+y} = e^x\cdot e^y\).

For more properties like these, take a glance at Chapter 4 of the freely available handbook [39].

For the logarithm to a different base, say \(\log_{10} x\), we can call:

log(c(0, 1, 10, 100, 1000, 1e10), 10)  # or log(..., base=10)
## [1] -Inf    0    1    2    3   10

Note that if \(\log_{b} x=y\), then \(x=b^y\), for any \(1\neq b>0\).

Note

Commonly, a logarithmic scale is used for variables that grow rapidly when expressed as functions of each other; see Figure 2.2.

x <- seq(0, 10, length.out=1001)
par(mfrow=c(1, 2))  # two plots in one figure (1 row, 2 columns)
plot(x, exp(x), type="l")
plot(x, exp(x), type="l", log="y")  # log-scale on the y-axis
../_images/ex-log-scale-1.png

Figure 2.2 Linear- vs log-scale on the y-axis

Note that \(e^x\) on the log-scale is just a straight line. Also, keep in mind that such a transformation of the axes can only be applied in the case of values strictly greater than 0.

2.3.4. Probability Distributions (*)

It should come as no surprise that R offers an extensive support for many univariate probability distribution families, including:

  • continuous distributions, which take values being arbitrary real numbers (over the whole possible range or in some interval):

    • *unif (uniform),

    • *norm (normal),

    • *exp (exponential),

    • *gamma (gamma, \(\Gamma\)),

    • *beta (beta, \(\mathrm{B}\)),

    • *lnorm (log-normal),

    • *t (Student),

    • *cauchy (Cauchy–Lorentz),

    • *chisq (chi-squared, \(\chi^2\)),

    • *f (Snedecor–Fisher),

    • *weibull (Weibull);

    with the prefix “*” being one of:

    • d” (probability density function, PDF),

    • p” (cumulative distribution function, CDF; or survival function, SF),

    • q” (quantile function, being the inverse of the CDF),

    • r” (generation of random deviates; already mentioned);

  • discrete distributions, i.e., those whose possible outcomes can be easily enumerated (e.g., some integers).

    • *binom (binomial),

    • *geom (geometric),

    • *pois (Poisson),

    • *hyper (hypergeometric),

    • *nbinom (negative binomial);

    here, prefixes “p” and “r” have the same meaning as above, however:

    • d” now gives the probability mass function (PMF),

    • q” yields the quantile function, but one that is defined as a generalised inverse of the CDF.

Each distribution is characterised by a set of underlying parameters. For instance, a normal distribution \(\mathrm{N}(\mu, \sigma)\) can be pinpointed by setting its expected value \(\mu\in\mathbb{R}\) and standard deviation \(\sigma>0\). In R, these two have been named mean and sd, respectively; see help("dnorm").

Note

The parametrisations assumed in R can be subtly different from what we know from statistical textbooks or probability courses. For example, the normal distribution can be parameterised based on either standard deviation or variance, and the exponential distribution can be defined via its expected value or the reciprocal thereof. We thus advise the reader to study carefully the documentation of help("dnorm"), help("dunif"), help("dexp"), help("dbinom"), and the like.

It is also worth to know the typical use cases of each of the distribution listed, e.g., a Poisson distribution can describe the probability of observing the number of independent events in a fixed time interval (e.g., the number of users downloading a copy of R from CRAN per hour), and an exponential distribution can model the time between such events; compare [17].

Exercise 2.4

A call to hist(x) draws a histogram, which can serve as an estimator of the underlying continuous probability density function of a given sample; see Figure 2.3 for an illustration.

par(mfrow=c(1, 2))  # 2 plots in 1 figure
# Uniform U(0, 1)
hist(runif(10000, 0, 1), col="white", probability=TRUE, main="")
x <- seq(0, 1, length.out=101)
lines(x, dunif(x, 0, 1), lwd=2)  # draw the true density function (PDF)
# Normal N(0, 1)
hist(rnorm(10000, 0, 1), col="white", probability=TRUE, main="")
x <- seq(-4, 4, length.out=101)
lines(x, dnorm(x, 0, 1), lwd=2)  # draw the PDF
../_images/ex-hist-1.png

Figure 2.3 Example histograms of some pseudorandom samples and the true underlying probability density functions: the uniform distribution on the unit interval (left) and the standard normal distribution (right)

Draw a histogram of some random samples of different sizes n from the following distributions:

  • rnorm(n, µ, σ) — normal \(\mathrm{N}(\mu, \sigma)\) with expected values \(\mu\in\{-1, 0, 5\}\) (i.e., \(\mu\) being equal to either \(-1\), \(0\), or \(5\); read “\(\in\)” as “belongs to the given set” or “in”) and standard deviations \(\sigma\in\{0.5, 1, 5\}\);

  • runif(n, a, b) — uniform \(\mathrm{U}(a, b)\) on the interval \((a, b)\) with \(a=0\) and \(b=1\) as well as \(a=-1\) and \(b=1\);

  • rbeta(n, α, β) — beta \(\mathrm{B}(\alpha, \beta)\) with \(\alpha,\beta\in\{0.5, 1, 2\}\);

  • rexp(n, λ) — exponential \(\mathrm{E}(\lambda)\) with rates \(\lambda\in\{0.5, 1, 10\}\);

Moreover, read about and play with the breaks, main, xlab, ylab, xlim, ylim, and col parameters; see help("hist").

Example 2.5

We roll a six-sided dice 12 times. Let \(C\) be a random variable denoting the number of cases where the “1” face is thrown. \(C\) follows a binomial distribution \(\mathrm{Bin}(n, p)\) with parameters \(n=12\) (the number of Bernoulli trials) and \(p=1/6\) (the probability of success in a single roll).

The probabilities that the number of “1”s rolled will be equal to 0, 1, …, 4 , i.e., \(P(C=0)\), \(P(C=1)\), …, \(P(C=4)\), respectively, can be computed based on the probability mass function (dbinom):

dbinom(0:4, 12, 1/6)  # probability mass function at 5 different points
## [1] 0.112157 0.269176 0.296094 0.197396 0.088828

On the other hand, the probability that we throw more than three “1”s, \(P(C > 3) = 1-P(C\le 3)\), can be determined by means of the cumulative distribution function (pbinom) or survival function (pbinom(..., lower.tail=FALSE)):

1-pbinom(3, 12, 1/6)  # pbinom(3, 12, 1/6, lower.tail=FALSE)
## [1] 0.12518

The smallest \(c\) such that \(P(C \le c)\ge 0.95\) can be computed based on the quantile function:

qbinom(0.5, 12, 1/6)
## [1] 2
pbinom(3:4, 12, 1/6)   # for comparison - 0.95 is in-between
## [1] 0.87482 0.96365

In other words, at least 95% of the time we will be observing no more than 4 successes.

Also here are some pseudorandom realisations of \(C\) – the number of “1”s in 30 simulations of 12 independent dice rolls each:

rbinom(30, 12, 1/6)
##  [1] 1 3 2 4 4 0 2 4 2 2 4 2 3 2 0 4 1 0 1 4 4 3 2 6 2 3 2 2 1 1

2.3.5. Special Functions (*)

Within mathematical formulae and across assorted application areas, certain functions appear more frequently than others. Hence, for the sake of notational brevity and computational precision, many of them have been assigned special names. For instance, the following may be mentioned in the definitions related to some of the probability distributions listed above:

  • gamma(x) for \(x>0\) computes \(\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \,dt\),

  • beta(a, b) for \(a,b>0\) yields \(B(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)} = \int_0^1 t^{a-1} (1-t)^{b-1}\,dt\).

Why do we have beta if it is merely a mix of gammas? A specific, tailored function should be faster and more precise than its DIY version; its underlying implementation does not have to involve any calls to gamma at all.

beta(0.25, 250)  # okay
## [1] 0.91213
gamma(0.25)*gamma(250)/gamma(250.25)  # not okay
## [1] NaN

The \(\Gamma\) function grows so rapidly that already gamma(172) yields Inf. It is due to the fact that a computer’s arithmetic is not infinitely precise; compare Section 3.2.3.

Special functions are plentiful; see the open-access [39] for one of the most definitive references (and also [2] for its predecessor). R package gsl [28] provides a vectorised interface to the famous GNU GSL [23] library, which implements many of them.

Exercise 2.6

The Pochhammer symbol, \((a)_x = \Gamma(a+x)/\Gamma(a)\), can be computed via a call to gsl::poch(a, x) (i.e., the poch function from the gsl package; see Section 7.3.1):

# call install.packages("gsl") first
library("gsl")  # load the package
poch(10, 3:6)   # calls gsl_sf_poch() from GNU GSL
## [1]    1320   17160  240240 3603600

Read the documentation of the corresponding gsl_sf_poch function in the GNU GSL manual available here.

And since you are there, do not hesitate to go through the list of all the other functions, including those related to statistics, permutations, combinations, and so forth.

Many functions also have their logarithm-of versions; see, e.g., lgamma and lbeta. Also, for instance, dnorm and dbeta has the log parameter. Its classical use case is the (numerical) maximum likelihood estimation, which involves the sums of the logarithms of densities.

2.4. Arithmetic Operations

2.4.1. Vectorised Arithmetic Operators

R features the following arithmetic operators:

  • `+` (addition) and `-` (subtraction),

  • `*` (multiplication) and `/` (division),

  • `%/%` (integer division) and `%%` (modulo, division remainder),

  • `^` (exponentiation; synonym: `**`).

They are all vectorised: they take two vectors on input and yield another vector in result.

c(1, 2, 3) * c(10, 100, 1000)
## [1]   10  200 3000

We note that the multiplication was performed in an elementwise fashion: the 1st element in the left vector was multiplied by the corresponding element in the right vector and the result has been stored in the 1st element of the output, then the 2nd element in the left… all right, we get the point.

Other operators are vectorised in the same manner:

0:10 + seq(0, 1, 0.1)
##  [1]  0.0  1.1  2.2  3.3  4.4  5.5  6.6  7.7  8.8  9.9 11.0
0:7 / rep(3, length.out=8)   # division by 3
## [1] 0.00000 0.33333 0.66667 1.00000 1.33333 1.66667 2.00000 2.33333
0:7 %/% rep(3, length.out=8) # integer division
## [1] 0 0 0 1 1 1 2 2
0:7 %% rep(3, length.out=8)  # division remainder
## [1] 0 1 2 0 1 2 0 1

Note that operations involving missing values also yield NAs:

c(1, NA_real_, 3, NA_real_) + c(NA_real_, 2, 3, NA_real_)
## [1] NA NA  6 NA

2.4.2. Recycling Rule

Some of the above statements can be written more concisely. When the operands are of different lengths, the shorter one is recycled (think: rep(y, length.out=length(x))) as many times as necessary.

0:7 / 3
## [1] 0.00000 0.33333 0.66667 1.00000 1.33333 1.66667 2.00000 2.33333
1:10 * c(-1, 1)
##  [1] -1  2 -3  4 -5  6 -7  8 -9 10
2 ^ (0:10)
##  [1]    1    2    4    8   16   32   64  128  256  512 1024

We call this the recycling rule.

If an operand cannot be recycled in its entirety, a warning[4] is generated, but the output is still available.

c(1, 10, 100) * 1:8
## Warning in c(1, 10, 100) * 1:8: longer object length is not a multiple of
## shorter object length
## [1]   1  20 300   4  50 600   7  80

Note

Some functions are also deeply vectorised, i.e., with respect to multiple arguments. For example,

runif(3, c(10, 20, 30), c(11, 22, 33))
## [1] 10.288 21.577 31.227

generates three random numbers uniformly distributed over the intervals \((10, 11)\), \((20, 22)\), and \((30, 33)\), respectively.

Also, pmin and pmax return the parallel minimum and maximum of the corresponding elements of the input vectors:

pmin(c(1, 2, 3, 4), c(4, 2, 3, 1))
## [1] 1 2 3 1
pmin(3, 1:5)
## [1] 1 2 3 3 3
pmax(0, pmin(1, c(0.25, -2, 5, -0.5, 0, 1.3, 0.99)))  # clipping to [0, 1]
## [1] 0.25 0.00 1.00 0.00 0.00 1.00 0.99

Note

Vectorisation and the recycling rule are perhaps most useful when applying binary operators on sequences of identical lengths or when performing vector-scalar (i.e., a sequence vs a single value) operations. However, there is much more: schemes like “every k-th element” appear in Taylor series expansions (multiply by c(-1, 1)), k-fold cross validation, etc.; see also Section 11.3.4 for use cases in matrix/tensor processing.

2.4.3. Operator Precedence

Apart from the seven binary arithmetic operators, other noteworthy, already mentioned ones include: the unary `-` (change of sign), `:` (sequence generation), and `<-` (assignment).

Expressions involving multiple operations need a set of rules governing the order of computations (unless we enforce it using round brackets). We have said that “-1:10” means “(-1):10” rather than “-(1:10)”. But what about, say, “1+1+1+1+1*0” or “3*2^0:5+10”?

Let us list the aforementioned operators in their order of precedence, from the least to the most binding (see also help("Syntax")):

  1. `<-` (right-to-left),

  2. `+` and `-`,

  3. `*` and `/`,

  4. `%%` and `%/%`,

  5. `:`,

  6. `+` and `-` (unary),

  7. `^` (right-to-left).

Hence, “-2^2/3+3*4” means “((-(2^2))/3)+(3*4)” and not, for example, -((2^(2/(3+3)))*4).

Note that `+` and `-`, `*` and `/`, as well as `%%` and `%/%` have the same priority. Expressions involving a series of operations in the same group, are evaluated left-to-right, with the exception of `^` and `<-`, which are performed from right to left.

Therefore:

  • 2*3/4*5” is equivalent to “((2*3)/4)*5”,

  • 2^3^4” is the same as “2^(3^4)” (which, mathematically, we would write as \(2^{3^4}=2^{81}\)),

  • x <- y <- 4*3%%8/2” binds both y and x with 6 and not x with the previous value of y.

And let us remember: when in doubt, we can always bracket a subexpression to make sure it is executed in the intended order (which can also increase readability of the code).

2.4.4. Accumulating

The `+` and `*` operators as well as the pmin and pmax functions implement elementwise operations that are applied on the corresponding elements taken from two given vectors:

\[\begin{split} \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{array} \right) + \left( \begin{array}{c} y_1 \\ y_2 \\ y_3 \\ \vdots \\ y_n \end{array} \right) = \left( \begin{array}{c} x_1+y_1 \\ x_2+y_2 \\ x_3+y_3 \\ \vdots \\ x_n+y_n \end{array} \right). \end{split}\]

However, we can also scan through all the values in a single vector and combine the successive elements that we inspect using the corresponding operation:

  • cumsum(x) gives the cumulative sum of the elements in a vector,

  • cumprod(x) computes the cumulative product,

  • cummin(x) yields the cumulative minimum,

  • cummax(x) generates the cumulative maximum.

The i-th element in the output vector will consist of the sum/product/min/max of the first i inputs:

\[\begin{split} \mathrm{cumsum}\left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{array} \right) = \left( \begin{array}{l} x_1 \\ x_1+x_2 \\ x_1+x_2+x_3 \\ \ \vdots\qquad\qquad\quad\ddots \\ x_1+x_2+x_3+\dots+x_n \end{array} \right). \end{split}\]

For example:

cumsum(1:8)
## [1]  1  3  6 10 15 21 28 36
cumprod(1:8)
## [1]     1     2     6    24   120   720  5040 40320
cummin(c(3, 2, 4, 5, 1, 6, 0))
## [1] 3 2 2 2 1 1 0
cummax(c(3, 2, 4, 5, 1, 6, 0))
## [1] 3 3 4 5 5 6 6

If we are interested only in the last cumulant, summarising all the inputs, we have the following functions at our disposal:

  • sum(x) computes the sum of elements in a vector, \(\sum_{i=1}^n x_i=x_1+x_2+\cdots+x_n\),

  • prod(x) outputs the product of all elements, \(\prod_{i=1}^n x_i=x_1 x_2 \cdots x_n\),

  • min(x) computes the minimum,

  • max(x) reckons the greatest value.

For example:

sum(1:8)
## [1] 36
prod(1:8)
## [1] 40320
min(c(3, 2, 4, 5, 1, 6, 0))
## [1] 0
max(c(3, 2, 4, 5, 1, 6, 0))
## [1] 6

Note

In Chapter 7, we will discuss the Reduce function, which generalises the above by allowing any binary operation to be propagated over a given vector.

Example 2.7

diff can be considered an inverse to cumsum: it computes the iterative difference. Namely, it subtracts the first two elements, then the 2nd from the 3rd one, the 3rd from the 4th, and so on. In other words, diff(x) gives \(\boldsymbol{y}\) such that \(y_i=x_{i+1}-x_i\).

x <- c(-2, 3, 6, 2, 15)
diff(x)
## [1]  5  3 -4 13
cumsum(diff(x))
## [1]  5  8  4 17
cumsum(c(-2, diff(x)))  # recreates x
## [1] -2  3  6  2 15

Thanks to diff, we can compute the daily changes to the EUR/AUD forex rates; see Figure 2.4.

aud <- scan(paste0("https://github.com/gagolews/teaching-data/raw/",
    "master/marek/euraud-20200101-20200630.csv"), comment.char="#")
aud_all <- na.omit(aud)  # remove all missing values
plot(diff(aud_all), type="s", ylab="Daily change [EUR/AUD]")
abline(h=0, lty="dotted")  # draw a horizontal line at y=0
../_images/euraud-diff-1.png

Figure 2.4 Iterative differences of the exchange rates (non-missing values only)

2.4.5. Aggregating

The above functions form the basis for some popular summary statistics[5] (sample aggregates), such as:

  • mean(x) gives the arithmetic mean, sum(x)/length(x),

  • var(x) yields the (unbiased) sample variance, sum((x-mean(x))^2)/(length(x)-1),

  • sd(x) is the standard deviation, sqrt(var(x)),

  • median(x) computes the sample median, i.e., the middle value in the sorted version of x.

For instance[6]:

x <- runif(1000)
c(min(x), mean(x), median(x), max(x), sd(x))
## [1] 0.00046535 0.49727780 0.48995025 0.99940453 0.28748391
Exercise 2.8

Let \(\boldsymbol{x}\) be any vector of length \(n\) with positive elements. Compute its geometric and harmonic mean, which are given by, respectively,

\[ \sqrt[n]{\prod_{i=1}^n x_i}=e^{\frac{1}{n} \sum_{i=1}^n \log x_i} \quad\text{ and }\quad \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}. \]

When solving exercises like this one, it does not really matter what data you apply these functions on (see, however, Section 9.3.4.2 for discussion). We are being abstract in the sense that the \(\boldsymbol{x}\) vector can be anything: from the one that features very accurate financial predictions that will help minimise inequity and make this world less miserable, through the data you have been collecting for the last the years in relation to your definitely-super-important PhD research, whatever your company asked you to crunch today, to something related to your hobby project that you enjoy doing after hours. Therefore, just test the above on something like “x <- runif(10)”, and move on.

All the aforementioned functions return a missing value if any of the input elements is unavailable. Luckily, they are equipped with the na.rm parameter on behalf of which we can request the removal of NAs.

aud <- scan(paste0("https://github.com/gagolews/teaching-data/raw/",
    "master/marek/euraud-20200101-20200630.csv"), comment.char="#")
c(min(aud), mean(aud), max(aud))
## [1] NA NA NA
c(min(aud, na.rm=TRUE), mean(aud, na.rm=TRUE), max(aud, na.rm=TRUE))
## [1] 1.6006 1.6775 1.8635

Note

In the documentation, we read that the usage of some of the aforementioned functions is like sum(..., na.rm=FALSE). prod, min, and max are defined similarly. They accept any number of input vectors, each of them can be of arbitrary length. Therefore, min(1, 2, 3), min(c(1,2,3)) as well as min(c(1,2),3) all return the same result.

However, we can also read that we have mean(x, trim=0, na.rm=FALSE, ...). This time, only one vector can be aggregated and any further arguments (except trim and na.rm) are ignored.

The extra flexibility (which we do not have to rely upon, ever) of the former group is due their being associative operations: it holds, e.g., \((2+3)+4 = 2+(3+4)\). Hence, the operations can be performed in any order, in any groups.

Also note that they are more primitive operations: it is mean that is based on sum, not vice versa.

2.5. Exercises

Exercise 2.9

Answer the following questions:

  • What is the meaning of the dot-dot-dot parameter in the definition of the c function?

  • We say that the round function is vectorised: what does that mean?

  • What do we mean by saying that multiplication operates element-by-element?

  • How does the recycling rule work when applying `+`?

  • How to (and why) set the seed of the pseudorandom number generator?

  • What is the difference between NA_real_ and NaN?

  • How are default arguments specified in the manual of, e.g., the round function?

  • Is a call to rep(times=4, x=1:5)” equivalent to rep(4, 1:5)?

  • List a few ways to generate a sequence like (-1, -0.75, -0.5, …, 0.75, 1).

  • Is “-3:5” the same as "-(3:5)"? What about the precedence of operators in expressions such as “2^3/4*5^6”, “5*6+4/17%%8”, and “1+-2^3:4”?

  • If x is a numeric vector of length \(n\) (for some \(n\ge 0\)), how many values will sample(x) output?

  • Does scan support reading directly from compressed archives, e.g., .csv.gz files?

When in doubt, refer back to the material discussed in this chapter and/or the R manual.

Exercise 2.10

The following code generates an example graph of arcsine and arccosine, whose preparation – thanks to vectorisation – is quite straightforward.

x <- seq(-1, 1, length.out=11)  # increase length.out for a smoother curve
plot(x, asin(x),                # asin() computed for 11 points
    type="l",                   # lines
    ylim=c(-pi/2, pi),          # y axis limits like c(y_min, y_max)
    ylab="asin(x), acos(x)")    # y axis label
lines(x, acos(x), col="red", lty="dashed")  # adds to the current plot
legend("topright", c("asin(x)", "acos(x)"),
    lty=c("solid", "dashed"), col=c("black", "red"), bg="white")

Inspired by the above, plot the following functions: \(|\sin x^2|\), \(\left|\sin |x|\right|\), \(\sqrt{\lfloor x\rfloor}\), and \(1/(1+e^{-x})\). Recall that the documentation of plot can be viewed by calling help("plot.default").

Exercise 2.11

It can be shown that:

\[ 4 \sum_{i=1}^n \frac{(-1)^{i+1}}{2i-1} = 4\left( \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \right) \]

slowly converges to \(\pi\) as \(n\) approaches \(\infty\). Compute the above for \(n=1{,}000{,}000\) and \(n=1{,}000{,}000{,}000\) using the vectorised functions and operators discussed in this chapter, making use of the recycling rule as much as possible.

Exercise 2.12

Let x and y be two vectors of identical lengths \(n\), say:

x <- rnorm(100)
y <- 2*x+10+rnorm(100, 0, 0.5)

Compute the Pearson linear correlation coefficient given by:

\[ r = \frac{ \sum_{i=1}^n \left(x_i-\frac{1}{n}\sum_{j=1}^n x_j \right) \left(y_i-\frac{1}{n}\sum_{j=1}^n y_j \right) }{ \sqrt{\sum_{i=1}^n \left(x_i-\frac{1}{n}\sum_{j=1}^n x_j \right)^2}\, \sqrt{\sum_{i=1}^n \left(y_i-\frac{1}{n}\sum_{j=1}^n y_j \right)^2} }. \]

To make sure you have come up with a correct implementation, compare your result to a call to the built-in cor(x, y).

Exercise 2.13

(*) Look up on the internet an R package that features functions to compute the 5-day moving (rolling) average and median of a given vector. Apply them on the EUR/AUD currency exchange data and plot thus obtained smoothened versions of the time series.

Exercise 2.14

(**) Compute the \(k\)-moving average using a call to convolve(..., type="filter").

In the next chapter we will study operations that involve logical values.