3  Logical vectors

3.1 Simplifying predicates

3.3.5. Simplifying predicates

Assuming that a, b, and c are numeric vectors, simplify the following expressions:

!(b>a & b<c),

One of the sub-conditions has to be FALSE for the result to be TRUE:

!b>a | !b<c

!(a>=b & b>=c & a>=c),

Same again:

!a>=b | !b>=c | !a>=c

a>b & a<c | a<c & a>d,

I think I complicated rather than simplified, but:

a<c & (a<b | a>d)

a>b | a<=b,

Uhh, TRUE?

a<=b & a>c | a>b & a<=c,

Hmm, not sure this one can be simplified.

a<=b & (a>c | a>b) & a<=c,

a is less than or equal to b AND less than or equal to c, but must be greater than at least one of b OR c, so always: FALSE.

!all(a > b & b < c).

If all(p) is equivalent to !any(!p) then isn’t !all(p) equal to any(!p)?

any(!a>b | !b<c)

3.2 Nesting ifelse

3.4. Choosing elements with ifelse

3.3 ifelse() to generate Guassian mixtures

we generated a variate from the normal distribution that has the expected value of −2 with probability 20%, and from the one with the expectation of 3 otherwise. Thus inspired, generate the Gaussian mixtures:

\(\frac{2}{3}X + \frac{1}{3}Y\), where \(X ~ N(100, 16)\) and \(Y ~ N(116, 8)\)

n <- 10000

z <- ifelse(runif(n) <= (2/3), rnorm(n , 100, 16), rnorm(n, 116, 8))

hist(z, breaks = 101, probability = TRUE, main = "", col = "white")

\(0.3X + 0.4Y + 0.3Z\), where \(X\) ~ \(N(-10, 2)\), \(Y\) ~ \(N(0,2)\), and $Z $ ~ \(N(10,2)\)

r <- runif(n)

d <- ifelse(
  r <= 0.2,
  rnorm(n, -10, 2),
  ifelse(r <= 0.7, rnorm(n, 0, 2), rnorm(n, 10, 2))
)

hist(d, breaks = 101, probability = TRUE, main = "", col = "white")

3.4 Exercises

3.5. Exercises

Answer the following questions.

Why the statement “The Earth is flat or the smallpox vaccine is proven effective” is obviously true?

Because at least one of those two statements is true, which is all you need with an OR.

What is the difference between NA and NA_real_?

NA is of type logical wheras NA_real_ is of type numeric, i.e. they indicate a missing value within those respective vector types.

Why is “FALSE & NA” equal to FALSE, but “TRUE & NA” is NA?

As long as one side of an & is FALSE, the expression will evaluate to FALSE. With “TRUE & NA” the expression could evaluate to either TRUE or FALSE depending on what the NA value is.

Why has ifelse(x>=0, sqrt(x), NA_real_) a tendency to generate warnings and how to rewrite it so as to prevent that from happening?

What is the interpretation of mean(x >= 0 & x <= 1)?

This calculates the proportion of vector x which lies between 0 and 1. mean coerces the logical result of the expression to numeric i.e. TRUE to 1 and FALSE to 0.

For some integer x and y, how to verify whether 0<x<100, 0<y<100, and x<y, all at the same time?

0<x & x<y & y<100

Mathematically, for all real \(x,y>0\), we have \(logxy=logx+logy\). Why then all(log(x*y) == log(x)+log(y)) can sometimes return FALSE? How to fix this?

Differences can occur in the results due to over or under accuracy in calculations using the double-precision floating point system.

Is x/y/z always equal to x/(y/z)? How to fix this?

# 1 / 0.1
0.1 / 0.1 / 0.1

[1] 10

# 0.1 / 1
0.1 / (0.1 / 0.1)

[1] 0.1

Clearly not as a result of order of operations. I’m not sure what is meant by fixing this.

What is the purpose of very specific functions such as log1p and expm1 (see their help page) and many others listed in, e.g., the GNU GSL library? Is our referring to them a violation of the beloved “do not multiply entities without necessity” rule?

Hmm. Well it covers a large number of common mathematical operations. In a certain sense it does some to conflict with the idea of being independent in the R environment. On the other hand it makes a lot of sense to me to rely on more libraries which are, presumably, thoroughly tested and more robust then trying to replicate all these maths functions. Using them makes it easy to write simpler vectorised R code and I’m sure it’s more efficient having the iteration at the C/C++ level than the R level.

If we know that x may be subject to error, how to test whether x>0 in a robust manner?

We can for instance use abs(x - 0) < e-8 to check for within an assumed error margin or more simply round round(x) > 0).

Is “y<-5” the same as “y <- 5” or rather “y < -5”?

It’s the same as the former, evaluated as assignment rather than less than negative 5.

3.5 Equality functions

What is the difference between all and isTRUE? What about ==, identical, and all.equal? Is the last one properly vectorised?

all checks if every element of a vector is TRUE, including coercion to TRUE, and isTRUE checks if the value passed to it is equal to the value TRUE.

# Are all values TRUE?
all(c(TRUE, TRUE))

[1] TRUE

# Is it a TRUE value?
isTRUE(c(TRUE, TRUE))

[1] FALSE

== checks for equality between each respective element of two vectors, return a result for each pair. identical checks if two objects are exactly equal, a single result for whether the entire thing is the same. all.equal() is similar to identical, but instead tests for near equality.

c(1, 2, 3) == c(1, 2, 3)

[1] TRUE TRUE TRUE

identical(c(1, 2, 3), c(1, 2, 3))

[1] TRUE

identical(c(1, 2, 3), c(1.1, 2.1, 3.1))

[1] FALSE

all.equal(c(1, 2, 3), c(1.1, 2.1, 3.1), tolerance = 0.1)

[1] TRUE

3.6 Entropy loss

Compute the cross-entropy loss between a numeric vector p with values in the interval (0,1) and a logical vector y, both of length n (you can generate them randomly or manually, it does not matter, it is just an exercise):

\(L(p, y) = \frac{1}{n} \sum_{i=1}^{n}l_i\)

where

\[ \begin{equation} l_i = \begin{cases} - log (p_i) & \text{ if } y_i \text{ is TRUE} \\ - log (1 - p_i) & \text{ if } y_i \text{ is FALSE} \end{cases} \end{equation} \]

cross_entropy_loss <- function(p, y) {
  stopifnot(is.numeric(p))
  stopifnot(is.logical(y))
  
  n <- length(p)
  
  stopifnot(n == length(y))
  
  (1 / n) * sum(ifelse(y, -log(p), -log(1-p)))
}

Interpretation: in classification problems, \(y_i ∈{FALSE,TRUE}\) denotes the true class of the i-th object (say, whether the i-th hospital patient is symptomatic) and \(p_i ∈(0,1)\) is a machine learning algorithm’s confidence that i belongs to class TRUE (e.g., how sure a decision tree model is that the corresponding person is unwell). Ideally, if \(y_i\) is TRUE, \(p_i\) should be close to 1 and to 0 otherwise. The cross-entropy loss quantifies by how much a classifier differs from the omniscient one. The use of the logarithm penalises strong beliefs in the wrong answer.

Interesting, so there should be no loss here:

p <- c(rep(0, 50), rep(1, 50))
y <- c(rep(FALSE, 50), rep(TRUE, 50))

cross_entropy_loss(p, y)

[1] 0

There should be some loss here, were particularly the middle of the vector will be “unsure” with \(p_i\) being far from the expected integer for the corresponding \(y_i\):

p <- seq(0, 1, length.out = 100)
y <- as.logical(round(p))

cross_entropy_loss(p, y)

[1] 0.3037506

And something completely off:

p <- 1
y <- FALSE

cross_entropy_loss(p, y)

[1] Inf

We did it! Marek advises to make sure the exercises have been tackled with ifelse() and base R vectorised operations, as opposed to if and using indexing to work on individual elements of vectors. So far so good, and I like it, it really does make things simple but effective. 🫡👍