## domingo, 31 de julio de 2016

### Reading R Code. The function Reduce. Folds

Among the essential higher-order functions in functional programming (we have already seen `map` and `filter`) `reduce` (aka. `fold`) is probably the most amazing and powerful. It is the Swiss-Army-knife in functional programmers' hands, sort of. I highly recommend this excellent exposition, about the power and flexibility of `fold`:

http://www.cs.nott.ac.uk/~pszgmh/fold.pdf

### What is `fold`?

Although `fold` deserves and probably requires several blog posts, I'll try an elementary presentation suitable to our purposes.

`fold` is essentially the functional abstraction that corresponds to the very structure of lists.

A list of whatever kind of values, say, type `X`, is either the empty list or a list constructed from an `X` and a list of `X`s, where by construction I mean the operation of `cons`ing `X` and a list of `X`s. For instance (in Lisp):

``````[1]> (cons 1 (list 2 3))
(1 2 3)``````

or in Haskell, where `:` is the Haskell equivalent to the Lispier `cons`:

``````Prelude> 1 : [2, 3]
[1,2,3]``````

Note the self-referential nature of the type description for lists. Any function over lists relies on this self-referential structure, and it will have a natural form, where the recursive call emanates naturally from the type description. So the sketch or template of a function for list of `X` looks like this [written now in Racket]:

``````(define (f-for-lox lox)
(cond [(empty? lox) (...)]
[else
(... (first lox)
(f-for-lox (rest lox)))]))
``````

On the other hand, and by stack space efficiency reasons, functions over lists can be written in such a way that the recursive call is in tail position (in a position that ensures that is the last call in the procedure). Observe that `f-for-lox`, the recursive call in the template above, is not in tail position; instead, the function ending up in place of the last three dots in that template will be in tail position.

I heartily recommend to watch these excellent videos, on which this exposition is mostly based, for a better understanding:

Such functions where the recursive call is in tail positions are known as tail-recursive functions and normally written with the aid of a local function that supplies an accumulator. A tail-recursive template for lists has this form:

``````(define (f-for-lox lox0)
(local [(define (f-for-lox-acc acc lox)
(cond [(empty? lox) (... acc)]
[else
(f-for-lox-acc (... acc (first lox))
(rest lox))]))]
(f-for-lox-acc ... lox0)))``````

Or this one, if we change the order of components in the recursive call:

``````(define (f-for-lox lox0)
(local [(define (f-for-lox-acc acc lox)
(cond [(empty? lox) (... acc)]
[else
(f-for-lox-acc (... (first lox) acc)
(rest lox))]))]
(f-for-lox-acc ... lox0)))``````

`fold` is the function that capture these two abstractions. Accordingly, there are two possible `fold` functions (the last one with two variants), `fold-right` that captures the structural recursive procedure, and `fold-left` that captures the tail-recursive one.

Let's see this. If we fill those templates with more meaningful placeholders, `i` standing for the initial value, and `f` standing for a function to combine the contribution of the first element and the contribution of the recursion, and we add them as parameters to the functions, we have this function for the natural recursive template:

``````(define (fold-right f i lox)
(cond [(empty? lox) i]
[else
(f (first lox)
(f-for-lox (rest lox)))]))``````

and this two variants for the two tail-recursive templates:

``````(define (fold-left-1 f i lox0)
(local [(define (f-for-lox-acc acc lox)
(cond [(empty? lox) acc]
[else
(f-for-lox-acc (f acc (first lox))
(rest lox))]))]
(f-for-lox-acc i lox0)))

(define (fold-left-2 f i lox0)
(local [(define (f-for-lox-acc acc lox)
(cond [(empty? lox) acc]
[else
(f-for-lox-acc (f (first lox) acc)
(rest lox))]))]
(f-for-lox-acc i lox0)))``````

that are precisely `fold-right` and `fold-left`, with two variants, respectively.

A careful analysis of those functions enable us to infer their respective signatures.

``````;; fold-right                :: (X Y -> Y) Y (listof X) -> Y
;; fold-left-1 (1st variant) :: (Y X -> Y) Y (listof X) -> Y
;; fold-left-2 (2nd variant) :: (X Y -> Y) Y (listof X) -> Y``````

Note, in particular, that `fold-right` and the second variant of `fold-left` share the same signature, while the signature for the first variant of `fold-left` differs due to the different order of arguments in the call to `f`.

`fold-right` is basically the same in all functional programming languages, while different languages pick the first version of `fold-left` (Lisp, Haskell, OCaml, ...) or the second one (SML, Racket, ...) for its implementation of this function.

The difference between the two variants of `fold-left` have an impact in many cases. Just choose as `f` a function for which the order of operands matters (a non-commutative function). For instance if `f` is `+` the order doesn't matter but if it is `-` it absolutely matters. Let's consider this for `-` in all the languages we have mentioned:

``````# Lisp [reduce is fold-left by default]
[1]> (reduce #'- '(1 2 3 4) :initial-value 0)
-10

Prelude> foldl (-) 0 [1, 2, 3, 4]
-10

# OCaml
# List.fold_left (-) 0 [1; 2; 3; 4] ;;
- : int = -10``````

While:

``````# Racket
> (foldl - 0 '(1 2 3 4))
2

# SML
- List.foldl op- 0 [1, 2, 3, 4] ;
val it = 2 : int``````

[The R's `Reduce` will be the subject of the next post.]

## jueves, 28 de julio de 2016

### Reading R code. The functions Map and mapply

In this post I'm going to consider the implementation of the function `Map` in R. As in the previous post in the series, I first introduce the typical `map` in functional languages. Then I go into the R's `Map` implementation. Along the way I explore some other R functions to the extent needed for a basic understanding of the implementation.

### What is `map`

`map` applies a function to each element in a given list to get a new list with the result of the application. For instance:

``map sqrt [1, 2, 3]``

produces a list of the square roots of each number:

``[1.0,1.4142135623730951,1.7320508075688772]``

In general, `map` takes a function and a list of elements of some type, say `a`, and produces a list of elements of some other type, say `b`, where `a` and `b` are the types of, respectively, the input and output values of the function that `map` takes. The signature for the example above could be written as:

``(Num -> Double) [Num] -> [Double]``

meaning that the first argument, `sqrt`, takes a number (in general) and produces a double-precision floating point number; the second argument is a list of numbers, and the result a list of `Double`s.

So in general `map` has this signature:

``(a -> b) [a] -> [b]``

As with `filter` a natural implementation of `map` is a math-like recursive definition:

``````map f []     = []
map f (x:xs) = f x : map f xs``````

Or, in Racket:

``````(define (map f lox)
(cond [(empty? lox) empty]
[else
(cons (f (first lox))
(map f (rest lox)))))``````

This is the basic usage. The function `map` can usually handle multiple lists too. One of the variants works as follows:

It applies the function to the first element of given lists, then to the second, and so on till the last element of the shortest list is processed, the rest of elements of longer lists are ignored, and the result is the list produced by the successive application. This is a possible example with two lists as input. I use the name `zipWith` to refer to this kind of `map` that can take in two lists:

``zipWith (+) [1, 2, 3] [4, 5, 6, 7]``

produces:

``[5, 7, 9] -- so [1 + 4, 2 + 5, 3 + 6]``

The signature would be:

``zipWith :: (a b -> c) [a] [b] -> [c]``

Generalizations to cope with an arbitrary number of lists are also available in functional languages.

### Map in R

The R `Map` function provides, as `?Map` states, a generalization of the sort described:

‘Map’ applies a function to the corresponding elements of given vectors.

(Note that like `Filter` the consumed objects are R vectors.)

However, unlike the generalized `map` mentioned, `Map` doesn't discard remaining elements of longer lists. Instead, it uses recycling:

‘Map’ ... is similar to Common Lisp's ‘mapcar’ (with arguments being recycled, however)

Common-Lisp `mapcar` is a function that behaves as explained above (the Haskell's `zipWith` example). Indeed, executing the corresponding Lisp code on the `clisp` interpreter gives the same result:

``````[1]> (mapcar #'+ '(1 2 3) '(4 5 6 7))
(5 7 9)``````

while in R recycling is at work, though:

``````> Map(`+`, 1:3, 4:7)
[[1]]
[1] 5

[[2]]
[1] 7

[[3]]
[1] 9

[[4]]
[1] 8``````

As you can see, when the shortest list is exhausted, R recycles it as needed: the last element is the result of 1 + 7, where 1 comes here from recycling the shorter list. Note also that the result is not an atomic vector as one might expect but a list. This is also documented:

‘Map’ ... does not attempt to simplify the result ... Future versions may allow some control of the result type.

As for the implementation `?Map` also tells us what it is:

‘Map’ is a simple wrapper to ‘mapply’.

It is not uncommon in R to find traces of implementation details in its docs.

The actual implementation expresses in code all of these points:

``````function (f, ...)
{
f <- match.fun(f)
mapply(FUN = f, ..., SIMPLIFY = FALSE)
}``````

Let's have a closer look into the definition.

The function takes a first argument `f`, the mapping function to be applied, and an undetermined number of extra arguments. The three dots construct allows to catch those extra arguments and pass them on later to `mapply`. Vectors on which `f` will be applied are among the arguments that the caller will pass; eventually, more arguments might be passed by the caller, arguments that `mapply` can receive.

In the first line, the function passed as first argument is extracted and saved into a local variable `f`, or discarded if it cannot be interpreted in any way as a function, that's what `match.fun` basically does.

### The function `mapply`

Once this first argument has been checked, it is passed as the `FUN` argument to `mapply` that in turn calls the underlying C code to carry out the computation. Additionally, `Map` sets the `SIMPLIFY` argument of `mapply` to `FALSE` to avoid the simplification, as documented.

We can see this better if we take a look at the implementation of `mapply`:

``````function(FUN,..., MoreArgs = NULL, SIMPLIFY = TRUE, USE.NAMES = TRUE)
{
FUN <- match.fun(FUN)
dots <- list(...)

if (USE.NAMES && length(dots)) {
if (is.null(names1 <- names(dots[[1L]])) && is.character(dots[[1L]]))
else if (!is.null(names1))
}
simplify2array(answer, higher = (SIMPLIFY == "array"))
}``````

Note that `answer`, the returned value, is the result produced by the internal function, written in C, which is invoked via the call to `.Internal`. We come to this construct over and over while reading R base functions. Many R base functions just wrap the call to an underlying function, eventually preparing things for it, and/or adapting the result of it according to the arguments passed in the first place.

`Map` sets `mapply`'s `SIMPLIFY` to `FALSE` so that the simplification that `mapply` could do otherwise will never be executed.

However, despite the fact that the intended arguments for the three dots are just the vectors, as `?Map` documents, nothing prevents us from passing other possible `mapply` arguments, `USE.NAMES` and `MoreArgs`.

`MoreArgs` allows for passing extra arguments to the `f` function. For instance, to get a list of vectors where 42 is repeated from 1 to 4 times, we can use `MoreArgs` in this way [example borrowed from the `mapply` doc]:

``````> Map(rep, times = 1:4, MoreArgs = list(x = 42))
[[1]]
[1] 42

[[2]]
[1] 42 42

[[3]]
[1] 42 42 42

[[4]]
[1] 42 42 42 42``````

As R allows us to pass extra arguments just by naming them after the function,

``> Map(rep, times = 1:4, x = 42)``

`MoreArgs` seems to be of little use in this regard.

Furthermore, one can always resort to the commonly-used idiom in functional programming: supplying an anonymous function in place of `f`:

``> Map(function(x) rep(42, times = x), 1:4)``

The other extra option that we may pass to `Map` is `USE.NAMES`. `mapply` sets it by default to `TRUE`. In such a setting, and if more arguments apart form the initial function `f` are passed (`length(dots)` is not 0) this code in `mapply` will be executed:

``````if (USE.NAMES && length(dots)) {
if (is.null(names1 <- names(dots[[1L]])) && is.character(dots[[1L]]))
else if (!is.null(names1))
}``````

Two cases are especially handled:

• The second argument passed to `mapply` (`dots[[1L]]`) doesn't have names but it is of character type. (Recall that the first argument is always the function `f`).
• The second argument passed to `mapply` has names.

In the first case, the character object is used to set the names of the result of `mapply`, as in this example taken from `?mapply`:

``> mapply(function(C, k) paste0(rep.int(C, k)), LETTERS[1:6], 6:1)``

In the second case, the names of the argument become the names of the result. For instance [again from `?mapply`]:

``````> mapply(function(x, y) seq_len(x) + y,
> +             c(a =  1, b = 2, c = 3),  # names from first
> +             c(A = 10, B = 0, C = -10))``````

This is simply put on the doc:

use names if the first ... argument has names, or if it is a character vector, use that character vector as the names.

If we want no names handling in `Map` we can just set `USE.NAMES` to `FALSE` overriding the default behavior of `mapply`, or viewed from the implementation, falling back to the default behavior of the internal C code.

A couple of points about R idioms that we see in the `mapply`:

Assignments can be sub-expressions. Here:

``is.null(names1 <- names(dots[[1L]]))``

The assignment expression `names1 <- names(dots[[1L]])` evaluates to the value of `names1`, once assigned to. The value is passed to `is.null` as a part of the `if` condition. The value of `names1` is later re-used in the last branch.

Secondly, in a logical context, the number 0 evaluates to `FALSE`, any other number evaluates to `TRUE`. This allows for concise conditions as the previous one:

``if (USE.NAMES && length(dots))``

The second part of the relational expression check whether `dots` is empty or not. Hence, there is no need to write `length(dots) != 0`. This is a common idiom in many languages supporting boolean evaluation of different classes of objects.

To complete the basic understanding of these functions I should inspect more closely `match.fun` and `simplify2array` (the function responsible of simplifying the result). `match.fun` appears frequently in R code and I pin it on top of my task list, `simplify2array` is a helper function currently used only by `mapply` and `sapply`. We can live without studying it for the time being.

## martes, 26 de julio de 2016

### Reading R code. The function Filter

R is functional language, however not a pure one like Haskell, R is fond of the functional paradigm. As John Chambers has recently remind us, in R everything that happens is a function call [John Chambers, Extending R, CRC Press]. R functions are first-class citizens and the functional programming style is prominent.

`filter`, `map`, and `reduce` (aka. `fold`) are among the most wanted functions in the toolset of every functional programmer. R provides them too under the names `Filter`, `Map`, `Reduce`. Even though the apply family and vectorization are preferred, R is kind enough to give these functions to seasoned functional programmers. It couldn't have been otherwise.

The source code of base R, currently at `/src/library/base/R` contains a file with the implementation of these functions among others, `funprog.R`.

In my exploration of R source code I have to begin with something. `Filter` may be a good candidate. It is brief, easier to read, and a sensible choice for a fan of functional programming as I am.

### What is filter?

`filter`, as its name suggests, serves the purpose of filtering elements in a list according to certain given function, called predicate, so that elements in this list for which the predicate holds are selected and the rest discarded, where predicate is the technical term for any function that produces true or false (a boolean value).

Some examples will make this clear.

Let's suppose we have this list of numbers `[1, 2, 3, 4, 5]`, and a function `odd` that takes a number and determines whether it is odd or not. Selecting all odd numbers in `[1, 2, 3, 4, 5]` is a matter of filtering them by their oddity

``filter odd [1, 2, 3, 4, 5]``

will produce `[1, 3, 5]`

Another example. We want to select words in `["hi", "world", "bye", "hell"]` whose first letter is 'h'.

``filter start_with_h ["hi", "world", "bye", "hell"]``

will produce the desired list `["hi", "hell"]`

In general, `filter` takes a predicate and a list of elements of some type, say type `a`, and produces a list of elements of the same type. Formally expressed:

``filter :: (a -> Bool) [a] -> [a]``

where `[a]` stands for list of elements of type `a`, and the arrow stands for a function whose arguments are to the left and the result to the right of the arrow. Note that the predicate, the first argument, is also a function that consumes values of type `a` and produces a boolean.

This type description is called the signature of the function. In some functional programming languages it is customary to apply the so called currying operation that translates the evaluation of a function taking multiple arguments into the evaluation of a sequence of functions, with a single argument each. Under currying, the signature of filter would be:

``filter :: (a -> Bool) -> [a] -> [a]``

Although this is a mathematically more appealing description, I'll stick here and in what follows to the first uncurried form.

What about implementing `filter`? A natural implementation of `filter` would take the form of a mathematical definition.

Mathematics is plenty of inductive or recursive definitions. Do you recall factorial? In Math, it is defined as a two-part function:

``````factorial 0 = 1
factorial n = (factorial n - 1) * n``````

In words, the factorial of 0 is equal to 1, the factorial of n is equal to the factorial of n - 1 times n.

Similarly, a math-like definition for `filter` could be expressed as a two-part function with two branches for the second part:

• The filter of a predicate and an empty list is the empty list.
• The filter of a predicate and a non-empty list is
• either (if the predicate holds for the first element of the given list) the list consisting of that first element and the result of the filter of the predicate and the rest of the given list,
• or (otherwise) the filter of the predicate and the rest of the given list.

This wording is exact but verbose. Formalizing it a bit turns out to be easier to read. What follows is the actual Haskell implementation of `filter`, that hopefully is almost self-explanatory. Actually the signature before is also the real Haskell signature.

``````filter p [] = []
filter p (x:xs) | p x       = x : filter p xs
| otherwise = filter p xs``````

Another possible syntax, now written in friendly Racket, that you may find even more readable, looks as follows [signature included as an initial comment]:

``````;; (X -> Boolean) (listof X) -> (listof X)
(define (filter p lox)
(cond [(empty? lox) empty]
[else
(if (p (first lox))
(cons p (filter p (rest lox)))
(filter p (rest lox)))))``````

### Filter in R

What about the R implementation? Here it is:

``````Filter <-
function(f, x)
{
ind <- as.logical(unlist(lapply(x, f)))
x[which(ind)]
}``````

It looks quite different. Obviously, it is not recursive. This is not surprising, recursion is the main technique in pure functional languages that are well prepared to handle recursive implementations without performance penalties.

Having noted that, let's explore this code in detail.

The very first thing we always have to do when studying a function definition is to be sure about the value(s) that the function consumes and the value that the function produces, the signature of the function. By the way, as you may know, some languages check signatures at compile time, like Haskell, others don't, like R or the Racket version above [though Racket has variants for type checking]. Whatever the case the signature is critical for programmers and for users, since it tells what kind of values are involved.

The documentation for `Filter` reveals the assumed signature. Omitting for the moment some nuances and summarizing the 'Arguments' and 'Details' section , `?Filter` says this:

‘Filter’ applies the unary predicate function ‘f’ to each element of ‘x’ [which is a vector] ..., and returns the subset of ‘x’ for which this gives true.

Recall the header of `Filter`

``function(f, x)``

The doc says that `f`, the first argument, is a unary predicate function, meaning a function that takes a single argument (unary) [of any type, say `a` as before], and produces a boolean (`TRUE` or `FALSE`). The signature of the predicate is therefore:

``(a -> Bool)``

The second argument, `x`, is in turn a vector, instead of a list. A vector in R can be an atomic vector, a list, or an expression. The main R data type to represent arbitrarily large collections of objects, possibly nested, is vector and in this regard is a natural choice to represent what in functional languages is typically represented by lists.

Let us symbolize, just by convention, vector of some type with `[a]`, that's the type of the second argument of `Filter`.

The result of `Filter` is in turn a subset of `x`, hence again a vector of type `[a]`.

Therefore the whole signature could be specified as follows:

``Filter :: (a -> Bool) [a] -> [a]``

As it's easy to see, this is the same signature of `filter` we figure out before, just that `[a]` stands for list in the former and for vector (atomic or not) in the latter.

Let's go on with the implementation. In order to understand an implementation I find really helpful trying first to design our own version.

Before starting off with the implementation itself we should write examples of the function usage, at least as many as the function signature requires, and wrap them as test cases. Examples will guide the implementation, and at the same time, wrapped as test cases, provide for the indispensable testing workhorse. For the latter I will use `testthat` as explained in the final part of the first post in this series: https://los-pajaros-de-hogano.blogspot.com.es/2016/07/reading-r-code-introduction.html

So save these examples into the file `test_my_filter.R`:

``````source("my_filter.R")

# Some predicates for testing filter
odd <- function(x) { x %% 2 != 0 }
starts_with_h <- function(x) { any(grepl(pattern = "^h.*", x = x)) }
numeric_expression <- function(x) { is.numeric(eval(x)) }

test_that("test my_filter", {
expect_equal(my_filter(odd, integer(0)), integer(0))
expect_equal(my_filter(is.atomic, list()), list())
expect_equal(my_filter(is.expression, expression()), expression())
expect_equal(my_filter(odd, 1:5), c(1, 3, 5))
expect_equal(my_filter(starts_with_h, c("hi", "world", "bye", "hell")),
c("hi", "hell"))
expect_equal(my_filter(is.atomic, list(1, list(3, 4), 5)), list(1, 5))
expect_equal(my_filter(numeric_expression, expression(c, "a", 1, 3)),
expression(1, 3))
})
``````

A quick glance at `Filter` gives us a first hint as to the way we could follow in the implementation. The last line is a typical subsetting operation. We could try to use this idea to delineate our code.

You may figure it out after a bit of thinking. Let's begin with one of our examples:

``my_filter(odd, 1:5)``

Well, I have something like the vector `c(1, 2, 3, 4, 5)` and I want to filter odd elements in it. I also have a predicate `odd` that allows me to determine whether each element there is odd or not.

How should `my_filter` be implemented to produce the expected `c(1, 3, 5)`?

It could apply `odd` to each element in `c(1, 2, 3, 4, 5)` to get this: `c(TRUE, FALSE, TRUE, FALSE, TRUE)`, and then, in a typical R way, use the logical vector for indexing `c(1, 2, 3, 4, 5)`:

``c(1, 2, 3, 4, 5)[c(TRUE, FALSE, TRUE, FALSE, TRUE)]``

I can sketch a general version of this idea. Instead of `odd` and `1:5` I generalize to whatever predicate, `f`, and whatever vector, `x`. Also, and just for readability, I create a local variable, `ind`, to name the logical vector.

``````my_filter <-
function(f, x) {
ind <- # apply f to each element in x
x[ind]
}``````

Only one thing remains to be filled here, the code that applies `f` to each element in `x`.

If you have an imperative programming background you will come up with some kind of loop to process each element in the vector and check whether the predicate holds for it. Something along these lines:

``````apply_f <-
function(f, x) {
fx <- vector("logical")
for (i in seq_along(x)) {
fx <- c(fx, f(x[[i]]))
}
fx
}``````

Let's add this helper function to `my_filter` as a nested (local) function

``````my_filter <-
function(f, x) {
apply_f <-
function() {
fx <- vector("logical")
for (i in seq_along(x)) {
fx <- c(fx, f(x[[i]]))
}
fx
}

ind <- apply_f()
x[ind]
}``````

Wait a minute! Shouldn't `apply_f` be tested before doing anything else? and definitely before converting it into a local (and as such untestable) function? This is a great question. Let's claim that it is not needed because it is a too easy function and we trust it will work without further testing. In a moment we will return to this and see the consequences of this assumption.

It's time to run tests:

``````> library(testthat)
> test_file("test_my_filter.R")``````

Great! All tests passed.

However, instead of explicit loops you may know that idiomatic R usually favors the apply family of functions.

The immediate candidate for this task seems to be `sapply`, that is typically used when we want a vector as a result of applying the function to each element in a given vector `x`. `apply_f` could be written with `sapply` as follows:

``````apply_f <-
function(f, x) {
sapply(x, f)
}``````

Being so simple, it doesn't make sense to wrap the `sapply` call in a function. So our more idiomatic and concise filter implementation would be:

``````my_filter <-
function(f, x) {
ind <- sapply(x, f)
x[ind]
}``````

Run tests ... and, oops! one test fails! The case for the empty vector as input.

Looking into `?sapply` more carefully, we see that our initial expectation was based in a partial understanding of `sapply`:

Simplification in ‘sapply’ is only attempted if ‘X’ has length greater than zero ...

It turns out that `sapply` does not always produce a vector when a vector is given. In other words, our current implementation breaks the signature of `my_filter`.

We have different alternatives to fix the bug. One would be to use a conditional with one branch for the empty vector and another one for the rest of cases. Another option is to use `lapply`, that produces a list instead of a vector, and unlist the result to get the vector we need. This second strategy has an advantage. Core R code shouldn't use user friendly wrappers like `sapply` but primitive functions. It is not only a matter of style, it is usually a matter of performance (core functions are more efficient), and it is above all a matter of reliability. User wrappers are for helping users write quickly, mainly in an interactive setting, but they may change or even become defunct in future releases. In general, we shouldn't trust user wrapper functions when we want to write enduring code.

With `lapply` and `unlist` we get this solution:

``````my_filter <-
function(f, x) {
ind <- unlist(lapply(x, f))
x[ind]``````

}

and it passes all tests.

If we compare our last version with the official implementation we'll observe that there are still some differences.

Starting from the end, the official implementation prefers `which(ind)` instead of just `ind` for indexing. This is a subtle divergence. One thing it handles differently than our implementation is `NA` values. In our implementation, if a `NA` appears in `x` it will appear also as `NA` in `ind`, and if we subset `x` directly via `ind` the `NA` ends up in the result too. However, `?Filter` has to say something about this (just we omitted it in the first reading to keep things simple):

Note that possible ‘NA’ values are currently always taken as false.

This implies that `NA` values in `x` will be discarded from the result, whatever `f` and `x`, and this is exactly what `which` can nicely achieve. `which` takes a logical object and produces the indices, as integers, of `TRUE` values. In other words, for our case, where a vector is given as input, it has this signature:

``which :: [Bool] -> [Integer]``

For instance,

``````> which(c(TRUE, FALSE, FALSE, TRUE))
[1] 1 4``````

Additionally, `which` omits `NA` values in the logical vector, in other words, it treats them as if they were `FALSE`. For example:

``````> which(c(TRUE, FALSE, FALSE, TRUE, NA))
[1] 1 4``````

and that's precisely the documented behavior that `Filter` should show.

The second difference lies in the first line, the explicit conversion to logical absent from our solution:

``as.logical(unlist(lapply(x, f)))``

Someone could argue that `as.logical` is redundant. After all, the predicate `f` always produces a boolean value and the result of the composition of `lapply` and `unlist` should produce a logical vector. It should do, but it actually doesn't. Passing an empty vector to `unlist(lapply(...))` produces `NULL`:

``````> unlist(lapply(integer(0), is.integer))
NULL``````

Inadvertently our previous implementation worked, and it worked because when `NULL` is an index it is treated as `integer(0)` and indexing an empty vector with `integer(0)` produces an empty vector, logical by default.

``````> vector()[integer(0)]
logical(0)``````

Once we substitute direct indexing by `which(ind)` things stop working, `which(NULL)` would raise error because `which` requires a logical object as argument, and `NULL` is neither a logical value nor converted by `which` to logical. That's the reason for introducing `as.logical`. In general, `as.logical` makes a lot of sense by making sure that the signature of `which` will never be broken.

Let's go back to the moment when we hesitated about why not to test the helper function `apply_f`. Under the new light the question appears pretty reasonable. Had we tested `apply_f`, in other words, had we left it as a non-local function and verified it independently, we would have caught this bug soon.

For instance, we could have written `apply_f` as:

``````apply_f <-
function(f, x) {
unlist(lapply(x, f))
}``````

``````test_that("test apply_f", {
expect_equal(apply_f(is.integer, integer(0)), logical(0))
expect_equal(apply_f(odd, 1:5), c(TRUE, FALSE, TRUE, FALSE, TRUE))
# ... (more tests here)
})``````

Running tests would have uncovered the bug in the very beginning. Once `apply_f` passes all tests we can convert it into a local function, in this case just the call we have seen in the final definition. The morale is that untested code, which we lazily assume being too easy to test is often the source of unexpected bugs and countless wasted hours of debugging. Better to always test everything that is not absolutely trivial.

To finish this exploration and for completeness just add this new test case, which verifies the `NA` discarding behavior, to the `test_my_filter.R` file:

``expect_equal(my_filter(odd, c(1:4, NA)), c(1, 3))``

and confirm that the final implementation with `as.logical`, that is exactly the R version of `Filter`, passes all tests.

## lunes, 25 de julio de 2016

One of the most important things I learned when I started my higher education was that in order to really know deeply about a subject one has to dive into the original sources as soon as possible rather than solely sticking to secondary literature. Better to read Plato's and Kant's works in their original language than countless books about Critiques and Dialogues; better to analyze Mozart's and Beethoven's Sonatas (as Charles Rosen does) than skimming over a lot of general articles on Classical music.

When I was put in charge of developing a web site for my institution I hadn't any programming background at all. After some discouraging attempts to learning from intro books and tutorials, I decided to go to the sources and I studied the entire HTML and CSS specifications directly. It was hard, but everything was clear at last [Remark: nowadays it would be overwhelming. HTML and CSS specs are currently huge].

In learning programming languages the next step to the initial exposure to syntax, concepts and techniques might be well studying core libraries written by main developers.

How could one understand better a programming language than by trying to read the code that its core developers have created?

I'm always surprised by the scarcity of commentaries about exemplary code in any language. There is somewhat like a gap between introductory expositions and the source code itself. The latter is silently digested by the people who develop it or who create libraries on top of it, while beginners live always in the other much narrower side, their little friendly universe of tutorials and simple recipes, kind enough, for sure, but maybe a bit tasteless [though great exceptions can always be found out] when their time, the absolute beginners' time, has passed.

One reason for this lack might be that core libraries are very complex and abstract beasts, and an understanding of parts of it is just hopeless without a firm grasp on the whole architecture and design, something unreachable to anyone else but experts.

This is not always the case, though. At least it is not the case for a good bunch of functions in R base code. Many are written in R itself and they are almost self-contained in the sense that a preliminary comprehension doesn't depend on a complete acquaintance with the abstract underlying architecture.

So I've thought that I can give this idea a try, picking some R functions, and reading them with the aim of understanding R better. The goal is educational, self-educational mainly. Along the way I'll try to make things perhaps easier to others with a bit less programming background than mine hoping that in doing so I not only reinforce my understanding but also help others deepen their own.

I'm not an R expert. This means that while reading and trying to make sense of official implementations I probably will make some more or less educated guesses and I could (and surely will) make mistakes. So if someone more knowledgeable than I am (there should be many) read this, please let me know to fix any error, misleading step, or gratuitous deduction.

The intended audience is people with a basic working understanding of R data structures and programming constructs, including conditionals, loops and function definitions. The only required tool is the R console. And for the moment only one extension package will be used, the package `testthat` for unit testing support. It can be installed as usual via:

``> install.packages(testthat)``

To use `testthat` in a simple way (not the best one for real projects but enough for our purposes now) proceed as follows:

1. Save the function you want to test in a file.
2. Create a new file on the same directory for testing that function. The name of this file should start with 'test'.
3. Source the code of the function to be tested.
4. Write tests.
5. Run tests from the console.

An example of this workflow.

Save the following function in `foo.R`:

``````foo <-
function(x) {
ifelse((identical(x, 1)), "I'm 1", "I'm not 1")
}``````

Create `test_foo.R` with this content:

``````source("foo.R")

test_that("foo is 1 or not 1", {
expect_equal(foo(1), "I'm 1")
expect_equal(foo(0), "I'm not 1")
expect_equal(foo("hi"), "I'm not 1")
})``````

Load `testthat` on your session and run tests from the console:

``````> library(testthat)
> test_file("test_foo.R")``````

Introducing unit testing from the very beginning may seem unnecessary. On the contrary, testing is of paramount importance, and writing first a minimal set of tests guides the implementation. Since we probably will write our own code here and there, it is crucial to be equipped with a tool that enable us to always write those tests. This is just common-place and fundamental practice whatever the programming language.

By the way, for a perfect introduction to programming fundamentals, where programming stand here for "programming well" rather than "just coding", read this book:

http://www.ccs.neu.edu/home/matthias/HtDP2e/

and/or take this course:

https://www.edx.org/xseries/how-code-systematic-program-design

They both are gems that no one should miss.

One last point about the R source code. Apart from interactively getting the code as usual by typing the function name, for instance:

``> which``

and its documentation:

``> ?which``

https://cran.r-project.org/sources.html

Also, you can access to the official current snapshot on Subversion if you know how to do so, or browse over (or clone) a non-official github mirror. I'm aware of these two:

## miércoles, 20 de julio de 2016

### How to read the R documentation. An example with plot().

From my experience as teaching assistant on several R intro MOOCs I'm getting the impression that beginners, and even intermediate users assume that the R documentation is only for experts and as a consequence don't read the doc in the first place.

This is an unfortunate prejudice since the R built-in documentation is one of its most remarkable features and it is there to help precisely the users. Even though (I admit) some docs are pretty technical, many others are perfectly readable, even for beginners.

In this post I'll try to show the strategy I typically follow when dealing with R doc pages.

Let's take as excuse the following question posted in one of those MOOCs:

What does the function plot do when its input is a data frame?

### Getting the right doc

The first thing we need to do is to call the help for plot [I put the first few lines of the result]:

`> ?plot`

```# -------------------

plot                 package:graphics                  R Documentation

Generic X-Y Plotting

Description:

Generic function for plotting of R objects.  For more details
about the graphical parameter arguments, see ‘par’.

...

# -------------------
```

Note the first sentence in the 'Description' section. It tells us that `plot` is a generic function for plotting R objects..

Not too much, isn't it? but still something. For our purposes generic function means that we need to search for a method (= another function), which will be actually called depending on the class of object we pass to `plot`. [Since my intent is to guide beginners I omit all discussions regarding technicalities, as that about the exact meaning of method and function in R, as well as the difference between non-generic and generic functions. For the moment take those terms and others of this sort just as jargon we are liberally using to talk about these things].

To know more about `plot` methods we just type this:

` > methods(plot)`

This is what I get on my installation:

``` [1] plot.acf*           plot.data.frame*    plot.decomposed.ts*
[4] plot.default        plot.dendrogram*    plot.density*
[7] plot.ecdf           plot.factor*        plot.formula*
[10] plot.function       plot.hclust*        plot.histogram*
[13] plot.HoltWinters*   plot.isoreg*        plot.lm*
[16] plot.medpolish*     plot.mlm*           plot.ppr*
[19] plot.prcomp*        plot.princomp*      plot.profile.nls*
[22] plot.raster*        plot.spec*          plot.stepfun
[25] plot.stl*           plot.table*         plot.ts
[28] plot.tskernel*      plot.TukeyHSD*
```

Among these methods `plot.data.frame` is naturally the one which we are interested in. And R helps here too:

` > ?plot.data.frame `

```# -------------------

plot.data.frame            package:graphics            R Documentation

Plot Method for Data Frames

Description:

‘plot.data.frame’, a method for the ‘plot’ generic.  It is
designed for a quick look at numeric data frames.

Usage:

## S3 method for class 'data.frame'
plot(x, ...)

Arguments:

x: object of class ‘data.frame’.

...: further arguments to ‘stripchart’, ‘plot.default’ or ‘pairs’.

Details:

This is intended for data frames with _numeric_ columns. For more
than two columns it first calls ‘data.matrix’ to convert the data
frame to a numeric matrix and then calls ‘pairs’ to produce a
scatterplot matrix).  This can fail and may well be inappropriate:
for example numerical conversion of dates will lose their special
meaning and a warning will be given.

For a two-column data frame it plots the second column against the
first by the most appropriate method for the first column.

For a single numeric column it uses ‘stripchart’, and for other
single-column data frames tries to find a plot method for the
single column.

‘data.frame’

Examples:

plot(OrchardSprays[1], method = "jitter")
plot(OrchardSprays[c(4,1)])
plot(OrchardSprays)

plot(iris)
plot(iris[5:4])
plot(women)

# -------------------
```

Most R help pages share the same structure consisting of different sections (Title, Description, Usage, etc)

Some sections may be more important than others depending on what we want to know. For instance, if we only want to know what this function is about in general terms it might be enough to read the terse 'Description'. Or if we already know what the function does but we forget some particular use of certain argument we can look into the 'Arguments' section. If we really want to know what the function exactly does we will need to read 'Details' and 'Examples'.

A superficial reading just doesn't work, we have to take the time to read thoroughly. Let's do it now. Even without fully understanding everything a careful reading allows us to outline what this function does:

• If the given data frame consists of more than two columns, it ideally displays a scatterplot matrix.
• If the data frame has two columns, it displays a suitable y versus x plot, where x is the first column and y the second.
• If the data frame has a single numeric column it generates a stripchart; if that column is not numeric a suitable plot [not specified] is displayed.

So far so good but an image, an example, worths thousands of words. And R usally excels in providing ready-made examples for us. This is critically important when reading the R documentation. If given, please don't skim examples, work them out. Even more, rather than just running them via `example(function_name)`, explore them on the console, one by one, at least till you reach a good enough understanding. Let's go on the 'Examples' section.

The first three examples apply `plot` to the data set `OrchardSprays`, which a basic R installation provides by default. This data set has the following structure:

```> str(OrchardSprays)
'data.frame': 64 obs. of  4 variables:
\$ decrease : num  57 95 8 69 92 90 15 2 84 6 ...
\$ rowpos   : num  1 2 3 4 5 6 7 8 1 2 ...
\$ colpos   : num  1 1 1 1 1 1 1 1 2 2 ...
\$ treatment: Factor w/ 8 levels "A","B","C","D",..: 4 5 2 8 7 6 3 1 3 2 ...
```

So four columns, the first three numeric, and the last one categorical (a factor in R parlance).

The first example passes a data frame with a single column, a one-column subset drawn from the `OrchardSprays` data frame. Therefore, it's an example for the case where the data frame is made of a single numeric variable. We should expect a stripchart, as documented, and we get that:

` > plot(OrchardSprays[1], method = "jitter")`

The second example passes this data frame:

```> str(OrchardSprays[c(4, 1)])
'data.frame': 64 obs. of  2 variables:
\$ treatment: Factor w/ 8 levels "A","B","C","D",..: 4 5 2 8 7 6 3 1 3 2 ...
\$ decrease : num  57 95 8 69 92 90 15 2 84 6 ...
```
So an example of the second case described above, where the first column (x) is categorical and the second (y) is numerical. A suitable plot would be a series of boxplots, one for level of the categorical variable. And that's exactly what we obtain:

` > plot(OrchardSprays[c(4, 1)])`

The third example passes the whole data frame. An illustration of the first case mentioned in the doc. And we get the corresponding scatterplot matrix:

` > plot(OrchardSprays)`

I leave the reader to investigate the last three examples. The only new case is `plot(women)`, where the input is a data frame with two columns but both numeric in this case.

I honestly believe that reading this doc (as many other R docs) gives more reliable information about the function at hand than googling during hours or skimming over dozens of books.

### Getting and reading the source code

One more thing is still available to users, the source code itself, that obviously is the definitive answer to all questions.

Many R functions are implemented in R itself, and an intermediate R user should be able to read and understand the implementation, to some extent at least. Yes, many core function are written in C and those are beyond the level of expertise of a non professional programmer with sufficient time to invest in navigating over the entire C basis and making sense of it. But we can always give a try, just in case.

As for the function `plot.data.frame` we are lucky.

There is an initial difficulty, though, locating the source. Methods listed by the above mentioned `methods` function that come suffixed with * are functions whose code cannot be reached by just typing the function name, as usual. The source code is still accessible. In particular, when the function is an S3 method, as plot.data.frame is [I omit commenting about S3 vs S4 methods. See the R manuals in cran.r-project.org for more info] we have among maybe others any of these two instructions:

` > getS3method("plot", "data.frame")`

or

` > getAnywhere("plot.data.frame")`

that displays this code [line numbers added for commenting below]:

```1 function (x, ...)
2 {
3      plot2 <- function(x, xlab = names(x)[1L], ylab = names(x)[2L],
...) plot(x[[1L]], x[[2L]], xlab = xlab, ylab = ylab,
...)
4      if (!is.data.frame(x))
5          stop("'plot.data.frame' applied to non data frame")
6      if (ncol(x) == 1) {
7          x1 <- x[[1L]]
8          cl <- class(x1)
9          if (cl %in% c("integer", "numeric"))
10             stripchart(x1, ...)
11         else plot(x1, ...)
12     }
13     else if (ncol(x) == 2) {
14         plot2(x, ...)
15     }
16     else {
17         pairs(data.matrix(x), ...)
19     }
19 }
```

A concise commentary:

The function takes a mandatory argument, a data frame by assumption, and an arbitrary number of optional arguments passed by possibly inner function calls to other graphic functions [line 1].

It defines an inner function, `plot2`, that in turn calls plot with the first and second column of the given data frame and sets appropriate titles and labels for the resulting plot. This function will be used later for one of the possible cases mentioned in the documentation, where the data frame has two columns [line 3].

The function exits with an error message if the argument passed is not a data frame [line 4-5]. This is just the usual defensive line to handle arguments that don't follow the assumed type of input.

Next the main part [lines 6ff.] goes, that conditionally selects a kind of graph depending on the number and, if required, the class of columns in the data frame.

If the data frame has one column and it is "numeric" or "integer" a stripchart is displayed; otherwise (the column is of another class) it relies on the generic plot again to generate the appropriate plot. [lines 6-11].

If the data frame has two columns the previously defined `plot2` function is called, so that a suitable y vs. x plot is obtained [lines 13-14].

Finally, if the number of columns is greater than 2, the another possible case, the data frame is coerced to a `data.matrix` and the function `pairs` is applied to the result of the coercion [lines 16-17].

To get a grasp on this last thing, here is my challenge, read now `?data.matrix` and `?pairs`. `Have fun and happy R reading!`