Name:
Andrew ID:
Collaborated with:

This lab is to be done in class (completed outside of class if need be). You can collaborate with your classmates, but you must identify their names above, and you must submit your own lab as an knitted HTML file on Canvas, by Tuesday 10pm, this week.

This week’s agenda: creating and updating functions; understanding argument and return structures; revisiting Shakespeare’s plays; code refactoring.

# Huber loss function

The Huber loss function (or just Huber function, for short) is defined as: $\psi(x) = \begin{cases} x^2 & \text{if |x| \leq 1} \\ 2|x| - 1 & \text{if |x| > 1} \end{cases}$ This function is quadratic on the interval [-1,1], and linear outside of this interval. It transitions from quadratic to linear “smoothly”, and looks like this:

It is often used in place of the usual squared error loss for robust estimation. The sample average, $$\bar{X}$$—which given a sample $$X_1,\ldots,X_n$$ minimizes the squared error loss $$\sum_{i=1}^n (X_i-m)^2$$ over all choices of $$m$$—can be inaccurate as an estimate of $$\mathbb{E}(X)$$ if the distribution of $$X$$ is heavy-tailed. In such cases, minimizing Huber loss can give a better estimate. (Interested in hearing more? Come ask Tudor or I!)

• 1a. Write a function huber() that takes as an input a number $$x$$, and returns the Huber value $$\psi(x)$$, as defined above. Hint: the body of a function is just a block of R code, i.e., in this code you can use if() and else() statements. Check that huber(1) returns 1, and huber(4) returns 7.

• 1b. The Huber function can be modified so that the transition from quadratic to linear happens at an arbitrary cutoff value $$a$$, as in: $\psi_a(x) = \begin{cases} x^2 & \text{if |x| \leq a} \\ 2a|x| - a^2 & \text{if |x| > a} \end{cases}$ Starting with your solution code to the last question, update your huber() function so that it takes two arguments: $$x$$, a number at which to evaluate the loss, and $$a$$ a number representing the cutoff value. It should now return $$\psi_a(x)$$, as defined above. Check that huber(3, 2) returns 8, and huber(3, 4) returns 9.

• 1c. Update your huber() function so that the default value of the cutoff $$a$$ is 1. Check that huber(3) returns 5.

• 1d. Check that huber(a = 1, x = 3) returns 5. Check that huber(1, 3) returns 1. Explain why these are different.

• 1e. Vectorize your huber() function, so that the first input can actually be a vector of numbers, and what is returned is a vector whose elements give the Huber evaluated at each of these numbers. Hint: you might try using ifelse(), if you haven’t already, to vectorize nicely. Check that huber(x = 1:6, a = 3) returns the vector of numbers (1, 4, 9, 15, 21, 27).

• Challenge. Your instructor computed the Huber function values $$\psi_a(x)$$ over a bunch of different $$x$$ values, stored in huber_vals and x_vals, respectively. However, the cutoff $$a$$ was, let’s say, lost. Using huber_vals, x_vals columns of oops_df, and the definition of the Huber function, you should be able to figure out the cutoff value $$a$$, at least roughly. Estimate $$a$$ and explain how you got there. Hint: one way to estimate $$a$$ is to do so visually, using plotting tools (if you do - please use ggplot); there are other ways too.

oops_df <- data.frame(
x_vals = seq(0, 5, length=21),
huber_vals = c(0.0000, 0.0625, 0.2500, 0.5625, 1.0000, 1.5625, 2.2500,
3.0625, 4.0000, 5.0625, 6.2500, 7.5625, 9.0000, 10.5000,
12.0000, 13.5000, 15.0000, 16.5000, 18.0000, 19.5000,
21.0000))

# Shakespeare’s complete works

Recall, as in lab/hw from Week 1, that the complete works of William Shakespeare are available freely from Project Gutenberg. We’ve put this text file up at https://raw.githubusercontent.com/benjaminleroy/36-350-summer-data/master/Week1/shakespeare.txt.

# Getting lines of text play-by-play

• 2a. Below is the get_wordtab_from_url() from lecture. Modify this function so that the string vectors lines and words are both included as named components in the returned list. For good practice, update the documentation in comments to reflect your changes. Then call this function on the URL for the Shakespeare’s complete works (with the rest of the arguments at their default values) and save the result as shakespeare_wordobj. Using head(), display the first several elements of (definitely not all of!) the lines, words, and wordtab components of shakespeare_wordobj, just to check that the output makes sense to you.
#' Get a word table from text on the web
#'
#' @param str_url string, specifying URL of a web page
#' @param split string, specifying what to split on. Default is the regex
#' pattern "[[:space:]]|[[:punct:]]"
#' @param tolower Boolean, TRUE if words should be converted to lower case
#' before the word table is computed. Default is TRUE
#' @param keep_nums Boolean, TRUE if words containing numbers should be kept in
#' the word table. Default is FALSE
#'
#' @return list, containing word table, and then some basic numeric summaries
#'
#' @examples
#' endgame_speech_list <- get_wordtab_from_url(
#'  paste0("https://raw.githubusercontent.com/benjaminleroy/",
#'  "36-350-summer-data/master/Week1/endgame.txt"))
get_wordtab_from_url <- function(str_url, split = "[[:space:]]|[[:punct:]]",
tolower = TRUE,
keep_nums = FALSE) {
text <- paste(lines, collapse = " ")
words <- strsplit(text, split = split)[[1]]
words <- words[words != ""]

# Convert to lower case, if we're asked to
if (tolower) {
words <- tolower(words)
}

# Get rid of words with numbers, if we're asked to
if (!keep_nums) {
words <- grep("[0-9]", words, inv = TRUE, val = TRUE)
}

# Compute the word table
wordtab <- table(words)

return(list(wordtab = wordtab,
number_unique_words = length(wordtab),
number_total_words = sum(wordtab),
longest_word = words[which.max(nchar(words))]))
}
• 2b. Go back and look Q5 of Homework 1, where you located Shakespeare’s plays in the lines of text for Shakespeare’s complete works. Set shakespeare_lines <- shakespeare_wordobj$lines, and then rerun your solution code (or the rerun the official solution code, if you’d like) for questions Q5a–Q5f of Homework 1, on the lines of text stored in shakespeare_wordobj$lines. You should end up with two vectors titles_start and titles_end, containing the start and end positions of each of Shakespeare’s plays in shakespeare_lines. Print out titles_start and titles_end to the console.

• 2c. Create a list shakespeare_lines_by_play of length equal to the number of Shakespeare’s plays (a number you should have already computed in the solution to the last question). Using a for() loop, and relying on titles_start and titles_end, extract the subvector of shakespeare_lines for each of Shakespeare’s plays, and store it as a component of shakespeare_lines_by_play. That is, shakespeare_lines_by_play[[1]] should contain the lines for Shakespeare’s first play, shakespeare_lines_by_play[[2]] should contain the lines for Shakespeare’s second play, and so on. Name the components of shakespeare_lines_by_play according to the titles of the plays.

# Getting word tables play-by-play

• 3a. Define a function get_wordtabfrom_lines() to have the same argument structure as get_wordtab_from_url(), which recall you last updated in Q2a, except that the first argument of get_wordtab_from_lines() should be lines, a string vector passed by the user that contains lines of text to be processed. The body of get_wordtab_from_lines() should be the same as get_wordtab_from_url(), except that lines is passed and does not need to be computed using readlines(). The output of get_wordtab_from_lines() should be the same as get_wordtab_from_url(), except that lines does not need to be returned as a component. For good practice, incude documentation for your get_wordtab_from_lines() function in comments (no need to include an example).

• 3b. Using a for() loop or one of the apply functions (your choice here), run the get_wordtab_from_lines() function on each of the components of shakespeare_lines_by_play, (with the rest of the arguments at their default values). Store the result in a list called shakespeare_wordobj_by_play. That is, shakespeare_wordobj_by_play[[1]] should contain the result of calling this function on the lines for the first play, shakespeare_wordobj_by_play[[2]] should contain the result of calling this function on the lines for the second play, and so on.

• 3c. Using one of the apply functions, compute numeric vectors shakespeare_total_words_by_play and shakespeare_unique_words_by_play, that contain the number of total words and number of unique words, respectively, for each of Shakespeare’s plays. Each vector should only require one line of code to compute. Hint: "[["() is actually a function that allows you to do extract a named component of a list; e.g., try "[["(shakespeare_wordobj, "number_total_words"), and you’ll see this is the same as shakespeare_wordobj[["number_total_words"]]; you should take advantage of this functionality in your apply call. What are the 5 longest plays, in terms of total word count? The 5 shortest plays?

# Refactoring the word table functions

• 4. Look back at get_wordtab_from_lines() and get_wordtab_from_url(). Note that they overlap heavily, i.e., their bodies contain a lot of the same code. Redefine get_wordtab_from_url() so that it just calls get_wordtab_from_lines() in its body. Your new get_wordtab_from_url() function should have the same inputs as before, and produce the same output as before. So externally, nothing will have changed; we are just changing the internal structure of get_wordtab_from_url() to clean up our code base (specifically, to avoid code duplication in our case). This is an example of code refactoring.

Call your new get_wordtab_from_url() function on the URL for Shakespeare’s complete works, saving the result as shakespeare_wordobj2. Compare some of the components of shakespeare_wordobj2 to those of shakespeare_wordobj (which was computed using the old function definition) to check that your new implementation works as it should.

• Challenge. Check using all.equal() whether shakespeare_wordobj and shakespeare_wordobj2 are the same. Likely, this will not return TRUE. (If it does, then you’ve already solved this challenge question!) Modify your get_wordtab_from_url() function from the last question, so that it still calls get_wordtab_from_lines() to do the hard work, but produces an output exactly the same as the original shakespeare_wordobj object. Demonstrate your success by calling all.equal() once again.