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 Wednesday 10pm, this week.

## For reproducibility --- don't change this!
set.seed(07012019)

The binomial distribution

The binomial distribution $$\mathrm{Bin}(m,p)$$ is defined by the number of successes in $$m$$ independent trials, each have probability $$p$$ of success. Think of flipping a coin $$m$$ times, where the coin is weighted to have probability $$p$$ of landing on heads.

The R function rbinom() generates random variables with a binomial distribution. E.g.,

rbinom(n=20, size=10, prob=0.5)

produces 20 observations from $$\mathrm{Bin}(10,0.5)$$.

Some simple manipulations

• 1a. Generate 200 random values from the $$\mathrm{Bin}(10,0.5)$$ distribution, and store them in a vector called bin_draws_0.5. Extract and display the first 10 elements. Extract and display all but the first 175 elements.

• 1b. Add the first element of bin_draws_0.5 to the seventh. Compare the second element to the fifth, which is larger? A bit more tricky: print the indices of the elements of bin_draws_0.5 that are equal to 5. How many such elements are there? Challenge: theoretically, how many such elements would you expect there to be?

• 1c. Find the mean and standard deviation of bin_draws_0.5. Is the mean close what you’d expect? The standard deviation?

• 1d. Call summary() on bin_draws_0.5 and describe the result.

• 1e. Find the data type of the elements in bin_draws_0.5 using typeof(). Then convert bin_draws_0.5 to a vector of characters, storing the result as bin_draws_0.5_char, and use typeof() again to verify that you’ve done the conversion correctly. Call summary() on bin_draws_0.5_char. Is the result formatted differently from what you saw above? Why?

Some simple plots

• 2a. The function plot() is a generic function in R for the visual display of data. The function hist() specifically produces a histogram display. Use hist() to produce a histogram of your random draws from the binomial distribution, stored in bin_draws_0.5.

• 2b. Call tabulate() on bin_draws_0.5. What is being shown? Does it roughly match the histogram you produced in the last question?

• 2c. Call plot() on bin_draws_0.5 to display your random values from the binomial distribution. Can you guess what the plot() function is doing here?

• 2d. Call plot() with two arguments, the first being 1:200, and the second being bin_draws_0.5. This creates a scatterplot of bin_draws_0.5 (on the y-axis) versus the indices 1 through 200 (on the x-axis). Does this match your plot from the last question?

Working with matrices and lists

• 3a. Create a matrix of dimension 5 x 10, called mat which contains the numbers 1 through 50, column-wise. That is, reading from top-to-bottom, the first (left-most) column of mat should read 1:5. Then, create a matrix of the same dimension, called tmat, except the numbers 1 through 50 are now stored row-wise. That is, the first row from left-to-right should read 1:10. Print out both matrices.

• 3b. Change the element in the second row, fourth column of mat into the string asdf. Print out mat afterwards. What happened to all the remaining entries of mat? (Hint: Use typeof().) What can you conclude about how R treats matrices of numerics when a string is included?

• 3c. Create a list called example_list where the first element is the string “36-350” (named “Course”), the second element is a vector of three TRUE values and five FALSE values (named “Boolean”), and the third element is tmat (from the previous question, named “Matrix”). Print out example_list.

• 3d. Using the class() function, determine the class of each of the three elements in example_list. (Hint: Your answer should not be lists. If it is, you are accessing the list incorrectly.)

Prostate cancer data set

We’re going to look at a data set on 97 men who have prostate cancer (from the book The Elements of Statistical Learning). There are 9 variables measured on these 97 men:

1. lpsa: log PSA score
2. lcavol: log cancer volume
3. lweight: log prostate weight
4. age: age of patient
5. lbph: log of the amount of benign prostatic hyperplasia
6. svi: seminal vesicle invasion
7. lcp: log of capsular penetration
8. gleason: Gleason score
9. pgg45: percent of Gleason scores 4 or 5

To load this prostate cancer data set into your R session, and store it as a matrix pros_data:

pros_data <-
as.matrix(read.table("https://raw.githubusercontent.com/linnylin92/36-350_public/master/dat/pros.dat"))

Basic indexing and calculations

• 4a. What are the dimensions of pros_data (i.e., how many rows and how many columns)? Using integer indexing, print the first 6 rows and all columns; again using integer indexing, print the last 6 rows and all columns.

• 4b. Using the built-in R functions head() and tail() (i.e., do not use integer indexing), print the first 6 rows and all columns, and also the last 6 rows and all columns.

• 4c. Does the matrix pros_data have names assigned to its rows and columns, and if so, what are they? Use rownames() and colnames() to find out. Note: these would have been automatically created by the read.table() function that we used above to read the data file into our R session. To see where read.table() would have gotten these names from, open up the data file: https://raw.githubusercontent.com/linnylin92/36-350_public/master/dat/pros.dat in your web browser. Only the column names here are actually informative.

• 4d. Using named indexing, pull out the two columns of pros_data that measure the log cancer volume and the log cancer weight, and store the result as a matrix pros_data_sub. (Recall the explanation of variables at the top of this lab.) Check that its dimensions make sense to you, and that its first 6 rows are what you’d expect. Did R automatically assign column names to pros_data_sub?

• 4e. Using the log cancer weights and log cancer volumes, calculate the log cancer density for the 97 men in the data set (note: by density here we mean weight divided by volume). There are in fact two different ways to do this; the first uses three function calls and one arithmetic operation; the second just uses one arithmetic operation. Note: in either case, you should be able to perform this computation for all 97 men with a single line of code, taking advantage of R’s ability to vectorize. Write code to do it both ways, and show that both ways lead to the same answer, using all.equal().

• 4f. Append the log cancer density to the columns of pros_data, using cbind(). The new pros_data matrix should now have 10 columns. Set the last column name to be ldens. Print its first 6 rows, to check that you’ve done all this right.

Exploratory data analysis with plots

• 5a. Using hist(), produce a histogram of the log cancer volume measurements of the 97 men in the data set; also produce a histogram of the log cancer weight. In each case, use breaks=20 as an arugment to hist(). Comment just briefly on the distributions you see. Then, using plot(), produce a scatterplot of the log cancer volume (y-axis) versus the log cancer weight (x-axis). Do you see any kind of relationship? Would you expect to? Challenge: how would you measure the strength of this relationship formally? Note that there is certainly more than one way to do so.

• 5b. Produce scatterplots of log cancer weight versus age, and log cancer volume versus age. Do you see relationships here between the age of a patient and the volume/weight of his cancer?

• 5c. Produce a histogram of the log cancer density, and a scatterplot of the log cancer density versus age. Comment on any similarities/differences you see between these plots, and the corresponding ones you produced above for log cancer volume/weight.

• 5d. Delete the last column, corresponding to the log cancer density, from the pros_data matrix, using negative integer indexing.

A bit of Boolean indexing never hurt anyone

• 6a. The svi variable in the pros_data matrix is binary: 1 if the patient had a condition called “seminal vesicle invasion” or SVI, and 0 otherwise. SVI (which means, roughly speaking, that the cancer invaded into the muscular wall of the seminal vesicle) is bad: if it occurs, then it is believed the prognosis for the patient is poorer, and even once/if recovered, the patient is more likely to have prostate cancer return in the future. Compute a Boolean vector called has_svi, of length 97, that has a TRUE element if a row (patient) in pros_data has SVI, and FALSE otherwise. Then using sum(), figure out how many patients have SVI.

• 6b. Extract the rows of pros_data that correspond to patients with SVI, and the rows that correspond to patients without it. Call the resulting matrices pros_data_svi and pros_data_no_svi, respectively, and print both matrices. You can do this in two ways: using the has_svi Boolean vector created above, or using on-the-fly Boolean indexing, it’s up to you. Check that the dimensions of pros_data_svi and pros_data_no_svi make sense to you.

• 6c. Using the two matrices pros_data_svi and pros_data_no_svi that you created above, compute and print the means of each variable in our data set for patients with SVI, and for patients without it. Store the resulting means into vectors called pros_data_svi_avg and pros_data_no_svi_avg, respectively. Hint: for each matrix, you can compute the means with a single call to a built-in R function. What variables appear to have different means between the two groups?

Some string basics

• 7a. Define two strings variables, equal to “Statistical Computing” and ‘Statistical Computing’, and check whether they are equal. What do you conclude about the use of double versus single quotation marks for creating strings in R? Give an example that shows why might we prefer to use double quotation marks as the standard (think of apostrophes).

• 7b. The functions tolower() and toupper() do as you’d expect: they convert strings to all lower case characters, and all upper case characters, respectively. Apply them to the strings below, as directed by the comments, to observe their behavior.

c(“I’M NOT ANGRY I SWEAR”) # Convert to lower case c(“Mom, I don’t want my veggies”) # Convert to upper case c(“Hulk, sMasH”) # Convert to upper case c(“R2-D2 is in prime condition, a real bargain!”) # Convert to lower case

• 7c. Consider the string vector presidents of length 5 below, containing the last names of past US presidents. Define a string vector first_letters to contain the first letters of each of these 5 last names. Hint: use substr(), and take advantage of vectorization; this should only require one line of code. Define first_letters_scrambled to be the output of sample(first_letters) (the sample() function can be used to perform random permutations, we’ll learn more about it later in the course). Lastly, reset the first letter of each last name stored in presidents according to the scrambled letters in first_letters_scrambled. Hint: use substr() again, and take advantage of vectorization; this should only take one line of code. Display these new last names.
presidents <- c("Clinton", "Bush", "Reagan", "Carter", "Ford")
• 7d. Now consider the string phrase defined below. Using substr(), replace the first four characters in phrase by “Provide”. Print phrase to the console, and describe the behavior you are observing. Using substr() again, replace the last five characters in phrase by “kit” (don’t use the length of phrase as magic constant in the call to substr(), instead, compute the length using nchar()). Print phrase to the console, and describe the behavior you are observing.
phrase <- "Give me a break"
• 7e. Consider the string ingredients defined below. Using strsplit(), split this string up into a string vector of length 5, with elements “chickpeas”, “tahini”, “olive oil”, “garlic”, and “salt.” Using paste(), combine this string vector into a single string “chickpeas + tahini + olive oil + garlic + salt”. Then produce a final string of the same format, but where the ingredients are sorted in alphabetical (increasing) order.
ingredients <- "chickpeas, tahini, olive oil, garlic, salt"

Shakespeare’s complete works

Project Gutenberg offers over 50,000 free online books, especially old books (classic literature), for which copyright has expired. We’re going to look at the complete works of William Shakespeare, taken from the Project Gutenberg website.

To avoid hitting the Project Gutenberg server over and over again, we’ve grabbed a text file from them that contains the complete works of William Shakespeare and put it on our course website. Visit https://raw.githubusercontent.com/linnylin92/36-350_public/master/dat/shakespeare.txt in your web browser and just skim through this text file a little bit to get a sense of what it contains (a whole lot!).

• 8a. Read in the Shakespeare data linked above into your R session with readLines(). Make sure you are reading the data file directly from the web (rather than locally, from a downloaded file on your computer). Call the result shakespeare_lines. This should be a vector of strings, each element representing a “line” of text. Print the first 5 lines. How many lines are there? How many characters in the longest line? What is the average number of characters per line? How many lines are there with zero characters (empty lines)? Hint: each of these queries should only require one line of code; for the last one, use an on-the-fly Boolean comparison and sum().

• 8b. Remove the lines in shakespeare_lines that have zero characters. Hint: use Boolean indexing. Check that the new length of shakespeare_lines makes sense to you.

• 8c. Collapse the lines in shakespeare_lines into one big string, separating each line by a space in doing so, using paste(). Call the resulting string shakespeare_all. How many characters does this string have? How does this compare to the sum of characters in shakespeare_lines, and does this make sense to you?

• 8d. Split up shakespeare_all into words, using strsplit() with split=" ". Call the resulting string vector (note: here we are asking you for a vector, not a list) shakespeare_words. How long is this vector, i.e., how many words are there? Using the unique() function, compute and store the unique words as shakespeare_words_unique. How many unique words are there?

• 8e. Plot a histogram of the number of characters of the words in shakespeare_words_unique. You will have to set a large value of the breaks argument (say, breaks=50) in order to see in more detail what is going on. What does the bulk of this distribution look like to you? Why is the x-axis on the histogram extended so far to the right (what does this tell you about the right tail of the distribution)?

• 8f. Reminder: the sort() function sorts a given vector into increasing order; its close friend, the order() function, returns the indices that put the vector into increasing order. Both functions can take decreasing=TRUE as an argument, to sort/find indices according to decreasing order. See the code below for an example.

set.seed(0)
x <- round(runif(5, -1, 1), 2)
sort(x, decreasing = TRUE)
## [1]  0.82  0.79  0.15 -0.26 -0.47
order(x, decreasing = TRUE)
## [1] 5 1 4 3 2

Using the order() function, find the indices that correspond to the top 5 longest words in shakespeare_words_unique. Then, print the top 5 longest words themselves. Do you recognize any of these as actual words? Challenge: try to pronounce the fourth longest word! What does it mean?

Computing word counts

• 9a. Using table(), compute counts for the words in shakespeare_words, and save the result as shakespeare_wordtab. How long is shakespeare_wordtab, and is this equal to the number of unique words (as computed above)? Using named indexing, answer: how many times does the word “thou” appear? The word “rumour”? The word “gloomy”? The word “assassination”?

• 9b. How many words did Shakespeare use just once? Twice? At least 10 times? More than 100 times?

• 9c. Sort shakespeare_wordtab so that its entries (counts) are in decreasing order, and save the result as shakespeare_wordtab_sorted. Print the 25 most commonly used words, along with their counts. What is the most common word? Second and third most common words?

• 9d. What you should have seen in the last question is that the most common word is the empty string "". This is just an artifact of splitting shakespeare_all by spaces, using strsplit(). Redefine shakespeare_words so that all empty strings are deleted from this vector. Then recompute shakespeare_wordtab and shakespeare_wordtab_sorted. Check that you have done this right by printing out the new 25 most commonly used words, and verifying (just visually) that is overlaps with your solution to the last question.

A tiny bit of regular expressions

• 10a. There are a couple of issues with the way we’ve built our words in shakespeare_words. The first is that capitalization matters; from Q9c, you should have seen that “and” and “And” are counted as separate words. The second is that many words contain punctuation marks (and so, aren’t really words in the first place); to see this, retrieve the count corresponding to “and,” in your word table shakespeare_wordtab.

The fix for the first issue is to convert shakespeare_all to all lower case characters. Hint: recall tolower(). The fix for the second issue is to use the argument split="[[:space:]]|[[:punct:]]" in the call to strsplit(), when defining the words. In words, this means: split on spaces or on punctuation marks (more precisely, it uses what we call a regular expression for the split argument). Carry out both of these fixes to define new words shakespeare_words_new. Then, delete all empty strings from this vector, and compute word table from it, called shakespeare_wordtab_new.

• 10b. Compare the length of shakespeare_words_new to that of shakespeare_words; also compare the length of shakespeare_wordtab_new to that of shakespeare_wordtab. Explain what you are observing.

• 10c. Compute the unique words in shakespeare_words_new, calling the result shakespeare_words_new_unique. Then repeat the queries in Q8e and Q8f on shakespeare_words_new_unique. Comment on the histogram—is it different in any way than before? How about the top 5 longest words?

• 10d. Sort shakespeare_wordtab_new so that its entries (counts) are in decreasing order, and save the result as shakespeare_wordtab_sorted_new. Print out the 25 most common words and their counts, and compare them (informally) to what you saw in Q9d.