Your homework must be submitted in R Markdown format. We will not (indeed, cannot) grade homeworks in other formats. Your responses must be supported by both textual explanations and the code you generate to produce your result. (Just examining your various objects in the “Environment” section of R Studio is insufficient—you must use scripted commands.)

  1. Decathlon with Superheroes. In your R console, run the following line: install.packages('ade4') in order to install the package ade4 (you only have to do this once). Then, the following code:

    library(ade4)
data(olympic)
    will load an object called olympic into your current R workspace, containing data about records of 33 athletes in the 10 events of a decathlon: 100 meters (100), long jump (long), shotput (poid), high jump (haut), 400 meters (400), 110-meter hurdles (110), discus throw (disq), pole vault (perc), javelin (jave) and 1500 meters (1500).
    1. olympic is a list. How many objects does it hold? What are the types and names of these objects?
    2. Take the first object in olympic and copy it into a new object called olympicmat. Cast it into a matrix, then back to a data frame. Did anything change?
    3. Change the names of the olympicmat into something more human-readable (although in general, usage of succinct variable names is good practice): replace the column names with their longer versions shown above (e.g., 100 into 100 meters). Show the first 10 rows after doing so.
    4. Suppose we have found out that the first three contestants are superheros. Replace just these names with ironman, wolverine and hulk. Again, show the first 10 rows after doing so.
    5. Now add a new datapoint to olympicmat, by appending the row shown below. Make sure to change the row name too. Show the last ten rows after doing so. Is thor an extraordinary discus thrower? Produce a plot of your choice to justify.
    thor = c(8.52, 10.31, 16.28, 4.51, 30.12, 13.62, 50.5, 10.1, 100.24, 200.12)
    1. Add the above changes back to olympic, by assigning your olypicmat back into the first object of olympic. Make sure you refer to objects of a list by its name (e.g., mylist[["mykey"]]), and not the index (e.g., mylist[[3]]).
    2. Now we will add a few objects to the list olympic. Add year and sporttype to the list in that order, with those same names.
    year = 1998
sporttype = "decathlon"
  2. Fun with Linear Regression. We will get more practice with matrix and vector data types, and also take a glimpse at linear regression. When asked to plot, always label and title the plots clearly.
    1. First generate a matrix as follows (the seed is to ensure we all get the same X1).
    onevec = rep(1,10)
set.seed(0)
X1 = rnorm(10,2.5,0.3)
X = cbind(X1, onevec)
    1. The “hat” matrix for linear regression onto the matrix \(X\) is defined by \(X(X^TX)^{-1}X^T\). Using functions for matrices like %*%, t() and solve(), calculate this matrix and store it as P.
    2. Compute the eigendecomposition of P (see example in lecture notes). Write a one line command to produce how many eigenvalues are larger than 1e-3. (Some insight: this is one way to find the rank of the matrix X.)
    3. Now right-multiply P by the vector y given below, and assign the resulting numeric vector to an object named yhat.
    y = c( 9.01, 8.39, 8.86, 11.2,  9.2, 6.29, 8.15, 8.97, 9.07, 10.4)
    1. Congratulations, you have computed your first linear regression! yhat is the best linear model prediction (of the form \(y = a X_1 + b\)) you could have made, using your predictor matrix X. Now, calculate \((X^TX)^{-1}X^Ty\), and cast it as a numeric vector called lincoef.
    2. These are your regression coefficients \((a,b)\)! A large positive value of a indicates a positive correlation between X1 and y (You will learn more about this in later courses). You may also use it to make future predictions at unobserved, new values of \(X_1\).
    3. Create two scatter plots, one of yhat and y, and one of y and X1. Do entries in yhat and y closely follow? Do you see a linear trend in the latter plot?
    4. Use abline(lincoef[2], lincoef[1], col = 'red') in the latter plot to see your regression model (and pat yourself on the back; a linear regression seems like an excellent choice).
    5. Lastly, gather all this information (P,X,y,lincoef, and yhat) into a list object called linregoutput. Show this object.