Instructor: Colin Grudzien

## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ## Activities: We will use the following package for the multivariate normal distribution: ```{r} require(mnormt) ``` ### Activity 1: The multivariate normal and the Mahalanobis transformation #### Question 1: Create a 2x2 matrix $\mathbf{A}$ with the columns given by the integers 1 through 4 in order. Evaluate if this has any zero eigenvalues. Then construct the symmetric matrix by $\mathbf{B} =\mathbf{A}\mathbf{A}^\top$. Evaluate if this has any zero eigenvalues. #### Question 2: Use the following seed and the `rmnorm` function to generate a random sample. Set the mean equal to mu below, but try each of $\mathbf{A}$ and $\mathbf{B}$ for the `varcov` parameter. What do you notice? Plot the sample and the lines formed by the eigenvectors for the `varcov` parameter using the supplied plotting code below. What do you notice about the plot? ```{r} set.seed(123) sample_size <- 10e3 mu = matrix(34:35, 2) ``` ```{r, eval=FALSE} eigen_decomp <- vecs <- vals <- eigScl <- vecs %*% diag(sqrt(vals)) # scale eigenvectors to length = square-root of eigenvalue xMat <- rbind(mu[1] + eigScl[1, ], mu[1] - eigScl[1, ]) yMat <- rbind(mu[2] + eigScl[2, ], mu[2] - eigScl[2, ]) matlines(xMat, yMat, lty=1, lwd=2, col="red") ``` #### Question 3: Now try subtracting the value `mu` from the sample in all entries. Compute the mean of the sample. Plot the transformed sample, and the eigen vectors as before with the code below: ```{r, eval=FALSE} plot(transformed_sample[,1], transformed_sample[,2], asp=1) xMat <- rbind(0 + eigScl[1, ], 0 - eigScl[1, ]) yMat <- rbind(0+ eigScl[2, ], 0 - eigScl[2, ]) matlines(xMat, yMat, lty=1, lwd=2, col="red") ``` #### Question 4: Now, try transforming the sample by the inverse of `B` and the inverse of `A`. Plot the results and see what you get. What do you notice about the results?