# Activity 11/08/2021

## STAT 445 / 645 -- Section 1001 <br> Instructor: Colin Grudzien<br>

## 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?

