Both Part I and II have to be done by all sections. Differences have been specified by CSCI4390 and CSCI6390 labels.
Part I: Principal Components Analysis (50 points)
You will implement the PCA algorithm as described in Algorithm 7.1 (Chapter 7). You need to compute the eigenvectors, and then project visualize the data. To compute the principal components (PCs), you may use the inbuilt numpy function eigh.
Run PCA on the Appliances energy prediction data set You should ignore the first attribute, which is a date-time variable, and you should a the last attribute, which is a duplicate of the previous one.
Next, determine and print how many dimensions are required to capture α = 0.975 fraction of the total variance?
Also print the mean squared error in the approximation using the first three components.
Plot the PCs
CSCI4390 Only: Project the points along the first two PCs, and create a scatter plot of the projected points.
CSCI64390 Only: Project the points along the first three PCs, and create a 3D scatter plot of the projected points.
Part II: Diagonals in High Dimensions (50 points)
Your goal is the compute the probability mass function for the random variable X that represents the angle (in degrees) between any two d high dimensions.
Assume that there are d primary dimensions (the standard axes in cartesian coordinates), with each of them ranging from -1 to 1. There are additional half-diagonals in this space, one for each corner of the d-dimensional hypercube.
Randomly generate n = 100,000 pairs of half-diagonals in the d-dimensional hypercube (random d-dimensional vectors with elements and compute the angle between them (in degrees).
Plot the probability mass function (PMF) for three different values of d, as follows d = 10,100,1000. Recall that PMF is simply the plo angle versus the probability of observing that angle in the sample of n points for a given value of d. What is the min, max, value range, me variance of X for each value of d?
What would expect analytically? In other words, derive formulas for what should happen to angle between half-diagonal ∞. Does the PMF conform to this trend? Explain why? or why not?
