Task: Mixture of Gaussians & EM

1. Assume we have two spherical Gaussians. For the first Gaussian, the mean is (1, 1), and standard deviation is 1. For the second Gaussian, the mean is (0, 4), and standard deviation is 2. Please generate 100 2-D points for the first Gaussian, and 50 points for the second Gaussian, and plot them in one figure.

2. Now, given the 150 points, implement the K-means algorithm discussed in class and partition the 150 points into 2 clusters. Compute the cluster centers and the standard deviation of each cluster. Discuss how you initialize the model and when to terminate the iterations.

3. Now, instead of using the K-means algorithm, let us use the soft EM algorithm. You could use the cluster centers and deviations calculated by K-means as initialization for the two Gaussians in EM. Compute the parameters for the two Gaussians. Discuss how you initialize the model and when to terminate.