Linear Algebra, Python is being used in this class. Numpy, sklearn, pandas can be used, below are notes that might contextualize the problems above.

II. K-Means The following are vectors in R2 that we wish to group using the K-Means algorithm. x0=0,1,x1=1,1,x2=0,0,x3=4,5,x4=5,3,x5=0,7 Exercise 5. Suppose group representative vectors are randomly assigned to be z0=x0,z1=x1, and z2=x4. This implies that there are total groups. Exercise 6. Calculate and report the initial J0clust that results from the representative vectors in the previous exercise. (Hint: do this by hand like we did in class.) Exercise 7. Now, run the K-means algorithm once (that is, complete the steps a single time). The updated version of the third group representative is now z2=[,]. Exercise 8. The final optimal grouping once the K-Means algorithm has converged has a Jclust value of . (Hint: this is lowest value Jclust can achieve for this initial grouping.) Exercise 9. If we wanted a Jclust of 0 we could make this happen by choosing k=. . (Hint: look at the numerator of Jclust and think about how to get each piece to be 0 . Also, just because this is possible doesn't mean it's a good idea.) More formally, if z1,,zk are the group centroid vectors with j1,,k, and ci=j, we define the objective function as Jclust=(x1zc12+x2zc22++xnzcn2) We have found an "optimal" grouping when Jclust is minimized. We can miminize Jclust by minimizing each inidividual xizci. Thus, the optimal grouping solution is found by minj=1,2,,kx1zj+minj=1,2,,kx2zj++minj=1,2,,kxnzj K-Means Clustering Algorithm: 1. Choose k, the number of groups. Randomly initialize k centroids. 2. Compute the Euclidian distance between each data observation (vector) and each of the k centroids. 3. Assign each vector to the group with the closest centroid. 4. Update each centroid as the average of all vectors assigned to that centroid's group. 5. Repeat steps 2-4 until convergence