Problem 1 For the K-means clustering problem, when the binary indicators (responsibilities) 𝑟𝑘𝑛’s are fixed for k=1, 2, …, K and n=1, 2, …, N, derive for the cluster centers 𝐦𝑘, k=1, 2, …, K, such that the following objective function J is minimized:
Problem 2 Iris.xls contains 150 data samples of three Iris categories, labeled by outcome values 0, 1, and 2. Each data sample has four attributes: sepal length, sepal width, petal length, and petal width.
Implement the K-means clustering algorithm to group the samples into K=3 clusters. Randomly choose three samples as the initial cluster centers. Calculate the objective function value J as defined in Problem 1 after the assignment step in each iteration. Exit the iterations if the following criterion is met: 𝐽(Iter−1)−𝐽(Iter) < ε, where ε = 10−5, and Iter is the iteration number. Plot the objective function value J versus the iteration number Iter. Comment on the result. Attach the code at the end of the homework.
Problem 3 Assume a data sample 𝐱 ∈ ℝ𝐷 comes from one of two classes, 𝐶1 and 𝐶2. Use logistic regression to do classification.
- Write the math expression of the logistic regression output, and the criterion used for the final classification.
- How many parameters (weights) need to be calculated/trained in this method?
Problem 4 Assume a data sample 𝐱 ∈ ℝ𝐷 comes from one of 𝐾 classes, 𝐶1, 𝐶2, …, 𝐶𝐾. Use logistic regression to do classification.
- Write the math expression of the logistic regression output, and the criterion used for the final classification.
- How many parameters (weights) need to be calculated/trained in this method?
