COSC380 - K-Means Algorithm by Matlab - Computer Science Assignment Help

Assignment Task

Question 1: [20 marks] Given the exponential probability density function normalized over the half-interval [0,∞): P(x|λ) = 1 λ e −x/λ , find the value of λ that maximizes the likelihood function for N i.i.d. data points {x1, ..., xN }. Simplify your expression and show all working, giving justifications along the way. A pdf file can be uploaded for this question.

Question 2: [60 marks] Continue on from Lab 2, and use Matlab to implement the following extensions to the K-means algorithm. Use two bivariate (two-dimensional) Gaussians, and test your implementations on the data from old faithful.dat by plotting your Gaussians for each question part. Label your script files my mix1.m to my mix3.m.

(a) Implement the soft K-means algorithm given in lectures for two Gaussians, each with a diagonal covariance matrix, and update the mean and responsibilities for each Gaussian. To find the appropriate diagonal covariance matrix, note that the simplest choice: Σ = 1 0 0 1 is not a good choice because the data in old faithful.dat has a range of 3.5 along one dimension, and 53 along the other. The squared ratio of these ranges is (53/3.5)2 ≈ 229. The variance along each dimension of the data should therefore maintain this ratio. A choice given by, Σ = 0.1 0 0 22.9 is one possibility. Please feel free to experiment with other possibilities that also maintain a ratio of σ 02 /σ2 = 229.

(b) Now develop a soft K-means algorithm where each of the Gaussians has a diagonal covariance matrix given by, Σk = σ 2 k 0 0 σ 0 k 2 and the four variance parameters σ 2 1 , σ0 1 2 , σ2 2 and σ 0 2 2 , as well as the two mixing coefficients π1 and π2 are now updated at each iteration using the maximum likelihood results derived in lectures for the univariate Gaussian. In this case, simply replace the scalar values xn and µk in each of the formulas, with the vector values xn and µk. Note that the scalar variance ˆσ 2 k then becomes a vector: σˆ 2 k = [σ 2 k , σ0 k 2 ]. 1

(c) Now implement the full Gaussian mixture model given in lectures (The EM algorithm for Gaussian mixtures). Each Gaussian now has a full covariance matrix: where each covariance matrix now has four parameters that must be learned from the data.

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