PS#10

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California Lutheran University *

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IDS575

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Computer Science

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Apr 3, 2024

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Q1 Mixture of Gaussian 45 Points Q1.1 8 Points Select all correct about GMM. Q1.2 8 Points Choose all of the following correctly describes the differences between GMM and k-means. Gaussian mixture moel is supervised The number of Gaussian distribution functions used is equal to the number of clusters  Gaussian Mixture model is equivalent to Quadratic Discriminant Analysis of GMM can be decomposed similarly to PCA  Σ k-means often gets stuck in a local minimum, while GMM with EM tends not to GMM is better at capturing clusters of different sizes and orientations  GMM is better at capturing clusters with overlaps  GMM is less prone to overfitting

Q1.3 15 Points Suppose we have a Gaussian mixture of dimensional data with 3 dimensions, and we use a model with full covariance matrices. How many parameters are in the model if we set the number of cluster as 4? 39 Q1.4 7 Points Which of the following contour plots describes a Gaussian distribution with diagonal covariance? Select all that apply.

Q1.5 7 Points The following four clusterings of points into 2 clusters.Appling GMM to which is better than k-means? select all correct A  B C D 

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Q2 Expectation and Maximization 55 Points Q2.1 7 Points The EM algorithm optimizes a lower bound on its objective function, which is the marginal likelihood of the observed data points . Q2.2 7 Points In the EM algorithm, what are the E step and M step? A B  C  D P ( x ) ∏ i i x , x , ... x 1 2 N True  False 

Q2.3 7 Points The EM algorithm is guaranteed to never decrease the value of its objective function on any iteration. Q2.4 7 Points Suppose we have data that come from a mixture of 6 Gaussians. Which model would we expect to have the highest log-likelihood after fitting via the EM algorithm? Estimate cluster responsibilities, Maximize likelihood over parameters  Estimate likelihood over parameters, Maximize cluster responsibilities  Estimate number of parameters, Maximize likelihood over parameters  Estimate likelihood over parameters, Maximize number of parameters  True  False  a mixture of Gaussians with 2 components  a mixture of Gaussians with 4 components  a mixture of Gaussians with 6 components  a mixture of Gaussians with 8 components  a mixture of Gaussians with 10 components 

Q2.5 7 Points EM algorithm can only be used for parameter estimation of mixture models. Q2.6 20 Points Suppose that we are fitting a Gaussian mixture model for data consisting of a single real value x, using K=2 components. We have m=5 training cases. to are 5,15,25,30,40. We use the EM algorithm to find the maximum likelihood estimates for the model parameters, , , and . The standard deviations for the two components are fixed at 10. Suppose that at some point in the EM algorithm, the E step found that the weights of the two components for the five data samples were as follows: What values for the parameters , , and will be found in the next M step of the algorithm? Q2.6.pdf  Download True  False  x (1) x (5) ϕ 1 ϕ 2 μ 1 μ 2 ϕ 1 ϕ 2 μ 1 μ 2 

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1 of 1 GRADED Problem Set (PS) #10 STUDENT 

Urvashiben Patel TOTAL POINTS 100 / 100 pts QUESTION 1 Mixture of Gaussian 45 / 45 pts 1.1 (no title) 8 / 8 pts 1.2 (no title) 8 / 8 pts 1.3 (no title) 15 / 15 pts 1.4 (no title) 7 / 7 pts 1.5 (no title) 7 / 7 pts QUESTION 2 Expectation and Maximization 55 / 55 pts 2.1 (no title) 7 / 7 pts 2.2 (no title) 7 / 7 pts 2.3 (no title) 7 / 7 pts 2.4 (no title) 7 / 7 pts 2.5 (no title) 7 / 7 pts 2.6 (no title) 20 / 20 pts

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