In this homework, you will implement an expectation-maximization (EM) clustering algorithm in Python. Here are the steps you need to follow:
1. You are given a two-dimensional data set in the file named hw08_data_set.csv, which contains 1000 data points generated randomly from nine bivariate Gaussian
densities with the following parameters.
𝜇! = #++55..00( ,
Σ! = #+−00..86
−0.6( ,
+0.8
𝑁! = 100
𝜇" = #−+55..00( ,
Σ" = #++00..86
+0.6( ,
+0.8
𝑁" = 100
𝜇# = #−−55..00( ,
Σ# = #+−00..86
−0.6( ,
+0.8
𝑁# = 100
𝜇$ = #+−55..00( ,
Σ$ = #++00..86
+0.6( ,
+0.8
𝑁$ = 100
𝜇% = #++50..00( ,
Σ% = #++00..20
+0.0( ,
+1.2
𝑁% = 100
𝜇& = #++05..00( ,
Σ& = #++10..20
+0.0( ,
+0.2
𝑁& = 100
𝜇’ = #−+50..00( ,
Σ’ = #++00..20
+0.0( ,
+1.2
𝑁’ = 100
𝜇( = #+−05..00( ,
Σ( = #++10..20
+0.0( ,
+0.2
𝑁( = 100
𝜇) = #++00..00( ,
Σ) = #++10..60
+0.0( ,
+1.6
𝑁) = 200
The given data points are shown in the following figure.
2. To initialize your EM algorithm, you should take the centroids given in the file named hw08_initial_centroids.csv as the initial values for the mean vectors. By assigning the data points to the nearest center, estimate the initial covariance matrices and prior probabilities in your EM algorithm. (20 points)
3. After the initialization step, run your EM algorithm for 100 iterations. Report the mean vectors your EM algorithm finds. Your results should be like the following matrix. (50 points)
print(means)
[[-4.9508988 -4.98464367]
[-4.85629614 0.0404331 ]
[-4.96379877 4.984647 ]
[ 0.02477868 -5.09014979]
[-0.09548618 -0.116943 ]
[-0.03701877 4.91812108]
[ 5.00933942 -5.02595861]
[ 4.99839618 0.13777844]
[ 4.96705774 4.97185503]]
4. Draw the clustering result obtained by your EM algorithm by coloring each cluster with a different color. You should also draw the original Gaussian densities you use to generate data points and the Gaussian densities your EM algorithm finds with dashed and solid lines, respectively. Draw these Gaussian densities where their values are equal to 0.05. Your figure should be like the following figure. (30 points
