Expectation Maximization

The EM algorithm will be implemented, and GMM estimation will be performed on an image dataset containing images of Manchester United and Chelsea football clubs in order to classify them.

from sklearn.mixture import GaussianMixture

Preprocessing

1. All the labels have been replaced with 'm' for Manchester United and 'c' for Chelsea. The dataset consists of 64 images with the label 'c' and 58 images with the label 'm'.
2. Only the blue and red channels have been extracted as features.
3. The dataset is divided into a training set and a test set.

Data Exploration

Please find the scatter plot depicting the data points below.

The table below provides an analytical description of these features.

Metric	Red Feature	Blue Feature
count	122	122
mean	91.975741	78.674581
std	29.686593	30.071529
min	36.829846	23.841212
25%	70.142986	55.459667
50%	90.938226	72.883534
75%	109.730649	96.326040
max	171.014902	188.230243

n_components = 2

Selecting the right number of Gaussian models is essential for achieving the best results. In this study, the focus is on examining the case where n_components = 2.

Parameter	First Gaussian Model	Second Gaussian Model
Mean	[62.90, 55.94726292]	[104.27, 86.41]
Covariance Matrix	[[209.36, 42.49], [ 42.49, 214.85]]	[[716.86, 177.62], [177.62, 792.93]]

The contour plot is shown below.

Best n_components

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are statistical measures commonly used for model selection and evaluation in the context of Gaussian Mixture Models (GMMs) and other statistical models.

Both AIC and BIC are based on the principle of parsimony, which favors simpler models that explain the data well. The goal is to strike a balance between model complexity and goodness of fit. Lower values of AIC or BIC indicate better model performance. The model with the lowest AIC or BIC score is considered the best choice.

Here are the AIC and BIC scores for n_components ranging from 1 to 10. Based on these scores, the best n_components for this classification problem, considering the aforementioned features, is 1.

Course Description

• Course: Machine Learning [ECE 501]
• Semester: Spring 2023
• Institution: School of Electrical & Computer Engineering, College of Engineering, University of Tehran
• Instructors: Dr. A. Dehaqani, Dr. Tavassolipour