7 add more explanatory comments for all scripts matlab and python in gmm hmm lecture directory

GMM_HMM_Lecture/MATLAB_code/gmm_2D_simulated.m

@@ -1,56 +1,89 @@
-%% Clustering with 2D Gaussian Mixture Models
-clear all
-close all
-%rng(0)
-%% Generate Data
-% Hardcoded to generate 2D variables
-
-% Create distribution parameters
-components = 3;
-mu_s = {};  % Mean vectors
-covs = {};  % Covariance matrices
-for i=1:components
-   mu_s{end+1} = rand(2,1).*10;  % Scaled to clearly separate clusters
-   covs{end+1} = gen_cov_mat(); 
-end
-
-% Generate data for each component
-sample_num = 1000; % Samples per component
-samples = [];
-for i=1:components    
-    samples = [samples;mvnrnd(mu_s{i},covs{i},sample_num)];
-end
-
-% Plot samples
-scatter(samples(:,1),samples(:,2));
-
-%% Fitting GMM
-gm = fitgmdist(samples,components);
-
-% Plot output
-% FIgure out plot range
-minx =  min(samples(:,1));
-maxx =  max(samples(:,1));
-miny =  min(samples(:,2));
-maxy =  max(samples(:,2));
-% Create meshgrid
-x1 = minx:.1:maxx;
-x2 = miny:.1:maxy;
-[X1,X2] = meshgrid(x1,x2);
-X = [X1(:) X2(:)];
-% Evaluate PDF
-y = pdf(gm,X);
-y = reshape(y,length(x2),length(x1));
-
-% Plot!
-figure
-scatter(samples(:,1),samples(:,2),10,'.'); % Scatter plot with points of size 10
-hold on
-contour(x1,x2,y,100)
-
-%% Functions
-function cov_mat = gen_cov_mat()
-    cross_cov = rand();
-    temp_cov_mat = [rand() cross_cov; cross_cov rand()];
-    cov_mat = temp_cov_mat*temp_cov_mat'; % Make matrix symmetric by multiplying with transpose 
-end
+%% Clustering with 2D Gaussian Mixture Models
+% This script demonstrates:
+% 1. Generation of synthetic 2D data from multiple Gaussian distributions
+% 2. Fitting a Gaussian Mixture Model (GMM) to the data
+% 3. Visualization of the fitted model using contour plots
+%
+% GMMs are probabilistic models that assume data points are generated from a 
+% mixture of several Gaussian distributions with unknown parameters.
+
+clear all
+close all
+%rng(0)  % Uncomment to set random seed for reproducibility
+
+%% Generate Data
+% Hardcoded to generate 2D variables with 3 distinct clusters
+
+% Create distribution parameters
+components = 3;  % Number of Gaussian components (clusters)
+mu_s = {};       % Mean vectors for each component
+covs = {};       % Covariance matrices for each component
+
+% Generate random means and covariance matrices for each component
+for i=1:components
+   mu_s{end+1} = rand(2,1).*10;  % Scaled to clearly separate clusters
+   covs{end+1} = gen_cov_mat();  % Generate random covariance matrix
+end
+
+% Generate data points for each component (cluster)
+sample_num = 1000; % Samples per component
+samples = [];
+for i=1:components    
+    % Generate multivariate normal samples using the specified mean and covariance
+    component_samples = mvnrnd(mu_s{i}, covs{i}, sample_num);
+    % Stack the new samples with existing ones
+    samples = [samples; component_samples];
+end
+
+% Plot the generated samples
+figure;
+scatter(samples(:,1), samples(:,2));
+title('Generated 2D Gaussian Mixture Data');
+xlabel('Dimension 1');
+ylabel('Dimension 2');
+
+%% Fitting a Gaussian Mixture Model to the data
+% The model will try to identify the original components that generated the data
+gm = fitgmdist(samples, components);  % Create GMM with same number of components as generated data
+
+% Plot the fitted GMM with probability contours
+% First, determine the plot range based on data extents
+minx = min(samples(:,1));
+maxx = max(samples(:,1));
+miny = min(samples(:,2));
+maxy = max(samples(:,2));
+
+% Create a grid of points to evaluate the GMM probability density
+x1 = minx:.1:maxx;  % Grid points along x-axis
+x2 = miny:.1:maxy;  % Grid points along y-axis
+[X1,X2] = meshgrid(x1,x2);  % Create 2D grid
+X = [X1(:) X2(:)];  % Reshape to 2D array of points
+
+% Evaluate the probability density function (PDF) at each grid point
+y = pdf(gm,X);  % Calculate probability density
+y = reshape(y,length(x2),length(x1));  % Reshape to match grid dimensions
+
+% Create visualization with data points and probability contours
+figure;
+scatter(samples(:,1),samples(:,2),10,'.'); % Scatter plot with points of size 10
+hold on;
+contour(x1,x2,y,100);  % Plot probability contours
+title('2D GMM with Probability Density Contours');
+xlabel('Dimension 1');
+ylabel('Dimension 2');
+colorbar;  % Add colorbar to show density scale
+
+%% Functions
+function cov_mat = gen_cov_mat()
+    % Generate a random covariance matrix that is guaranteed to be 
+    % positive semi-definite (a requirement for covariance matrices).
+    %
+    % Returns:
+    %   cov_mat: A 2x2 random covariance matrix
+
+    cross_cov = rand();  % Random correlation between dimensions
+    temp_cov_mat = [rand() cross_cov; cross_cov rand()];
+    % Make matrix symmetric and positive semi-definite by multiplying with transpose
+    % This is a common technique to ensure valid covariance matrices
+    cov_mat = temp_cov_mat*temp_cov_mat'; 
+end