Hello Murat,
To plot the linear decision boundary between two Gaussian-distributed datasets with given means and covariances, you can use the concept of discriminant functions. The decision boundary is where these discriminant functions are equal, i.e., 
.
.Given your datasets and parameters, you can follow these steps:
- Define the Discriminant Functions:
For each class 
, the linear discriminant function 
can be expressed as:
, the linear discriminant function can be expressed as:
where:
- Calculate the Coefficients:
Since your covariance matrices are identity matrices, the inverse is simply the same matrix, and the calculations simplify.
- Find the Decision Boundary:
The decision boundary is found by setting 
, which simplifies to solving:
, which simplifies to solving:
- Plot the Decision Boundary:
Solve the above equation for 
in terms of 
to get the equation of a line, and plot this line.
in terms of to get the equation of a line, and plot this line.Here is the updated code:
% Given data and parameters
u1 = [1; 1];
u2 = [3; 3];
s1 = [1 0; 0 1];
s2 = [1 0; 0 1];
Pr1 = 1/2;
Pr2 = 1/2;
% Generate random samples
r1 = mvnrnd(u1, s1, 500);
r2 = mvnrnd(u2, s2, 500);
% Plot the samples
plot(r1(:,1), r1(:,2), '+r');
hold on;
plot(r2(:,1), r2(:,2), '+b');
grid on;
% Calculate discriminant coefficients
W1 = inv(s1) * u1;
W10 = -0.5 * (u1' * inv(s1) * u1) + log(Pr1);
W2 = inv(s2) * u2;
W20 = -0.5 * (u2' * inv(s2) * u2) + log(Pr2);
% Calculate the decision boundary
W_diff = W1 - W2;
W0_diff = W10 - W20;
% Decision boundary line: W_diff' * X + W0_diff = 0
% Solve for X2 in terms of X1: W_diff(1)*X1 + W_diff(2)*X2 + W0_diff = 0
% X2 = -(W_diff(1)/W_diff(2)) * X1 - (W0_diff/W_diff(2))
x1_vals = linspace(min([r1(:,1); r2(:,1)]), max([r1(:,1); r2(:,1)]), 100);
x2_vals = -(W_diff(1)/W_diff(2)) * x1_vals - (W0_diff/W_diff(2));
% Plot the decision boundary
plot(x1_vals, x2_vals, '-k', 'LineWidth', 2);
xlabel('X1');
ylabel('X2');
title('Linear Decision Boundary');
legend('Class 1', 'Class 2', 'Decision Boundary');
hold off;
The output of this code looks like:
I hope this helps!
