How do I plot linear decision boundary in 2 classes

I have 2 different dataset that have different mu and sigma, and X vector such as [1.8; 1.8]. And also I know probability of each classes that P(ω1)= P(ω2) = 1/2

I want to ploting linear decision boundary between these two dataset but I don't have any idea how to do. My code is down below, here

X = [1.8; 1.8];
u1 = [1;1]; u2 = [3;3];
s1 = [1 0;0 1]; s2 = [1 0;0 1];
Pr1 = 1/2;
Pr2 = 1/2;
r = mvnrnd(u1,s1,500);
plot(r(:,1), r(:,2), '+r');
hold on
r = mvnrnd(u2,s2,500);
plot(r(:,1), r(:,2), '+b');
hold on
grid on
W1 = (u1')/(s1(1,1))^2;
W10 = (u1'*u1)/(-2*s1(1,1)) + log(Pr1);
g1 = W1'.*X + W10;
W2 = (u2')/(s2(1,1))^2;
W20 = (u2'*u2)/(-2*s2(1,1)) + log(Pr2);
g2 = W2'.*X + W20;

Is there someone who can give any idea to me please?

Answer 1

[Prateekshya]

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:

where:

1. 
2. 

• 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:

• Plot the Decision Boundary:

Solve the above equation for 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!