Decision Boundary for Gaussian Distribution

The two distributions (blue and red) are represented by three iso-lines of the probability density functions p₁(x) and p₂(x), being con-central similar ellipses. We assume the iso-lines represent the same density for the two classes. Thus, the decision boundary must pass through the intersection points of corresponding iso-lines. This decision boundary generally is a conic.

The initial configuration shows two Gaussian distribution which have the same covariance matrix. The resulting decision boundary is a straight line.

You may change the two distributions by moving the centres (means), changing the semi-axes of the ellipses (left) and by rotating one of the axes. The animation then shows the resulting decision boundary, specifically the coordinates x where |ln p₁-ln p₂| < tol.

In addition the Bhattacharyya-distance with its translation component (the Mahalanobis distance) and its covariance component, and the Hellinger distance are given.

Explore the configuration:

• - Move the means of the two distributions and observe the change of the change of the decision boundary. Can you give a simple rule for the position of this decision line?
• - Change both covariances matrices such that the iso-lines are circular and of the same size. Again, can you give a simple rule for the position of this decision line?
• - Change the mean and covariance of both distributions such that the decision boundary is an ellipse/hyperbola?
• - Explore under which condition the decision boundary is an ellipse and the two means are close together?
• - Explore under which condition the two classes can be distinguished if the mean values are the same?