quadratic discriminant function linear discriminant function linear and quadratic discriminant analysis andaccording to the bayes optimal decision rule ( )we classify to come from class if ( th ( th orequivalently (by taking logarithmsif ln | ( ) - ln ln | ( ) - ( ln ( the function rp ( dy (xln ay ln |sy ( uy ) - ( uy ) is called the quadratic discriminant function for class point is classified to class for which dy (xis largest the function is quadratic in and so the decision boundary { (xd ( )is quadratic as well an important simplification arises for the case where the assumption is made that nowthe decision boundary is the set of for which ln ( ) - ( ln ( ) - ( expanding the above expression shows that the quadratic term in is eliminatedgiving linear decision boundary in ln > - xs- ln > - xs- the corresponding linear discriminant function for class is dy (xln ay > - uy xs- uy rp ( example (linear discriminant analysisconsider the case where / and su the distribution of is mixture of two bivariate normal distributions its pdf ( thy ( thy ) is depicted in figure figure gaussian mixture density where the two mixture components have the same covariance matrix