Page 38 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN CLASSIFICATION 27

with:

w k ¼ ln jC k jþ 2ln Pð! k Þ m C m k
T
1
k
k
w k ¼ 2C m k ð2:21Þ
1
k
W k ¼ C 1
k
A classifier according to (2.20) is called a quadratic classifier and the
decision function is a quadratic decision function. The boundaries
between the compartments of such a decision function are pieces of
quadratic hypersurfaces in the N-dimensional space. To see this, it
suffices to examine the boundary between the compartments of two
different classes, e.g. ! i and ! j . According to (2.20) the boundary
between the compartments of these two classes must satisfy the follow-
ing equation:

T
T
T
T
w i þ z w i þ z W i z ¼ w j þ z w j þ z W j z ð2:22Þ
or:
T
T
w i w j þ z ðw i w j Þþ z ðW i W j Þz ¼ 0 ð2:23Þ
Equation (2.23) is quadratic in z. In the case that the sensory system has
only two sensors, i.e. N ¼ 2, then the solution of (2.23) is a quadratic
curve in the measurement space (an ellipse, a parabola, an hyperbola, or
a degenerated case: a circle, a straight line, or a pair of lines). Examples
will follow in subsequent sections. If we have three sensors, N ¼ 3, then
the solution of (2.23) is a quadratic surface (ellipsoid, paraboloid,
hyperboloid, etc.). If N > 3, the solutions are hyperquadrics (hyperellip-
soids, etc.).
If the number of classes is more than two, K > 2, then (2.23) is a
necessary condition for the boundaries between compartments, but not
a sufficient one. This is because the boundary between two classes may be
intersected by a compartment of a third class. Thus, only pieces of the
surfaces found by (2.23) are part of the boundary. The pieces of the sur-
face that are part of the boundary are called decision boundaries. The
assignment of a class to a vector exactly on the decision boundary is
ambiguous. The class assigned to such a vector can be arbitrarily selected
from the classes involved.