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## Linear Discriminant Functions

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**Linear Discriminant Functions**Chapter 3, pp. 77 -**Linear Discriminant Functions**• Chapter 1 introduced the concept of a discriminant function y(x) • The vector x is assigned to class C1 if y(x)>0 and C2 if y(x)<0. • Simplest choice of such a function is linear in the components of x, and therefore can be written as**terminology**• w is weight vector, d-dimensional • w0 is bias, -w0 is threshold**Geometric interpretation of**(3.1) • Decision boundary y(x)=0 corresponds to (d-1)-dimensional hyperplane in d-dimensional x-space. • For d=2 (plane), decision boundary is a straight line**Geometry (cont’d)**• If xA and xB are 2 pts on the hyperplane, then y(xA) and y(xB) are 0. • So, using (3.1), we have Thus w is normal to any vector lying in the hyperplane!**More on the nature of the hyperplane**• We’ve seen that w is normal to any vector lying in the hyperplane! • Thus w determines the orientation of the hyperplane • But how far is the hyperplane to the origin? • If x is any point on the hyperplane, then the normal dist from the origin to the hyperplane is… So the bias w0 determines the position of hyperplane**Classifying several classes**• For each class Ck, define the discriminant function • A new point x is then assigned to class Ck if**How far is the classification boundary from the origin?**• The boundary separating class Ck from class Cj is given by • Which correspond to (partial) hyperplanes of the form • By analogy to the 2-class case, the perpendicular distance of the decision boundary from the origin is given by**Expressing multiclass linear discriminat function as a**neural network diagram