Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

Support vector machines 271

see, e .g., Cristianini and Shawe–Taylor (2000), Chapter 5. We shall see how

to do this in the sequel, however le t us for now suppose that we have found

the solution α

∗

which maximizes

˜

L(α). Then

w

∗

=

X

ν

ℓ(ν)α

∗

ν

g(ν)

determines the maximal margin hyperplane and it has geometric margin

kw

∗

k

−1

. In order to classify a new observation g, we simply evaluate the

sign of

I

∗

(g) = w

∗

⊤

g + w

∗

0

=

X

ν

ℓ(ν)α

∗

ν

(g(ν)

⊤

g) + w

∗

0

, (6.58)

that is, we ascertain on which side of the hyperplane the observation lies. (We

still need an expression for w

∗

0

. This is described below.) Note that both the

training phase , i.e., the solution of the dua l problem