Week 9 Solution

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Question 1 Corrected (For Assessment)

Show that, for the maximum margin classifier, the correct value of β0 is

β0=−

maxi:yi=−1(β)T xi + mini:yi=1

(β)T xi

,

where β = P

2

n

λiyixi is the optimal value for

β.

i=1

Question 2

Work through labs 9.6.2 and 9.6.5 in the text book. This should give you a feeling for how support vector classifiers work in R.

Question 3

Consider the support vector classifier with the Lagrangian

1

n

n

n

X

X

X

L(β, β0, ξ, λ, µ) =

βTβ+C

ξi λi yi(xiT β + β0) 1 + ξi µiξi

2

i=1

i=1

i=1

Using the KKT equations, show that the optimal β can be written as

n

β =

X

λiyixi.

i=1

Show that λ solves

n

1

n n

X

X X

max

λ

y y

λ

λ

xT x

i 2

λ

i j

i

j

i

j

i=1

i=1 j=1

Subject to:

0 ≤ λiC, i = 1, . . . , n

n

X

λiyi = 0.

i=1

Argue that

λi = 0 yi(βT xi + β0) 1

λi = C yi(βT xi + β0) 1

0 < λi < C yi(βT xi + β0) = 1.

1