Méthodes des k-plus proches voisins

k
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YL
Y={1, . . . , L}L
L= 2
x= (x1, . . . , xp)>∈ X ⊂ Rp
jxj
Dn={(xi, yi), i = 1,...n}n
X Y ∀i∈ {1, . . . , n},(xi, yi)i.i.d ∼(X, Y )
Dn
ˆ
f:X 7→ Y xnew ˆ f(xnew )
p= 2
k

k k k
x
k
Vk(x) 1 ≤k≤n
xVk(x)
Y=R
ˆ
fk(x)=1
x
y= 3
y= 1
y= 2
k= 5
ˆ
fk(x)=2
x
y= 3
y= 1
y= 2
k= 11
k
k= 5 k= 11 L= 3
y= 1 y= 2 y= 3
d:Rp×Rp7→ R
xk Vk(x)
x∈Rdi= 1, . . . , n
di(x)x xidi(x) = d(xi,x)r1(x)
x x1,...,xn
r1(x) = i∗di∗(x) = min
1≤i≤ndi(x).
rk(x) 1 ≤k≤n
rk(x) = i∗di∗(x) = min
1≤i≤n
i6=r1,...,rk−1
di(x).
kxVk(x) = {xr1,...,xrk}
x
ˆ
fk(x)∈arg max
y∈Y
k
X
j=1
1{Yrj=y}
.

d(x,v) = kx−vk2
k
k
j e−d2
j/h h
ˆ
fk(x)∈arg max
y∈Y
k
X
j=1
exp(−d2
j/h)1{Yrj=y}
.
h
k
n100 500 1000 k
k≥1
(P{Y=i, Ck(X) = j})i, j Ck
k
k
k= 1