2016/June/23 トップカンファレンス読み会資料

2. non-transformed input space.

3. non-transformed input space.

4. non-transformed input space.

5. non-transformed input space.

8. l
l
n P
Zn − bn
an
≥ z = − log G(z)
(1)
bn ∈ R
(2)
(3)
(i) : i (4)
(5)
lim
n→∞
n P
Zn − bn
an
≥ z = − log G(z)
lim
n→∞
n P
Zn − bn
an
≥ z = (1 + γz)−1/γ
1 + γz > 0, γ ∈ R
bn ∈ R
ˆI : (1)
N(i) : i (2)
(3)
bn ∈ Rd
Zn = max{X1, . . . , Xn}
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
. or
Zd
n − bd
n
ad
n
≥ zd = (1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
(1)
bn ∈ Rd
Zn = max{X1, . . . , Xn}
G(z) = (1 + γz)−1/γ
Gj(z) = (1 + γjz)−1/γj
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
b1
n
≥ z or . . . or
Zd
n − bd
n
≥ z = (1 + γ z)−1/γj

9. 1.5 2.0 2.5 3.0 3.5 4.0
010203040
Block Maximum from N(0, 1)
maximum
120
Block Maximum from U(0, 1)
4 6 8
maximum
1.5 2.0 2.5 3.0 3.5 4.0
0
maximum
20 40 60 80 100
k Maximum from Par(2)
maximum
0.94 0.96 0.98 1.00
020406080120
Block Maximum from U(0, 1)
maximum
200

10. l
l Zn = max{X1, . . . , Xn}
lim
n→∞
n P
Zn − bn
an
≥ z = − log G(z)
or . . . or
Zd
n − bd
n
ad
n
≥ zd = (1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
(1)
(2)
(3)
bn ∈ Rd
Zn = max{X1, . . . , Xn}
m
∞
n P
Zn − bn
an
≥ z = − log G(z)
. or
Zd
n − bd
n
ad
n
≥ zd = (1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
(1)
(2)
bn ∈ Rd
Zn = max{X1, . . . , Xn}
G(z) = (1 + γz)−1/γ
Gj(z) = (1 + γjz)−1/γj
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
lim
n→∞
n P
Z1
n − b1
n
a1
n
≥ z1 or . . . or
Zd
n − bd
n
ad
n
≥ zd = G(Z)
G()(1 + γjz)−1/γj
bn ∈ Rd
Zn = max{X1, . . . , Xn}
G(z) = (1 + γz)−1/γ
Gj(z) = (1 + γjz)−1/γj
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
lim
n→∞
n P
Z1
n − b1
n
a1
n
≥ z1 or . . . or
Zd
n − bd
n
ad
n
≥ zd = G(Z)
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
ˆI : (1)
N(i) : i (2)
(3)

11. ● ●● ●●● ●●●● ●●●● ●●● ●●● ●●●● ●●●● ● ● ●● ●●● ●● ● ●●●● ●●●● ● ●● ●●● ●● ● ●●● ●● ●● ●●● ●● ●● ●●● ●●● ●● ●●●● ● ●● ●● ●● ●●● ●●● ●●● ●● ●●
l
bn ∈ Rd
Zn = max{X1, . . . , Xn}
G(z) = (1 + γz)−1/γ
Gj(z) = (1 + γjz)−1/γj
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
− b1
n
1
n
≥ z1 or . . . or
Zd
n − bd
n
ad
n
≥ zd = G(Z)
lim
n→∞
n P
V 1
n
≥ v1 or . . . or
V d
n
≥ vd =
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
(1)
(2)
Zn = max{X1, . . . , Xn}
G(z) = (1 + γz)−1/γ
Gj(z) = (1 + γjz)−1/γj
lim
n→∞
n P
Zn − bn
an
≥ z = G(z)
lim
n→∞
n P
Z1
n − b1
n
a1
n
≥ z1 or . . . or
Zd
n − bd
n
ad
n
≥ zd = G(Z)
lim
n→∞
n P
V 1
n
≥ v1 or . . . or
V d
n
≥ vd =
[0, v1] × · · · × [0, vd]
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
ˆI : (1)
N(i) : i (2)
(3)
≥ v1 or . . . or
n
≥ vd =
[0, v1] × · · · × [0, vd]
V j
= 1/(1 − Fj(Xj))
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
(1)
(2)
(3)
V = 1/(1 − Fj(Xj)
G()(1 + γjz)−1/
1 + γz > 0, γ ∈ R
bn ∈ R
1 − ˆFj(Xj)
ˆI : (1)
N (i) : i (2)
(3)
V = 1/(1 − Fj(Xj))
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
1 − ˆFj(Xj)
ˆI : (1)
N (i) : i (2)
(3)
1
1 − ˆFj(Xj)
1/(1 − ˆFj(Xj))
ˆI :
N (i) : i

13. t nonzero mass on C↵ is the same as nonzero mass
ll.
cones in 3D Figure 2: Truncated ✏-rectangles in 2

14. Figure 3: Estimation procedure
equivalently the corresponding sub-cones are low-dimensional compa

15. Figure 3: Estimation procedure
equivalently the corresponding sub-cones are low-dimensional compa
1 + γz > 0, γ ∈ R
bn ∈ R
µα,ϵ
n
µ{1},ϵ
n
µ{1,2},ϵ
n
µ{2},ϵ
n

17. Figure 3: Estimation procedure
equivalently the corresponding sub-cones are low-dimensional compa
G()(1 + γjz)
1 + γz > 0, γ ∈ R
bn ∈ R
µα,ϵ
n
µ{1},ϵ
n
µ{1,2},ϵ
n
µ{2},ϵ
n
1 − ˆFj(Xj)
1/(1 − ˆFj(Xj))
n→∞ n
1
n
d
[0, v1] × · · · × [0, vd]
V j
= 1/(1 − Fj(Xj))
G()(1 + γjz)−1/γj
1 + γz > 0, γ ∈ R
bn ∈ R
µα,ϵ
n
µ{1},ϵ
n
µ{1,2},ϵ
n
µ{2},ϵ
n
1 − ˆFj(Xj)
d
− Fj(Xj))
+ γjz)−1/γj
> 0, γ ∈ R
bn ∈ R
µα,ϵ
n
µ{1},ϵ
n
µ{1,2},ϵ
n
µ{2},ϵ
n
α
µα,ϵ
n
ϵ
1 + γz > 0, γ ∈ R
bn ∈ R
µα,ϵ
n
µ{1},ϵ
n
µ{1,2},ϵ
n
µ{2},ϵ
n
C↵ is the same as nonzero mass on
ure 2: Truncated ✏-rectangles in 2D
et ; 6= ↵ ⇢ {1, . . . , d}, the exponent
µ(R✏
↵).
d}. Then R✏
↵’s forms an increasing
0
↵ = [✏>0,✏2Q R✏
↵. The result follows
y of the measure µ. Now, for ✏ 0

20. non-transformed input space.
Figure 3: Level sets of sn on simulated 2D data

21. 5: Combination of any AD algorithm with DA
ain purpose of Algorithm 1 is to deal with ex
In this section we show that it may be com

22. ta
-1) (Phe-
ntrated in
b-cones is
ely 93%,
-cones of
ely (to be
the mass
Fig. 4.
xtreme
obtained with the combined method ‘iForest + DAMEX’
above described, to those obtained with iForest alone on
the whole input space.
number of samples number of features
shuttle 85849 9
forestcover 286048 54
SA 976158 41
SF 699691 4
http 619052 3
smtp 95373 3
Table 3: Datasets characteristics
Six reference datasets in AD are considered: shuttle, forest-
cover, http, smtp, SF and SA. The experiments are per-
formed in a semi-supervised framework (the training set
consists of normal data). In a non-supervised framework
(training set including abnormal data), the improvements
brought by the use of DAMEX are less significant, but the
precision score is still increased when the recall is high
Sparse Representation of Multivariate Extre
Dataset iForest only iForest + DAMEX
ROC PR ROC PR
shuttle 0.996 0.974 0.997 0.987
forestcov. 0.964 0.193 0.976 0.363
http 0.993 0.185 0.999 0.500
smtp 0.900 0.004 0.898 0.003
SF 0.941 0.041 0.980 0.694
SA 0.990 0.387 0.999 0.892

23. l
l