!
" #$% &!'
& ! '
' ( )
*
+
'
, +
(
+
- . / )
-0
- 1
- 2
- 3 - 4
5
&
'
"
*
+ & 6 $&
7
'8 8
s
i
+b
j
w
s
i
w
b
j
x
i
'EPJ Web of Conferences
4
, 02003 (2010)
DOI:10.1051/epjconf/20100402003
© Owned by the authors, published by EDP Sciences, 2010
- (
$
. -
x
0 6
+ - .1
1 -
2 & $
& (
(
,
+ + 6 $
(
p
T
, +M
t
p
T
9 ,H
T
& + +:
"
+
;
)
p
6 (
<
p
s
1
T
≤
p
b
T
34
≤ · · · ≤
p
T
b
2
≤
p
s
T
12
<
H
b
5
T
≤
H
T
b
3
≤ · · · ≤
H
T
s
67
≤
H
T
s
43
<
M
b
6
t
≤
M
t
s
8
≤ · · · ≤
M
t
s
12
≤
M
t
b
9
s
i
b
j
+i
j
=(
<
p
T
<
56
:> ? 0 <H
T
<
242
:> ?2<
M
t
<
105
:> ?@4& ))
H
T
<
242
:> &
M
t
>
162
:> ' ))6 +
p
= 0
.
82
'
i
x
i
($ -
x
i
. *
0 &
1 ! 0
2 '
i
'
p
=
s
s
+
b
s
b
+
[0
,
1]
$ @ ++
+1
@−
1
++1
p >
1
2
−
1
A+ 6$
p
= 0
.
82
@B.+1
@2
' '
( +
C +
@2 +
+ + D
x
i
<
cut
i
CE+ "
-.1 .2
'
(
< F &
N
min
√
N
min
8
(
N
eff
=
N
i
=1
w
i
2
N
i
=1
w
2
i
,
N
w
i
N
eff
=
N
'<
< G
<
+)F
6 '
'
i
(
t
)
t
+(
< +
< +
< + +
< '
&
S
t
t
P
t
F
(Δ
i
(
S, t
) =
i
(
t
)
−
p
P
·
i
(
t
P
)
−
p
F
·
i
(
t
F
)
,
p
P
p
F
S
'
S
∗
(
Δ
i
(
S
∗
, t
) =
max
S
∈{
splits
}
Δ
i
(
S, t
)
.
- #.% (
Δ
i
(
S, t
) =
i
(
t
)
−
min
p
P
·
i
(
t
P
)
, p
F
·
i
(
t
F
)
,
& (
Δ
i
(
S
∗
, t
)
) +
6 +
w
i
s
w
i
b
(p
=
i
∈
signal
w
i
s
i
∈
signal
w
i
s
+
j
∈
bkg
w
j
b
.
-
p
s
=
p
=
s
s
+
b
+
p
b
=
b
s
+
b
= 1
−
p
s
= 1
−
p
6. (
< (
1
−
max(
p,
1
−
p
)
< #$% (
−
i
=
s,b
p
i
log
p
i
< :
6 : #0% (
Gini =
i
=
j
i,j
∈{
classes
}
p
i
p
j
.
' ,
i
p
i
j
p
j
:+
i
=
s
j
=
b
p
s
=
p
= 1
−
p
b
:(
Gini = 1
−
i
=
s,b
p
2
i
= 2
p
(1
−
p
) =
signal purity
0
0.2
0.4
0.6
0.8
1
ar
bi
trary un
it
0
0.05
0.1
0.15
0.2
0.25
Split criterion
Misclas. error
Entropy
Gini
>
)
' : )
(
< (
−
s
2
s
+
b
< (
−
s
2
b
&
' 6
,
'
8D E
6 H=
nN
log
N
n
N
& (
'
( '
' 8 H=
& +
+ + '
8 '
A
f
:
x
i
→
f
(
x
i
)
(x > y
f
(
x
)
> f
(
y
)
'
x
i
f
(
x
i
)
x
i
f
(
x
i
)
'
= D
x
i
< c
i
CE+ + D
i
a
i
x
i
< c
i
CEa
= (
a
1
, .., a
n
)
G||
a
||
2
=
i
a
2
i
= 1
S
∗
(
a
∗
)
G
a
∗
Δ
i
(
S
(
a
)
, t
)
' +H='
7
+ ' +
x
i
x
i
'
' + ( +>
x
j
,x
i
+>
x
j
+ "x
i
x
j
' #$% 6
' +
+ '(
+
+
' - 0$
& 8
6 G
8 '
9
*
' F
' +
6 0
±
1
- 0.
+)
)
@
+ ' )
)
5
'
/
& 8 ) )
- . '
'
F
' )
5 C
(
6
' F "
+ )F
H 8
' 8
/
I#1 % 0 &
A
'
+ (
I #1 %
-
'
&!' #$% '
J
T
max
6T
N
T
T
max
R
(
T
)
R
α
(
T
)
(R
α
(
T
) =
R
(
T
) +
αN
T
,
α
5R
α
(
T
)
α
+T
max
α
+T
max
'
t
P
t
F
t
R
(
t
)
≥
R
(
t
P
) +
R
(
t
F
)
8t
P
t
F
K
t
T
t
R
(
t
)
> R
(
T
t
)
t
) ' )
t
(R
α
(
{
t
}
) =
R
α
(
t
) =
R
(
t
) +
α
N
t
= 1
R
α
(
T
t
)
< R
α
(
t
)
T
t
)
{
t
}
+ "α
=
ρ
t
R
ρ
t
(
T
t
) =
R
ρ
t
(
t
)
(
ρ
t
=
R
(
t
N
)
−
R
(
T
t
)
T
−
1
,
+
t
'ρ
t
+ + '
' ) 6
T
max
'
H
9 +
&
/
#2%
- )
V
) ) #$%&
L
V
FL
=
v
=1
..V
L
v
T
v
L − L
v
L
v
'V
(
1
V
v
=1
..V
T
v
.
6 )
"
&
&
- 1$ - 1.
&" -10" -11 6
- 12
" ' -)
$LL@ #3% + (
<
T
1
K<
T
2
KT
1
<
T
3
T
1
T
2
' ,
T
1
T
2
T
3
$LL2 6 #4 % , )
'
6 M-
&" #B%
"
'*"K 8
+ #L % ' @
+
#$@%
!
"
+ +
+ )
- + #2% '
+
T
k
N
'i
th
w
k
i
x
i
y
i
= +1
−
1
+ '(
T
1
k
N
tree
T
k
T
k
α
k
T
k
T
k
T
k
+1
'
F
(
T
1
, .., T
N
tree
)
(F
(
i
) =
N
tree
k
=1
α
k
T
k
(
x
i
)
.
'+ )
!
&"
6 M- #B%&" )
, '
"
& &" + &
T
k
+T
k
T
k
R
(
T
k
)
I
:
X
→
I
(
X
)
I
(
X
) = 1
X
@
T
k
N±
1
O (isMisclassified
k
(
i
) =
I
y
i
×
T
k
(
i
)
≤
0
,
@2 (
isMisclassified
k
(
i
) =
I
y
i
×
(
T
k
(
i
)
−
0
.
5)
≤
0
.
' (
R
(
T
k
) =
k
=
N
i
=1
w
k
i
×
isMisclassified
k
(
i
)
N
i
=1
w
k
i
.
'
T
k
(α
k
=
β
×
ln
1
−
k
k
,
β
, $' &" (
T
k
T
k
+1
(w
k
i
→
w
k
i
+1
=
w
i
k
×
e
α
k·
isMisclassified
k
(
i
)
.
'
T
k
T
k
+1
e
α
k
'
T
k
+1
T
k
+1
' 8&"
i
(T
(
i
) =
N
1
tree
k
=1
α
k
N
tree
k
=1
α
k
T
k
(
i
)
.
&
β
= 1
& ))= 40
Pα
= ln
1
−
0
.
4
0
.
4
= 0
.
4
&
e
0
.
4
= 1
.
5
+
K
= 5
Pα
= ln
1
−
0
.
05
0
.
05
= 2
.
9
K
e
2
.
9
= 19
Q'
(G
#$$%
(
≤
N
tree
k
=1
2
k
(1
−
k
)
.
k
= 0
.
5
+( F GN
tree
Q&
CK 5
'
& 6 1
& G + G
6 1 '
"
8
H
K
' 61
' (
-
5
'
'
8
Number of trees
Significance
3
3.5
4
4.5
5
5.5
Cross section significance
Background Fraction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background Fraction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Efficiency vs. background fraction
Single tree on testing sample
Boosted trees on testing sample
Single tree on training sample
Boosted trees on training sample
testing
training
Number of trees
Misclassification rate
0
0.1
0.2
0.3
0.4
0.5
Misclassification rate for each tree
Number of trees
k
α
0
0.05
0.1
0.15
0.2
0.25
k
α
Tree weight
" ' ( -
' (-G+ G
"(*
" (5
T
/ (e
T
(
i
)
=
p
(
S
|
i
)
p
(
B
|
i
)
=
BD
(
i
)
,
" #$0 % ' &"
"
! "
' '*>& + #$1 %
: #$2 %
x
y
62
x
:
y
)
+ -
+
" 63
x
:
y
'
+1
−
1
+ +−
1
+1
& +' 6 3 '
+ G
6
'
+ G + ,
& R!
+ 645
+ 6
6 4 K
, 6 "
+ ' +
#
&" &"
&"/ '!&" #$.%
(
T
k
(
i
) = 0
.
5
×
ln
p
k
(
i
)
BDT response
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalized
0
1
2
3
4
5
6
7
Signal
Background
BDT response
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalized
0
1
2
3
4
5
6
7
U/O-flow (S,B): (0.0, 0.0)/ (0.0, 0.0)
TMVA response for classifier: BDT
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MVA Method:
BDT
Fisher
DT
Background rejection versus Signal efficiency
' (6
x
y
' (
x
:
y
; - " + " (" (+ G
+ 6
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2
0.4
0.6
0.8
1
y:x
x
0
0.2
0.4
0.6
0.8
1
y
0
0.2
0.4
0.6
0.8
1
y:x
BDT response
-0.6
-0.4
-0.2
0
0.2
0.4
Normalized
0
5
10
15
20
25
30
Signal
Background
BDT response
-0.6
-0.4
-0.2
0
0.2
0.4
Normalized
0
5
10
15
20
25
30
U/O-flow (S,B): (0.0, 0.0)/ (0.0, 0.0)
TMVA response for classifier: BDT
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MVA Method:
BDT
DT
Fisher
Background rejection versus Signal efficiency
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal efficiency
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Background rejection
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MVA Method:
BDT
Fisher
DT
Background rejection versus Signal efficiency
!" !" #
p
k
(
i
)
i
7 (w
k
i
→
w
k
i
+1
=
w
k
i
×
e
−
y
i
T
k
(
i
)
T
(
i
) =
N
tree
k
=1
T
k
(
i
)
: &" A" )#$. %
)" + #$3%
e
2
)
α
k
&")A"#L%
e
−
yiTk
(
i
)
1+
e
−
yiTk
(
i
)
)S"#L% (
w
i
k
→
w
k
i
+1
=
I
(
y
i
×
T
k
(
i
)
≤
0)
.
6 D , E #4% " )
"#$4% + +
8 "
! " # "$
& - 0. +
/ &
#2%
" &:: K: $LL1 #$B % 8
7
+ D E '
#$L % "
'
,
&
+
' "
%
)
) ' +
& ( )
+
8
'
& &"
" 6
*"K @ +
AS '
*
"
" + +
+ ' 6 B
+ @ #$@%
" +
'
+
$ % &' % (
)*'+ , - (
. %
-/ !
p
(+
p
(( " .
6
'
AS
& '(
* + S + &!' #$%
0 #.@% 12 #.$ % 8 '
*" #L% (JJ)J
∼
J -H )! #.% (JJ J ,J AS
8 )
/
5+ T
S7H
(JJ+FJJ+J
!' +
#..% (JJ !'
) *+
+ + '
,
,-(-
#$% A " TS 6 !& T - 5)
- $LB1
#.% K +D-H ! (& UUH+ - & S7 H
E R(J@2@4$10$ .@@2
#0% :D> V *VE $L$.
7 HF ' -! (A 7 > > $L22
#1% T!ID- E .40(..$<
.01$LB4
#2% SH D7 E
#3% !7- D' + E 2. ($L4<..4$LL@
#4% W 6 D" + , E
$.$.(.23<.B2 $LL2
#B% W6 !7 - D7 K " & E
A - * X
-6 $1B$LL3
#L% "H ! S)T W T Y W A - : *: K * H
!-& 210244 .@@2;S)T W"H!T Y K * H!
- &222 04@.@@2
#$@% > * &F # @ % D7 +
|
V
tb
|
E H! A $B$B@..@@4;> * &F D7 +E H ! @$.@@2.@@B;> * &F D+ E H !A
#$$% W 6 !7 - D& ) F ) )
E 22$ ($$L<$0L$LL4
#$.% TS6 ' S!' D& ()
E ! .B. 044<0B3.@@@
#$0% SH D E
#$1% &SZ+ D'*>&('+ EH-&@1@.@@4# R()
J@4@0@0L%
#$2% : D* E "# !
T .@@B
(JJ J C?.1 B.4
#$3% TS6 D: ( E !
.L 2 $$BL<$.0. .@@$
#$4% W 6 D& , E 10 0
.L0<0$B .@@$
#$B% A" D" H E .1 . $.0<$1@ $LL3
#.@% T! I D E $$ (B$<$@3$LB3
#.$% T! I $% * X H - 6 )
& $LL0
