Multivariate discriminants

(1)

!

"

f

(x)

n

x

#$%& #%%&

f

(x)

< q

f

(x)

≥

q

'(%

( ) *

+"'

n

x

= (x

1

,

· · ·

, x

n

)

,

- .

/ + .

0

L

+

12

,33

4 3

" ! !

' 5- 3

6

EPJ Web of Conferences

, 0

(2010)

DOI:10.1051/epjconf/2010040001

© Owned by the authors, published by EDP Sciences, 2010

(2)

' 7

8 78

f

(x)

x

n

f

. #% )&

#9&

! : !

y

=

f

(x)

'

f

3

f

(x)

#%) &

y

=

f

(x)

' .

;

F

=

{

f

(x, w)

}

w

'

0

;

Q(w)

F

;

L(y, f)

-

f(x, w)

F

f

(x, w)

y

=

f

(x)

y

f(x)

f

(x, w)

'

L(y, f

)

<

f

(x, w)

6'

0

78 6

.

'

' '

T

6 512

L(y, f(x, w))

R(w)

R(w) =

1 N

N

i

=1

L(y

_i

, f

(x

_i

, w)),

1%2

N

'

T

f

(x, w)

'

R(w)

Q(w)

'

C(w) =

R(w) +

λQ(w),

1)2

λ

+

χ

2

+

C(w)

f

(x, w

∗

)

N

→ ∞

f

(x, w

0

)

(3)

'

R(w) =

L(y, f

(x, w))

p(y, x)

dydx,

192

p(y, x)

y

x

"-1%2

"- 192

#% &

: #=&

y

=

f

(x)

#9& : .

; '

F

=

{

f

(x, w)

}

0

;

π(f

)

;

p(y

|

x, w)

y

n

x

π(f

)

f

-

π(w)

{

w

}

>

?:

π(w)

p(y

|

x, w)

:@

p(w

|

T

) =

p(T

|

w)π(w)

p(T

)

,

=

p(y, x

|

w)π(w)

p(y, x)

,

=

p(y

|

x, w)p(x

|

w)π(w)

p(y

|

x)p(x)

,

∼

p(y

|

x, w)π(w),

1=2

p(w

|

T

)

w

{

f

(x, w)

}

w

f(x, w)

p(w

|

T

)

p(w

1

|

T

)

> p(w

2

|

T

)

w

1

w

2

f

(x, w

1

)

T

f

(x, w

2

)

"- 1=2

n

x

w

p(x

|

w) =

p(x)

p(w

|

T

)

: : -4- #=&

p(y

|

x, T

) =

p(y

|

x, w)p(w

|

T

)dw,

1A2

f

(x)

n

x

T

x

y

p(y

|

x, T

)dy

y

=

f

(x)

ˆ

f

'

f

(x)

≈

f

ˆ

(x) = arg

_f

_ˆ

min

L(y,

f

ˆ

)p(y

|

x, T

)dy,

1B2

(4)

ˆ

f

L(y, f

)

+

L(a, b) = (a

−

b)

2

_,

1C2

ˆ

f

(x) =

yp(y

|

x, T

)dy,

1D2

y

=

f

(x)

-

y

=

f

(x)

"-1D2 4

* : !

R(w)

∼

ln

p(y

|

x, w),

=

N

i

=1

ln

p(y

_i

|

x

_i

, w),

1E2

λQ(w)

∼

ln

π(w).

1%$2

16+(2

y

=

f

(x)

' "-1)2 ' "-1=2 ,

f

(x, w

∗

)

y

=

f

(x)

: : !

f

(x, w)

p(w

|

T

)

( )

#AB C&

F 1 2 !

y

L

x

- "-1C2

R(w) =

1 N

N

i

=1

[y

_i

−

f

(x

_i

, w)]

2

,

1%%2

-

p(y

|

x, w) = exp(

−

N R(w)/2σ

2

_)/σ

√

_2π.

1%)2

,

S

B

+

0

, ' %

x

'

n

x

(5)

p(x, B) =

p(x

|

B

)p(B

)

p(x, S) =

p(x

|

S)p(S)

k

=

p(S)/p(B

)

p(x

|

S)

p(x

|

B

)

x

:

x

p(x) =

p(x, S) +

p(x, B)

S

B

x

=

x

0

4

x

≥

x

0

x < x

0

,%

S

B

3'

G

C

S

C

B

*

C

=

C

_S

H(x

₀

−

x)

p(x, S)

dx,

+

C

_B

H(x

−

x

₀

)

p(x, B)

dx,

1%92

H(x)

3 12.

H(x) = 1

x >

0

-

C

S

C

B

:

x

=

x

0

f

(x)

"-1%92 6'

C

x

=

x

0

0 =

C

_S

δ(x

₀

−

x)

p(x, S)

dx,

−

C

_B

δ(x

−

x

₀

)

p(x, B)

dx,

1%=2

BD(x)

BD(x

₀

) =

C

B

C

_S

=

p(x

₀

, S)

p(x

₀

, B)

=

p(x

₀

|

S)

p(S)

p(x

₀

|

B

)

p(B

)

.

1%A2

+

x

:

BD(x) =

B

(x)

p(S)

p(B)

=

p(S

|

x)

p(B

|

x)

,

1%B2

B(x)

B(x) =

p(x

|

S)

p(x

|

B

)

.

1%C2

"- 1%B2 :@

p(K

|

x) =

p(x

|

K)p(K)/p(x)

K

=

S

B

H

C

B

/C

S

x

=

x

0

:

x

H

p(S

|

x) +

p(B

|

x) = 1

:

(6)

p(S

|

x) =

p(x

|

S)p(S)/p(x),

=

p(x

|

S)p(S)/[p(x

|

S)p(S) +

p(x

|

B)p(B)]

1%D2

'

n

x

S

-

q

=

p(S

|

x)

q

=

C

B

/(C

S

+

C

B

)

1

2

x

p(S

|

x)

≥

q

p(S

|

x)

< q

+

f

(x, w)

p(S

|

x)

-

p(S

|

x)

125-

:

k

=

p(S)/p(B

)

=

p(S)

p(S

|

x)

k

,

p(S

|

x)

D(x) =

s(x)/[s(x) +

b(x)],

1%E2

s(x)

≡

p(x

|

S)

b(x)

≡

p(x

|

B)

+

D(x)

p(S

|

x) =

D(x)/[D(x) + (1

−

D(x))/k],

1)$2

+()

+

D(x)

@ '

3

D(x)

>

f

(x)

f

(x)

D(x)

p(S

|

x)

!

p(S

|

x)

#D&

w(x)

=

p(S)

d(x)

s(x)

b(x)

d(x) =

s(x) + (1

−

)

b(x).

1)%2

"

w(x)

(7)

H

¯

w

_d

=

dx w(x)

d(x),

1)92

¯

w

_s

=

dx w(x)

s(x),

1)=2

¯

w

_b

=

dx w(x)

b(x),

1)A2

= ( ¯

w

_d

−

w

¯

_b

)/( ¯

w

_s

−

w

¯

_b

),

1)B2

I

() =

1 N

dx[(w

_d

(x)

−

w

¯

_b

)/( ¯

w

_s

−

w

¯

_b

)]

2

d(x),

1)C2

N

:#D& '

w(x) =

p(S

|

x),

=

s(x)/[s(x) +

b(x)/k],

=

D(x)/[D(x) + (1

−

D(x))/k],

1)D2

k

=

p(S)/p(B) =

/(1

−

)

+

k

3

w(x)

k

>

k

*

?

D(x) =

s(x)/[s(x) +

b(x)],

1)E2

s(x)

b(x)

6

D(x)

12

p(S

|

x)

'

n

x

S

J J 6

D(x)

+

#E&

4-

*

6

- #%$& 3

?

*

K #%% &

L5#%) &6

(8)

; FM 1FM 2

; N GO

; I6

; HP :

; QO"

; : H H

" # $#

n

x

, 1 2 -

n

M ,

3

n

K

n

K

* 7 8

3 ?

,) )

x

= (x

1

, x

2

)

?

65

6 5

0 . #%9&

+

1

2 6 5

1

2

-

?

S

= #(

)/#(

)

B

= #(

)/#(

)

7R128

? !

,9: 12

%= I

>MF+ #%=& 1 .

m

0

= 3280

MI

m

1 /

2

= 300

MI

A

0

= 0

tan

β

= 10

(μ) = +1

m

top

= 175

MI 2

(9)

k

=

p(S)/p(B)

k

∼

1/3000

+ FM A 1

2 ,9

" A

D(x)

: .

D(x)

+

D(x)

:

>MF+

6%= I

%

−

1

% # #

6

3 , 9

+ ( %

-

D(x) =

s(x)/[s(x) +

b(x)]

:

B(x) =

s(x)/b(x)

4,1

F #%$&2 4

(%

s(x)

b(x)

M

(x

|

μ,

Σ) = exp[

−

(x

−

μ)

T

_Σ

−

1

_(x

₋

_μ)/2]/(2π)

n/

2

_|

_Σ

_|

_,

19$2

μ

Σ

|

Σ

|

D(x)

1

B(x)

2 :

B(x)

+

λ(x) =

χ

2

(μ

_B

,

Σ

_B

)

−

χ

2

(μ

_S

,

Σ

_S

),

19%2

χ

2

_(μ,

_{Σ) = (x}

₋

_μ)

T

_Σ

−

1

_(x

₋

_μ)

S

B

- -"-19%2

+

λ(x)

+

λ(x)

s(x)

b(x)

M

λ(x)

, !

0

Σ = Σ

S

+ Σ

B

-

1 2 ,@

(10)

,=

w

= Σ

−

1

(μ

_S

−

μ

_B

).

1992

- * ,

9

,

w

x

-

w

· x

≥

q

w

· < q

q

&

,

3 M %EE$

'#%& ,

M

h

:

x

∈

R

n

_→

_z

_∈

_R

H

n

R

n

x

R

H

5 9 +: 5

w

· h(x

₁

) +

b

= +1

19=2

w

· h(x) +

b

= 0

19A2

w

· h(x

₂

) +

b

=

−

1,

19B2

( : , A6 +

h(x

1

)

5

h(x

2

)

, + 5

- 5+

w

· [h(x

1

)

−

h(x

2

)] = 2

ˆ

w

=

w/

|

w

|

w

|

w

ˆ

· [h(x

1

)

−

h(x

2

)] = 2/

|

w

|

( :

'

+ 5

m

= 2/

|

w

|

'

R(w)

∝ |

w

|

2

3 '

78 +

y

= +1

78 5

y

=

−

1

( 78 +

w

· h(x) +

b >

1

78

5

w

· h(x) +

b <

−

1

5-

y

_i

[w

· h(x

_i

) +

b]

≥

1,

∀

i

= 1,

· · ·

, N

N

'

C(w, b, α) =

1

2 |

w

|

2

−

N

i

=1

α

_i

[y

_i

(w

· h(x

_i

) +

b)

−

1],

19C2

α

(11)

( + : 5

w

ˆ

+

5

h(x

1

)

−

h(x

2

)

w

ˆ

( 78 +

y

= +1

78 5

y

=

−

1 w

b

C(α) =

N

i

=1

α

_i

−

1

2

N

i

=1

N

j

=1

α

_i

α

_j

y

_i

y

_j

[h(x

_i

)

· h(x

_j

)].

19D2

' ' -

G #% &+

C(α)

G

α

' '

A

C

0 #% &

> "-19D2 ,

h

x

?

h(x

i

)

h(x

_j

)

h(x

i

)

· h(x

j

)

-

K(x

i

, x

j

)

x

i

x

j

h(x

i

)

h(x

j

)

K(x

_i

, x

_j

)

'

C(α) =

N

i

=1

α

_i

−

1

2

N

i

=1

N

j

=1

α

_i

α

_j

y

_i

y

_j

K(x

_i

, x

_j

).

19E2

-

* .

' !

6

12

y

i

(w

· x

i

+

b)

≥

1 −

ζ

i

ζ

_i

"-19D2

?

α

i

ζ

i

'

:3?

' ()

"'

D(x) =

s(x)/[s(x) +

b(x)]

(12)

s(x)

≈

n

i

=1

s

₁

_D

(x

_i

),

b(x)

≈

n

i

=1

b

₁

_D

(x

_i

),

1=$2

x

= (x

1

, x

2

,

· · ·

, x

n

)

s

1 D

b

1 D

%

s(x)

b(x)

0

n

x

1

x

n

x

i

"-1=$2 - H

I

H

6

s(x)

b(x)

*

+

s(x)

b(x)

?U

('1 F#) %$ &2 %EB$ -

1QO"2 , B +

+ %

QO" +

p(x)

x

1 2

K(x, μ)

μ

p(x)

x

p(x)

≈

p(x) =

ˆ

1 N

N

j

=1

K(x, μ

_j

),

1=%2

N

M

K(x, μ) =

_n

i

=1

M

(x

i

|

μ, h

i

)

! 1 2

h

i

% P :

h

1 2

4

+ , M

1 F#)&2

h

=

σ

₄

(n

+ 2)N

₁

_/

₍

_n

₊₄₎

1=)2

(13)

' !

3

M

, C

p

T

η

φ

p

T

(

)

p

T

p

T

(

)

"-1=)2

[GeV/c]

T

p

0

100

200

300

400

0

200

400

600 η

-4

-2

0

2

4

0

100

200

300 φ

-4

-2

0

2

4

0

100

200 (gen)

T

p

0

100

200

300

400

0

500

( = 1 2

= QO" 1 2 QO" "-1=)2 5

:

, D 4 =

p(p

_T

, η, φ, p

_T

(

))

p(p

T

, η, φ)

3

p(p

T

(

)

|

p

T

, η, φ) =

p(p

T

, η, φ, p

T

(

))/p(p

T

, η, φ)

(14)

[GeV/c]

T

p

0

100

200

300

400

0

200

400

600 η

-4

-2

0

2

4

0

100

200

300 φ

-4

-2

0

2

4

0

100

200 (true)

T

p

0

100

200

300

400

0

500

( = 1 2

= QO"12 QO"

(15)

N

→ ∞

ˆ

p(x) =

K(x, μ)

p(μ)

dμ,

1=92

p(μ)

x

(

p(x)

ˆ

"- 1=)2 1 2

N

→ ∞

δ

.

K(x, μ)

→

δ

n

_(x

₋

_μ)

-

p(x) =

ˆ

p(x)

QO"

'

3 ' -QO"

0 4

.

0 H

3

+

>

! : 3 %EE$

', #%A &

,(#A&

f

(x) =

K

i

=1

a

_k

f

(x, w

_k

).

1==2

"'

f

(x, w

k

)

?

a

k

#B& #C & #%A &

#%) %B%C %D%E& .

; 1 2 ?

a

k

= 1/N

N

!

0

; '

; ?

a

k

!

+ :

'

:

!

O #%)&

M

n

.

+

1 2 ,

(16)

y

=

−

1 y

= +1

4

y

=

s/(s

+

b)

D(x)

x

s

b

O 6

- 3

D(x)

' ' +

<

1! #%A&2

'

f

(x, w)

±

1 y

=

±

1

,

f

(x, w)y

M

'

w

k

K

"

%

f

k

=

f

(x, w

k

)

k

9

a

k

= ln[(1

−

k

)/

k

]

=

n

exp(

−

a

_k

f

(x

_n

, w

_k

)y

_n

/2)/

N

i

=1

ω

_i

exp(

−

a

_k

f

(x

_i

, w

_k

)y

_i

/2),

f

(x)

"- 1==2

+

(f

(x))

f

(x)

"-1==2 , E

>MF+ #%= &

t

¯

G35 '

(17)

,E

'<+

'

4 *

: @ 1

F #)$&2

, ' '-.

: 1:HH2 #9 )% &

:

f

(x)

: - #%C %D&

:HH "-1A2

3 #)%&

:HH

y(x) =

z p(z

|

x, T

)dz,

=

f

(x, w)p(w

|

T

)dw.

1=A2

:HH

p(y

|

x, w) =

f

(x, w)

y

_[1

₋

_f

_{(x, w)]}

1 −

y

∈ {

0,

1 }

:

y

= 1

y

= 0

*

y

= 1

p(y

|

x, w) =

f

(x, w)

1 −

f

(x, w)

y

= 0

0

f

(x, w)

"- 1=A2 F #%C& #%D &

12H#9&

f

(x, w) =

1 1 + exp[

−

g(x, w)]

,

1=B2

g(x, w) =

b

+

H

j

=1

v

_j

tanh(a

_j

+

n

i

=1

u

_ji

x

_i

),

1=C2

n

x

H

w

= (b, v, a, u)

f

(x, w)

12

+

"-1=A2 6 5 6 5 165652

#9&6565

w

1

, w

2

,

· · ·

, w

M

p(w

|

T

)

"-1=A2

y(x)

≈

1 M

M

m

=1

f

(x, w

_m

),

1=D2

M

p(w

|

T

)

6

{

w

j

}

p(w

|

T

)

"- =D 3 6565

(18)

"-1=A2 "-1=D2

* ?

π(w)

>?

*

' M

O

1

H

T

2

-

) 1

F #%D &2

'

* % :HH,%$

12 1

H

T

2 -

) #%D&

H

= 20

%$$$ %$$$ +

M

= 50

"-1=D2 , %%

A$

f

(x, w

k

)

A$

w

k

"

f

(x, w

k

)

k

6565

x

H(x

|

S)/[H(x

|

S) +

H(x

|

B)]

H(x

|

S)

H(x

|

B)

D(x) =

p(x

|

S)/[p(x

|

S) +

p(x

|

B)]

:

f

(x, w

k

)

D(x)

-

f

(x, w

k

)

D(x) =

p(x

|

S)/[p(x

|

S) +

p(x

|

B

)]

%

D(x)

H(x

|

S)/[H(x

|

S) +

H(x

|

B

)]

H(x

|

S)

H(x

|

B

)

x

(19)

x

:HH

D(x)

6

D(x)

x

f

(x, w

k

)

6

G35 G

-

! 3

-

; 3

f

:/

; + - .

n

s(x)

/

;

s(x)

'

n

/

; 3

/

; -

/

;

/

+ - 4

D(x) =

s(x)/[s(x) +

b(x)]

f

ˆ

(x)

D(x)

s(x)

k

w(x) = ˆ

f

(x)/[ ˆ

f

(x) + (1

−

ˆ

f(x))/k]

6

k

= 1

: ?

H #%D &

*

D(x) =

s(x)/[s(x)+

b(x)]

+

- *

' .

D(x)

' + '

'

D(x)

'

D(x)

':

*: !

. :

3

(20)

!"

'

#$#

#%& II H L )" )$$$

#)& 3 F K , ! "

# $ IH L)" )$$E

#9& F6H% I HL%EEB

#=& +4@3& ' ()%" % # 4>

( HL)$$)

#A& K3, :" ( $ * * * #192 E%B

1)$$D20

∼

#B& G:% $ ! #%)9 1%EEB2

#C& G:+ ,! $%A 1)$$%2

#D& FK : -. / 0 ! 1*-* $*&#% 1%EDC2

#E& K ' !( !

#%$& 56: $ + ! HL)$$C

#%%& K 23

#%)& L5 2

#%9& 3:( + 4 0 , 5- 5 3

" ( 153"( EA2 FK :' %EEA

#%=& 3 : ! ,

16$% ' )$ 1)$$A2

#%A& L , F" 7

1*-* **%%1%2 %%E 1%EEC2

#%B& :F% " . *# *!*

%$ ACC 1)$$A2

#%C& OV5 1I6+'2 8 .

|

V tb

|

$* +* *()%D%D$) 1)$$C2 !"#"$

#%D& OV 5 1I6 +' 2 8 $* +*

&)$%)$$A1)$$D2 % & '()(*+ $

#%E& 5O, 5 1 + 2 ! 9 -

- ,$* +* * ' )A)$$%1)$$D2 % & '*"#'!+ $

#)$& K3, 3F

* *#)1)2 9CC 1)$$$2

#)%& (5 : 3: ( % $ $"