Page 1 Page 2 fi(t) gi(t) gi(t) Îi(t) fn(t) Im (t) Im(t) f (t) g(t) = [ A(1,795) g(t

Page 3 ... Page 4 ... 2. P(=) x exp -> 2(s) ((=) = + tanh(s) p(2) Exp [-(2 – a)?]. + exp[(z + a)?]. ©()= ax - a tanh(aa) ... S FII = arg max f1..tp (Â(k-1), f1..1/9...T ). | Â(k) ...
253KB taille 0 téléchargements 70 vues
 

 

    

 

   

 

 

!

& +

$* &  %

&



"



%$&

%



"

,

)('

(

$  !

#

! 



  









.





( & & 

'

'

! '

( % &

+  



! '

-

,

/

7

8

;: 0 09

6 06 3 3 0 54 3 012

M

BD

AL E

DK BA J=I

?

BC

BD

E

HGF 2

@ 2 =A =>

@?
?X

B

=

VK

2D



Z

U

@U 0 B X

D =

B

>

G= 1 M

2

@a



% &

−→ B = Σ Λ A−1 Λ f1 (t), . . . , fn (t)

yn (t) gm (t)

y1 (t) B g1 (t) g1 (t) A f1 (t)

gm (t) fn (t)

−→ −→

$ '

 

 

    

 

   

 

 































































































c



BD X >

G= 1 AB

B

JBI @ A D V

V

GB

1 @ @? X G? G A

G? D























B X

G= @2= a ? X B BC X 2@=A M = 1B C G? 2=A M 2DA b 

@ 12 G

BD L

U

IF

D

2D @ IB 1a= GB G? 2A

X V

?

2?D M

6







D

@L U

B

X

A

I

=X

IB

C

V

1B

1



@

_

G



BA

G?



V

B

I

B

?A

2=A

@

G

G 2M

@B A V C A BD

B 1=

A

?

G?



@? B X

I

B

2@=A M G

BX a =

F



% &

1X = gk (t) gl (t) T t [Rgg ]kl

Θ AΘΛΘt At = AΛAt −→ A −→ AΘ

Rgg = AΛAt −→ fb(t) = (Λ+ )1/2 At g(t) A

 







f1 (t), . . . , fn (t)

$ '

Rf f = E f f t = Λ

Rgg = E gg t = E Af f t At = ARf f At = AΛAt



 













@

M

BI =A 2BDA

U





B C D A

G=

M

?D

?

X

BD

BC

X

G=

C

C

V

GB

G

2

I

I

X

AL

BC

B

BD

BC

1B

@

M

@ 2@ G @

2?D 2 

S

U

U

+



















1 @ ?I B I A X I A D B

U



φ(.) 



D

@B 2M

2A U 4





@

@

D

I

AE

I

X

A

BC

2A U ?

2M

2=A

G?

fb B

 

    

   

 









@

@

G I ?D

@B

M ;

b b b (B, b) = arg max (B, b)S(B, b) g S









i

















S=−



 

  







i

X pi (fbi ) ln pi (fbi ) = −

X

D

C D

I

V

B

=I

B

BI

G? @ G

@ G D G= C







   



% &

pi (φi ([Bg]i + bi )) ln pi (φi ([Bg]i + bi ))

fb fbi = φi ([Bg + b]i ) −→ fb = φ(Bg + b)

$ '

 

 

    

 



















 



 

 

























 









1 @ M2 G2 2M @

@

= BI D

?I = =I 2I

@

B A =D A

G? X B C

G? 2XA

I

B

G? _ G @ G2

V

U



c

G? 2=A

DV

X

BC

B

@=

@

2=A

U

B M



B

B

1U M J b

C

i

  

 

 

  







6

pi (yi ) c(B) = KL p(y) :

Y







 















 

 











 









 







 

















B I 2DA @2 V A =X @= X G? 2XA G?

C

DV

BC

B 2@ GB A GB M B ? V

1a V



U

1= K

@?

1B

=

D

I

_

A

U

G= 1 M I X BD D @=

B C G? 2=A M 2? J

U

@U 0 6

1 M

C

I X

I

G=

A

D

B

(yi )



% p0i (yi ) ∂c(B) ∝E ∝ ∂B pi (yi )

&

pi (yi )

p(y) y i pi (yi ) p(y) ln Q =

Z ! Ã

c(y) = c (Bg)

$ '

 

 

    

 

   

 

 







































 























A 2I @ > B

@

B V @= X B G= 1 MV B BC 2@=A

X

B M

I

1B

=

I BD B C A A

GB B

2I

1 I BD

D

G?

AB

BD D




@

A

1

B

A



BD

M A 2X 12 U B

?

U

AU

JB M X GB =D A

GB G GB

M

GB

@



C?

B

@U G



MV BD X B C A

@=

@

BD

BI

U @

@B =

D I 1U
2D a

@B G 2BA

c c cC 2 2A

?I

X

BD

D

G?

@ D BD 1B I A

I

A

C?

BD

D

?D

GB



MV BD

U U ;S 0 ?I YS 0

A G= a =

2 



B X 1=

G

A

B C @=

=

D I 1U


B

?I

2I

@

1

@



























=

λ = σn2 /σs2

$

fb

G

G2 B

@=

1

T

T F

w 

&

%

















 







 











 









'

$

















 











f = (At A + λI)−1 At g

f = (At A + λD t D)−1 At g

−→













































































B

C

G?

D

V

C?

B

V

A

I

X

BC

2=A

12

3



1

2

1=

2X

G

2@

Y U

p(fj (1), . . . , fj (T ))

 































ln p(fj (1), . . . , fj (T )) =

X

ln p(fj (t)|fj (t − 1))

t

 





































t

£

t

2 =

C

F

[λ1 , . . . , λn ] f (t − 1) + A g(t)

w



&

f (t) = (A A + λI)

−1

¤

%

















 







 

































 







'

J(A, f )kg − Af k2 + λφ(f ) + µψ(A)

BX

¸

a

1 p(A, f |g) ∝ exp − 2 J(A, f ) 2σn

g = Af + ² =

·

$

p(A|g) =

p(A, f |g) f C

Z



@ B

@C?

C

G? BX

D

I

C

G?

2=A

M

2?

J U

@U

0

³

C

G=

´¯ ¯ Pbs−1 ¯ − J(A, fb) 

BX

a

=

B

BC

I

A

Pbs = (At A + λI)−1 ; T:

2M



2@?

=

V

X

D

B

B

1=

M



M

M

1 A

GB

?>

2A

G4



F

−→

fb = (At A + λI)−1 At g

BA

C

¯ ¯ − ln p(A|g) ∝ − ln ¯

∆A ∝ At (At A + λI)−1 + gf + µψ 0 (A) 7 

3

b

G?

D

2=A

2M

2=

J

3 B

X

VU

G=

@ D

A

bB

M



2@?

01 F 



g

c

BD

?

X

A 1L U

M

G

2AB

BD

?

X

BD

V

A 1L U

M

G

G?

6

w 

&

(g, f )

%

















 







 











 









'

$



































 



 













7 B

DV

V

VF

12=

@

G B





p(f j |f j−1 ) ∝ exp [−α|f j − f j−1 |p ] i h P p(f ) ∝ exp −α j |f j − f j−1 |p

3

2

X X˛ ˛2 ˛ 4 f (r) − β p(f ) ∝ exp −α f (s)˛ 5

3

X X˛ ˛p ˛ 4 p(f ) ∝ exp −α f (r) − β f (s)˛ 5 r∈R

s∈V(r)

w

&

G

^

^

2

r∈R

GB

=I

a

?

2DD



D

D

=I

B

@=

3

G

GB

2DD =I ^

p(f j |f j−1 ) = N (f j−1 , σf2 ) i h P p(f ) ∝ exp −α j |f j − f j−1 |2

s∈V(r)

%

 



















































w 

&



 



 











  









´ ` p(f j |zj = k) = N mk , σk2 i h P P p(f |z) ∝ exp −α k j∈Rk (f j − mk /σk )2 h P P i p(z) ∝ exp α j i∈V(j) δ(zj − zi )

P (qj = 1) = α, P (qj = 0) = 1 − α

$





























































































´ ` p(f j |qj , f j−1 ) = N (1 − qj )f j−1 , σf2 i h P p(f |q) ∝ exp −α j (1 − qj )|f j − f j−1 |2



















 











 









'

%























 









 















































































'

$







 





2 p(fj |zj = k) = N (mjk , σjk ) −→ p(fj ) =

Kj X

2 αjk N (mjk , σjk )

k=1 























zj ∈ Zj = (1, . . . , Kj ) αjk = p(zj = k) Y Y 2 2 N (mjk , σjk ), θ = {(mjk , σjk )} p(f |z, θ) = X

a



j zj ∈Zj

p(f |θ) =



 





 





X

p(z) p(f |z, θ)

X

z∈Z 

p(f |g, θ) =

 





#



!"





 

p(z|g, θ) p(f |g, z, θ)

z∈Z 









































%$

 





'

, &

*'

&'

()

b 

'

&

,

+ *'

&'

()

.

'

&

,

*

+'

&





)1'

)

&



)



+

&'

()

-

0

0/

p(z|g, θ) p(f |g, z, θ) θ (

2

&

+

z f

p(θ|g, z, f )

%



 

 

   

   



 

  







 

 





 

.



, ' , +' & 

* &

) 

' 

, ) & 0 

& 1

( + 

)& 0 + 

 &



+

0

 )0

'



,

&

/ 



)' , , + 

,*



+

.





&  '

0

' (0 ,

,

,

' &   



& 

* ' ' ) 

, & , 

&



+& ,

,



' '

,

( 

+

,

)

(





& 

'





'

,



 







 )0

,

,

' (0



+0



+'



' )

  , + ) , (  

'

(



) , ,





,



0

/



(

& 

* ' ' &

) ,

& 

 









)0 0

,



.

 )0 , + & ) ,

' 

,&  0

*0 & & * & & 

 +' ,

& 

'

 ' 

+0 

(

+



( & 





.



 '

&

&





&'

 

,





0

 )0 + (

&

 *



*



) '

* 

)&' + * & &  

+ ,

)

  

 

 * '

* ' 



'

)&'

, , & 

  

0



/





 

 * 



)' , 

, & & 

' 



 * *

)

' )0 0 



, &

'  

+ 

 

(

,



,

+& ' 







&

+



( &  +& , 



.

 

* ' & 



'

,



(



&'  & 



)0 0 &









&

&'





 

,

)



,

/



+ *  +  + +

) , -,





'

, ( 

)0 0

 +

, 

/ 

&





, ( &





(0 

,



'

, ( 



)0 0 + +

(0 





)

2

% 0

&

A

$ '