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Utilisation d'approches probabilistes basées sur les critères entropiques pour la recherche d'information sur supports multimédia

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R × R

+

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f

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2

(x) =

¡

2πσ

2

¢

−1/2

exp

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(x

− µ)

2

2

, θ = (µ, σ

2

)

∈ R × R

+

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n

Y

i=1

f

µ,σ

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2

¢

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exp

Ã

1

2

n

X

i=1

(x

i

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2

!

.

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(0, 1)

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Θ

:

 

N (0, 1)

Θ

1

:

B B

N (µ, 1)

5

µ

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)

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2

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:

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N (µ, σ

2

)

5

µ

6= 0, σ

2

6= 1.

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θ

?

w ‚  BB   4  5 JB   K4q  K ¥ KK  w BK5B B   B C  B 5rB CK }‚    B KrB 

R × R

+

BK4B  BB  K ¥ KK  BK5 B

ˆ

µ =

1

n

n

X

i=1

x

i

, ˆ

σ

2

=

1

n

n

X

i=1

(x

i

− ˆµ)

2

.

ƒ B J

ˆ

µ

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2

6= 1

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S

⊂ [[1, 2]]

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S

, ˆ

σ

2

S

)

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Θ

S

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µ, ˆ

σ

2

)

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Θ

{1,2}

  5 K K B   445  C45q B

N (0, 1)

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n

B   4BB 

(16)

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P

np

(X = j) =

1

6

,

∀j = 1, . . . , 6

J  5 5 Kr  

P

p

(X = j) = α

j

,

∀j = 1, . . . , 6

5

P

α

j

= 1

J 5 V KrB B } ~ B KrB BK4B  BB  K ¥ KK  BK5   ¥ rK B

b

α

j

= n

j

/n

5

n

j

KC B5q 

j

BB4B } 

b

P

p

4 5B }“  4  CBK BB  K ¥ KK  BK5 

n

Y

i=1

b

P

p

(X = x

i

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n

Y

i=1

1

6

,

C44 B B JK B

n

B K   J 5qJ

n

j



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n

B  

(x

i

, y

i

)

 B

x

i

B 4KBB B5B  B

y

i

B 4B 

y

i

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?

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i

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i

, i = 1, . . . , n

” 5

A

?

B K45B

ε

i

4B 

n

w45qC

ε

∼ N (0, 1)

} v qBBBB

A

0

, . . . , A

n−1

 B 

R

n−1

[X]

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B =

P

b

i

A

i

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S

⊂ [[0, n − 1]]

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b

i

= 0

⇔ i /

∈ S

} ~ B KrB B4B   }   5K4 B

Θ

S

=

{L(A(x) + ε),

B

(A) = S

} , S ⊂ [[0, n − 1]].

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ε

B BBCB4wBK5C45q K    K

A

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n

X

i=1

(A(x

i

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− y

i

)

2

.

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S

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n

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Θ

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n

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A

?

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(17)

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(18)
(19)

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Θ

i

⊂ Θ, i ∈ I

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i

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b

θ

i

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θ

i

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i

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i

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n

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(20)

X ~ C4 B 5rB CK B B  B 6BB  5qJ Kr  5K4  B

p(Θ

i

)

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θ

i

)

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l(b

θ

i

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i

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(21)

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i

∈ I

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i

|

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Θ

i

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: I

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θ

i

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i

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l(b

θ

i

)

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n

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θ

i

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(22)

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(i)

W }  KB C4J B 4B 

Θ

i

44   B  5 Kr } €  44    ”v

(i)

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(j)

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i

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θ

i

)

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i

|α(n)

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Θ

j

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(i)

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(j)

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Θ

i

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Θ

j

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bi =

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(

”v

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∈ I) .

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Θ

bi

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α(n)

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α(n) = K(1 +

p

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i

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K > 1



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i

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Θ

i

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α(n)



n

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n

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(23)

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log log n,

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α(n) = 2

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α(n)

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f

θ

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∈ Θ

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Θ =

R

m

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Θ

i

, i

I

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i

|

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x

n

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f

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θ

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2

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θ

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r

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s

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¸

(24)

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h., .i

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θ

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θ

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θ

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θ



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θ

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i

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n

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n

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i

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n

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i

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i

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°

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°

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n→∞

0.

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θ

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f

θ

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f

θ

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(25)

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S

         

S

?



(26)

v  4B K B CB C K  5rB  5q ¥   44 C  C B5 J  K  55 J C Bq  } “   ¥ K 5qB

α(n) = (log log log n)

α

, α > 0

B  BB  55  4   BK  B } ~  5q ¥

α(n) = (log n)

γ

, γ > 0

 J   55 BJwB } = & A-+/-&

ϕ

β

EB 5 J ‚ w œ     }  ‚œ   B CB  5r

ϕ

β

  44 B 4 

0 < β < 1



α(n) = n

β

log log n.

 W } X ~  5q ¥ B B  

β

K4BB 44B BB   q4rK W } }   55 BJwB  B BK4 B   B } ‚  K  B   

β = 0

  5r

ϕ

       } v  K4B CB  B B  B BB } 

n

 ¥ 4  BK    

β

K    44 B B 5rB BB  BJB B  B   W } }“  5 B B BB B  5r

ϕ

β

B J B B 

β

} ~ B B 5rB 

β

B 6

β



= (log 2

− log log log n)/ log n

β



= (log log n

− log log log n)/ log n

 W }  ƒ B J

β



< 0

rB J

n

≥ 1619

} EB   œ  B B  5r

ϕ

β

B BB B B 5rB 

β

BB 6

β



=

log log n

log n

= 1

− β

  W } W  5B CB  5r

ϕ

β



β



< β < β

 } ƒ B 5BB   J JC 5q ¥ 5q 

β

  B B KB 4BB  B  BK } ƒ B J

β



< β

  CB KB  5r

ϕ

β

4B B J  5r |”v}  STROQN  M NON NT     B5  CB B 5rB B B 4BBB B 5 K4K   5K ¥ 4  5 B } “  5K ¥ 4 B B

(27)

 54  B45  KrB } EB  5B B KrB B 45B   }   K4q W } WJ B JB   J BC45 5 6

b

S =

 K

(

”v

(S), S

⊂ [[1, m]])

W }    5K ¥ 4 

2

m

} v  5K ¥ 4  ¥  B B  44  B   ¥ 45  B K5q } ƒ B 45B 5 B K4qB CB B 5rB CK B   K4q  }   k\haci gape`f`\] i v  K4q B B4  ƒ Bq B  ƒ BYY }  ¥ B    5r BJ B B KrB B B 5KK 445  BKB  B

S

?



b

S

4 5KK B 6

(

”v 

=

”v

([[1, m]])

b

S =

©

j

∈ [[1, m]],

”v 

”v

([[1, m]]

\ {j})

ª

.

W }   B

b

S

5 B 5KBB J  5r  KB  ”v 

”v

([[1, m]]

\ {j})

} v  K4q 45BB  55 

m + 1

5rB }    k\haci gape`f`\] i ]b if_ki ~  K4q 5K B4 5 B5KK 445    5r BJ B B KrB B 4B  BK

S

?



(

”v 

=

”v

(

∅)

b

S =

©

j

∈ [[1, m]],

”v

(

{j}) ≤

”v 

ª

.

W } V v  B5 BB5KBB J5rKB } v  K4q 45BB 4K  55 

m + 1

5rB } ƒ B B  5q J  K4q 5K B4 B KB w B J  454 K4q } v   K   J B J4B ”v 

,

”v

(

{j}), j = 1, . . . , m,

B BJB B B BB  CBK  B 5 KB CKB J 5B B4B   K4q W }  }     k\haci gape`f`\] i ci_gibc`b\i v  K4q CB   W } 4KB 5KBB4B  B   5r } ~  54 B 4 5 B   K 4 C4B } “  5KK5 ¥ B

S

(0)

= [[1, m]]

”v

(0)



=

”v

(S

(0)

).

  Kr 4 B B45B B 5B  C4K 

C

(1)

=

n

j

∈ S

(0)

,

”v

¡

S

(0)

\ {j}

¢

”v

(0)



o

.

(28)

Y  W } W  ~ B K4qB 5KB   5K ¥ 4 } œ 4q v K ¥ 4 ¦  W } 

2

m

v K  W } 

m + 1

v K B4 W } V

m + 1

v K B5 W }  }

≤ m(m + 1)/2

‚ 5B B  5KB

J

(1)

4  KB    5r  5BB  445     5q 4 6

J

(1)

=

 K

¡

”v

¡

S

(0)

\ {j}

¢

, j

∈ C

(1)

¢

S

(1)

= S

(0)

\

©

J

(1)

ª

”v

(1)



=

”v

(S

(1)

).

~ C4

k

≥ 1

  44  54  5r  C4

k + 1

6

C

(k+1)

=

n

j

∈ S

(k)

,

”v

¡

S

(k)

\ {j}

¢

”v

(k)



o

J

(k+1)

=

 K

¡

”v

¡

S

(k)

\ {j}

¢

, j

∈ C

(k+1)

¢

S

(k+1)

= S

(k)

\

©

J

(k+1)

ª

”v

(k+1)



=

”v

(S

(k+1)

).

  5 4

k

f

+ 1

  

C

(k

f

+1)

=

} v  B J  5r  JC C  B  5KBB B B

S

(k

f

)

} ƒ B  B 5  54  5qBBBB 5  B 5KK BK 

S

?

} ƒ B J B B J4B

C

(.)

, J

(.)

, S

(.)

,

”v

(.)



, k

f

B   K4q 5K B5 B 4B } ~  5K ¥ 4  5 K4q B 5  BB 4KB  B 4 

m(m + 1)/2

}   ape^i]\k_ ci_ pk\haci_ ~   W } W4BK B 5K ¥ 4B B K4qB 45B }~ B K4qB 5Kw B 4B B  5K ¥ 4  K   K  w KrB 5qBB   B45  KrB }v BB J  BB B  rB K4K B K5q } E B B K  B45w   CK J B 5 B

[[1, m]]

  5  B5  Kr BB  J  K4q  }  MR   5B  5 5q B B 4B4 C4 44 B 5rB Cw K  BB4  qBJ  B B  BB } ƒ B B 54 B KrB B5B  5B 5rB  ”v  W }   

ϕ

β

W } X B4B B B 5B4B B5B  KKB  BJB   B4 }

(29)

EB K4qB 5KB B K B   W }  } ‚ B   B4B 5 CK J 5r  4   B5  5Kw  ¥ 4 CB }  5B  5 K4KB B  5B  B B   K J  554 4 5 B K4qB 5BBJB B 5B  KB  55 }

(30)
(31)

        C B  B 4B 4B

X = (X

t

)

t∈N

J 5CB  w 5BBB w4BB KB B  

t

∈ N

 4  

X

t

=

X

i≥1

a

i

X

t−i

+ E

t

,

 }  

(a

i

)

i≥1

BB4KB4B BJ 5BBB

E = (E

t

)

t∈N

B   5 BB  5

σ

2

4  ¥ 5   } “  5  B

X

t

= 0



t < 0

} ~   

k

 J B

(a)

 BC B B 4 C Cw 4BB B

X

t

4 4K 

X

t−1

, . . . , X

t−k

KB B 

X

t−k−1

} ~ B

a

i

, i = 1, . . . , k

B 4B B 55B C 4BB } “ B 45B4K   BC4BB  B C4BB   Cw BK

S =

{j ∈ N

?

, a

j

6= 0} ⊂ [[1, k]].

 } W ~  4  5 B B B 45B J  BK 4  C BJC K  B  ¥ 5K  JB B BB4B

X

t

4 4K } EB  B B  ¥ B B KrB 4BBB   B BB  BB ‚ 4w | 4 3 w | 4  J B  B  C 44w   |     }  ||  4B   ¥ 5  5

σ

2

= 1

6

€   C W  55B wX } V wX } XV €   C Y  55B  X } W X } W wX } Y X X X } V X } W X } W €   C V  55B  X } VX } X X

. . .

XX } V  }  v B KrB B  JK 5qBB B  4    K } ~  Kr  W  5K  5 5  Kr C  5B   B  ¥ rKKr } V J   5 5  Kr C  W 4 4 JC  W 55B B rB B  ¥ KB wB }   BB   K4B B4J5 4B4B

x

n

 5BBB w4BB } EB  K KB  4K  C  5 5BBB B  rK  B45  KrB J B B  C  B 5rB CK } EB   ¥ rK KB B BBB  4K  B  4BB }

Figure

Figure 2 presents, for m ranging from 1 to 256 the esti- esti-mated bit-rate CRIT(x n |k, P(m))/n for k = 0, 1,2
Fig. 1. Rayleigh RR using optimal histograms, empirical his- his-togram and KS methods.
Table 1. KL distances from optimal histograms to the laws in competition in LOS(a) and NLOS(b) cases.

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