• Aucun résultat trouvé

Constrained covariance (COCO)

N/A
N/A
Protected

Academic year: 2022

Partager "Constrained covariance (COCO)"

Copied!
8
0
0

Texte intégral

(1)

! "# $ % &

! "# $ "

% $&'( "(

)" *(

%++' * (

" * , ! )

++"

$& * ,

" -

" "

* , .(

" *

++) /)

(,

# ( "

*, ! -(

" * ) "" (

" ++, 0 )

++ 1 *(

)*-

2# "

,

! )3 *(

,

" "

* )

, & ""2

"%4567 ')

" )*

)"*

"4587,

"" ) "" (" "(

) (

* "

4% '7

4% '79

4% '% '7 %5,5'

%

45: 5;7', "" )# -(

")"

<$&$& %5,5'*"("=

" "(

*-"

) ) " (

-", !

*%++',

" )# "(

1# "(

, " )

"# $ "% '

= 457 * # 45>7

" ",

# ?)

/ )

", &6 )(

" # $ " % $&'

) "" % ++

'@ 2 % $&

"

4887', &"= <" #

* -" (#

" # )* * ,

!-&:=-(/

=" * -(

* ", )(

)"*"" ++

)

"* )#

Æ " =(

" , #

" # (

% '4;7 # *

% A' 457 ) ) ) 4;7

"" " /)(

) ,

" 1

" *

"(")#"

-" = # (

,

(2)

) *)

"" " ++,=

"" ++ )"(

++ )"" (

++ )" ++

%"" )" ',

*

"@ )

" **(

= 4B , C7,

""""= ++

" #(

<& % # "

"@ 487', "

2"(

)* ) "

*" "(

" =-" /,

0 )"" )

, " %,,

467' * - )

,

+<. 2 #

= %:,CD C' 45C 5B7

" " (

)*"

* , & B ) "" ++

"(

)+<.") * (

-, ++ "(

) )4E7 )-# ((

"% A'

* , ! *)(

")* (

) "

,

" )(

) *=#

* "

$&,

<%F

' " ",

* @ %F ' % '

@ %F ' % ' ) "

" "* (

, *) =

),

'( ) *+,-

*% '@9

4 7 4 7

4 7 %8,5'

0"" " (

* -" )

@

!" #!

$

*%% ' % '' 9 G % '

) =

" ,

'( 3 *+ ,- *

"

)=

++%

'@9 "

4* %% ' % ''7 %8,8'

% "*"

1 * "(

"" @ 4E7

', * " * Þ @9

%% ' %

'' % '

" (

=

++

%Þ '@9

"

5

%

'%

'

5

%

'

%

'

) 5

5 ) *

++%

'9G (

",

) ++ ++

) )

%

* *',

)* (

) ++

) )

* ,

**)(/" *

) ++ /

", 3 (

)*

*++

++(

" *

%& >',

%& 6' ( (

* " # $ "

(* " (

, ) )

"" "48G7, "#

$ " $ " )

"* Æ

@ Ê

)" %' Ê ,

"# $ " "

2 "* = # @ Ê %

"# ' )"(

"@ (

/1,

% & &

(3)

"" "(

* (

* $& "

# * *,

""-" "-(

"++

%Þ ') (

"* "# $

" , " 4;7,

&)*4++

%Þ ',-

!"

!"

++

%Þ '9 5

H

H

%8,6'

H

#

H

9

#

9 Æ

$

9

%

'

H

% & '

(

) ) "

I ) "* "

# * ,

. 5 *6 ,- ) % '

%

'

% '

J

%

'@9

J

%

' %

'

*

J

0 )* # ("

Æ -"

%' I(

,

& 1 * # Æ7

,- ) %' @9

J

%'

% J

'

% J

'

Æ

G

J

+

@9

J

J

,

"( 2 < : (

= # %

)%%

J

'

'-"%

''4857, !

# =2(

! ) # )

++ (* ",

*B ))

++)

$&) * # 4887,

'(8*9,- %

% ' " " % '

* $&

# % ' )" "

=

0 4887) ))#

* "Ê

@

%

'9-"

%

'9-"%

' G

!) ,

. : *++ ; #7

,-

!" %

!" ,

5

5

++%

'9G

+ ++%

' /

", ! "* (

* "

++%

% ' %'' 9 G %

% '% ' (

5

% ' %'=

' ++%

' 9G,

-)2

% '

%'

5

5

)

*%

% '

% ''9

" - )

*%

£

% '

£% ''

8, !#)

"* % '%'@

56G ) = %

'

£

9

,

! J

% ' @9 % '

£% 'K

£% ' %)

J

% '=' )

*% % ' % ''

9

J

% 'J% '

% J

% ''

%J% ''

* %

£%

'

£% ''

8 %

£%

''

8

%

£% ''

8

8 B

8:

9

: G

"

*% % ' % ''9G " ",

(4)

' 4; 57

* "*")"(

,

1-) #

) ) " -"

= " - (

" ) %,, ) " (

' " ",

" -(

))-" "

* ) Æ " ++

", !

) ""

2 ",

! ) ,

"

Ü

) (

, " ) @

(

Ü

" (

*

Ü 9

(

Ü

" * , !

- L%Ü' ) #

Ü

Ü

L%Ü'95 @ Ü

Ü

L%Ü'9G @ Ü

Ü

* ="

) " )

)

Ü

Ü

, ! L()

"

È

Ü ÜÈ

Ü

%L%Ü'95'

) "" "

", + )@

. < *9 #

,- - Æ

5G

Ü

ÜÈ

Ü

%L%Ü'9G'5

*

**

+ !

Ü

@9

Ü K%5

'

Ü

) G 5, *

)

Ü

", "

"-

ÜÈ

Ü

%L%Ü'9G' 9

Ü

Ü %Ü'Á

Ü

9

Ü

Ü

%

Ü 9

Ü

Ü

%

Ü

9

ÜÈ

Ü

%L%Ü'9G' 9

%5'

ÜÜ

' $

Ü &Ü Ü $ Ü

Ü

Ü

Á

* !, # * 5 %,, #

" * # -'

"*,

1-3 = #

!"*-"" (

) ++ )

*) )

A, "" ++

/ / ) #

" (

" * , ! "

" " 1

%) (

'* %

% '

% ''

% * (* (

"' ++%

' , (

()

++ $&

++%

'9 "

*%% ' % ''

%:,5'

) * ) $&

*

@ )) *

, "= (

#

%

%'

%' '

* $& ) -"

" 4857,

"

%,,

( ' ) -" ++

* ) - )

*%

% '

% '' ,

. > *'#

++,-

*%

% '

% '' "

" G

++%

' G

+ " # @

* 45G7, !

)*%

% '

% ''", )

% '9

K"

%'

%' %:,8'

)

% ' G

% ' 9 5,

=2

"

%

%'

%''G

( ) $ * $ '$+

,

'$+ '$

- + $ '$+

'+ +'+ +

'

$ $ + .

+' $+ ,

/0. .

$ '

(5)

,

(

*)#

9 *%

%'

%'' %:,6'

@9

%

% '

% '' %

% ''

%

% ''

+ * #

@9

@9"K

%:,:'

)

2 ) * (

% " 2

Æ @

# < :',

! - -" ) (

= ++ %,, "* $&

) " ' %' 9

J

%' %' 9

J

%' % -"(

Æ ) * J

',J (

-" %:,5'

*%%' %''9 J

+J

J

+J

J

J

K

J

J

"

9 J

J K J

J

"

))"

+)-"

%:,:'

) * ,

< : J

J

*,

J

J

" )*

++

J

J

"

J

J

J

J

"

J

J

"" /

# 6,

! )) 457 (

"" "

* $ 1

#! 2##" + .

+' 3 + .

4 $ $'+ +'

563 +

$ Æ 7

7

+ 44

' $ "#

' '$+ +

$ 44 +

$ ' '*

+

%

'

@9 "

*% % ' % ''

*% % ' 'K$

*%% ''K$

$

*% % ' % ''

$

++%

'

) "=

) ,

0 ) 4;7 A

457 ) "* - ++

*"", )

)

%

%5 %

''

) %

"" ++

A

%

%5

'' )

"(

" , & ++ ) (

)

A,

! * ) * ,

= % " ' ""

++)" ++

)* **

" ++ ""

++ , " * #

* "* 45G7,

2 "

++ * "" ++ "

5

, )="

L%Þ'%&:,5'++

%5' )

L%Þ' ( ) "

""""/5

,

+ = # )

)"" (#)

*,

. ? *!

# 97 0)) #- 382,- .

%

' ,

/

@9

5

% 5'% &K5'

'

%

'

&

5

' # / * (')

%

%'*'-"

8*

&

%)('

!),

. )@ *9## ##

++,-

(6)

"

*%% ' % '' "

*%%' %''KM

M 9

Æ

Þ

*%%' %''@9 5

% 5'

5

@9%

'

@9%

'

+ 0 )

"

*%% ' % '' "

*%%' %''

"

% *%% ' % ''*%%' %'''

! "

* @9 % ' @9 % ',

" )#"

%*% '*% '*'

5

K

% '%5'*

K

5

% 5'

%'

% '*

=) $(

3I 2 4557,

)=*

@9%

'

# '

%

'@9

, (

;"" , !" "

98

8,

) "" ++

" )* )",

. )) *&A ##

++,-

5

% ' 5

% '5 5

ÆG 5Æ

"

*% ' "

*% 'K 56:

K

5E 8Æ

+ !)"

"

*%%' %'' "

*

%% ' % ''

"

% '% ' 5

%

'%

'

K

"

5

% 5'

%

'%

' % '

% '

),

)).

"

5

% 5'

"

5

% 5'

K*

+  

! -" @ (

)="" $3"

" ) 4>7, ""

-"%'),

! "

* "",

" )

%'68 "

5

% 5'

%

'%

'

%

'%

'%

'%

'K%

'%

'

K :

"

%

%

'%

''9%)'

)

*

# * 5 5 ) 2 "

""

""

""

, (

" )

-" ) "@

)"" " %45G7'

%)' 58E

% 5'

!

%

'

%

'

K :

¼

"

#

#

$

%

'

%

'

% '5

% '5,

! "* "" ++ - (

" " %' 4;7 ) "(

# 457,

++ "

, " )

*" ) ++

) "

(7)

* 3 )@ %'

" ) %'

%' (

, " ) (

" )

-" *

2 2 45C7 (

" " 3 "

, + ) # (

:,C )" 2 :$/

;B-;B= *)%0+A' ) (

8>B-8>BG,> #

) " 45B7, )

" N&!O ) ) -(

*)* -,

.")**- "

* %A5' 8>G*- (

) 8>G %5GGG

" ', *

6>-",

* ) )

", & 2 #

) 2

) (

2""- G,:$/, !

" (

""",

* ) "(

"# 2

) "

* ) 2

3* , ) )

*

*- ) )

" "#2 *

*- *

, $ (

" 2)

), -2

) *"* *-

" *- ) -

"1(

, + *- * )(

(

)"" "*-

) " )

* , ! ) "

"2")"1

% *

2 = (

', " ) *

= ), ! (" =

* ) *

" " "(

*,

" ) " *- )

"" ) *-

++ % $& ) # ',

*")*- )(

) *

)*- % ) ) "

"*- 2

) " )(

*- ") ',

%+' ) *-

* , &"=(

P * (

= 3 - (

, ! "(

" )

) * " )

A5

* ", "

" ) * *(

* " )A5

, " 0B,5,

" " (

* )=3 (

"

"*,

* %++ '@ = (

- )

"(

)" " ) *

"2" , (

* ++ * (

" ) *- % (

'

-,

-" " ) (

* 3*- )

* 3

", "

* Q >

* ))

", "

* % ) *(

'8 )(

* )* ")Q

Æ , 0 ")

++ "

* "

) #++

", +

*

* )

++ , -"

1 )

**,

0 )# )

"(

) * " )# Q(

+

441.

$

(8)

0 2 4 6 8 10

−2 0 2 4 6 8 10

Constrained covariance (COCO)

Between−voxel distance(mm)

Dependence

0 2 4 6 8 10

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Mutual information

Between−voxel distance(mm)

Dependence

0 2 4 6 8 10

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Correlation

Between−voxel distance(mm)

Dependence

0 B,5@ % #' %' * ++ %'

, 3 " -,

/ -" *)*- ",

" 4C7, ." #

,,, )

" ) " "", (

* " # ( "(

(,,, ,

! 361 8 .

) !9":#;;#

# <%6=>< 1 306 (? $

1+' $ .

5 !;!1/+$ 6 '

#;;"

) 6+ 3 1 + $ .

+ .

+++ @+ '

$15 A AB:9AA: !A

" <> C DE 2 $ .

$ ' $ .

"F" !)!F9!)#! !

F D/72<G

(',H !

B IC I<? *<I+'

+)! $

(', H !B

A CD2%C+(8 113+.

++' $+ .

+ +$ !"

" # $#F A::9AF!:

: 3++0+35618 C.

+ $ +' .

+' F A)9#;;"

> < 5% > (

++$ 1/ $ 6 .

'#;;)

!; >< >( 46 +@+5%

6 + ' $ .

1/$ 6 ' .

#;;"

!! % -' / @+ $ + $

+

# F: !)9); !B)

!# > %? 8 + 2 4E

8 ( ,

H #;;!

!) H+' > $

$ "#%""" "BB9

"AB !:

!" 8 8 / /

(',H #;;;

!F ( 2 I+' 5 > 1 6 (.

+ "" #"#"&FF) A#F9A";!)

!B , I ' % <+''' 8 /+

( / 3+ '' $

!! # FFF9FB# !

!A , I ' 8/+1 >+'

> 4 ,+ ' ' $

$ $15 ' ! "!# !F;9!FA

#;;!

!: 1C4 $ +-

#% '!":9!::!:.

'J/

! > 5G 4+ $ "

"#"'"!; ""!9"F!!F

#; (( (

I '(*%

J!::

#! 6 (? $ > ( ) (

1'1>#;;#

## ( 4 K+ $

$ + #

#;;!

Références

Documents relatifs

l’algèbre de covariance possédait une unité approchée, résultat fondamental dans le paragraphe III où l’on montre que ~1~ est une A*-algèbre et que, pour tout

Mélanger tous les ingrédients dans le saladier puis laisser reposer 30 minutes au frais.. Faire des petits rochers sur une plaque de

Pour Ecoflo ® Filtre Coco en béton avec pompe intégrée, s’assurer que le niveau maximal saisonnier de la nappe d’eau souterraine se trouve sous le radier de

Depending on the ability of the class and the complexity and length of the story on the video, the teacher may wish to stop during the first viewing in order to aid comprehension

La présence de sauces et condiments à partir de lait de coco et d’eau de mer était omniprésente dans l’alimentation polynésienne (Beaglehole 1988, Wilson

Or, dire cela, qui est une évidence pour qui s’est un tout petit peu penché sur la fabrication des opinions dans les deux états-majors, c’est dire que l’enjeu ne se joue pas

Index Terms—Adaptive signal processing, covariance matrix estima- tion, robust estimation, Majorization-Minimization, Kronecker product, low rank..

Analogous results apply for two correlation-based dependence functionals introduced earlier: we show the kernel canonical correlation and the kernel generalised variance to be