UZHjpfister/papers/Pfister_02.pdf · 2015. 9. 24. · \ x tatax 1h4:x, 8 lw q* 9t (4 'v 7*1s4 'x...

39

Transcript of UZHjpfister/papers/Pfister_02.pdf · 2015. 9. 24. · \ x tatax 1h4:x, 8 lw q* 9t (4 'v 7*1s4 'x...

  • � �������������������� ��� �����������! " "$# %

    &('�) *,+.- /�0�120�3542+.' )6'.7819):=?:A@ +.1B3C7ED(- ) FHGJI

    )K7 : )L4NMPOL4B)Q0�&HR ST3VUW&�483VX�4

    Y[Z]\_^9`ba�\_cedf\_gKaih2c5jkZ]lm

  • ���������������������������������������������� �!��"��#�$�%"������'&�����(��'&%)+*�)+��&%��"��,���-�/. �0�0)�1����-23. �'4��#5��0����6���71���������8��"'�9�6�:����(;1����

  • ���������� n

  • ��������������������������� ]

    � ������� !�#" $ �&%('�)+*(�,"

  • ��� ��� � � ������� � � ����� � ������ `

    � ������������� �

    �! % *#" %$%& %('�"

    ')( \+* (-, W�.�XRW0/cV21-354�.�X , 1�V76 , XRW U51-1-8LXLW:9RTAXLW;4'XL\= ( XLW�.A@ V2B U539X , VAXDCE*�1�\F4'V7*�1-1#XLT=TAXG. ( \�X , /9XLU (*51H4N\+*�1-1 ( .#XLW�I , *53 , 6RWKW:V2351#V7J \�UK4:V7CW�LNMD1�I U , 4:V\ ( TAV=X , TO@ V2B U539X , VAX�CE*�1�\F4'V7*�1-1#XLT=TAX�I�U ,P, 8LWQ*51-1 UK1�\�XB%UK3�1-8+4'V2> ( XSRTCVUXW)Y[Z X�4 , XR\�X�B\B X�1H4 TAU]4Q*�B\*53 , UKI^ ( X�TO@ 8�TAXL\+4 , *�b X�1�\�8�I- ( 8LWOX:1 , 8�I�*51�W:Xv ( 1 W[4:V2B ( T ( W�.N*�1-1-8bREMDW YGZFL- XRWQW (~y X�4'W , 8�I�*51^.�X:1H4wI�U , TUgB%U�V213cU ( \= ( X ( 1#XKW ( V4'XD.�X \LU , 4:XRW.�V l 8 , X:1�4'XLW> ( X�1-* ( W�1-*5B\B X , *51�WP������"���FL

    �ED ]5LA]�^48���$��2=� ([�J��"��0)+�

    Y+T^W�@ UK3[V , Ut.N*�1�\s.#X , XL\�*51-1�U~24 , XJTAU \�TU9W'W:Xsv TU5> ( XLT=TAX UKI-I�U , 4'V=X:1H4 ( 1#Xs4 | \= ( X , *51�Wm> ( @ ( 1c4'X�TK4 , U/�U9V=T9W:V2B�VATU�V , X�U�.N8 y vP8�4Q8OX l XR\F4 ( 8wv Tf@ U�V?.�XD.�XOTf@ U�T235* , V74Q

  • ��� ��� � � ������� � � ����� � ������ a

    �!UK1�WO\�XPI , * y X+4r9K1-* ( Wk/5* ( T2*51�WZ\�TU9W'W:VJ�X , \= ( X XRW:W'U�V�V21^.�V2/8V2. ( X�TAT=X:B X�1H4:9c\�X)> ( V#XRW4D1�X+4[4'X�B X�1H4I#T ( W).�V z \�VAT=X�X:1 , U9VAWQ*51 .�X�TU> ( UK1H4:V74Q8g.#Xce ,Q( V4�.#U51�W TAXLW).-*51-1-8LXLW:L^`@ UKB\I#TAV74 ( .�Xg. ( W3V2351�U9T/X+4. ( e ,Q( V4PWQ*51H4�XRW4'V7B\8LW , XLWQI XR\F4'V7/[X�B X�1H4cv

    10µVX�4

    200µV9/\�X\> ( V`.N*�1-1#X ( 1 , UKI-I�* , 4!W:V2351�U9T

    W (-, e ,}( V4).�X�Tf@ * , . , X�.�X0.05 �Y+T XLW[4 X�1�\�* , XivbB X�1H4'V7*�1-1#X , > ( X TU \�TU9W'W3V7J \�UK4:V2*51�.�X�W3V2351�U (Nu M`Mwx XRW4P8�3cU�TAX�B X�1H4 ( 4:VATAVAWQ8�X

    .#U51�W ( 1 \+*�1�4'X u 4:X�.�V l 8 , X�1H4\.�X \�X�T ( V�.�XLWsId*54:X:1�4'V=XLTAWs8�/�*N> ( 8LW:L��!UK1�W�W[*�17U , 4:V\�TAX590U VAT=TUK1 5B\*51H4 , Xc> ( @ VAT XRW4PI�*[W'W3V2e#TAXg.�XP\�TU9W'W3V7J�X , .#XLWPM`MGx , 8LW ( T74'U51�4�W�.�X�4 | \=

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � �

    � �������������

    � " �,� $� '�� $ "'�� "

    ����� ���������! ����"�$#&% XI , V21�\�V2I�U�T�V71H4Q8 ,(' 4s.�X \�X\4 , U/�U9V=T�XLW[4n.A@ X u I#T7* , X ,c( 1#Xt1^* ( /[X�TAT=Xb4:XR\= ( X.�X \�TAU[W:W:VJ�\�U�b

    4:V2*51{.�XRWPM`MwxbL h X�TAT=X�b \�VI , X�1^. X:1 \�*5B\IN4'X�TAU%W[4 ,}( \F4 (-, XPW[I�UK4:V2*�bf4:X:BbI�* , X�TATAX�.#XLW).N*�1-1-8LXLW�M`Mwx\+*�1H4 , U�V , X:B�X:1H4DvPTf@ U9T73�* , V4} ( V/\+*51 W3V?.N6 , XNTAXLWG.N*51^1-8�XRWw.�X�CU *51�W[4'UK4:V?> ( X5Lj)* ( WD/�* ( bT2*51�W\.N*�1�\/�*9V , W:VN\�XRW , 8RW3XRU (Nu .�X 1#X (-, *�1#XLWb.�V l�( W3V7CW%W[*�1H4 U5I-I#TAVA\LUKe#TAXLW�X+4%X z \LU9\�XLWsId* (-, TAU\�UK4Q8:35* , VW:UK4:V2*51.�X�W:V73�1�U (-u M`MwxbL

    a* ( 4:XP\�X+4Q4:X U5I-I , *8\= ( @ ( 1�\= ( V U{I�* (^, X l X+4�TAU.�VATAUK4'UK4:V2*51 .A@ ( 1 W:V2351�U9T�I U ,P, UKI-I�* , 4)v ( 1 U ( 4 , X5L

    �ED `NLA]c*)���2A"�) ����7����) �:�J2=�0���:5�����"��94?(�� + �8�0� ,'�%��) 2=����"���) �:��� �%�L(.-�2A�0�0/+����) )+��1�29���:� ����) ) ���:5��(9�43V6587

    �UK1 W X�1H4 , X , .�UK1�W TAXLW�.N8+4�U�VATAW:9N1-* ( WPId* ( /5*�1�WP.�V , Xg> ( X�TAXLW , 8RW3XRU (Nu .#V lm( W:VCWNWQ*51H4 , XLWQI XR\F4'V b/9X:B X�1H4 ( 1�Xi3�8�1-8 , U9T=VW'U~4:V2*51t.#XLWwJ�T74 , XLW�.�X�9�U�T2B U519j.�X TAUsB%U9\=

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � �

    .N8�4:X , B V71�VAW[4:XR\5L C�J^3 (-, XcNNZ+L

    ��� � ����%��� ������� ����������

    U\.Nj1�U5B�V?> ( Xc.�XLW , 8LW:XLU (Nu .�V l�( W3V7CW XLW[4 V71 W[I#V , 8�X�I�U ,�( 1 B\*N.N6�TAXc.�X��)*�INJ�XLT2.59�U ( > ( X�T

    XLW[4KU y * ( 4}8. ( e ,}( V4rLN`@ U , \= ( X TAXBb*N.N6LT=X%WQ*9V74P\+*5B\I#TAX+4r9/V=T`CU ( 4PX:1�\+* , X\.N8L\ , V , XJTAXbCE*51�\+4:V2*51-1#X:B X�1H4n.�X%\= ( X

    1#X (-, *�1#XZV71d.�V7/8V?. ( XLT=TAX�B X�1H4rL h XLT ( V=b+\�V�XRW4 V21�WQI#V , 8kI�U ,A( 1#X`B\*N.N8LT=VW:UK4:V2*51�8�TAXL\+4 , V2> ( X`.@ ( 1c1�X (-, *51#Xe#V2*9T2*53[V2> ( X\R\5L CLNJd3 (-, XnNjZFL

    GEw6`NL an��/�'��5��%" 5�2=����) �0�1�����( . �����6�����'�����j*9V74

    xi9#TAXcId*54:X:1�4'V=XLTB X�Bse , U51�U�V , X�.#X�TAU

    i ����� .�X�1d. , V74:X�9#X+4 vi = ϕ−1(xi) 9�TAX�\=

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � _

    Cj

    .vj(t).

    t+

    vj(t)

    Rj=

    N∑

    i=1

    Wjiϕ(vi(t)) + IjR `NLA]rZ

    Y�* (-, W:V7B\I#TAV7J�X , > ( XLT2> ( X\I�X ( T=XB\*N.N6LT=X�9A1-* ( W�I�* ( /5*51 W�Id*cW3X ,Ij = 0

    9Cj = C = 1 ∀j

    9\�Xs> ( V;1-* ( W)I�X , B X+4�.A@ X u I , V2B X , T=Xn4:X�B\I�W.�XnB\8:Bb*[V , XgI�U ,

    τj = RjCL� XnB\*N.N6�TAXn> ( Xn1^* ( W

    U�TAT2*51�W ( 4'V=TAVAW:X , \+*51 \�X , 1#XKT=XRWO\= ( X�1-* ( W , X�B\I#TU9\�X , *51�W

    I�U , TU\1-*54'U~4'V7*�1xjL ' V21�W:VO9�TAU\.Nj1�U5B�V?> ( Xc. ( , 8RW3XRU ( .�X��)*�INJ�X�T?. XLW[4).N8L\ , V74:XcI�U ,

    ẋ(t) = µ(x(t),λ)R `-L `�Z

    * � TAU.-8 , V7/[X!W:@ X u I , V2B X�I�U ,

    µj(x(t),λ) = −xj(t)

    τj+

    N∑

    i=1

    ϕ(xi(t))Wij ∀j = 1, . . . ,NR `NL a�Z

    X+4λ = {W,γ},γi = 1τi

    , X:I , 8LW:X�1H4:X�TO@ X�1 W3X:Bne#TAXb.#XLW�I�U , U5Bb6�4 , XLW�. ( B\*j.-6�TAX5L��)1 X u X:BbI�T=X.�XiCE*51 \F4:V2*51{.A@ U9\+4:V2/�U~4:V2*51

    ϕ(x)XLW[4�.N*51-1^8iI U ,

    ϕ(x) = −1 + 21 + e−θx

    R `NL �HZ)* ( Wc1-*54:X , *51 W

    µo ≡ (µ1, . . . ,µD)X+4µh ≡ (µD+1, . . . ,µN )

    9 , XLWQI XR\F4'V7/[X�B X�1H4PTAXLWcI�U , 4'V=X*5e W3X , /�UKe�T=XNX�4Q\LU9\= ( X5LjY+T6XLW[4 V2BbI�* , 4'U51H4�.�Xi1-*54:X , > ( X!T=Xe ,Q( V4�1@ XLW[4GI�U[WQU y * ( 4Q8vTU�WQ*9T ( 4'V7*�1�.�X�Tf@ 8:> ( U~4'V7*�1pRNjZF9-B%U9VAW�e#VAX�1{v�TUb.-H1 UKB V2> ( Xg.#X \=< U5> ( X�1#X (^, *�1#X��q

    dXλ(t)

    dt= µ(Xλ(t),λ) + σ

    dB(t)

    dt

    R `NL � Z* �

    σXLW[4 ( 1#X \+*�1�W4�UK1H4:X X+4^\�U , U[\F4Q8 , VW:XOTO@ UKB\I#TAV74 ( .�XP. ( e ,}( V74:L~)* ( W01-*K4'X , *51�W0> ( X TU�CE* , B ( b

    TU~4:V2*51dB(t)/dt

    XLW[4 ( 1 UKe ( W).�Xc1-*54'UK4:V2*51 I ( VWQ> ( X�TAXcB\* ( /[X�B X�1H4Pe , *�P1#VAX�1B(t)

    9�> ( Xc1-* ( W

    ���������������������

    Xλ(t) ��� ��� ����� � � �"! ��#�$&%'���(� $ � $&)*#,+ ��-.� �/� + ��0�� � ! � ��1 + #2� �3� ��% ! - �&4 ���2��������� 1 #5� � % !2�76 %8��� �9 � � � 1 + #�� ! $��:� �;��� # �/� ���*� �� � ����)*# �

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � �

    .N8�J^1#V , *�1�W�.#UK1�WZTAX�I , *8\=< U�V21�\= ( U~4:V2*51RfNNZ+L�Y0* (-, > ( X�TUJWQ*9T ( 4'V7*�1I ( VW'W3X ' 4 , X�.N8+4'X , B V71-8LX598VATmCU ( 4X�1 \+* , X!V7B\I�*[W:X ,P( 1#XP\�*51^.#V4'V7*�1 V21#V74:VU�TAXg> ( Xi1^* ( W�1-*54:X , *�1�W

    νL

    Xλ(t) = Xλ(0) +

    ∫ t

    0

    µ(Xλ(s),λ)ds + σB(t)R `-L _HZ

    Xλ(0) ∼ ν)* ( W1-*54:X , *51 W

    Oλ(t)X�4Hλ(t)

    9 , XLWQI�XL\F4'V7/[X�B X�1H4 TAXLWI U , 4:VAXLW*�e�W3X , /�U5e#T=X X�4 \�U[\= ( U~4:V2*51 WcRfNjZKX+4nRNjZFL

    � � �"!$#&%'#�()!=5 �����'�L�����'�0�8���0)+�L���#"���) �1����B = {B(t),t ∈ [0,∞)} ���0)K��� �%���������%��� )N��:�8: 4������-��"�� �:"��9�����0) - 2=" � ���0���:�

    P��)�������) �:5����+?,����2A�0��1���"�) �:�������6(���) ������ �0������"�� ) ��� ������)

    ��"�) �� ,�"���)+�0��C@ C

    P (B(0) = 0) = 1C

    A C ∀t ∈ [0,∞),∆t > 0,B(t+∆t)−B(t) ���0) ���6� ��"��0�"9�2=��"�2=5�"�)+�����:��2c"��8���0���� ���^�#"�� �:"'�[�#���0)-P(9� � �879��� ���

    0��) (9� ��"��0�"��6���

    ∆tC

    B C ∀tk,0 ≤ t0 < t1 < · · · < tl < ∞&N2=��� ���6���'5��%��� )� ��:�;:^� ������

    B(tk) − B(tk−1),∀k =1, . . . ,l

    ������) (9������"��0�"9�2=����"�2=5�"�) �����:�0�P���6(95+�#����(["�� ) ���K�#"��P�:"'�[�#���0) -PC

    C CB(t)

    �0�0)K������) ������C5 � �%���������%����)���:�;:^� ���� -

    N(����%�����0����#�������#�0��0)+�P���

    N�%���������%��� )� ��:�8:^������#�!����(95+����� 4(9"���) ��C

    � � �"!$#&%'#�()!EDF, "��%�0�0���:��(9�^���:�['"[���2 ��) 5Qλ : F → [0,1] ���0) 2="P� ���0���:�!(9�^���:�['"[���2 ��) 5!���6(�����)+���"���2=� ���:�R���0���0�8� /VN 7

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � 9

    Qλ(A), X:I , 8RW3X:1H4:X TUcI , *5e�U5e#V=TAV74Q8)> ( XNTAX , 8LW:XLU ( 3�8�1-6 , X ( 1 \=��) 5P�0��� (Ω,F) ���6(�����)+�0���#"�� 2=�0��%���������%��� )�P��:�;:^� ������P������) �:5������

    uC

    Y^U , TU%W ( V74:X�9N1-* ( WP1-*K4'X , *51�WP9#TUbB XLW (-, Xc.�XcI , *�e�UKe#VATAV4}8!V21^. ( V4'XgI�U , TU\e�U9W:Xc.�Xg.N*�1-1-8�X

    .A@ X�1H4 , UT1#X:B�X:1H4:LA)*54 , Xbe ( 4 W3X , U�.�X\4 , * ( /9X ,λ4'X�T`> ( X

    QλoU5I-I , * u V2B XJU ( B VAX (Nu TU�B XLW (-, X

    .�X�I , *5e�U5e#VAT=V74Q8PL

    ��������� ����� � l�� Z��Z�2� l ce\_^������U51�WK\�X�I U , UK3 , UKI^ ( X , X:1.N8�4'U�VATWk4Q* ( 4:XRW�TAXLWDe�U9W:XLWD1-8L\�XLW'W:U9V , XLW

    v TU�\�*5B\I , 8� ( UK1H4 v \�X]4}

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � �

    exp

    [n∑

    i=1

    µiZi −1

    2

    n∑

    i=1

    µ2i

    ]P [Z1 ∈ dz1, . . . ,Zn ∈ dzn]

    = exp

    [n∑

    i=1

    µiZi −1

    2

    n∑

    i=1

    µ2i

    ](2π)−n/2 exp

    [−1

    2

    n∑

    i=1

    z2i

    ]dz1 . . . dzn

    = (2π)−n/2 exp

    [−1

    2

    n∑

    i=1

    (zi − µi)2]

    dz1 . . . dznR `NLA]La�Z

    9 ���#�0�&c�#"��Q�:"'�[�#���0) - 2A" � ���0���:�P̃&/2=���K��"��0�"[�2A�0�$"�2A5'"�) �����:���

    Z1, . . . ,Zn������) ���6(95+�#����(["���)+���

    ��)N�����0� "�2=�0� "������Ẽ(Zi) = µi

    ��)Ẽ[(Zi − µi)2

    ]= 1

    C 9 ��) �:��� ����)$(���)&Q�0�Q2=�0����"��0�"9�2=��� "�2=5�" 4)+�����:���Z̃i = Zi − µi

    (95 1O� ��0�J�����(Ω,F ,P̃ ) ������) ���6(95+�#����(["�� ) ��� ��)$(9� (���0) �0����) ����,�����0� "�2=�0&"�2A������2="�� ���0���:� ([�Q�6�:�9�"9���2>��)+5

    P̃�0�0) (95 1O���� ��"�� /VN 79C

    X�4Q

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � ]*�

    Lλo (xo) =

    Ωh

    Lλ(xo,xh)dRh

    dSλ,xoh(xh)dS

    λ,xoh (xh)

    =

    Ωh

    exp{ 1

    σ2

    ∫ T

    0

    µo(x(t),λ) · dxo(t)

    − 12σ2

    ∫ T

    0

    ‖µo(x(t),λ)‖2dt}

    dSλ,xoh (xh)R `NLA] � Z

    Y�* (-, UKI-I , * u V7B X , TAU.�VAW[4 , V2e ( 4:V2*51Sλ,xoh

    9�VAT W (Nz 4i.�Xn358�1^8 , X ,M\= ( X�TAX

    m������ \= ( X TAX , 8LW:XLU ( W[*[V4*5I-4:V2B U9TOLY+T#W:@ U539V74`.�XP.N8+J^1�V ,�( 1#XGCE*�1�\F4'V7*�1b> ( @ V=TjCU ( . , U�B%U u V2B VAW:X , I�* (-, U5I-I , * u V7B X , U ( B V=X (NuTAXLW \=

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � ]9]

    ∂ log Lλ(x)

    ∂Wij=

    1

    σ2

    ∫ T

    0

    ∂µ(x(t),λ)

    ∂Wij(dx(t)− µ(x(t),λ)dt)

    =1

    σ2

    (bji(x)−

    N∑

    n=1

    Wjnain(x)

    ) R `NL `9`5ZU/9XR\

    bij(x) =

    ∫ T

    0

    ϕ(xi(t))dxj (t) +1

    τj

    ∫ T

    0

    ϕ(xi(t))xj(t)dtR `-L `9a�Z

    aij(x) =

    ∫ T

    0

    ϕ(xi(t))ϕ(xj (t))dtR `-L ` �HZ

    X+4

    ∂ log Lλ(x)

    ∂γi=

    1

    σ2

    ∫ T

    0

    ∂µ(x(t),λ)

    ∂γi(dx(t)− µ(x(t),λ)dt)

    =1

    σ2

    −γi

    ∫ T

    0

    xi(t)2dt +

    N∑

    j=1

    ∫ T

    0

    ϕ(xj(t))Wjixi(t)dt

    −∫ T

    0

    xi(t)dxi(t)

    ) R `NL ` � Z

    Y�* (-, Id* ( /5*[V ,w( 4:VATAVAW:X , TAXLW , 8RW ( T74'UK4'WiRfNjZ�X+4cRfNjZ+9cVAT�CU ( 4�I�* ( /�*9V , X u I , V2B�X , ∇λ log L̂λo (xo)X�1�CE*51 \F4:V2*51{.�X ∇λ log Lλ(x) � q

    ∇λ log L̂λo (xso) =M∑

    m=1

    pλh|o(hm,s|xso)∇λ log Lλ(xso,hm,s) ∀hm,s ∈ H̃

    R `NL `�_�ZU/9XR\

    pλh|o(hm,s|xso) =

    pλ(xso,hm,s)

    Lo(xso)

    R `NL ` � Z Xc3 , U�.�V=X:1H4�.�X�TUsCE*51�\+4:V2*51 .�Xc/ , U9VAW:X�Bse#TUK1�\�Xi.�X:/8V=X:1�4rq

    ∇λφ(λ) =1

    S

    S∑

    s=1

    M∑

    m=1

    pλh|o(hm,s|xso)∇λ log Lλ(xso,hm,s)

    R `NL `89�Z

    ' /[XL\P\�X , 8LW ( T74'U~4r9^1-* ( W)I�* ( /5*51 W X u I , V7B X , TUt.N8 , V2/58�Xs.�Xφ(λ)

    I�U ,�, UKI-I�* , 4�U ( Id*[V2.#WWL

    j*[V4λ∗ = (W ∗,τ)

    4'X�T> ( X

    b̃λ∗ − ãλ∗W ∗ = 0 R `-L ` ��Z

    * ��2�� ��1 ������- 1�� � � �7��-���#�-��0�#�$ � 1 ��� � - + ���2� � � ����� �

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � ]R`

    b̃λ∗

    =

    S∑

    s=1

    M∑

    m=1

    pλ∗

    h|o(hm,s|xso)b(xso,hm,s)

    R `NL a ��Z

    ãλ∗

    =S∑

    s=1

    M∑

    m=1

    pλ∗

    h|o(hm,s|xso)a(xso,hm,s)

    R `NL a#]rZ

    WP8LWQ* ( . , X�TO@ 8:> ( UK4:V2*51 RfNjZPXLW[4 ( 1oI , *�e#T76:B Xt1-*51S4 , V2/8VAU9Twv , 8LWQ* ( . , X�9 \�U , VAT W:@ U539V74s.A@ ( 1#X8:> ( UK4:V2*51 1-*51#b TAV71^8LU�V , X X:1

    W ∗LOY^U , U9V=TATAX (-, W�9kB ' B�X?W:VNTUCE*51�\+4:V2*51 .A@ U9\F4'V7/�UK4:V2*51

    ϕ(x)8�4'U�V74

    TAV71-8RU�V , X�9�\�X+4Q4:Xc8r> ( U~4:V2*51 , XLW[4:X , U�V74P1-*51#b TAV71^8LU�V , X5L��������� � g�� � l � j � Z����Y�* (-,w, 8RW[* ( . , XKTf@ 8:> ( U~4'V7*�1XRfNjZ+9�1-* ( W ( 4:VAT=VW3X , *�1�W^Tf@ U9T73�* , V4}

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � ]La

    ∇W Ĝ(λ,λ(k−1)) =1

    σ2

    (b̃λ

    (k−1) − ãλ(k−1)W)

    = 0R `-L a[_�Z

    X+4$U9V71�W:V/X u I , V2B�X ,W (k)

    X�1�CE*51 \F4:V2*51 .�XW (k−1)

    W (k) =(ãλ

    (k−1))−1

    b̃λ(k−1) R `NL a � Z

    Y�* (-, \�Xg> ( V XLW[4.�XRWN\+*�1�W[4'UK1H4'XLWP.�X�4'X�B\I�W:9�Tf@ ( 4'V=TAVW:UK4:V2*51{.�X�Tf@ 8r> ( U~4:V2*51!RNNZGI X , B X�4.A@ *5e�b4:X:1#V , q

    ∇γiĜ(λ,λ(k−1)) =1

    S

    S∑

    s=1

    M∑

    m=1

    pλ(k−1)

    h|o (hm,s|xso)∇γi log Lλ(xso,hm,s)

    =1

    S

    S∑

    s=1

    M∑

    m=1

    pλh|o(hm,s|xso)

    1

    σ2

    (−γi

    ∫ T

    0

    xm,si (t)2dt

    R `-L a 9�Z

    +

    N∑

    j=1

    ∫ T

    0

    ϕ(xm,sj (t))Wjixm,si (t)dt −

    ∫ T

    0

    xm,si (t)dxm,si (t)

    = 0

    X+4�.N*�1�\

    γ(k)i =

    S,M∑

    s,m=1

    pλ(k−1)

    h|o (hm,s|xso)

    N∑

    j=1

    ∫ T

    0

    ϕ(xm,sj (t))W(k−1)ji x

    m,si (t)dt−

    ∫ T

    0

    xm,si (t)dxm,si (t)

    S∑

    s=1

    M∑

    m=1

    pλ(k−1)

    h|o (hm,s|xso)

    ∫ T

    0

    xi(t)2dt

    * �xm,s(t) = (xs(t),hm,s(t))

    LQY�* (-, *5eN4'X�1#V , TAX]I U , UKB\6+4 , Xλ∗

    *5IN4'V7B%U9TO9 V=TCU ( 4 V4}8 , X ,W ( \�\�XLW'W3V2/9X:B�X:1H4KTAXLW�8�4'UKI�XLWP]!X�4�`b.�X TO@ U�T235* , V74Q ( @ v \�Xg> ( X

    λ\+*51j/[X , 39X�/[X , W

    λ∗L

    ��������� � Z�d���� j.c �$��g ���U51�W T=X)I U , UK3 , UKI^ ( X TAX , 8LW:XLU ( 3�8�1-6 , X TAUe�U9W:X .A@ X:1�b

    4 , UT1#X:B�X:1H4 E LDY+T , XLW[4:X X:1�\+* , X T=X I�U , U5B\6+4 , X σ vJ u X , q^\�X�T ( V�> ( VN\�*51H4 , �9TAX�Tf@ UKB\I#TAV4 ( .�XX. (e ,}( V74:L8V�TAXe ,}( V74 XLW[4c4 , *5IpCE* , 4:9 T=X , 8LW:XLU ( I�X ( 4n358:1-8 , X , .�XRW \+* (-, e XRWiCE* , 4:X�B X:1�4n.#V l 8 , X�1H4:XRWg.�X

    \�XLT=TAXLWG> ( X�1^* ( Ww/5* ( . , V7*�1�W�U5I-I , * u V2B X , L�8Vf9�U ( \�*51H4 , U�V , X�99VAT XRW4G4 , *5ItCU�V2e#T=X�9[VAT/W3XI X ( 4P> ( X�TAXLW, 8LW:XLU ( 1#XI�U , /8V=X:1-1#X)I U9WDvPX l XL\F4 ( X , .�XRW`8L\LU , 4'W W (Nz W'UKB\B X�1H4D3 , U51^.#WkI�* (-, Id* ( /5*[V , W ( V7/ , XNTAU.Nj1�U5B�V?> ( X5LNY+T W�@ UK3[V4P.N*51 \c.�Xi4 , * ( /[X ,�( 1

    σ*�IN4:V2B%U�Tfq-1#VA4 , *�I�8LT=X:/58�9N1#Vm4 , *5I�CU�V2e#TAX5L

    U�B\8+4} ( V9XRW4 TAU�4'X�B\I�8 , U~4 (^, X�L ��U51�W1-*54 , X!\LU9W:9N1-* ( WGI�* ( /�*51�WGCU9V , X!TO@ UK1�U9T7*�39VAX�X:1H4 , X�TUsCE*51�\+4:V2*51

    φ(λ)X�4 TAUsCE*51 \F4:V2*51 .A@ 8�1#X , 3[V=X�9�\�X

    > ( XiCU�V74�.�X�Tf@ U5BbI�T=V74 ( .#Xg. ( e ,Q( V4σT=XP\�* ,}, XRW[I�*51^.�UK1H4G.�X�TUs4:X:BbI�8 , U~4 (-, X5L

    `@ V2.-8�X�XRW4 TAU$W ( V2/�UK1H4:X�q:I�* (^, 4 , * ( /9X , T=XDB U u V7B ( B�39T2*5e�U9T�.#U51�W ( 1 XLWQI�U9\�XD.�XDI U , UKB\6+4 , XRW�9�VATCU ( 4DId* ( /5*[V , X l XR\F4 ( X , I , X:B�V26 , X�B X:1�4D.#X)3 , UK1^.�Wk.N8�I#TU9\�X�B X�1H4�WkId* (-, .N8�4:X , B V21#X , TUcB X�VATAT=X (-, X, 8�39V2*519kI ( VW , XRW4 , XLV71d. , X Tf@ 8�4:X:1^. ( X].#X TAU , XL\=

  • ��� ��� � � ��� � � ��� � ���\�� ������i�� ��� � ] �

    .�Xσ> ( V0.#V7B V21 ( XJX:1XCE*51 \F4:V2*51. ( 1-*�Bne , XLW.A@ V4}8 , U~4:V2*51 W�L YZU , TU�W ( V74:X�9�1^* ( W ( 4:VATAVAW:X , *51�W ( 1#X

    W:V7B\I#TAX�W ( V74:Xc3�8�*�Bb8�4 , V2> ( X�.N*�1-1-8�XcI U , q

    σ(k) = ασ(k−1), α ≤ 1 R `-L a+��Z' \= ( X V74Q8 , UK4:V2*51S.�X%TO@ U�T235* , V74Q ( VA.N8:I X:1^.t.�Xλ(k−1)

    X�4σ(k−1)

    L �n@ U ( 4 , XRWD/�U , VUK1H4:XRWOWQ*51H4wI�*[W'W3V2e#TAXLW:q[VAT�XLW[4G8:3[U�TAX�bB X�1H4NX�1j/8VW:U539XLU5e#TAX�.@ V74Q8 , X ,P( 1?\�X , 4�U�V21�1^*5Bne , Xc.�XcCE*[VAWNTO@ U�T235* , V74Q ( XLT=TAX%TU{/ , U9VAW:X�Bse#TAU51�\�X�XLW[4nB%U u V2B%U�TAX5LM`1 .A@ U ( 4 , XLWc4:X , B XLW:9 W:V

    λ∗c, X�I , 8LW:X�1H4:X Tf@ X:1�b

    W:X�Bne�T=Xi.�XLWGI�U , U5Bb6�4 , XLWD*5IN4'V7B VW[8RWDI�* (-, TUJ\�TAU[W:W:Xc9cTAX1-* ( /9XRU ( \=

  • ��� ��� � � ��� � ����;�`�������� � � �P��� ��� ��� ��� ] �

    � ������������� �

    � $���� " * % � � & % " * � & ' �

    Y ( VWQ> ( X Tf@ U�T235* , V74Q ( X�9cVAT�CU ( 4 \+*�B\I , X:1^. , X$TAXLWG/9XR\0b4:X (-, W

    xs ∈ RTE 9�\+*5B\B X TO@ X�1 W3X:Bne#TAXi.#XLW�I�*K4:X:1H4:VAX�TW�Id* (-,�( 1 XRW:W'U�VA.N*�1-1-8�L�YOT ( W�I , 8R\�VW[8:B X�1H4:9

    xs(t−1)E+e, X:I , 8RW3X:1H4:X%TAXI�*K4:X:1H4:VAX�TQU ( 4'X�B\I�W

    t.�X�Tf@ 8LT=XR\F4 , *N.�X

    e.�X TO@ XLW'W:U9V

    s9;4'UK1d.�VAWs> ( X

    E\+* ,Q, XLWQI�*51^. U ( 1-*5Bse , X�.A@ 8�TAXL\+4 , *j.#XLWQX+4TU ( 1-*�Bne , Xc.�X�I�U9W).#X4'X�B\I�W:L

    `@ V2.-8�Xg.�XRWjiU XRW4i.�X�4 , * ( /[X ,P( 1

  • ��� ��� � � ��� � ����;�`�������� � � �P��� ��� ��� ��� ]L_

    *5I-4:V2B U9T^.�X$WQ8�I U , U~4:V2*51LK)* ( WkId* ( /5*�1�W , X�B%U , > ( X , > ( XNTU�B%U , 39XρR \KL CLKJ^3 (-, XNjZ^XLW[4w.N*51^1-8�X

    I�U , Tf@ X u I , XLW'W3V2*51 W ( V7/�UK1H4'X5q

    ρ =2

    ‖w0‖Ra-L `5Z

    8V01-* ( W�/5* ( T7*�1�W�4 , * ( /9X , TO@ ( VmI X ( 4 ' 4 , X�CU9V4QX�1B�V21#V2B VAW'UK1H4

    TUtCE*�1�\F4'V7*�1.�X \+* � 4φ(w) = 12w

    TwW[* ( W$TU \+*51H4 , U9V71H4:X�RfNjZ+L�UK1�W!X:1�4 , X , .#UK1�W�TAXLWc.N8�4'U�VATW�9

    1-* ( WPI�* ( /�*51�WPB X:1�4'V7*�1-1#X , > ( XP\�XgI , *�e#T76:B XcI X ( 4 ' 4 , X , 8RW[*[T ( U ( B\*~9X:1�.#XPTAU\B\8�4Q ( X�TAX�I�U , UKB\6�4 , X

    b0q

    w0 =

    S∑

    s=1

    α0,sdsψ(xs)

    Ra-L �HZ

    b0 = 1−wT0 x ��� Ra-L � Z UsCE*51 \F4:V2*51{ UK3 , UK1^39VAX�1-1#Xi.�X�/8VAX�1H4NU�T2* , W�q

    Q(α) =

    S∑

    s=1

    αs −1

    2

    S∑

    s=1

    S∑

    s′=1

    αsαs′dsds

    ψ(xs)Tψ(xs′

    )︸ ︷︷ ︸=K(xs,xs′ )

    Ra-L _�Z

    YOT ( 4��54;> ( XD.�XO\�*51�W:V2.-8 , X , TAXkI , *N. ( V4ψ(xn)

    Tψ(xs′

    )9+1-* ( W ( 4'V=TAVAW:X , *51 W ( 1c1-*~cU (

    K(xs,xs′

    )L

    XRW�1-*~[U (Nu TAXLW�I#T ( WN\�* (-, UKB\B X�1H4 ( 4:VAT=VWQ8LWNWQ*51H4 T=XRW W ( V7/�UK1H4'XLW:q

    • TAV21-8LU9V , X5q K � (xs,xs′) = xsT xs′• I�*9T2j1-*5B VU�TAX5q K � (xs,xs′ ) = (xsT xs′ + c)d• 3cU ( W:VAX�1-1#X�q K � (xs,xs′) = exp(− 12σ2 ‖xs − xs′‖2

    )

    )* ( WAI�* ( /�*51�W U9VAWQ8�B X:1�4A3�8�1-8 , U9T=VW3X , T=X`I , *�e#T76:B X^X�1 ( 4:VATAVAW'UK1H4M\�TU9W'W3XRWkR

    ds ∈ {1, . . . ,M} ZFL�!UK1�W ( 1�I , *�e#T76:B XgvM\�TAU[W:W:XLW:9N1-* ( WNU/5*�1�W

    M

  • ��� ��� � � ��� � ����;�`�������� � � �P��� ��� ��� ��� ] �

    e

    e e

    hhhhhhhhhhh

    �����������

    AAAAAAAAAA

    ((((�� � ]

    �� � `�� �Ba CE*�1�\F4'V7*�1{.�X.#VAW'\ , V7B V21�U~4'V7*�1

    �Ew a^L `g MZ2="��������P�J��2>) � ��2A�0�

  • ��� ��� � � ��� � � ����� � �� \��� ���� ]'9

    � ������������� �

    � ��" � !� " % � �#"

    � ��� � ��% � ��� � � �"� # � ��� � � � ��� ����# � �h *5B\B X�1 *51�W�I U , .N8R\ , V , X TO@ X u I�8 , V=X:1�\�Xg> ( V UbCE* (-, 1#V/TAXLW).N*�1-1-8LXLW ( 4'V=TAVW[8LXLWPI�* (-, TAU%\�TAU[W:W:V b

    J \LU~4:V2*51L� ( U~4 , XJW (~y X�4'W!X�1]ed*�1-1#XJW'UK1H4}8g*51H48�4Q8n4:XLW[4Q8RW�LmY+T T=X (-, U8+4}8s.�X:B U51^.N8b.�XnJ u X ,�( 1�XJ\ , *9V u

    \�X:1H4 , 8�X%W (^,c( 1S8L\ , UK1 X+4b.�X , 8:Id*�1^. , X%U ( W'W3Vk/8V74:Xt> ( XI�*[W'W3V2e#TAXv ( 1,W4'V7B ( T ( Wc/8VAW ( X�T^TU~4}8 , U�TREI�*9V21H4k1-*9V , .�XQTAU , 3[X (-, \+* ,Q, XLWQI�*51^.#U51H4;v ( 1�U51-39TAX�.�X

    0.50Z U ( B\*~9X:1s.A@ ( 1#X U[\F4:V2*51bI , 8:.N8�J^1#VAX5L')( \�* (-, W�.�XLWPW:XLW'W:V7*�1�Wc.A@ X�1 , X�3[VAW[4 , X�B X�1H4r9/T=XRW�W4'V7B ( T=VKW[*�1H4�U5I-I�U ,Q( Wi4}* ( 4'XLW�TAXLW � v _ W:X�b

    \+*�1^.�XRWn.�XCU *�1 U�T28LUK4Q*9V , Xt.#UK1�W T=X \=

  • ��� ��� � � ��� � � ����� � �� \��� ���� ]*�

    � ��� ������ � ��� � # �� �" &�'� ��#&%��"� #0 ��"� ��� ��� �>� � �`@ V2.-8LU�T W3X , U9V4c.�XI�* ( /�*9V , X l XL\F4 ( X , TU \�TU9W'W:VJ \LU~4'V7*�1.�V , XR\F4'X�B X�1H4PW (^, T=XRWc.N*51^1-8�XRWi4:X�TATAXLW

    > ( Xg1-* ( W)/[X�1-*�1�W).�XJT=XRW)1-* , B%U9T=VW3X , L^M`1�\�Xs> ( V \�*51�\�X , 1#X�T=XRW , 8LW:XLU (Nu .#V lm( W:VCW:9�\�X�TU%XLW[4)4}* ( 4W:V7B\I#TAX�B X�1H4%V2BbI�*[W'W:V7e#TAX . � U ( 4'X�B\I�W\.�X \�U9TA\ ( T X u 4 ,(' B�X:B X�1H4�V2B\Id* , 4�UK1H4:L�M`1;X l X�4 Tf@ U�T235*�b, V4} ( VZW:V73�1#VJ Xs> ( @ V=TZXLW[4�V2B\Id*cW:W:V2e#T=Xb.�XsCU�V , X \�* ,}, XRW[I�*51d. , X ( 1]1#X (-, *�1#Xsv

    \= ( X8LT=XR\F4 , *N.�X5LY�* (-,, 8:. ( V , X T=X 4'X�B\I�Wt.�X?\LU�T\ ( Tf9OVATPCU ( . , U9V4I�* ( /�*9V ,\, 8r. ( V , X TAX{1-*5Bse , X{.A@ 8�TAXL\+4 , *N.�XLW( 4'V=TAVAWQ8�XRWgI�U ,n( 1oCU9\F4'X (-,

    10L;U?U9VAWJ\+*5B\B Xt1-* ( Ws1#X�W'U/5*51 WcI�U9Ws> ( X�TAT=XRWJWQ*51H4 TAXLWn8�TAXL\+4 , *j.#XLW

    > ( V�\�*51H4:VAX�1-1�X�1H4OT=XPI�T ( Ww.A@ V71NCE* , B%UK4:V2*51\> ( UK1H4Dv TAU�\�TAU[W:W:V7J \�UK4:V2*519~1-* ( WOU/�*51�W�.N8R\�V?.N8).A@ ( 4:VAT=VW:X ,( 1#X Y h ' /��Q�0������� �#"�2�MO���$�#��������) 9 ��"�2 7��0��87sId* (-,P, 8r. ( V , X TU\.�V7B X:1�W3V2*51 U�TAV4}8�.@ X:1�4 , 8LX5L

    j*9V74C9�TAUB%U~4 , V\�X�.�XQ\+*~/�U , VAU51�\�Xw.�XG4Q* ( 4:XLW TAXLW^\LU , 4:XLW0.�XKTAUie�U[W3XG.A@ X�1H4 , UT1#X�B X:1�4 E .-8+J^1#VAX\+*�B\B�X�W ( V74:q

    C =

    T̄∑

    t=1

    V (t)V (t)TR �^L a�Z

    * �V (t)

    XLW[4 ( 1 B%U~4 , V\�XE × S .N*�1�4NTf@ 8�T28�B X�1H4 Ves(t) = xse(t) , X:I , 8LW:X�1H4:X TAXcId*54:X�1H4'V=XLT.�XTf@ 8LT=XR\F4 , *N.�X

    e95. ( W (~y X+4

    sX�4 U ( 4:X:B\I�W

    tL�N*9VAX�1H4

    Γ1, . . . ,ΓP99T=XRW

    P/[XL\+4:X (^, W�I , *5I , XRW0.#X

    CU[W:WQ*�b

    \�V28LW$U (NuP/�U�TAX (^, W�I , *5I , XRWQT=XRWPI#T ( WP3 , UK1d.�XLW:L h XLWP/[XL\+4:X (^, W�I , *5I , XRW W[*�1�4$W:X�1 W[8RW , X:I , 8RW3X:1H4:X ,

    TAXLW�W[4 ,}( \+4 (-, XLWOWQI�UK4:VU�TAXLW)R U ( W:X�1�WD.�XLW \�U , 4:XRW}Z`.N*5B V21�UK1H4:XRW�L5Y+T XRW4 U9V71 W3VdId*cW:W:V7e�T=X�.#X�.N8+Jd1#V ,w( 1#X1-* ( /9XLT=TAXne�U[W3Xb.�X\.N*51-1^8�X\.A@ X:1H4 , UT1#X�B X�1H4 Ẽ X�4�.�Xs4:XRW4 T̃ .N*51H4 T=XRW!\�*5B\I�*[W'UK1H4:XRW�.�X \= ( X8�T28�B X:1�4NWQ*51H4).N*51^1-8�XRW�I�U , TAXLW�I , * y XR\F4'V7*�1�.�X

    xs(t)W (-,

    Γp, p = 1, . . . ,Pq

    x̃sp(t) = xs(t)Γp, ∀(s,p,t) ∈ NS × NP × NT̄

    R �^L �HZ XRW?]*�SI , X:B�V26 , XLW�\+*5B\I�*[W'UK1H4:XRW> ( XX1-* ( W�U/�*51�W%\�*51�W:X , /58LXLW%WQ*51H4 , X�I , 8LW:X�1H4Q8LXLW X:1 UK1�b

    1#X u XsNdL� U WQ*5B\B Xg.�XPTAX (^, /�U , VAU51�\�X , X�I , 8LW:X�1H4'X85.2%

    .#XPTAU\/�U , VUK1�\�Xi4Q*K4�U�TAX5L� UbJ^3 (-, XsNB\*51H4 , X)e#VAX�1�> ( X$W3V�1-* ( W`/�* ( TAV7*�1�W�U ( 35B X:1�4'X ,`( 1tI X ( TAX�I�* (-, \�X:1�4�UK3[X).�X�/�U , VUK1�\�XNX u I#TAV2> ( 8LX59VATmCU ( . , U�V74$U y * ( 4:X , e XRU ( \�* ( I .�XP\+*�B\Id*cW:U51�4'XLW:L

    U 4�U�VAT=TAX.�XRWg/[XL\F4'X (-, Wg.N8R\ , V7/�U51�4PTf@ 8+4'UK4bv ( 1S4:X�B\I�Wt.N*51^1-8 XLW[4nI�U[W:WQ8�Xt.#X

    E = 122v

    P = 10Lj)* ( WQU/5*�1�W�U9V71�W:V ( 1 , 8LW:XLU ( Id*cW:WQ8:.�UK1H4�]��g1#X (-, *�1#XLWw*5e W3X , /�UKe�T=XRWOX+4�1-* ( W U/�*51�W`J u 8

    v � T=Xc1-*�Bne , X�.�Xc1#X (^, *�1#XLWQ\�U[\=

  • ��� ��� � � ��� � � ����� � �� \��� ���� ` �

    0 5 10 15 20 25 30 35 400

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    Composantes principales

    % d

    e la

    var

    ianc

    e to

    tale

    �Ew �^LA]) , " �������%�J([� 2A"%��"��0�"��6����([�0�K�6�:����� -��:�0�J�����N������"�� ) ���

    0 20 40 60 80 100 120 140 160 180 200−0.2

    −0.1

    0

    0.1

    0.2

    neur

    one

    1

    0 20 40 60 80 100 120 140 160 180 200−0.2

    −0.1

    0

    0.1

    0.2

    neur

    one

    2

    0 20 40 60 80 100 120 140 160 180 200−0.1

    −0.05

    0

    0.05

    0.1

    neur

    one

    3

    �ED �dL `g , ��� B �6�:����� -��:�0� �����$������"�� ) ��� (9�x1, . . . ,x20 ∈ Ẽ "'�[�#"��0) ����"���) � - 2="���2="��'��� @ C

  • ��� ��� � � ��� � � ��� �� ������� � `�]

    � ������������� �

    � " $ ��% �&%("

    ����� � �" %�%�� ���� ���"� # ��� �� ��% � ��% �� � � &��� �0%���� %����� ��� � � � ��� �Z�c�\�l \ ��jkl Z�cY�* (-, I�* ( /�*9V , X�1H4 , UT1#X ,

    λW (^,P( 1#Xce�U[W3Xc.A@ X�1H4 , UT1#X:B�X:1H4 Ẽ = {xso}Ss=1 9#TO@ U�T235* , V74Q ( XPW:Ube�* , 1#X V71-CE8 , V=X (-, X

    σ � ��

    L

    Xs1-*5Bse , XK.A@ V74Q8 , UK4:V2*51�W$XLW[4)J u 8JU ( .N8�I U , 4�B%U�VWNVAT;I�* (-,}, U�V74 ' 4 , Xn.N8�4:X , B V21-8 X:1�\+* (-, W

    .�X�\�U9TA\ ( T T2* , WQ> ( X!TUnCE*�1�\F4'V7*�1{.�Xc/ , U9VAW:X�Bse#TAU51�\�XiI�* (-, TUbe�U9W:Xc.A@ X:1H4 , UT1#X�B X�1H4P1#Xg.�V l 6 , XcI�U9WI#T ( W\.A@ ( 1

    �.A@ ( 1#X V74Q8 , UK4:V2*51_v ( V�\+*�1�\�X , 1#Xh(1),θ,∆t

    X+4M9^1-* ( W ( 4'V=TAVW3X , *�1�W TAXLW)B ' B�XRW)/�U�TAX (^, W)> ( XJ\�X�TATAXLW ( 4:VAT=VW[8LXLW$T7* , W).#XPTO@ X�1H4 , U~?1#X�b

    B X�1H4:L � V21�U�TAX�B X�1H4r9HI�* (^, \�Xi> ( V6XLW[4P. ( I�U , U5B\6+4 , Xσ1-8L\�XLW'W:U9V , X$U ( \�U�T\ ( Tm.#X

    L̂λo9j1-* ( W U/�*51�W

    \=

  • ��� ��� � � ��� � � ��� �� ������� � `9`

    ����� ��� � Z �2c��Z df\_g d��Bg��U51�W TU W:V4 ( U~4'V7*�1* � Tf@ X:1H4 , UT1#X�B X�1H4 XLW[4�T=XbI#T ( W , U5I#V?.�X59/\K@ XLW[43b v�b .�V , XJT7* , W}> ( @ V=T�1@ XLW[4cI�U9W

    1-8R\�XLW'W'U�V , Xg.A@ U5.�UKIN4'X , TAXLW!\+*�1�W[4'UK1H4'XLW).�Xn4:X:BbI W�9�T=Xn4:X:BbI Wi.#X \�U9TA\ ( T0.#X Tf@ U9T73�* , V4} ( Xt.A@ U ( 4 , XRW X+4n*�1�4PU9V71�W:V ( 1�X%V7B\I�* , 4'UK1 \�X.�V l 8 , X:1�4'X5L Y�T ( 4 �K4s> ( Xt.#X \+*51 W3V?.N8 , X ,Tf@ 8:/5*9T ( 4'V7*�1].�X TUCE*51�\+4:V2*51].�Xb/ , U9VAW:X�Bse#TUK1�\�X59^1^* ( W8+4 ( .�VAX , *51�W�TO@ 8�/�*9T ( 4:V2*51. ( B X�VATAT=X (-, \=

  • ��� ��� � � ��� � � ��� �� ������� � `�a

    0 50 100 150 2000

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    4.5

    5x 10

    −3

    Itérations

    Impo

    rtanc

    e du

    chem

    in m

    ax.

    �ED � L=] MO��������0� ( . "'�9�6�:����) ��'��"�2[�%([�0� ��2="������0� @ & A & B ��) C "������ 2=" �'����1 2����:"�) ����ξ1

    ξ1 C1 C2 C3 C4R1

    ] ] ] ]

    R2_ ]R` � ]��

    R3` � a a

    R4� � � � �c_ �.�

    �����; � L=] MZ2="����0� 1 ��"�) ����,(9��� A'C @ ������"���J(9� ) ���0)K"������J2="�������1�29���:"�) ����ξ1

    Y^U , X u X�B\I#TAX59 W (-, TAXLW � � XLW'W'U�VW!U5I-I�U , 4'X�1�U51�4�v TAU \�TU9W'W3X aSR VfL X�L W4'V7B ( T ( W�/8VAW ( X�T`. , *[V4 X+4, 8�I�*51�W:Xnv Tf@ U�V?.�Xs.�XJTAUB%U�V21X3[U ( \= ( XGI�* (^, TAXLW�� \�TU9W'W:XLW:9R\LU , T=XRWZXLW'W:U9VAW WQ*51H4O\�TU9W'W3V7J^8LWI�* (-, TU\3 , UK1^.#XiB%U y * , V4}8�.�UK1�W TU%\�TU9W'W:X!�d9#\�Xg> ( V W3V2351#V7J�Xc> ( X�TUbCE*51�\+4:V2*51{.�Xc/ , U9VAW:X�Bse#TUK1�\�X.�X TU%\�TU9W'W3X!� XLW[4NX�1�Bb*~[X�1-1�XiI�T ( W)8�TAX�/58LXi> ( XP\�X�TAT=Xc.�XRWNU ( 4 , XLWQ\�TU9W'W3XRW�L

    h X�I- ( Xi1^* ( W/9X:1-*51 W;.�X�B X�1H4'V7*�1-1#X , LM`1 X l X+4r9�W3V�TAX \�TU9W'W3V7J \�UK4:X (-, TAV28�v�TU)4 | \= ( XJ\�X�T ( VZT=V28\v TU4 | \= ( @ ( 1]1-* ( /9XLT XRW:W'U�Vf9�V21^.N8:I X:1^.�X�B\B X�1H4g.�X�TAU\�TU9W'W:X\v TU5> ( XLT=TAX%V=T UKI-I�U , 4'V=X:1H4:9AI�*[W'W[8r.�X , U ( 1#X/�U�TAX (-, .#X/ , U�VW:X�Bne�TAU51�\�X W ( Id8 , VAX (-, I�* (^, TAU\�TU9W'W:XJ]c> ( XcI�* (-, TU \�TU9W'W3XP`NLY ( VWQ> ( X \= ( XJ\�TU9W'W:Xn.�X T̃ I�*[W'W[6r.�X ( 11-*5Bse , Xs.�V l 8 , X:1H4i.A@ XLW'W:U9VAWbR S1 = 56 9 S2 = 65 9

    S3 = 59X+4

    S4 = 61ZF9ZV=T X u VAW[4:X�I#T ( W3VAX (^, WbCU *51 Wb.#X .-8+J^1#V , T=X�4�U (Nu .#X \�TAU[W:W:VJ�\�U~4'V7*�1Lk U

    B%UK1#V26 , X TU�I#T ( W W3V2B\I#T=X�9/X+4P\K@ XLW[4�\�X�TATAXb> ( X\1-* ( W \=

  • ��� ��� � � ��� � � ��� �� ������� � ` �

    , UKI#V?.�Xc> ( X T=XcI , 8L\�8:.�X:1H4�I ( VAW}> ( XαX+4

    σ(0)W[*�1H4�I�T ( WPCU�V2e#T=XRW�L

    0 20 40 60 80 1000

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4x 10

    −3

    Itérations

    Impo

    rtanc

    e du c

    hemi

    n max

    .

    �ED � L `g MO��������0� ( . "'�9�6�:����) ��'��"�2[�%([�0� ��2="������0� @ & A & B ��) C "������ 2=" �'����1 2����:"�) ����ξ2

    8V�1-* ( W \+*�BbI U , *51�W^TAXLW \+* (-, e�XLWw.A@ UKI-I , X�1H4'VAW'W:U539X , X�I , 8LW:X�1H4Q8LXLWOW (-, TAU�Jd3 (-, XiN�U/9XR\ \�XLT=TAXLW.�X TUJ^3 (-, X�Nd9K1-* ( W�Id* ( /5*�1�W , X�B%U , > ( X , > ( X TU!\�*51NJd3 (-, U~4'V7*�1

    ξ2I�X , B X+4 ( 1�XNU ( 3�B�X:1H4'U~4'V7*�1

    I#T ( W , UKI#V?.�X.�X TU / , U�VW:X�Bne�TAU51�\�X�9�B%U9VAW XLT=TAX W:UK4 (-, X\8�3cU�TAX�B X�1H4cI#T ( W , UKI#V?.�X�B X:1�4r9 \�X> ( VOXRW4. � U ( W'\= ( X , > ( X \+*�1�4 , U9V , X�B X:1�4�U (Nu \�* (-, e XRWn.A@ UKI^I , X:1H4:VW:W'UK3[X.�X TUJd3 (-, XNd9 TAU�/ , U9VAW:X�Bse#TUK1�\�Xs.�XRW�\�TU9W'W3XRW ` X�4�a .�X TU \+*51-J^3 (-, U~4'V7*�1

    ξ21#X W:X%\ , *9VW3X:1H4

    I�U[W�9�\�Xn> ( V0/9X ( 4i.�V , Xn> ( X TU\CE* , B Xn.�XRW�\�* (-, e XRW�.A@ UKI^I , X:1H4:VW:W'UK3[Xi1@ XRW4iI�U9W�\+*51 W3X , /�8�Xg.A@ ( 1#X\+*�1NJ^3 (-, UK4:V2*51�vJTf@ U ( 4 , X�LY�* (-, 1�XtI�U9W�U9T7* (-, .�V , TU I , 8LW:X�1H4'UK4:V2*519 T=XRWg.N8+4�U�VATAWn.�X�\�TU9W'W:VJ \LU~4'V7*�1XJ^3 (-, X , *�1H4:9v I�U , 4'V ,

    .A@ VA\�VO9K.�UK1�W Tf@ U51-1#X u X�NdLK XG4�UKe#TAXLU ( N�XLW[4ZTf@ UK1�U9T7*�3 ( XG. ( 4�UKe#TAXLU ( NcId* (-, TU$TU \+*51-J^3 (-, U~4'V7*�1ξ2L� XPB ' B�X�I , *�e#T26�B X).�XN\�TU9W'W:VJ \LU~4'V7*�1�W:@ s4 , * ( /9X�B ' B X$W:@ VAT#XLW[4�T28�3�6 , X�B X�1H4kB\*9V21�WkB%U , > ( 85L XRWA4'U (Nu .�X \�TAU[W:W:V7J \�UK4:V2*51R\5L CL:4'U5e#T=XRU ( NjZ W[*�1H44Q* ( 4 U ( W:W:V9V21�W:V73�1#V7J UK1H4'W;> ( X�\�X (Nu *5eN4'X�1 ( W

    U/9XR\$TUbI�U , UKB\8�4 , VAW'U~4'V7*�1ξ1X:1 , U�VW[*�1�. ( e#VU�VWP.�X�TUsCE*51�\+4:V2*51{.�X�/ , U9VAW:X�Bse#TAU51�\�X�L

    � ()!�������'�% %'#�()!ξ3 UiI , V71 \�V2I�U�TAXG.#V l 8 , X�1�\�XQX�1H4 , X TU \+*�1NJ^3 (^, UK4:V2*51

    ξ3X�4OT=XRW0.#X (Nu I , 8R\+8:.#X�1H4:XRW W:XG4 , * ( /[XG.�UK1�W

    TAXLW^\+*�1�W[4'UK1H4'XLWk.�X�4'X�B\I�W:L9SKV=X:1\> ( XN\�X�TAT=XRW b+\�V#WQ*9VAX�1H4^TAXLWkB ' B XRW`.#UK1 WZT=XRWk1#X (^, *�1#XLW�*�e�W3X , /�U5e#T=XRW�9X�TATAXLWG.�V l 6 , X�1H4�.#UK1�W TAXLWK\+* ( \=

  • ��� ��� � � ��� � � ��� �� ������� � ` �

    0 10 20 30 40 500

    0.5

    1

    1.5

    2

    2.5x 10

    −3

    Itérations

    Impo

    rtanc

    e du c

    hemi

    n max

    .

    �ED � L an MO��������0� ( . "'�9�6�:����) ��'��"�2[�%([�0� ��2="������0� @ & A & B ��) C "������ 2=" �'����1 2����:"�) ����ξ3

    0 20 40 60 80 1000

    0.5

    1

    1.5

    2

    2.5x 10

    −3

    Itérations

    Impo

    rtanc

    e du

    chem

    in m

    ax.

    �ED � L �s MO��������0� ( . "'�9�6�:����) ��'��"�2[�%([�0� ��2="������0� @ & A & B ��) C "������ 2=" �'����1 2����:"�) ����ξ4

    M`17\�X> ( VK\�*51�\�X , 1#X%TAX4'U (Nu .#X�\�TU9W'W3V7J \LU~4:V2*519 T=Xt4'UKe�T=XRU ( N1-* ( WcB\*51H4 , X> ( X TAX�\�TAU[W bW:VJ \LU~4'X (-, 3[U ( \= ( Xξ2v Tf@ X u \�X�I-4:V2*51S. ( I�U , UKB\6+4 , X

    Kc> ( VOW'U~4'VAW[CU�V74�TAU \�*51^.�V74:V2*51 RfN-ZFL

    UgJ^3 (^, XgNbB\*51H4 , X!T=XRWK\�* (-, e�XLWG.A@ U5I-I , X�1H4:VW'W:U539X)I�* (-, TAXLWQ\+*�1NJ^3 (-, UK4:V2*51�Wξ̂2X+4

    ξ̂3L ' V71 W3Vf9�TAU

    \�TU9W'W:VJ \LU~4'V7*�1 W:@ X l XR\F4 ( XPU/9XL\�TAXiI U , UKB\6+4 , Xλ4:XLT> ( X�TAX , 8LW:XLU (

    c( 4'V=TAVAW:X�TAXcI�U , UKB\6+4 , X

    λ(kc)L

    U ' B XJX�1 ( 4:VAT=VW:U51H4!\�X�4[4'Xg4:XR\= ( X59�T=XRW)4�U (Nu .�X�\�TU9W'W3V7J \LU~4:V2*51{1�XJW:@ U5Bb8LT=V2* , X�1H4)I�U[W�L�Y^U ,\+*�1H4 , X59K1-* ( W`I�* ( /5*51 W`/5*[V , W (-, T=X�4'U5e#T=XRU ( Nb> ( @ v�Tf@ X u \�X�IN4'V7*�1.�XNTU \�TAU[W:W:X�]59[T=XI , *5e#T26�B X�. (

  • ��� ��� � � ��� � � ��� �� ������� � `�_

    0 20 40 60 80 1000

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    4x 10

    −3

    Itérations

    Impo

    rtanc

    e du c

    hemi

    n max

    .

    0 10 20 30 40 500

    0.5

    1

    1.5

    2

    2.5x 10

    −3

    Itérations

    Impo

    rtanc

    e du c

    hemi

    n max

    .

    �ED � L � IMO�����������( . "'�[���:����) �����" 2c� ([�0����2="�������� @ & A & B ��) C "�������2="7�����61�29���:"�) ����ξ̂2

    -2c"������#����)ξ̂3

    - (��:����) � ��� �#. "��0�:*�)+"���) - 2A"%�%*��%�P��"�2A����� ([�q(k)

    e#VU�VWQXLW[4 , 8RW[*[T ( L

    ��� � � �" %�%�� ���� ���"� # ��� �� ��% ��� ���������� � g \_cc � h2df\wj����^ �Z ẼY�* (-, I�* ( /5*9V , \+*�B\I�U , X , TAXLW , 8LW ( T74'UK4'Wm*5e-4:X�1 ( W.#UK1�W T=XkI�U , U53 , UKI- ( VD1@ XLW[4nI�U[W�X u \�XLT=TAX�1H4r9B%U�VWg1#X�4[4:X:B X�1H4nB X�VAT=TAX (^, > ( X�\�X (Nu

    *5e-4:X�1 ( W U/[XL\�T=XRW , 8RW3XRU (Nu .#V lm( W:VCW:L1^*~[U (

    C1/C4 C2/C3 Cc,c = 1, . . . ,4%

    N%

    j%

    N ] ij!` j!a N �K �

    � �^L � ]L_#] � ] ]R_ � a9_-L � a#]�� a �+� ac`89 a � �K � ,d = 2

    � `NL � ` � _ � �^L ` `89+� a 9-L _ � � _ � ][] � �9a � 9 �K � ,σ = 0.5

    _[_-L � a �+� _9a^L � a9_+� a+�-L 9 � ]'9 � ] 9 � ]'9 � ]'9�����; � L `n7MZ2="������ 1K��"�) ���� "������ � M 9 - A )��9���#��� / ��2="������ @ ����� ) �:� C ��) ��2A"������ A ����� ) �:� B 7 ��)- C )��[�'���0�

    Y�* (-, \�X�> ( V/XRW4).�X TU \�TU9W'W3V7J \�UK4:V2*51�v�`n4 | \= ( XLW[4:V2*51 W:V6TAUn4:XR\= ( X�Y h ' XRW4 y( .�V\�VAX ( W3X�I�* (-, 1-*K4 , X W:V4 ( U�b4:V2*51L � , 9 \�*5B\B X VAT XRW4!V7B\I�*[W'W3V2e#TAXs.A@ X�1H4 , UT1#X ,�( 1 , 8LW:XLU ( .�V l�( W:V7C W (^, TUte�U[W3Xn.#Xs.N*�1-1-8�X E 91-* ( W�U/5*�1�W!X�1H4 , UT1-8%Tf@ U9T73�* , V4}

  • ��� ��� � � ��� � � ��� �� ������� � ` �

    1^*~[U (C1/C4 C2/C3 C1−4

    %N

    %j

    %N ] ij!` j!a N �

    K �� _-L � ` � � � ]5L 9 `#]*� � �^L � a � � � ]La �2� � a 9 9

    K � ,d = 2� �^L � ` 9 � � � a �c` �ca-L _ � ]*� � �.� � a � � `89

    K � ,σ = 0.5� �^L � a �+� � 9^L 9 a9_+� a � L � � ]'9 � ] 9 � ]'9 � ]'9

    �����; � L an7MZ2="��'�0� 1K�'"�) ���� ��"��#� �!M 9 - A )��[�'���0� / ��2="������ @ ����� ) �'� C ��) ��2="������ A ����� ) �'� B 7 ��)- C )��[�'���0�

    ����� ��� � ��*� �����" ������Y�* (-, 3cU , .�X ,�( 1�Xc/ ( XPW[j1H4Q ( Xn.�X�4}* ( W TAXLW , 8RW ( T74'UK4'W�*�eN4:X:1 ( W�9-1^* ( W U/�*51�W , X�I , 8LW:X�1H4}8

    W (-, TAXP4'U5e#TAXLU ( NJTAXLW`4'U (Nu .#X$\�TU9W'W3V7J \LU~4:V2*51\*�eN4:X:1 ( W`3 ,=| \�XNU (Nu\, 8LW:XLU (-u .�X)1�X (-, *51#XRWD.�V lm( W3V7CWR �i�iZQX+4$U (Nu jiU U/9XL\i.�V l 8 , X:1H4:XLW�I U , UKB\8+4 , VW:UK4:V2*51�W:L

    U9T73dL I , 8+4 , U�V74:L I�U , U5B LC1−4 C1/C4 C2/C3

    % % %�i� Y h 'ξ1

    ` �^L � � `NLA] � � L _�i� Y h '

    ξ2` � L � � _-L � �[_

    �i� Y h 'ξ3

    ` �^L � � `NLA] � ��i� Y h '

    ξ4` � � a � �

    �i� Y h 'ξ̂2

    ` � L � � 9-LA] ��]�L ��i� Y h '

    ξ̂3` �^L � � �-L � � � L �

    jU Y h 'K �

    a9_-L � � �^L � � ]jU Y h '

    K � ,d = 2a 9-L _ � `-L � � �^L `

    jU Y h 'K � ,σ = 0.5

    a+�-L 9 _9_^L � _9a^L �jU

    K �� �^L � � _-L � � ]�L 9

    jUK � ,d = 2

    �ca-L _ � �dL � � �jU

    K � ,σ = 0.5a � L � � �dL � � 9^L 9

    �����0 � L �s�� "9�2=��"�� �:5���"'�6��) ��2="�) � ,J(9��� )+"��94 (9����2="����0� 1 ��"�) ����)* ( W , X�B%U , > ( *�1�W�U9VAWQ8�B X�1H4�> ( X$TAXLW�jU W[*�1�4Q\�TU�V , X:B X�1H4 W ( Id8 , VAX (^, WKU (Nu, 8LW:XLU (Nu .#V lm( b

    W:VCWDId* (-, \�X> ( V�XRW4�.�X$TUP\�TAU[W:W:VJ�\�U~4'V7*�1v �c4 | \= ( X�\�X�T ( V

    v �b4 | \= ( XJT=Xg4'U (-u .�XJ\�TU9W'W3V7J \�UK4:V2*51{.�XPTf@ U�T235* , V74Q ( XTAXLW ]��!\�*5B\I�*[W'UK1H4:XRW�I , V21�\�V7I�U9T=XRW�1#X , X�I , 8LW:X�1H4:X:1H4k> ( X). ( e ,Q( V4OX�4 U�V21�W:V�X u I#TAV2> ( X , X�1H4:V26 , X�B X:1�4

    TAXiB%U ( /�U�VWG4'U (-u .�X�\�TU9W'W:VJ \LU~4'V7*�1�.�XRW , 8RW3XRU (Nu .#V lm( W:VCW:LY�* (-, \�X�> ( V/XRW4).�XRW \+*�1NJ^3 (^, UK4:V2*51�W

    ξ1v

    ξ49-1^* ( WQU/5*�1�Ww/ ( > ( @ V=T XRW4P1-8R\�XRW:W'U�V , Xi.A@ U ,},(' 4:X ,

    TAX \LU�T\ ( Twv�.#V l 8 , X�1H4:XRW V4}8 , U~4'V7*�1�Wc.�XbCU *51Sv \�X> ( Xq1(k1) = · · · = q4(k4)

    9 \�X> ( V`I�X , B X+4.A@ X�1#TAX�/[X , TAXce#VAU9VAWP.#Xc/ , U�VW:X�Bne�TAU51�\�X�X�1H4 , X T=XRW).�V l 8 , X:1�4'XLW \�TAU[W:W:XLW:L

    U , U9VAWQ*51 I�* (-, TAU�> ( X�TAT=X T=XRW�I�U , U5B\8+4 , VW:UK4:V2*51�Wξ̂2X+4

    ξ̂31@ *�1H4I�U[W�.N*�1-1-8JW'U~4'VAW[CU9\+4:V2*51 XRW4

    \�X , 4�U�V21#X�B X:1�4!T=V28�X%U ( I�X+4:V74g1-*�Bne , Xs.A@ V74Q8 , U~4'V7*�1�W:96\�Xb> ( Vk1�XsI�X , B X+4cI�U9W�.A@ *5e-4:X�1�V ,�( 1 , 8LX�TAT=X\+*�1j/9X , 39X:1�\�Xi.�X TAUbCE*�1�\F4'V7*�1�.#Xc/ , U�VW:X�Bne�TAU51�\�X�T7* , WP.�X Tf@ U5I-I , X�1H4:VW'W:U539X\R\KL CLNJ^3 (-, XgNjZ+L

    ' 4'V4 , X\.�X�\+*5B\I�U , U9VAWQ*519#T=Xs4'U (Nu .�X \�TU9W'W3V7J \�UK4:V2*51{*�eN4:X:1 ( W (-, .�XRW�B\*~9X:1-1#XRW.A@ ( 1#X \�V21�b> ( UK1H4�U�V21#X{.A@ XRW:W'U�VW\/�U ( 4

    54.2%I�* (^, T=X{I , *�e#T76:B XXv `4'U9\=

  • ��� ��� � � ��� � � � �� � �)�� ���� `89

    � ������������� �

    � " & �P$ "'�"

    M`1�\�*5B\B X�1 UK1H4�\�X)I , * y X+4:9[TAUc> ( XRW4'V7*�18+4�U�V74w.�X$W'U/5*[V , W:@ VAT^8�4'U9V4DI�*[W'W:V7e#TAX�.�X�\�TU9W'W3V7J�X , .�XLWXLW'W'U�VW�V21^.�V2/8V2. ( X�TWcI#T ( 4��54s> ( Xt.�XRW�Bb*~[X�1-1�XLW�W (-, .�V l 8 , X:1�4�W�XLW'W'U�VW�L/Y ( VAW}> ( X \�X�4[4:X\4 | \= ( XTf@ U9T73�* , V4} ( Xi4 , 6LWKX z \�U[\�X�9N1@ XRW4PI�U[WQU5.#U5IN4Q8cvJTAU \�TU9W'W:VJ \LU~4'V7*�1.�X�W[8r> ( X�1�\�XLW4:X:B\Id* , XLT=TAXLW:L

    h *51H4 , U�V , X�B X�1H4Gv \�X�> ( X�1-* ( W U~4Q4:X:1^.�V2*51�W:98T=XRW , 8RW3XRU (Nu .�V lm( W3V7CWP1@ *51H4�I�U9W�8�4Q8 TU�WQ*9T ( 4'V7*�1B%UK3[V2> ( XsId* (-, TU�\�TAU[W:W:V7J \�UK4:V2*51].�X�W3V2351 U (Nu M`Mwxb9. ( B\*9V21�W U/9XR\�T=XRW�\= ( Xb1-* ( W�U/�*51�WCU�V74'W:L

    XgI , V21�\�V2I�U9T V21�\+*�1H/�8�1#VAX�1H4�.�XJ\�Xc4jI�Xn.A@ U�T235* , V74Q ( X5LM`1 X l X+4r9#W�@ V=TA8�4'U9V4PI#T ( W , U�VW[*�1-1�UKe�T=X�951-* ( W�Id* (-,}, V2*51�WD4:XRW4'X ,�( 1�I#T ( W�3 , UK1^.t1-*5Bse , X�.�X�\+*�1NJ#b3 (^, UK4:V2*51�W�.�V l 8 , X:1H4:XLWP.#XcI�U , UKB\6+4 , XRW�X�4$U�V21�W:V/XLWQId8 , X , .�X�B�XLV=TATAX (-, W , 8LW ( T4�U~4�W�LY ( VWQ> ( X ( UKe#TAXsI ( VAW}> ( @ V=T�W:@ UK39V74n.A@ ( 1SI , X�B V=X ,g, 8LW ( T4�U~4

    \+*�1�\�X , 1 UK1H4KTU \�TU9W'W3V7J \LU~4:V2*51�.A@ XLW'W:U9VAW ( 1�V2> ( XRW�L� V71�U9T=X:B X�1H4:9�\�X�4 , U/�U�VATAB�X�4)e#VAX�1 X�1 8�/8V?.�X�1 \�X TAU%\�*5B\I#TAX u V74Q8g.�XRW W3V2351�U (Nu M`MGx U9V71 W3V> ( X

    1-*54 , X�.#V z \ ( T4}8gvJX:1 8�TUKe�* , X ,�( 1 B\*j.-6�TAX \�*5

  • � � ������� � ����� � ��� ` �

    � '�����'�"�� *(� � '��

    A]+PSiLmM)L/Sw*cW3X , 9dYFL�UoLAx ( 5*51�X+4�nLmgLmOUKI-1�V < 9Z]*�+�[`NL � ' 4 , U�V21#V21-3�U�T235* , V74Q

  • ���� ��� �n� 4��� ��� � � �� � ��� � �� a �

    � ��� ��� � �

    � � %(��'���" !� & ���P&$ �

    ' /�U51�4c.�X.N8:Bb*�1H4 , X , TU , XLTAUK4:V2*51_RNNZF9/\+*�B\B�X:1 *�1�WiI�U , 8+4�UKe#TAV , TAX , 8RW ( T74'UK4!W ( V2/�U51H4�> ( X1-* ( W U/5*�1�WG8�3[U9T=X:B X�1H4 ( 4:VAT=VW[8PU ( I U , UK3 , UKI^ ( UK4:V2*51SRfNjZ+L

    dQλh|o

    dSλ,xoh(h|xo) =

    Lλ(xo,h)

    Lλo (xo)

    dRhdSλ,xo

    (h)

    =pλ(xo,h)

    Lo(xo)

    = pλh|o(h|xo)R ' L � Z

    U , XLTAUK4:V2*51oRfNjZ�8+4�UK1H48�4'U5e#T=VAX59�1-* ( W�I�* ( /5*51 W$\�U9TA\ ( TAX , TAXn3 , U�.�VAX�1H4. ( T7*�3[U , V74Q

  • ���� ��� �n� 4��� ��� � � �� � ��� � �� a#]

    ∇λ log Lλo (xo) =∫

    Ωh

    ∇λ log Lλ(xo,xh)Lλo (xo)

    dRh(xh)

    =

    Ωh

    Lλ(xo,xh)

    Lλo (xo)∇λ log Lλ(xo,xh)dRh(xh)

    =

    Ωh

    dQλh|o

    dRh(xh|xo)∇λ log Lλ(xo,xh)dRh(xh)

    =

    Ωh

    dQλh|o

    dSλ,xoh(xh|xo)∇λ log Lλ(xo,xh)dSλ,xoh (xh)

    R ' L _�Z

    � V71�U9T=X:B X�1H4:9�Id* (-, *�eN4:X:1#V , Tf@ 8r> ( U~4:V2*51!RNNZF9�1-* ( W)I�* ( /�*51�WNU5I-I , * u V7B X , Tf@ X u I , XLW'W3V2*51 I , 8�b\+8r.�X�1H4'X59#\�Xg> ( VA1^* ( WP.N*�1-1#X�q

    ∇λ log L̂λo (xo) =M∑

    m=1

    pλh|o(hm|xo)∇λ log Lλ(xso,hm), hm ∈ H

    R ' L � Z

  • ���� ��� ��� ��� � ����� � � � � a[`

    � ��� ��� � �

    � � � " *('�% $ �

    )* ( WtI , 8LW:X�1H4}*51�W.#U51�W�\�X+4Q4:X�U51-1#X u X TAXLW U9T73�*K4'V7

  • ���� ��� ��� ��� � ����� � � � � a9a

    ' T235* , V74Q σ � ��

    %*0�� !σ = αkσ(0)

    �� !��

  • ���� ��� � � � ���{���� � ���������� � � � ����� � a �

    � ��� ��� � �

    � " $ � " "� % �#" � *(' & ' �������#"

    −0.2

    −0.1

    00.1

    0.2

    1. s

    ig: 3

    0.8

    −0.2

    −0.1

    00.1

    0.2

    2. s

    ig: 2

    2.4

    −0.2

    −0.1

    00.1

    0.2

    3. s

    ig: 1

    6

    −0.2

    −0.1

    00.1

    0.2

    4. s

    ig: 5

    .5

    −0.2

    −0.1

    00.1

    0.2

    5. s

    ig: 3

    .5

    −0.2

    −0.1

    00.1

    0.2

    6. s

    ig: 3

    −0.2

    −0.1

    00.1

    0.2

    7. s

    ig: 1

    .6

    −0.2

    −0.1

    00.1

    0.2

    8. s

    ig: 1

    .5

    −0.2

    −0.1

    00.1

    0.2

    9. s

    ig: 1

    .2

    −0.2

    −0.1

    00.1

    0.2

    10. s

    ig: 1

    .1

    ��������������������������������� !#"$�%��"$ �&$')(*�� +���,�-'�!�� �)&$./�� 102./&134&5./��67�98- #�;:)��?, ���')(*��.��@�)"$67�=!=��')(*&�:A�B �1./&C34&5�,��&$'�!#�D(*"$(*&$./�FE

  • ���� ��� s� 4������� � � ��� ��� �� ������� � a �

    � ��� ��� � �

    � � %(��'���" !�," * � " $ ��% �&%("

    ξ2 C1 C2 C3 C4R1

    a ] � �R2

    _ 9 ] � �R3

    ]L_ ] � ][] 9R4

    a#] �c` a[a �.�� � �; �sL=]i MZ2="����0� 1 ��"�) ����-([�0� A'C @ ������"���J(9� )+�0�0)K"������J2=" �'����1 2����:"�) ����

    ξ2

    ξ3 C1 C2 C3 C4R1

    ]*� ]R` ]�` ]��R2

    ` a a �R3

    ]9] ]*� ][] ]RaR4

    ` � a#] a[a ` �� � �; �sL `n MZ2="����0� 1 ��"�) ����-([�0� A'C @ ������"���J(9� )+�0�0)K"������J2=" �'����1 2����:"�) ����

    ξ3

    ξ4 C1 C2 C3 C4R1

    ` ] � aR2

    ]'9 `�_ ]Ra ] �R3

    ]'9 ]'9 `#] ` �R4

    ]R` ` � ` � `#]

    � � �; �sL as MZ2="����0� 1 ��"�) ����-([�0� A'C @ ������"���J(9� )+�0�0)K"������J2=" �'����1 2����:"�) ����ξ4

  • ���� ��� s� 4������� � � ��� ��� �� ������� � a9_

    ξ̂2 ξ̂3C1 C2 C3 C4 C1 C2 C3 C4

    R1]R_ ]�� � � ` � ][] ]�` ]R`

    R2` � `#] a � `�a ] � ]�� ]�� ` �

    R3_ � � � 9 ]R_ ]Ra �

    R4]�� ` � ] � ` � ] � ` � ` � ]La

    �����; �sL �G�M 2A"����0� 1 ��"�) �����([�0� A C)@ ���'��"���K(9�K)+�0�0)/"������K2=�0�Q�'����1 2����:"�) ����#�ξ2��)

    ξ3���J��. "��0�:*�) "���)

    - 2A"%�%*��%�P��"�2A����� ([�q(k)

    K � (x,x′) K � (x,x′),d = 2 K � (x,x′),σ = 0.5C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4

    R1] 9 ]*� 9 � ` � ]�� ]9] � ]�` ][] � _

    R2` � `�_ ]R` � `[` ` � ]La ]9] a[_ �.� ]'9 ]La

    R3][] ]R` ` � `9` 9 � `�] ` � 9 � ] � ] �

    R4a 9 ] � `�] ` ]�� ] � `�] � � `9` ` �

    N a#]�� a �2� a[`89 a � � � � _ � ][] � �[a � 9 � � ]'9 � ] 9 � ] 9 � ]'9

    �����0 �nL � 7M 2A"����0� 1 ��"�) ����,(9��� A'C)@ ���'��"���J(9� Ẽ - 23. "��(9�%([�0� ��� D

    K � (x,x′) K � (x,x′),d = 2 K � (x,x′),σ = 0.5C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4

    R1`#] ] � ]R` � `[` ] � 9 � � ` ` a

    R2`9a a[` ]R` ]R` ` � ac` ]9] 9 � � � � a9a `89

    R39 _ ` � ] � � 9 `�a ]'9 � a _ ]9]

    R4� ]R` ]9] ` � a ]�� ] � `89 ` a ]'9 ]*�

    N a � � ��]Ra �+� � a 9 9 � ]*� � �.� � a � � `89 � ]'9 � ] 9 � ] 9 � ]'9�����0 �nL _n7M 2A"����0� 1 ��"�) ����,(9��� A'C)@ ���'��"���J(9� E - 23. "��(9�%(9��� ��� D