particle filter
TRANSCRIPT
Mó�Ü�²
1�Ù âfÈÅ�{0�
§1.1 Introduction
G��mÛê��Å�.Jø��4Ù(¹�µe5é�mS�?1ï�"
�,ù�.é¯Käkr��£ãUå§�é¯K�¦)%Ø´é�BµØ�
Ü©4Ù{ü��¹§é�Ü©·�a,��¢S¯K§ÑéJ��¯Kíä�
)Û). �Ù0��“âfÈÅ”�{´�a61��AkÛ�{§å8®²k15c�{¤§§�JÑ�´^5éùJ±¦Ñ)Û)�íä¯Ké�Cq). §gl1993cùa�{�ÄgJѧ§�®²¤�é��5�pd�.?1�`�O��a61�¦)Cq)��{. ÚIO�Cq�{§X61�*Ðk�ùÈÅì?1'�§âfÈÅ�{�Ì�`:´§�Ø�6u?Û�ÛÜ�5zEâ½
?Û� crude functional approximation. ù«Cq¦)¯K�(¹5�¦�p�O�E,Ý", §�XO�Uå�FÃO�§ùa�{®²�^u�«+��
¢�A^§~X§zÆó§!O�ÅÀú!ã²!8I�lÚÅì<�. d§=¦éu@vk¢�5�¦�A^§ùa�{��±^5O�ê��Åó�Ak
Û(MCMC)�{§½ö��|^§�5�OÑ�~k��MCMC�{.
§1.2 Bayesian Inference in Hidden Markov Models
§1.2.1 Hidden Markov Models and Inference Aims
�Ä�� X ��lÑê��ÅL§ {Xn}n≥1 ÷v
X1 ∼ µ(x1) and Xn|(Xn−1 = xn−1) ∼ f(xn|xn−1) (1.2.1)
Ù¥ ∼ L«Ñl©Ù§µ(x) ´��VÇ�ݼê§f(x|x′) L«d x′ £Ä� x �
VÇ�ݼê. ·�a,��´3==®� Y �L§ {Yn}n≥1��OÑ {Xn}n≥1.·�b�§3�½ {Xn}n≥1 �§*� {Yn}n≥1 ´ÚOÕá�§¿�§��>�
©Ù�
Yn|(Xn = xn) ∼ g(yn|xn) (1.2.2)
�{zå�§3ùp·�==�Ä��5��¹§==£Ú*��ݼê��m
· 1 ·
Mó�Ü�²
2 1�Ù âfÈÅ�{0�
eI n Ã'. ��5�¹Ué�*�*Ð����5�¹. �Ù¥b½¤k��.ëêþ�®�.dúª (1.2.1)Ú (1.2.2)£ã��.�¡��Ûê��Å�. (HMM) ½ö
´���G��m�.. ùa�.�)éõa,���.. e¡�ÑA�{ü�~f.~1.2.1. k¡G��m HMMµk X = {1, . . . ,K}§Ïd
Pr(X1 = k) = µ(k), Pr(Xn = k|Xn−1 = l) = f(k|l)
*��.�÷vúª (1.2.2) �?¿�..
~1.2.2. �5pd�.µX = Rnx§Y = Rny§X1 ∼ N (0,Σ)§¿�
Xn = AXn−1 + BVn,
Yn = CXn + DWn
Ù¥ Vn ∼ N (0, Inv)§Wn ∼ N (0, Inw
)§A§B§C§D ´äk·��ê�Ý
. 3ù«�¹e µ(x) = N (x; 0,Σ)§f(x′|x = N (x′;Ax,BBT )) Ú g(y|x =N (y′;Cx,DDT )). ù«�.e§é¯K�íä´�±¦Ñ)Û)�. §�2�¦^38I�lÚ&Ò?n+�.
úª (1.2.1) Ú (1.2.2) ½Â����d�.§Ù¥úª (1.2.1) ½ÂL§ {Xn}n≥1 � prior distribution§úª (1.2.2) ½Â likelihood function. =µ
p(x1:n) = µ(x1)n∏
k=2
f(xk|xk−1) (1.2.3)
Ú
p(y1:n|x1:n) =n∏
k=1
g(yk|xk) (1.2.4)
3ù������dµee§��½*�S������ Y1:n = y1:n§é X1:n
�íä�6u posterior distribution
posterior distribution︷ ︸︸ ︷p(x1:n|y1:n) =
unnormalised posterior distribution︷ ︸︸ ︷p(x1:n, y1:n)
p(y1:n)︸ ︷︷ ︸marginal likelihoods
(1.2.5)
Ù¥
p(x1:n, y1:n) = p(x1:n)p(y1:n|x1:n) (1.2.6)
p(y1:n) =∫
p(x1:n, y1:n)dx1:n (1.2.7)
Mó�Ü�²
§1.2 Bayesian Inference in Hidden Markov Models 3
��§éuúª (1.2.5) L«���©Ù§©fdúª (1.2.6) �±N´�O�Ñ5 (�\ (1.2.3) Ú (1.2.4)). '�´úª (1.2.7) L«�©1Ø�BO�.
éu~ 1.2.1 ¥?Ø�k¡G��mÛê��Å�.§úª (1.2.7) ¥�È©�±ÏLk¡�¦Úö�5�¤§Ïdúª (1.2.5) ¥���©Ù�±°(/O�Ñ5. éu~ 1.2.2 ¥?Ø��5pd�.§éN´����©Ù p(x1:n|y1:n)�´��pd©Ù§§�þ�Ú��Udk�ùÈÅ�{¦). , §éu�õê��5�pd��.§Ø�U¦)��©Ù�4)§ÏdI�|^ê���{.âfÈÅ�{´�«(¹ qr��Äu�ý��{§UJøCqÑl��©Ù
p(x1:n|y1:n) ���§l UCq¦) p(y1:n). âfÈÅ�{´S��AkÛ�{��f8.
âfÈÅ�{�±^5¦)e¡�¯Kµ
• Filtering and Marginal likelihood computationµb½·�a,��´S�/Cq posterior distribution {p(x1:n|y1:n)}n≥1 Ú marginal likelihoods{p(y1:n)}n≥1. �=§·�F"31����:Cq p(x1|y1)Ú p(y1)§31����:Cq p(x1:2|y1:2) Ú p(y1:2)§�daí. ·�òrù�¯K¡���`ÈůK. 3�õê©z¥§ÈůK��´�O marginal distributions{p(xn|y1:n)}n≥1 Ø´éÜ©Ù {p(x1:n|y1:n)}n≥1.
• Smoothing: b½l��éÜ©Ù p(x1:T |y1:T ) æ�§F"Cq/��>�©Ù {p(xn|y1:T )}§Ù¥ n = 1, . . . , T .
§1.2.2 Filtering and Marginal Likelihood
·�a,��§�=�õê'uâfÈÅ�©z¤ïÄ�¯K§Ò´Èů
Kµ�½��c��¤Â8��¤k*�&E§�OÛê��Å�.�c���
G��©Ù. ÈÅk�ÿ���´�½��c��¤Â8��¤k*�&E§�OÛê��Å�.��c���¤kG�S�. /ªz£ã§=��½��*�S� {Y1:n = y1:n}§íäÑ X1:n �©Ù½ö´ Xn �©Ù.
posterior distribution p(x1:n|y1:n) dúª (1.2.5) ½Â§prior distributionp(x1:n)dúª (1.2.3)½Â§likelihood functiondúª (1.2.4)½Â.úª (1.2.5)¥�©f§= unnormalised posterior distribution p(x1:n, y1:n) ÷v
p(x1:n, y1:n) = p(x1:n−1, y1:n−1)p(xn, yn|x1:n−1, y1:n−1)
= p(x1:n−1, y1:n−1)p(xn|xn−1)p(yn|xn)
= p(x1:n−1, y1:n−1)f(xn|xn−1)g(yn|xn)
(1.2.8)
Mó�Ü�²
4 1�Ù âfÈÅ�{0�
Ïd posterior p(x1:n|y1:n) ÷ve¡�48ªf
p(x1:n|y1:n) =p(x1:n, y1:n)
p1:n
=p(x1:n−1, y1:n−1)f(xn|xn−1)g(yn|xn)
p(y1:n−1)p(yn|yn−1)
= p(x1:n−1|y1:n−1)f(xn|xn−1)g(yn|xn)
p(yn|y1:n−1)
(1.2.9)
Ù¥
p(yn|y1:n−1) =∫
p(yn, xn−1:n|y1:n−1)dxn−1:n
=∫
p(xn−1|y1:n−1)p(yn, xn|xn−1, y1:n−1)dxn−1:n
=∫
p(xn−1|y1:n−1)f(xn|xn−1)g(yn|xn)dxn−1:n
(1.2.10)
é��©Ù�O��ª�´�6uúª (1.2.10) ¥�¦È©. Ï�8céù�È©ªfØЦ§¤±��©Ù�éJ¦Ñ5.éúª (1.2.9)§È©K x1:n−1§Ò�±�� marginal distribution p(xn|y1:n)
p(xn|y1:n) =g(yn|xn)p(xn|y1:n−1)
p(yn|y1:n−1)(1.2.11)
Ù¥
p(xn|y1:n−1) =∫
f(xn|xn−1)p(xn−1|y1:n−1)dxn−1 (1.2.12)
úª (1.2.12) �¡�ýÿ§úª (1.2.11) �¡��#. , �õêâfÈÅ�{�6u48ª (1.2.9) �ê�Cq§ Ø´úª (1.2.11) Ú (1.2.12).XJ·�UO�Ñ {p(x1:n|y1:n)}§ÏdUS�/O�Ñ {p(xn|y1:n)}§?
marginal likelihood p(y1:n) �U�48/O�
p(y1:n) = p(y1)n∏
k=2
p(yk|y1:k−1) (1.2.13)
Ù¥ p(yk|y1:k−1) äkúª (1.2.10) �/ª.
§1.2.3 Summary
é��5�pdÄ��.���díä�6u��©ÙS�Ú§�>�©Ù.Ø{ü�¯K§X~ 1.2.1 Ú 1.2.2 §Ø�U��ù©Ù�4). ùp·�ò?Øù©Ù��AkÛCq. �AkÛCq�{´�aê��{§�Cq�©Ùd N ��Å���§¡�âf§5Cq. ù�{�Ì�`:´3éf�b�e§§�UJøé8I©ÙìC��O (~X� N →∞ �).
Mó�Ü�²
§1.3 Sequential Monte Carlo Methods 5
§1.3 Sequential Monte Carlo Methods
3L�15c��mp§^âf��{5?1ÈÅÚ²w¤� SMC �{A^�²~�~f. ¢Sþ§3�õê©z¥§râfÈÅÚ SMC ��Ó��Vg. ùp·�rN SMC ¢Sþ�¹����2. SMC �{´�AkÛ�{��a§§S�/l8IVÇ�ÝS� {πn(x1:n)} ¥æ�. z��©Ù πn(x1:n) ½Â3(k��m X n þ. L«�
πn(x1:n) =γn(x1:n)
Zn
(1.3.1)
·�==�¦ γn : X n → R+ ´®��§8�z~þ
Zn =∫
γn(x1:n)dx1:n (1.3.2)
�±´���. SMC Jø�� 1 � π1(x1) ���CqÚ Z1 ����O§,
�3�� 2§Jø π2(x1:2) ���CqÚ Z2 ����O§�daí. ~X§éuÈůK§·��±k γn(x1:n) = p(x1:n, y1:n)§Zn = p(y1:n)§Ïd πn(x1:n) =p(x1:n|y1:n).
§1.3.1 Basics of Monte Carlo Methods
Äk�ÄCq����� n ��½��VÇ�Ý πn(x1:n). XJ·�l§æ�N �Õá��ÅCþ§X i
1:n ∼ πn(x1:n)§Ò�±^�AkÛ�{5Cq πn(x1:n)
πn(x1:n) =1N
N∑i=1
δXi1:n
(x1:n)
Ù¥ δx0(x) L«3 x0 ?� Dirac delta mass. =
δx0(x) ={
+∞, x = x0
0, x 6= x0
�Ó�÷v ∫ +∞
−∞δx0(x)dx = 1.
Äuù�Cq§�±é�B/Cq?Û�>� πn(xk)
πn(xk) =1N
N∑i=1
δXik(xk)
?Û¼ê ϕn : X n → R �Ï"
In(ϕn) =∫
ϕn(x1:n)πn(x1:n)dx1:n
Mó�Ü�²
6 1�Ù âfÈÅ�{0�
�±��O¤
IMCn (ϕn) =
∫ϕn(x1:n)πn(x1:n)dx1:n =
1N
N∑i=1
ϕn(X i1:n)
5¿�Oþ IMCn (ϕn) E,´���ÅCþ§éN´�yù��Oþ´Ã �O§
¿�§����
V[IMC
n (ϕn)]
=1N
(∫ϕ2
n(x1:n)πn(x1:n)dx1:n − I2n(ϕn)
).
Ä���AkÛ�{�Ì�g�´§XJ·�®�k N ���ægu��©Ù§
K�±^ù��5CqT©Ù§±9½Â3�ÅCþþ�¼ê�êÆÏ". �'uIO�Cq�{§�AkÛ�{�Ì�`:�O�����m X n ��êÃ'§
´± O(1/N) ��Çeü. �§kü�Ì��¯Kµ
• ¯K1: XJ πn(x1:n) ´��E,�p�VÇ©Ù§·�Ø�U��l§æ�.
• ¯K2: =¦��XÛ°(/l πn(x1:n) æ�§æ��O�E,5��´ n
��5�. �S�/l πn(x1:n) æ��§O�E,ݬ�X n �O\ O\.
§1.3.2 Importance Sampling
IS �{�±^5)û¯K1§§�Ø%g�´Ú\���5©Ù½JÆ©
Ù qn(x1:n)§÷v
πn(x1:n) > 0⇒ qn(x1:n) > 0
3ù«�¹e§lúª (1.3.1) Ú (1.3.2) �� IS �Ý
πn(x1:n) =wn(x1:n)qn(x1:n)
Zn
(1.3.3)
Zn =∫
wn(x1:n)qn(x1:n)dx1:n (1.3.4)
Ù¥ wn(x1:n) ´�8�z��¼ê
wn(x1:n) =γn(x1:n)qn(x1:n)
AO/§·��±ÀJN´æ���5©Ù qn(x1:n)§~X��õ�pd©Ù.b½·�æ� N �Õá��� X i
1:n ∼ qn(x1:n)§,�ò qn(x1:n) ��AkÛ
Mó�Ü�²
§1.3 Sequential Monte Carlo Methods 7
Cq�\��§ (1.3.3) Ú (1.3.4) ¥§��
πn(x1:n) =n∑
i=1
W inδXi
1:n(x1:n) (1.3.5)
Zn =1N
N∑i=1
wn(X i1:n) (1.3.6)
Ù¥
W in =
wn(X i1:n)∑n
j=1 wn(Xj1:n)
(1.3.7)
·�a,��´O� In(ϕn)§�±¦^Xe��O
IISn (ϕn) =
∫ϕn(x1:n)πn(x1:n)dx1:n =
N∑i=1
W inϕn(X i
1:n)
§1.3.3 Sequential Importance Sampling
e¡5)û¯K2§=é��«)§¦�éu?¿��� n §æ���mE,
Ý´�½�. ¦)g´´�ÀJ��äkXe(���5©Ù
qn(x1:n) = qn−1(x1:n−1)qn(xn|x1:n−1)
= q1(x1)n∏
k=2
qk(xk|x1:k−1) (1.3.8)
þ¡�úª¿�X§XJ·�3�� n§�æ�âf X i1:n ∼ qn(x1:n)§�±3��
1 �§æ� X i1 ∼ q1(x1)§,�éu k = 2, . . . , n �§æ� X i
k ∼ qk(xk|X i1:k−1).
¦^e¡�©)
wn(x1:n) =γn(x1:n)qn(x1:n)
=γn−1(x1:n−1)qn−1(x1:n−1)
γn(x1:n)γn−1(x1:n−1)qn(xn|x1:n−1)
(1.3.9)
�âféA��8�z��U�48/O�
wn(x1:n) = wn−1(x1:n−1) · αn(x1:n)
= w1(x1)n∏
k=2
αk(x1:k) (1.3.10)
Ù¥ incremental importance weight ¼ê αn(x1:n) �
αn(x1:n) =γn(x1:n)
γn−1(x1:n−1)qn(xn|x1:n−1). (1.3.11)
Mó�Ü�²
8 1�Ù âfÈÅ�{0�
SIS�{Xe:Algorithm 1: Sequential Importance Sampling
3�� n = 11
for i = 1 to N do2
æ� X i1 ∼ q1(x1)3
O�� w1(X i1) Ú W i
1 ∝ w1(X i1)4
end5
for n = 2 to T do6
for i = 1 to N do7
æ� X in ∼ qn(xn|X i
1:n−1)8
O��9
wn(X i1:n) = wn−1(X i
1:n−1) · αn(X i1:n),
W in ∝ wn(X i
1:n).
end10
end11
éu?Û�� n§�{©O�� πn(x1:n) Ú Zn ��O πn(x1:n) (1.3.5)Ú Zn
(1.3.6). dù��8Ü�U�� Zn/Zn−1 ��O
Zn
Zn−1
=N∑
i=1
W in−1αn
(X i
1:n
).
3SISµee§3�� n§ r���gd´ÀJ qn(xn|x1:n−1). Ün�ÀJüÑ´¦� wn(x1:n) �����z. �ÀJ
qoptn (xn|x1:n−1) = πn(xn|x1:n−1)
�§3 x1:n−1 �^�e§wn(x1:n) ����"§¿��A� incremental weight�
αoptn (x1:n) =
γn(x1:n−1)γn−1(xn−1)
=∫
γn(x1:n)dxn
γn−1(x1:n−1).
, §·�Ø�Ul πn(xn|x1:n−1) æ�§�Ø�UO�Ñ αoptn (x1:n). Ïd·�
I�|^ qoptn (xn|x1:n−1) ���Cq.
�Ün�ÀJ qn ¿�^r'%�´~XÈÅùa¯K�§l qn(xn|x1:n−1)æ�ÚO� αn(x1:n) ¤I���m´� n Ã'�§l )û¯K2. , SIS�{k��":. =¦´éuIO� IS �{§����O����X n ¤�ê?
�O�. Ï� SIS ==´ IS ���A~§·�==���5©Ùäkúª(1.3.8) �/ª§Ï SIS �3Ó��¯K. ·�ÏL��{ü�~f5�ãù�¯K.
Mó�Ü�²
§1.3 Sequential Monte Carlo Methods 9
~1.3.1. �Ä��~f§X = R§¿�
πn(x1:n) =n∏
k=1
πn(xk) =n∏
k=1
N (xk; 0, 1), (1.3.12)
γn(x1:n) =n∏
k=1
exp(−x2
k
2
),
Zn = (2π)n/2.
ÀJ�5©Ù
qn(x1:n) =n∏
k=1
qk(xk) =n∏
k=1
N (xk; 0, σ2).
� σ2 > 12�§VIS
[Zn
]<∞§¿� relative variance
VIS
[Zn
]Z2
n
=1N
[(σ4
2σ2 − 1
)n/2
− 1
].
N´�y§éu?¿� σ ÷v 12
< σ2 6= 1§k σ4
2σ2−1> 1§d� relative variance
�X n�êO\. ~X§XJ·�ÀJ σ = 1.2§�,d�U����Ð��5©Ù qk(xk) ≈ πn(xk)§�´ N
VIS[Zn]Z2
n≈ (1.103)n/2. � n = 1000�§N
VIS[Zn]Z2
n≈
1.9 × 1021§d�§·�I�¦^ N ≈ 2 × 1023 �âf5¦� relative varianceVIS[Zn]
Z2n
= 0.01§ùA�´Ã{���.
§1.3.4 Resampling
dc¡�Qã��§IS Ú SIS �{Jø��O����X n �êO\§æ
�Eâ´ SMC �{¥���'�Ú½§5)ûù��¯K. æ�´��é�*��{§äk��¢SÚnØd�. Äk�Äé8I©Ù πn(x1:n) ��� IS Cq πn(x1:n)§ù�Cq´Äul©Ù qn(x1:n) æ������§ vkJøCqægu8I©Ù πn(x1:n) ���. ���ægu πn(x1:n) �Cq��§·��±l§� IS Cq πn(x1:n) ?1æ�§=·�±VÇ W i
n ÀJ X i1:n§ù�ö�¡
�� resampling§Ï�§´l����Ò´²Læ� ���Cq©Ù πn(x1:n)¥?1æ�. XJ·�él πn(x1:n) �� N ���a,�§K·��±{ü/
l πn(x1:n) æ� N g§ù�du4z��âf X i1:n �� N i
n ���§�¦÷
vXe�ªµN 1:Nn = (N 1
n, . . . , NNn ) Ñl��ëê�þ� (N,W 1:N
n ) �õ�ª©Ù§¿��z����D� 1/N . ·�^ resampled empirical measure 5Cqπn(x1:n)
πn(x1:n) =N∑
i=1
N in
NδXi
1:n(x1:n) (1.3.13)
Mó�Ü�²
10 1�Ù âfÈÅ�{0�
Ù¥ E [N in|W 1:N
n ] = NW in. Ïd πn(x1:n) ´ πn(x1:n) ���à Cq.
©z¥JÑn«æ��{µ
• Systematic Resampling æ� U1 ∼ U[0, 1
N
]§éu¤k� i = 2, . . . , N§
½Â Ui = U1 + i−1N§,��½ N i
n =∣∣∣{Uj :
∑i−1
k=1 W kn ≤ Uj ≤
∑i
k=1 W kn
}∣∣∣§Ù¥�½
∑0
k=1 = 0.
• Residual Resampling � N in = bNW i
nc§l��õ�ª©Ù(N,W
1:N
n
)¥æ� N
1:N
n §Ù¥ Wi
n ∝W in −N−1N i
n§,�� N in = N i
n + Ni
n.
• Multinomial Resampling l����ª©Ù (N,W 1:Nn ) ¥æ� N 1:N
n .
3 O(N) �mE,ÝS§�±k�/l��õ�ª©Ùæ�. ,¡ systematicresampling �{ÏÙ4´¢y§ ©z¥¦^�2���«�{§¿�3�õê
A^|Üe§Ù5U��LÙ§�æ��{.
æ�#N·���Cqægu πn(x1:n) ���§�7L�Ù�¿£�§XJ·�a,��´�O In(ϕn)§K¦^ πn(x1:n) ?1�O§�Oþ���'¦^πn(x1:n) ����Oþ�����. ÏLæ�§·�¢SþO\���D(. , §æ����²w�`:´§#N·�±�p�VÇ£Ø$��@âf§ù´4Ùk^�§Ï�3·�a,��S�µee§·�¿ØI�D4@
$VÇ�âf§ ´rO�å8¥3pVÇ�@âfþ¡. 7L�Ù§k�U¬Ñy§��� n ��$��âf§3�� n + 1 �U¬k�p��§3ù«�¹e§æ�ÒL¤. , éu·��ca,���O¯K ó§æ�®²�y²´k��. �*þ`§æ�±O\����d5¦XÚ3ò5�½.
§1.3.5 A Generic Sequential Monte Carlo Algorithm
SMC �{¢�þÒ´ò SIS Úæ�éÜå5. 3�� 1§·�O�Ñπ1(x1) � IS Cq π1(x1)µ��D��âf8Ü {W i
1, Xi1}. ,�·�¦^æ�
Ú½±�p�VÇ�Ø@$��âf§Ó�E�@p��âf. ·�^{ 1
N, X
i
1} L«²Læ�����âf8Ü. P4§z���k�âf X i1 k
N i1 ���§Ïd�3 N i
1 �ØÓ�eI j1 6= j2 6= · · · 6= jNi1¦� X
j1
1 = Xj2
1 =
· · · = XjNi
11 = X i
1. æ���§·�¦^ SIS �{æ� X i2 ∼ q2(x2|X
i
1). Ïd§(X
i
1, Xi2) Cq/Ñl©Ù π1(x1)q2(x2|x1). d�§�A��5�Ò�u
incremental weights α2(x1:2). ,�§æ�ùvk8�z��âf§�da
Mó�Ü�²
§1.3 Sequential Monte Carlo Methods 11
í. ����{Xe:Algorithm 2: Sequential Monte Carlo
3�� n = 11
for i = 1 to N do2
æ� X i1 ∼ q1(x1)3
O�� w1(X i1) Ú W i
1 ∝ w1(X i1)4
æ� {W i1, X
i1}§�� N ����âf { 1
N, X
i
1}5
end6
for n = 2 to T do7
for i = 1 to N do8
æ� X in ∼ qn(xn|X
i
1:n−1) ¿�� X i1:n ←
(X
i
1:n−1, Xin
)9
O�� αn(X i1:n) ¿� W i
n ∝ αn(X i1:n)10
æ� {W in, X i
1:n}§�� N ����âf { 1N
, Xi
1:n}11
end12
end13
é?Û�� n§�{�� πn(x1:n) �ü�Cq. æ����Cq:
πn(x1:n) =N∑
i=1
W inδXi
1:n(x1:n) (1.3.14)
Úæ����Cqµ
πn(x1:n) =1N
N∑i=1
W inδ
Xi1:n
(x1:n) (1.3.15)
Cq (1.3.14) 'Cq (1.3.15) �Ð. �{��� Zn/Zn−1 ���Cq
Zn
Zn−1
=1N
N∑i=1
αn
(X i
1:n
)�Xc¡®²J��§æ�U�ØK$��âf¿�E�p��âf§ù
´±Ú\������d�. XJvkæ��c§vk8�z���k�����§K���æ�Ú½´v7��. Ïd§3¢S�¥§�Ün�üÑ´§�vk²L8�z�����pu�½�K��§â?1æ�. ÏL*d¤¢� Effective Sample Size (ESS) IO½Â����C55�ä´Ä��æ��^�. 3�� n � ESS ½Â�
ESS =
(N∑
i=1
(W i
n
))−1
.
�±�ù��)ºµÄu N �D����?1íä�±Cq�d (l�O����Ý) uÄul8I©ÙæÑ� ESS �IÐ����íä. ESS ��3 1 � N
Mó�Ü�²
12 1�Ù âfÈÅ�{0�
�m§==�§$u�½�K� NT �â?1æ�. Ï~ NT = N/2. ��±r������ä´ÄI�æ��IO. � W i
n = 1N�§� {W i
n} �����. ���u��K��§?1æ�.
Algorithm 3: Sequential Monte Carlo with Adaptive Resampling
3�� n = 11
for i = 1 to N do2
æ� X i1 ∼ q1(x1)3
O�� w1(X i1) Ú W i
1 ∝ w1(X i1)4
if æ�IO then5
æ� {W i1, X
i1}§�� N ����âf { 1
N, X
i
1}6
� {W i
1, Xi
1} ← { 1N
, Xi
1}7
else8
� {W i
1, Xi
1} ← {W i1, X
i1}9
end10
end11
for n = 2 to T do12
for i = 1 to N do13
æ� X in ∼ qn(xn|X
i
1:n−1) ¿�� X i1:n ← (X
i
1:n−1, Xin14
O�� αn(X i1:n) ¿� W i
n ∝Wi
n−1αn(X i1:n)15
if æ�IO then16
æ� {W in, X i
1:n}§�� N ����âf { 1N
, Xi
1:n}17
� {W i
n, Xi
n} ← { 1N
, Xi
n}18
else19
� {W i
1, Xi
1} ← {W in, X i
n}20
end21
end22
end23
�{�� πn(x1:n) �ü�Cq.
πn(x1:n) =N∑
i=1
W inδXi
1:n(x1:n), (1.3.16)
πn(x1:n) =N∑
i=1
Wi
nδX
i1:n
(x1:n) (1.3.17)
XJ3�� n vk?1æ�§ùü�ªf´���. �{��� Zn/Zn−1 ��
�Cq
Zn
Zn−1
=N∑
i=1
Wi
n−1αn
(X i
1:n
)
Mó�Ü�²
§1.4 Particle Filter 13
UY�Ä~f 1.3.1§� σ2 > 12�§k asymptotic variance
VSMC
[Zn
]Z2
n
=n
N
[(σ4
2σ2 − 1
)1/2
− 1
]
�
VIS
[Zn
]Z2
n
=1N
[(σ4
2σ2 − 1
)n/2
− 1
].
?1'�§SMC�O�ìC���X n�5O\§ IS�O����X n�êO
\. ~X§XJÀJ σ2 = 1.2§K�±����Ð��5©Ù qk(xk) ≈ πn(xk).3ù«�¹e§� n = 1000 �§IS �{7L¦^ N ≈ 2 × 1023 �âf5��VIS[Zn]
Z2n
= 10−2. , ���Ó��5U§VSMC[Zn]
Z2n
= 10−2§SMC �{==I�N ≈ 104 �âf§�� 19 ��U?.
§1.3.6 Summary
JÑ����� SMC �{§ 5S�/Cq {πn(x1:n)} Ú {Zn}.
• ÃØ3�o�¹e§Ñ�±Õáu�m n§l©Ù qn(xn|x1:n−1) ?1æ�¿��O αn(x1:n)§�{��mE,Ýج�X n �O\ O\.
• é?Û� k§�3X n > k duëY�æ�Ú½§¬¦�é πn(x1:k) �SMC Cq==d��üÕ�âf�¤. Ïd� n ���§Ø�U��éÜ©
Ù {πn(x1:n)} ���Ð� SMC Cq. ù3¢S�¥,ÏL*Cq πn(x1)�ØÓâf��ê�±éN´�uyù�:.
§1.4 Particle Filter
c¡J� SMC �{¢�þÒ´ SIS Úæ��{�(ܧ �«|^�Å
��é©Ù½½Â3©Ùþ�¼ê�êÆÏ"?1�O��{. ��ÄÈůK�§·�F"S�/O���©Ù {p(x1:n|y1:n)}n≥1 �ê�Cq. �L«�{ü§e¡JÑ��{3z���Ñ?1æ�§ ¢S¥§�í���ª´�
ESS �u��K��§âI�?1æ�.
§1.4.1 SMC for Filtering
�ò SMC �{A^�Èŵee§=O�Ñ©Ù {p(x1:n|y1:n)}n≥1 �ê�
Mó�Ü�²
14 1�Ù âfÈÅ�{0�
Cq§�I�ÀJµ
πn(x1:n) = p(x1:n|y1:n)
γ(x1:n) = p(x1:n, y1:n)
Zn = p(y1:n) (1.4.1)
nonumber (1.4.2)
éu�5©Ù§·�v7�ÀJ qn(x1:n)§ �I�ÀJ qn(xn|x1:n−1). Ï�Xc¤ã§3 IS �{p§·�ÀJ��©Ù´ qn(x1:n) ù«/ª§ �3 SIS �{p§·��I��5©Ùäk qn(xn|x1:n−1) ù«/ªÒ. XJ���z�� n ���5����§·�ATÀJ�`��5©Ùµ
qoptn (xn|x1:n−1) = πn(xn|x1:n−1Z)
= p(xn|yn, xn−1)
=g(yn|xn)f(xn|xn−1)
p(yn|xn−1)(1.4.3)
éA� incremental importance weight ´
αn(x1:n) = p(yn|xn−1)
3¢S¥§Ø�Ul qoptn (xn|x1:n−1) ?1æ�§ ´ATÀJXe/ª��5
©Ùµ
qn(xn|x1:n−1) = q(xn|yn, xn−1) (1.4.4)
éÜúª(1.3.9)§(1.3.11) Ú (1.4.4)§�±�Ñ incremental weight
αn(x1:n) = αn(xn−1:n) =g(yn|xn)f(xn|xn−1)
q(xn|yn, xn−1).
Mó�Ü�²
§1.4 Particle Filter 15
Algorithm 4: SMC for Filtering
3�� n = 11
for i = 1 to N do2
æ� X i1 ∼ q1(x1|y1)3
O�� w1(X i1) = µ(xi
1)g(y1|Xi1)
q(Xii |y1)
Ú W i1 ∝ w1(X i
1)4
æ� {W i1, X
i1}§�� N ����âf { 1
N, X
i
1}5
end6
for n = 2 to T do7
for i = 1 to N do8
æ� X in ∼ qn(xn|yn, X
i
n−1) ¿�� X i1:n ← (X
i
1:n−1, Xin)9
O�� αn(X in−1:n) = g(yn|Xi
n)f(Xin|X
in−1)
q(Xin|yn,Xi
n−1)¿� W i
n ∝ αn(X in−1:n)10
æ� {W in, X i
1:n}§�� N ����âf { 1N
, Xi
1:n}11
end12
end13
�z
[1] A.D. and A. Johansen, Particle filtering and smoothing: Fifteen years later, in Hand-
book of Nonlinear Filtering (eds. D. Crisan et B. Rozovsky), Oxford University Press,
2009. See http://www.cs.ubc.ca/~arnaud