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Nonlinear Filtering and Path Integral Method (Paper Review)

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Nonlinear Filtering and Path Integral Method (Paper Review)

1. 1. Brief Paper ReviewB.Balaji (2009)“Nonlinear filtering and quantum physics A Feynman path integral perspective”Kohta IshikawaOct 30. 2011 3 1
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3. 3. xk = f (xk 1 , tk 1 )xk 1 + e(xk 1 , tk 1 ) System Modelyk = h(xk , tk ) + wk Observation Modeldx(t) = f (x(t), t)dt + e(x(t), t)dw(t) System Model(SDE)y(tk ) = h(x(tk ), tk ) + wk Observation Modeldx(t) = f (x(t), t)dt + e(x(t), t)dv(t) System Model(SDE)dy(t) = h(x(t), t)dt + dw(t) Observation Model(SDE) 3
4. 4. m X d (t, x) = LY (t, x)dt + hi (x) (t, x)dyi (t) i=1 n X @ n X @ 2 (t, x) m X 1 1LY (t, x) = (fi (x) (t, x)) + h2 (x) (t, x) i i=1 @xi 2 i=1 @x2 i 2 i=1 (0, x) = 0 (x) dy(t) 4
5. 5. dy(t) m ! X u(t, x) = exp hi (x)yi (t) (t, x) i=1Then, DMZ equation translates to 0 1 n X @ 2 u(t, x) n X m X @u(t, x) 1 @ fi (x) + @hj (x) A @u(t, x) = + yj (t) @t 2 i=1 @x2 i i=1 j=1 @xi @xi ✓X n m X m X @fi (x) 1 1 + h2 (x) i yi (t) hi (x) i=1 @xi 2 i=1 2i=1 m n XX m X X n ◆ @hi (x) 1 @hi (x) @hj (x) + yi (t)fj (x) yi (t)yj (t) u(t, x) i=1 j=1 @xj 2 i,j=1 @xk @xk k=1 u(0, x) = 0 (x) 5
6. 6. ⇢ y(⌧l ) post-measurement formy(t) ! y(⌧l 1 ) pre-measurement form 6
7. 7. ⌧l 1  t  ⌧l {⌧0 , ⌧1 , · · · } 0 1 n n m@ul (t, x) 1 X @ 2 ul (t, x) X @ X @hj (x) A @ul (t, x) = 2 + fi (x) + yj (⌧l ) @t 2 i=1 @xi i=1 j=1 @xi @xi ✓X n m X m X @fi (x) 1 1 + h2 (x) i yi (⌧l ) hi (x) i=1 @xi 2 i=1 2i=1 m n XX m X X n ◆ @hi (x) 1 @hi (x) @hj (x) + yi (⌧l )fj (x) yi (⌧l )yj (⌧l ) ul (t, x) i=1 j=1 @xj 2 i,j=1 @xk @xk k=1ul (⌧l , x) = ul 1 (⌧l 1 , x) 7
8. 8. m ! X (t, x)ul (t, x) = exp˜ yi (⌧l )hi (x) ul (t, x) i=1 n X @ 2 ul (t, x) n X@ ul (t, x) ˜ 1 ˜ @ ul (t, x) ˜ = fi (x) @t 2 i=1 @x2 i i=1 @xi n m ! X @fi (x) 1 X + h2 (x) ul (t, x) i ˜ i=1 @xi 2 i=1 m ! X ul (⌧l ˜ 1 , x) = exp yi (⌧l )hi (x) ul 1 (⌧l 1 , x) i=1 yi (⌧l 1) 8
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10. 10. Stochastic Processprobability density of state path measure, functional differentiation Fokker-Planck equation Path Integral representation green’s function 10
11. 11. SDE for continuous filtering xi (t) = fi (x(t)) + vi (t) ˙ yi (t) = hi (x(t)) + wi (t) ˙Probability density Z P (t, x, y|t0 , x0 .y0 ) = [d⇢(v(t))][d⇢(w(t))] (x(t) ˙ f (x(t)) v(t)) n ⇥ [dx(t)] (x(t) ˙ f (x(t)) v(t)) x(t) (y(t) ˙ h(x(t)) w(t)) m ⇥ [dy(t)] (y(t) ˙ h(y(t)) w(t)) y(t) n m ⇥ (x(t) x)|x(t0 )=x0 (y(t) y)|y(t0 ) = y0 11
12. 12. ! 1 XZ t n [d⇢(v(t))] = [Dv(t)] exp 2~v vi (t)2 dt ~v , ~w i=1 t0 ! 1 XZ t m [d⇢(w(t))] = [Dw(t)] exp vi (t)2 dt 2~w i=1 t0 Z y(t)=y Z x(t)=xP (t, x, y|t0 , x0 .y0 ) = [Dv(t)][Dw(t)][Dx(t)][Dy(t)] y(t0 )=y0 x(t0 )=x0 ! 1 XZ t n XZ t n @fi (x(t)) 1 XZ t m 2 2 ⇥ exp vi (t)dt dt wi dt 2~v i=1 t0 i=1 t0 @xi 2~w i=1 t0 n ⇥ (x(t) f (x(t)) v(t)) m (y(t) h(y(t)) w(t)) ˙ ˙ Z y(t)=y Z x(t)=x = [Dx(t)][Dy(t)] exp ( S) y(t0 )=y0 x(t0 )=x0 Z " n n m # 1 t 1 X X @fi (x(t)) 1 X 2S= dt (xi (t) ˙ fi (x(t))) + + ( yi ˙ hi (x(t)))2 2 t0 ~v i=1 i=1 @xi ~w i=1 12
13. 13. measurement term in the action S Z ti m X Z ti m X⇥ 1 1 ⇤ dt ( yi ˙ hi (x(t))) = dt ˙2 yi (t) + h2 (x(t)) i 2hi (x(t))yi (t) ˙ 2~w ti 1 i=1 2~w ti 1 i=1relevant term Z ti m X ⇢ 1 Pm 1 ~w j=1 hj ([x(ti ) + x(ti Pm 1 )]/2)[yj (tj ) yj (tj 1 )] dt hj (x(t))yj (t) ⇠ ˙ 1~w ti 1 j=1 ~w j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj 1 ) yj (tj 2 )] 13
14. 14. leads to approximated probability densityP (ti , xi , yi |ti 1 , xi 1 , yi 1 ) ˜ ⇠ P (ti , xi |ti 1 , xi 1 ) ⇣ Pm ⌘ ⇢ 1 exp j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj ) yj (tj 1 )] ⇥ ⇣ ~w Pm ⌘ 1 exp ~w j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj 1 ) yj (tj 2 )] Z x(ti )=xi ˜ P (ti , xi |ti 1 , xi 1 ) = [Dx(t)] exp( S(ti 1 , ti )) x(ti 1 )=xi 1 Z " n n m # 1 ti 1 X X @fi (x(t)) 1 X 2 S(ti 1 , ti ) = dt (xi (t) ˙ fi (x(t))) + + h2 (x(t)) i 2 ti 1 ~v i=1 i=1 @xi ~w i=1˜P (t, x|ti 1 , xi 1 ) 14
15. 15. Z x(t)=x ✓ ◆˜ 1P (t, x|t0 , x0 ) = [Dx(t)] exp S x(t0 )=x0 ~v Z " n n m # 1 t X @fi (x(t)) ~v X XS= dt + [x2 (t) ˙i ˙ fi2 (x) 2xi (t)fi (x(t))] + ~v + h2 (x(t)) i 2 t0 i=1 i=1 @xi ~w i=1 Z t "X n n X @fi (x(t)) m # 1 ~v X 2 = dt [x2 (t) + fi2 (x)] + ~v ˙i + hi (x(t)) 2 t0 i=1 i=1 @xi ~w i=1 X Z x(t) n dxi (t)fi (x(t)) i=1 x(t0 ) Z t XZ n x(t) L=T V 1 ⌘ dtL dxi (t)fi (x(t)) Z t n X 2 1 t0 i=1 x(t0 ) T = dt x2 (t) ˙i 2 t0 i=1 " # 1 Z t X n @fi (x(t)) ~v m X V = dt fi2 (x) + ~v + h2 (x(t)) i 2 t0 i=1 @xi ~w i=1 15
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18. 18. dx = (b(x, t) + Bu)dt + dv ⌧ Z tf ✓ ◆ 1C(xint , tint , u) = (x(tf )) + dt u(t)T Ru(t) + V (x(t), t) tint 2 xintJ(x, t) = min C(x, t, u(t ! tf )) u(t!tf ) 18
19. 19.  T  @J 1 @J @J @J 1 @2J = BR 1 B T + V + bT + Tr ⌫ 2 @t 2 @x @x @x 2 @x Z ✓ ◆ SJ(x, t) = log [Dx(t)] exp x(t)=x Z tf ✓ ◆ 1S = (x(tf )) + d⌧ (x(⌧ ) ˙ b(x(⌧ ), ⌧ ))T R(x(⌧ ) ˙ b(x(⌧ ), ⌧ )) + V (x(⌧ ), ⌧ ) t 2V (x(t), t) 19
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