Pattern Recognition and Machine Learning

13.3. Linear Dynamical Systems 643

μnew 0 = E[z 1 ] (13.110)

Vnew 0 = E[z 1 zT 1 ]−E[z 1 ]E[zT 1 ]. (13.111)

Similarly, to optimizeAandΓ, we substitute forp(zn|zn− 1 ,A,Γ)in (13.108)

using (13.75) giving

Q(θ,θold)=−

N− 1

2

ln|Γ|

−EZ|θold

[

1

2

∑N

n=2

(zn−Azn− 1 )TΓ−^1 (zn−Azn− 1 )

]

+const (13.112)

in which the constant comprises terms that are independent ofAandΓ. Maximizing
Exercise 13.33 with respect to these parameters then gives

Anew =

(N

∑

n=2

E

[

znzTn− 1

]

)(N

∑

n=2

E

[

zn− 1 zTn− 1

]

)− 1

(13.113)

Γnew =

1

N− 1

∑N

n=2

{

E

[

znzTn

]

−AnewE

[

zn− 1 zTn

]

−E

[

znzTn− 1

]

Anew+AnewE

[

zn− 1 zTn− 1

]

(Anew)T

}

. (13.114)

Note thatAnewmust be evaluated first, and the result can then be used to determine

Γnew.

Finally, in order to determine the new values ofCandΣ, we substitute for

p(xn|zn,C,Σ)in (13.108) using (13.76) giving

Q(θ,θold)=−

N

2

ln|Σ|

−EZ|θold

[

1

2

∑N

n=1

(xn−Czn)TΣ−^1 (xn−Czn)

]

+const.

Exercise 13.34 Maximizing with respect toCandΣthen gives

Cnew =

(N

∑

n=1

xnE

[

zTn

]

)(N

∑

n=1

E

[

znzTn

]

)− 1

(13.115)

Σnew =

1

N

∑N

n=1

{

xnxTn−CnewE[zn]xTn

−xnE

[

zTn

]

Cnew+CnewE

[

znzTn

]

Cnew

}

. (13.116)