PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 302

The method proceeds as follows. Let the random measure at time instant n 2 1be
given by x ( n 1) ¼ {x

( m ) ( n 1), 1 =M } m¼ 1 . In the first step, we draw samples x ( m )

c

( n )

from the prior, that is

( m )
c

( m ) ( n 1)) :

x ( n ) f (x( n ) j x

For the obtained samples, we compute their weights according to
w ( m ) ( n ) ¼ f ( y( n ) j x

( m )
c

( n ))

( m )

and then we normalize them. Now, the assumption is that the particles x

c ( n ) and their
weights w ( m ) ( n ) approximate a normal distribution whose moments are estimated by

m ( n ) ¼ X

M

w ( m ) ( n ) x

( m )
c

( n )

m¼ 1

S ( n ) ¼ X

M

( n ) m ( n )) ` :

w ( m ) ( n )(x

( m )
c

( m )
c

( n ) m ( n ))(x

m¼ 1

Finally, the particles that are used for propagation at the next time instant n þ 1are
generated by

( m ) ( n ) N ( m ( n ), S ( n ))

x

and they are all assigned the same weights. The method is summarized by Table 5.5.

5.5.4 Comparison of the Methods

In this Subsection, we show the performances of the presented methods (SIR, APF,
and GPF) on simulated data.

B EXAMPLE 5.7
The data are considered to be generated according to a stochastic volatility model,
which is defined by [6, 16, 31, 47]

x ( n ) ¼ ax ( n 1) þ v 1 ( n ) (5 : 29)

y ( n ) ¼ b v 2 ( n ) e x( n ) = 2 (5 : 30)

where the unknown state variable x ( n ) is called log-volatility, v 1 ( n ) N (0, s 2 ),
v 2 ( n ) N (0, 1), and a and b are parameters known as persistence in volatility
shocks and modal volatility, respectively. The Gaussian processes v 1 ( n ) and
v 2 ( n ) are assumed independent. This model is very much researched in the econo-
metrics literature. Given the observations y ( n ) and the model parameters, we want
to estimate the unobserved state x ( n ).