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K /7 = P n H n R- 1 = MX(H, 7 MX+R, 7 )~' (8)
5. Calculate the most likely-hood(or post-) estimation value of the state vector x by
** =x lf +K ll (z /f -H fl xJ " (9)
6. Return to 2. after renewal of the time step.
Monte Carlo Filter(MCF)
The MCF can be applied even if the state space model is non-linear and non-Gaussian. In the MCF, the
probabilistic nature of the state vector is described by many realizations instead of first and second
moments as defined in the KF. However, in the MCF, the observation noise vector v is assumed to be
uniquely determined by a function G differentiate with respect to the observation vector z n as
follows:
\ n =//"'(z /7 ,xJ = G(z /7 ,x /7 ) (10)
The MCF is an algorithm to identify the conditional distribution function p(\ n |Z,,)of the state
variable x n when observation values Z = {r, •• = n }up to the present time n are given. Then, we define
P(\ n |Z,,_,) as the predictor distribution and p(\ n \ Z /7 ) as the filter distribution. In this paper, each
probability density function of each distnbution is approximated by finite realizations composed of
m particles as follows:
The MCF algorithm is defined by the following steps.
1. Generate a initial set (j = l~m) of random number f^ obeying an arbitrary probability
density function p Q (f 0 )
2. Repeat the following steps until the end of time steps.
(a) Generate a set of random number w|/' obeying probability density function q(w n )
(b) Compute a state transfer of particles by
(c) Compute the likelihood value of each particle by
a ( n j] = r(G(z /7 ,b J/')) —— (14)
n
(d) Generate f ( n ]) by resampling of b (/) (A)
fu) = b a» with probability -^— (15)
(e) Return to (a) Yor (n