17 Modified Continuous-Time Particle Filter Algorithm …
247
dY (t) = c(t, X (t))dt + ζ(t)dV (t), Y (t 0 ) = Y 0 = 0,
(17.2)
where X is an n-dimensional state vector, Y is an m-dimensional measurement
vector, W (t) and V (t) are the s-dimensional and d-dimensional independent standard
Wiener processes, respectively, f (t, x), σ (t, x), c(t, x), ζ(t) are given vector-valued
and matrix-valued functions with corresponding dimensions (the matrix ζ(t)ζ
T
(t)
must be nondegenerate), t ∈ [t 0 , T ]. Distribution of the vector X 0 is determined
by the probability density function ϕ 0 (x). Equation 17.1 is the state equation while
Eq. 17.2 is the measurement equation.
Coefficients in Eqs. 17.1 or 17.2 should satisfy the conditions on the existence
and uniqueness of the solution of SDEs. According to [3], we assume that f (t, x),
σ (t, x), c(t, x) are the Lipschitz functions with respect to x. Moreover, E|X 0 |
2
< ∞,
where E denotes the mean.
We use MMSE criterion, this means ˆ
X (t) = E[X (t)|Y
t
0 ]. Such an estimate
provides the minimum value E|X (t) − ˆ
X (t)|
2 for all t ∈ [t 0 , T ].
Further, to form a sequence of relationships when solving the optimal filtering
problem it is convenient to express Eq. 17.2 in the Langevin form:
Z (t) = ˙
Y (t) = c(t, X (t)) + ζ(t)N (t),
(17.3)
where N (t) is a standard Gaussian white noise corresponding to the Wiener process
V (t). Since Z (t) and Y (t) are interchangeable in models of the considered type, the
estimate of trajectories X (t) can be found using measurements Z (t), i.e. ˆ
X (t) =
E[X (t)|Z
t
0 ].
17.3 Continuous-Time Particle Filter Based on the DMZ
Equation
Simulating an ensemble of continuous-time stochastic system trajectories underlies
the particle filter algorithm that will be given below. Moreover, it can be possible
to construct an algorithm, where it is required to simulate trajectories of an auxiliary stochastic system whose mathematical model is formed with the system model
defined by Eq. 17.1 and coefficients of Eqs. 17.2 or 17.3 [14]. Each trajectory is
assigned a weight function whose values are calculated using given measurements.
The estimation of a trajectory is carried out by applying the statistical treatment of
results to calculate the weighted mean. Other statistical characteristics, for example,
the mode, the posterior distribution function and the posterior probability density
function can also be found from the same simulation results. Such a group of methods
is named by particle filters. Behavior of a particle is determined by an ordered pair
(trajectory, weight function). The continuous-time particle filter is described in more
detail in [3].
247
dY (t) = c(t, X (t))dt + ζ(t)dV (t), Y (t 0 ) = Y 0 = 0,
(17.2)
where X is an n-dimensional state vector, Y is an m-dimensional measurement
vector, W (t) and V (t) are the s-dimensional and d-dimensional independent standard
Wiener processes, respectively, f (t, x), σ (t, x), c(t, x), ζ(t) are given vector-valued
and matrix-valued functions with corresponding dimensions (the matrix ζ(t)ζ
T
(t)
must be nondegenerate), t ∈ [t 0 , T ]. Distribution of the vector X 0 is determined
by the probability density function ϕ 0 (x). Equation 17.1 is the state equation while
Eq. 17.2 is the measurement equation.
Coefficients in Eqs. 17.1 or 17.2 should satisfy the conditions on the existence
and uniqueness of the solution of SDEs. According to [3], we assume that f (t, x),
σ (t, x), c(t, x) are the Lipschitz functions with respect to x. Moreover, E|X 0 |
2
< ∞,
where E denotes the mean.
We use MMSE criterion, this means ˆ
X (t) = E[X (t)|Y
t
0 ]. Such an estimate
provides the minimum value E|X (t) − ˆ
X (t)|
2 for all t ∈ [t 0 , T ].
Further, to form a sequence of relationships when solving the optimal filtering
problem it is convenient to express Eq. 17.2 in the Langevin form:
Z (t) = ˙
Y (t) = c(t, X (t)) + ζ(t)N (t),
(17.3)
where N (t) is a standard Gaussian white noise corresponding to the Wiener process
V (t). Since Z (t) and Y (t) are interchangeable in models of the considered type, the
estimate of trajectories X (t) can be found using measurements Z (t), i.e. ˆ
X (t) =
E[X (t)|Z
t
0 ].
17.3 Continuous-Time Particle Filter Based on the DMZ
Equation
Simulating an ensemble of continuous-time stochastic system trajectories underlies
the particle filter algorithm that will be given below. Moreover, it can be possible
to construct an algorithm, where it is required to simulate trajectories of an auxiliary stochastic system whose mathematical model is formed with the system model
defined by Eq. 17.1 and coefficients of Eqs. 17.2 or 17.3 [14]. Each trajectory is
assigned a weight function whose values are calculated using given measurements.
The estimation of a trajectory is carried out by applying the statistical treatment of
results to calculate the weighted mean. Other statistical characteristics, for example,
the mode, the posterior distribution function and the posterior probability density
function can also be found from the same simulation results. Such a group of methods
is named by particle filters. Behavior of a particle is determined by an ordered pair
(trajectory, weight function). The continuous-time particle filter is described in more
detail in [3].