246
I. A. Kudryavtseva and K. A. Rybakov
the solution of the Duncan–Mortensen–Zakai (DMZ) equation [3, 17] for the unnormalized posterior probability density function of a random process whose trajectories are under estimation. Similar modifications can be developed for the particle
filter algorithm based on the robust DMZ equation [14] and for the estimation of
jump-diffusion processes as well [18].
Methods for solving the DMZ equation and the robust DMZ equation, among
which the continuous-time particle filter algorithm can be particularly distinguished
[4, 14], are given in [19–22].
According to the algorithm, it is supposed to simulate pairs including process
trajectories themselves driven by a known state equation and associated weight functions that determine the importance of trajectories in an estimate needed to find. On
the program implementation stage the overflow errors can appear because weight
functions increase rapidly as exponential functions.
In the theory, the standard technique of the normalization of weights is a perfect
instrument to gain the goal [3], but it fails in practice and the overflow errors can
occur. To avoid appearing the mentioned errors it is offered to apply the procedure
of taking logarithms, i.e. to switch from exponential functions to their exponents and
as consequence from multiplying to summarizing, with additional customization of
the exponents. Since relative values of weights but not their absolute values count
for much in formulae for the optimal estimate, the customization does not affect the
resulting estimate of unobserved process trajectories or the filtering problem solution
in other words.
Note that the time discretization and the use of discrete-time particle filter algorithms [3, 23, 24] reduce the risk of overflow errors. However, if we need to apply the
continuous-time particle filter algorithm, it should provide a lack of overflow errors.
The offered modification of the continuous-time particle filter algorithm is applied
to solve the tracking problem to find coordinates and velocities of an aircraft executing
a maneuver in the horizontal plane [6].
The remainder of this chapter is organized as follows. The optimal filtering
problem is considered in Sect. 17.2. The known continuous-time particle filter algorithms and the offered modification that provides the lack of overflow errors are
given in Sect. 17.3. Section 17.4 is devoted to the approbation of the new filtering
algorithm. The chapter is summarized in Sect. 17.5.
17.2 Problem Formulation
The optimal filtering problem for continuous-time stochastic systems is to find an
estimate of trajectories of an unobserved Markov random process X (t) from given
trajectories of an observed random process Y (t) in accordance with the given quality
criterion. The random processes X (t) and Y (t) satisfy the following system of Itô
SDEs:
dX (t) = f (t, X (t))dt + σ (t, X (t))dW (t), X (t 0 ) = X 0 ,
(17.1)
I. A. Kudryavtseva and K. A. Rybakov
the solution of the Duncan–Mortensen–Zakai (DMZ) equation [3, 17] for the unnormalized posterior probability density function of a random process whose trajectories are under estimation. Similar modifications can be developed for the particle
filter algorithm based on the robust DMZ equation [14] and for the estimation of
jump-diffusion processes as well [18].
Methods for solving the DMZ equation and the robust DMZ equation, among
which the continuous-time particle filter algorithm can be particularly distinguished
[4, 14], are given in [19–22].
According to the algorithm, it is supposed to simulate pairs including process
trajectories themselves driven by a known state equation and associated weight functions that determine the importance of trajectories in an estimate needed to find. On
the program implementation stage the overflow errors can appear because weight
functions increase rapidly as exponential functions.
In the theory, the standard technique of the normalization of weights is a perfect
instrument to gain the goal [3], but it fails in practice and the overflow errors can
occur. To avoid appearing the mentioned errors it is offered to apply the procedure
of taking logarithms, i.e. to switch from exponential functions to their exponents and
as consequence from multiplying to summarizing, with additional customization of
the exponents. Since relative values of weights but not their absolute values count
for much in formulae for the optimal estimate, the customization does not affect the
resulting estimate of unobserved process trajectories or the filtering problem solution
in other words.
Note that the time discretization and the use of discrete-time particle filter algorithms [3, 23, 24] reduce the risk of overflow errors. However, if we need to apply the
continuous-time particle filter algorithm, it should provide a lack of overflow errors.
The offered modification of the continuous-time particle filter algorithm is applied
to solve the tracking problem to find coordinates and velocities of an aircraft executing
a maneuver in the horizontal plane [6].
The remainder of this chapter is organized as follows. The optimal filtering
problem is considered in Sect. 17.2. The known continuous-time particle filter algorithms and the offered modification that provides the lack of overflow errors are
given in Sect. 17.3. Section 17.4 is devoted to the approbation of the new filtering
algorithm. The chapter is summarized in Sect. 17.5.
17.2 Problem Formulation
The optimal filtering problem for continuous-time stochastic systems is to find an
estimate of trajectories of an unobserved Markov random process X (t) from given
trajectories of an observed random process Y (t) in accordance with the given quality
criterion. The random processes X (t) and Y (t) satisfy the following system of Itô
SDEs:
dX (t) = f (t, X (t))dt + σ (t, X (t))dW (t), X (t 0 ) = X 0 ,
(17.1)