252
I. A. Kudryavtseva and K. A. Rybakov
where γ k is determined sequentially for each time moment t k by the formula:
γ k = max
i=1,2,...,M
{ln ω
i
k + μ(t k , X
i
k , Z k )h}.
The described technique, as well as, the standard normalization does not affect
the final result because all weights are multiplied by the same factor e
−γ k .
Continuous-time particle filter algorithm based on taking logarithms (Algorithm 3)
1. See Step 1 of Algorithm 1.
2. See Step 2 of Algorithm 1.
3. Obtain a realization of the state vector at t k+1 and update the corresponding
weight:
X
i
k+1 = F(t k , X
i
k , h),
ω
i
k+1 = exp{ln ω
i
k + μ(t k , X
i
k , Z k )h − γ k },
where
γ k = max
i=1,2,...,M
{ln ω
i
k + μ(t k , X
i
k , Z k )h}.
4. See Step 4 of Algorithm 1.
Note that these algorithms include modeling the trajectory of a random process
X (t) as well as its measurements Y (t) and Z (t) on Step 2 (and also modeling the
initial state vector X 0 on Step 1). It is needed for the simulation purpose only. In the
real filtering problem, the random process X (t) is unobservable, and measurements
Y (t) or Z (t) are given.
In Algorithm 3, at least one particle has weight 1 for all k = 0, 1, . . . , N , and this
weight is maximal in the set W k , therefore, M k = 0. Thus, this algorithm provides
not only the lack of overflow errors but also the lack of underflow errors, when all
weights become zero in one step. It is important to emphasize that the proposed
modification of the continuous-time particle filter does not exclude the resampling
procedure, but complements it. In Algorithms 1–3, the resampling procedure is not
described only for briefness.
All algorithms mentioned above are implemented in Mathcad computer algebra
system and grouped in the software modules. This software permits to solve estimating problems (filtering, smoothing, and prediction problems) for multidimensional continuous-time stochastic systems. In addition to the described algorithms,
the various numerical methods for solving SDEs such as Euler–Maruyama method,
Runge–Kutta type methods, Milstein’s method, Kuznetsov’s method, Platen and
Rosenbrock type methods have been implemented in the developed software. Moreover, MAP estimator [14, 34] can also be used for solving the optimal filtering
I. A. Kudryavtseva and K. A. Rybakov
where γ k is determined sequentially for each time moment t k by the formula:
γ k = max
i=1,2,...,M
{ln ω
i
k + μ(t k , X
i
k , Z k )h}.
The described technique, as well as, the standard normalization does not affect
the final result because all weights are multiplied by the same factor e
−γ k .
Continuous-time particle filter algorithm based on taking logarithms (Algorithm 3)
1. See Step 1 of Algorithm 1.
2. See Step 2 of Algorithm 1.
3. Obtain a realization of the state vector at t k+1 and update the corresponding
weight:
X
i
k+1 = F(t k , X
i
k , h),
ω
i
k+1 = exp{ln ω
i
k + μ(t k , X
i
k , Z k )h − γ k },
where
γ k = max
i=1,2,...,M
{ln ω
i
k + μ(t k , X
i
k , Z k )h}.
4. See Step 4 of Algorithm 1.
Note that these algorithms include modeling the trajectory of a random process
X (t) as well as its measurements Y (t) and Z (t) on Step 2 (and also modeling the
initial state vector X 0 on Step 1). It is needed for the simulation purpose only. In the
real filtering problem, the random process X (t) is unobservable, and measurements
Y (t) or Z (t) are given.
In Algorithm 3, at least one particle has weight 1 for all k = 0, 1, . . . , N , and this
weight is maximal in the set W k , therefore, M k = 0. Thus, this algorithm provides
not only the lack of overflow errors but also the lack of underflow errors, when all
weights become zero in one step. It is important to emphasize that the proposed
modification of the continuous-time particle filter does not exclude the resampling
procedure, but complements it. In Algorithms 1–3, the resampling procedure is not
described only for briefness.
All algorithms mentioned above are implemented in Mathcad computer algebra
system and grouped in the software modules. This software permits to solve estimating problems (filtering, smoothing, and prediction problems) for multidimensional continuous-time stochastic systems. In addition to the described algorithms,
the various numerical methods for solving SDEs such as Euler–Maruyama method,
Runge–Kutta type methods, Milstein’s method, Kuznetsov’s method, Platen and
Rosenbrock type methods have been implemented in the developed software. Moreover, MAP estimator [14, 34] can also be used for solving the optimal filtering