Where
\( t \) represents a time index and
t = 1, 2, …;
x
t
∊
R
dx
represents the state of the hidden model (for instance, not observed);
y
t
∊
R
dy
represents the observation;
u
t
∊
R
du
and
v
t
∊
R
dv
represent the white noises that are not dependent on each other; and
\( g{:}\, R^{dx} \times R^{du} \to R^{dx} \) and
\( h{:}\, R^{dx} \times R^{dv} \to R^{dy} \) represent known functions. Alternatively, these equations can be represented by the state’s probability distributions,
\( p(x_{t} \left| {x_{t - 1} } \right.) \), and by the observation,
\( p(y_{t} \left| {x_{t} } \right.) \), which one can gather from (
1) and (
2) and the
u
t
and
\( v_{t} \) or probability distributions,
v
t
, respectively. The focus is on nonlinear models when the noises observed in (
1) and (
2) need not necessarily be Gaussian. The particle filter aims to sequentially estimate the state’s distributions, including the predictive distribution
\( p(x_{t} \left| {y_{1:t - 1} } \right.) \), the filtering distribution
\( p(x_{t} \left| {y_{1:t} } \right.) \), or the smoothing distribution,
\( p(x_{t} \left| {y_{1:T} } \right.) \), where
t <
T. The focus of this part is on the filtering distribution. The expression of this distribution can be based on the filtering distribution during time instant
\( t - 1, p(x_{t - 1} |y_{1:t - 1} ) \)—for instance, in a recursive form using;
$$ p\left( {x_{t} |y_{1:t} } \right) \propto \int p\left( {y_{t} |x_{t} } \right)p\left( {x_{t} |x_{t - 1} } \right)p\left( {x_{t - 1} |y_{1:t - 1} } \right) dx_{t - 1} , $$
(5)
where ∝ implies being ‘proportional to’. Except in rare cases, one cannot analytically implement this update. Thus, the authors have to approximate, while emphasising that with particle filter, the fundamental approximation is a representation of the continuous distributions by discrete random measures that are made up of particles
x
t
(m)
, which can be probable values of the unknown weights
w
t
(m)
and state
x
t
and were given to the particles. One can approximate the distribution
\( p(x_{t - 1} \left| {y_{1:t - 1} } \right.) \) using a random measure having the form
\( X_{t - 1} = \left\{ {x_{t - 1}^{\left( m \right)} , w_{t - 1}^{\left( m \right)} } \right\}_{m = 1}^{M} \), where
M represents the amount of particles, like in the example below:
$$ p\left( {x_{t - 1} |y_{1:t - 1} } \right) \approx \mathop \sum \limits_{m = 1}^{M} w_{t - 1}^{\left( m \right)} \delta \left(x_{t - 1} - x_{t - 1}^{\left( m \right)} \right) $$
(6)
where
δ(·) represents the Dirac delta impulse with all the weights adding up to one. Given this approximation, one can readily solve the integral in (
3) and express it as follows:
$$ p\left( {x_{t} |y_{1:t} } \right)\dot{ \propto }\,p\left( {y_{t} |x_{t} } \right)\mathop \sum \limits_{m = 1}^{M} w_{t - 1}^{\left( m \right)} \, p\left(x_{t} |x_{t - 1}^{\left( m \right)} \right) $$
(7)
where
\( \dot{ \propto } \) signifies ‘approximate proportionality’. The final expression is a demonstration of how the filtering distribution’s approximation
x
t
can be recursively obtained overtime. During time instant
t − 1, the development of
x
t
begins by the generation of particles
x
t
(m)
, which represent
\( p(x_{t} \left| {x_{t - 1} } \right.) \). This particle filter step is called particle propagation, since particle
x
t−1
(m)
moves forward through time and is considered the parent of
x
t
(m)
. The importance sampling concept is used for weight computation and particle propagation [
15]. Ideally, each of the propagated particles has to be taken from
\( p(x_{t} \left| {y_{1:t} } \right.) \) in order to obtain equal weights. However, this is not feasible for most cases, therefore necessitating the utilisation of an instrumental function
π(
x
t
) (as in [
32]), with the
p(
x
t
|
x
t−1) function. The particle filter’s second basic step is calculating particle weights. To find a right inference based on the generated particles, the theory shows how the generated particles from
π(
x
t
), are different from
p(
x
t
|
y
1:t
), and therefore need to be weighted [
33‐
35]. When working in mild conditions, one can demonstrate how these weights can be computed recursively based on:
$$ w_{t}^{(m)} \propto w_{t - 1}^{\left( m \right)} \frac{{p\left( {y_{t} |x_{t}^{\left( m \right)} } \right)p\left(x_{t}^{( m )} |x_{t - 1}^{(m)} \right)}}{{\pi \left( {x_{t}^{(m)} } \right)}}. $$
(8)