Fusion of Non-Simultaneous Elements of Information: Temporal Fusion - Information Fusion in Signal and Image Processing - page 239

( v + v − ) over

Note that the interval used to model the error has increased by Δ t · −

the course of the prediction:
x + ( t )

x − ( t ) <x + ( t +Δ t ) x − ( t +Δ t ) − −

< v + Δ t + x + ( t ) − v − · Δ t + x − ( t ) ·

< x + ( t ) x − ( t ) +Δ t · v + v −

− −

Evolutionary models come in very diverse forms, such as differential equations,
recursive equations, evolution graphs, logical rules and others. They are merely
approximations of the evolution of the observed variables. Therefore, they lead to
errors in the value of the predicted data, errors that need to be estimated and taken into
account during the fusion operation with the measured value. Generally speaking, the
evolutionary model M X of the variable X and of the confidence Conf X assigned to
it is defined in terms of accuracy and/or reliability, by equation [11.1], to obtain the
predicted values X of this variable and of the associated confidence Conf

X :

M X X ( t )

, Δ t = X ( t +Δ t )
Conf X ( t +Δ t )

[11.1]

Conf X ( t )

For example, let us assume that we are observing a discrete system described by
the following rule:

if X ( t )= E 1 , then X ( t +Δ t )= E 2
where E 1 and E 2 are two possible states. We can introduce uncertainty in this modeled
rule by using a probability distribution:
if X ( t )= E 1 , then p X ( t +Δ t )= E 1 = p 1 and p X ( t +Δ t )= E 2 = p 2

This rule can be interpreted as the definition of a conditional probability:
p X ( t +Δ t )= E 1 /X ( t )= E 1 = p 1
p X ( t +Δ t )= E 2 /X ( t )= E 1 = p 2

If we know the probability for the state to be E 1 at t , described by p ( X ( t )= E 1 )=
p , then, by conditioning, we infer:
p X ( t +Δ t )= E 1 = p X ( t +Δ t )= E 1 /X ( t )= E 1 ·

p X ( t )= E 1 = p 1 ·

p

p X ( t +Δ t )= E 2 = p X ( t +Δ t )= E 2 /X ( t )= E 1 · p ( X ( t )= E 1 = p 2 .p

In addition, we notice here that the uncertainty has increased, since the probability
initially assigned to the state E 1 was divided at the time t +Δ t between the states E 1
and E 2 .