Database Reference
In-Depth Information
Table 10.1.
Notation
Notation Explanation
x
t
Belief state (hidden) for at time
t
X
Location component of belief state
X
Velocity component of belief state
z
0:
t
Observations from time 0
..t
A
Deterministic transition matrix
H
Deterministic observation mask
Q
Covariance matrix for the movement dynamics. Determines the amount
of uncertainty we have in our transition model.
R
Observation covariance matrix. Determines the amount of uncertainty
we have in our observations.
N
(
μ,
Σ)
Gaussian distribution with expected value
μ
and covariance matrix Σ.
locity (
x
t
=[
X,Y, X, Y
]) and the location measurement,
z
t
is described
by a position (
z
t
=[
X,Y
]).
z
1:
t
)=
p
(
z
t
|
x
t
)
p
(
x
t−
1
|
z
1:
t−
1
)
p
(
x
t
|
p
(
z
1:
t
)
∝
p
(
z
t
|x
t
)
p
(
x
t−
1
|z
1:
t−
1
)
(10.3)
The first term in Eq. 10.2 is the likelihood function which describes
the relationship between
x
t
and
z
t
(i.e. describes the sensor error).
For example, GPS sensors are typically assumed to have error that is
normally distributed about the true location. In this case,
p
(
z
t
|
x
t
)=
N
(
z
t
;
x
t
,σ
), where
σ
describes how much variation we expect to see in
the measurement.
The second term,
p
(
x
t−
1
z
1:
t−
1
), is the posterior distribution of
x
t−
1
.
That is, this term is the result of filtering at the previous time step. From
this equation, we see that it is possible to recursively update our belief
about the state of a mobile object online (as new data arrive). All of
the information about
x
t−
1
is captured in
p
(
x
t−
1
|z
1:
t−
1
), thus there is no
need to revisit old datum. Lastly, the denominator is the marginal prob-
ability of the sequence of observations. Since the observations remain
fixed (this data is observed), this term may be considered a normaliz-
ing constant and ignored for our purposes. For a readable introduction
to Bayesian statistics in a more general context, we refer the interested
reader to [30].
While tracking is an
online
problem (i.e. updates must be made as
data arrive), in general there are three types of inference we may be in-
terested in for any DBN: (i) prediction, (ii) filtering and (iii) smoothing.
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