Database Reference
In-Depth Information
Algorithm 5
Kalman filtering algorithm
Input
:
μ
t−
1
,
Σ
t−
1
,z
t
// predict the current state from previous values
m
t
=
Aμ
t−
1
P
t
=
A
Σ
t−
1
A
T
+
Q
// compute Kalman gain
K
=(
P
t
H
T
)(
H ∗ P
t
∗ H
T
+
R
)
−
1
// apply correction to predictions using new observation
μ
t
=
m
t
+
K
(
z
t
− Hm
t
)
Σ
t
=(
I − KH
)
P
t
Conditioning on
z
t
, we have the measurement update step from 10.7
and 10.8.
p
(
x
t
|
z
0:
t
)=
(
μ,
Σ)
K
=
HP
t
(
HP
t
H
T
+
R
)
−
1
μ
=
m
t
+
K
(
z
t
−
N
Hm
t
)
(10.12)
K
(
HP
t
)
T
Σ=
P
t
−
(10.13)
Where
K
is referred to as the Kalman Gain. The inference algorithm for
the KFM proceeds by first predicting the
t
+1
st
state of
x
which updates
the parameters as described in Eq. 10.8. Then, upon observing a new
measurement,
z
t
,weperformthe
measurement correction
step, which
updates the parameters for
x
t
according to equations 10.12 and 10.13.
We show the complete filtering inference algorithm for the KFM in al-
gorithm 5 for completeness. From the update equations above, we see
how updating belief states upon the arrival of new observations can be
computed eciently, using matrix multiplications and a matrix inverse
operation.
3.2.3 Smoothing.
Before we get into the smoothing equations
for the KFM, we first provide some notation below to identify the differ-
ent versions of parameters. The first line shows the filtered probability
distribution for
x
t
(explained in the previous section) which we still iden-
tify with the parameters
μ
and Σ. The second line shows the smoothed
probability distribution, for which we use the parameters
ν
and Φ for
the mean and covariance to differentiate from the filtered parameters.
p
(
x
t
|
z
1:
t
)=
N
(
x
t
;
μ
t
,
Σ
t
)
p
(
x
t
|
z
1:
T
)=
N
(
x
t
+1
;
ν
t
+1
,
Φ
t
+1
)
Smoothing utilizes observations from the past, present, and future to
provide an improved estimate of the belief state. Inference for smoothing
