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saw the equivalence between Plackett formulation and KF formulation, see eq. 2
for EKF formulation, so every result, property, test, etc., from regression theory
is applicable. For example KF formulation is independent of
σ
2
and prediction
test from regression theory may be used as a outlier detection test. He gave a
procedure to sequentially compute the residual sum of squares (
RSS
), needed
for
σ
2
estimation, reaching a linear temporal complexity for the process.
To deal with non-constant variances weighted least squares may be used [7].
When
σ
n
=
σ
2
h
2
(
,being
y
n
the independent variable and
h
aknown
function, a two rounds procedure can be applied due to expected values are
usually unknown. Firstly, ordinary least squares, over the whole data set, are
applied to obtain estimated measures
E
[
y
n
])
y
i
and a second least squares round, over
the whole data set, is applied weighting data using
w
i
=
ˆ
1
|h
(
y
i
)
|
.Using
Peña's KF formulation and filter independence of
σ
2
, matrix covariance noise
remains
w
i
w
(ˆ
y
i
)=
, so data have not to be weighted if this matrix is used. In this work
we are going apply the two rounds procedure using KF, but rounds just involve
the new incoming data.
During initial experiments, with linear models, we observed that if weights
were significantly less than
at initial stages could lead to a bad result. In these
cases Kalman gains were too small, making filter response to innovation small
too. The solution was to scale weights, by a factor
ω
, in order to make initial
weight values near or over
1
,being
y
s
the seed.
Estimation of parameters and their covariances matrix are unaffected by this
scale factor. Noise deviation estimation is scaled by the same factor.
Equation 1 represents a non-linear model, so we need an EKF. The EKF
formulation, adapted from [10], is simply:
1
, for example,
ω
=max(
|
h
(
y
s
)
|
)
β
n
=
β
n−
1
ρ
n
r
n
=
θ
n
)
+
ε
n
(2)
cos(
ϕ
n
−
first equation is the state equation and second one is the measure equation. The
state parameters are
β
ρ
θ
and
ε
n
is gaussian distributed with zero mean
=
and covariances matrix
σ
n
=
σ
2
h
2
(
E
[
r
n
])
.
4 Clustering
The scale-space procedure, described in section 2, can be applied for range scan
segmentation. Fig. 3 a) shows a simulated world composed by plain walls and
a hedge, where noise was generated following the model explained in subsection
3.1 with
w
i
=
|
r
i
)
|
=
r
i
, so noise is proportional to measure. Measures on the
walls have a standard deviation proportional to the measure being of
h
(ˆ
2
cm.
at
10
m.
, standard deviation is 10 times higher on hedge.
In fig. 2 we can see super-imposed the polar representation of that world.
Segments edges are extrema and inflexions points of the curve, so first and sec-
ond derivative can be used to find them. Zero-crossing level curves for first
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