Geoscience Reference
In-Depth Information
median function g
used for modeling the respective estimator. More applications
to classification and regression are pointed out in Bertsimas and Shioda (
2007
).
7.3.2
Ingredients for Analyzing Hyperplane Location Problems
7.3.2.1
Distances Between Points and Hyperplanes
A hyperplane is given by its normal vector a
D
.a
1
;:::;a
D
/
2
D
and a real
R
number b
2
R
:
D
W
a
t
x
C
b
D
0
g
:
H
a;b
Df
x
2
R
D
and
a hyperplane H
a;b
is given as d.H
a;b
;v/
D
min
f
d.x;v/
W
a
t
x
C
b
D
0
g
.Forthe
vertical distance (see again the left part of Fig.
7.1
) the following formula can easily
be computed:
D
D
!
Given a distance d
W
R
R
R
, the distance between a point v
2
R
Lemma 7.1 (Schöbel
1999a
)
8
<
j
a
t
v
C
b
j
a
D
if
a
D
6D
0
0
if
a
D
D
0
and
a
t
v
C
b
D
0
1
if
a
D
D
0
and
a
t
v
C
b
6D
0
d
ver
.H
a;b
;v/
D
:
The second case and the third case comprise the case of a hyperplane which is
vertical itself. Its distance to a point v is defined as infinity unless the hyperplane
passes through v. If not all existing points lie in one common vertical hyperplane,
this means that a vertical hyperplane can never be an optimal solution to the
hyperplane location problem, hence without loss of generality we can assume the
hyperplane H
a;b
to be non-vertical if the vertical distance is used.
If d is derived from a norm or a gauge
W
D
!
, the following formula for
computing d.H
a;b
;v/has been derived in Plastria and Carrizosa (
2001
).
Lemma 7.2 (Plastria and Carrizosa
2001
)
d.H
a;b
;v/
D
(
a
t
v
C
b
R
R
if
a
t
v
C
b
0
ı
.a/
a
t
v
b
ı
.a/
if
a
t
v
C
b<0;
where
ı
W
R
D
!
R
is the dual (polar) norm common in convex analysis (e.g.,
Rockafellar
1970
), i.e.,
ı
.v/
D
sup
f
v
t
x
W
.x/
1
g
:
Note that d.H
a;b
;v/
D
j
a
t
v
C
b
j
if is a norm.
ı
.a/
