Geoscience Reference
In-Depth Information
to using the approximating distances. It is important to consider both cause and
effects in order to get the whole picture. There are a number of ways to measure
error effects; further, the magnitude of aggregation effects can depend on the model
structure—for the same aggregation, some models can have more error than others.
What is clear, in any case, is that the way to minimize DP aggregation error is to not
aggregate DPs—certainly this is what we recommend when it is feasible.
The ideal
way to aggregate DP data is not to aggregate it
.
If DP data must be aggregated, then we need to consider aggregation error
measures. We list and summarize ten such measures in Table
18.2
. All these error
measures have an ideal value of zero. One simple way to measure aggregation
error is to consider
ADP-DP distances
. If these distance values are all zero then
ADPs and DPs are identical, so there is no error. Later we establish a relationship
between ADP and DP distances and other error measures, including the
distance
difference error
. For the PMM, this distance difference error leads to an error we
call
DP error
. Like the difference error, the DP error can be negative or positive.
Still considering the PMM, note that the
total DP error e
(
S
)inTable
18.2
satisfies
e.S/
D
f.S
W
V
0
/
f.S
W
V/, the difference between the aggregated PMM and
the original model. Even though no DP error is zero, the total DP error can be zero or
nearly zero, since negative errors can cancel out positive errors—this is the concept
Table 18.2
Various demand point aggregation error measures for a location model
f
(
S
:
V
). Ideal
error measures have value zero for all
j
and all
S
No.
Error name
Error definition
d
v
j
;v
j
; j2J
1
ADP-DP distances
D
S;v
j
D
S;v
j
; j2J,all
S
2
Distance difference error
w
j
h
D
S;v
j
D
S;v
j
i
;j2J,all
S
3
DP error, PMM
e
j
.S/ D
e.S/ D
P˚
e
j
.S/ W j2J
,all
S
4
Total DP error, PMM
abc
i
.S/ D !
i
D
.
S;
i
/
P
˚
w
j
D
S;v
j
W j2J
i
,
all
S
5
ABC error for PMM:
J
1
, :::,
J
q
is a partition of
J
Df1, :::,
n
g;!
i
P
˚
w
j
W j2J
i
for
i
D 1, :::,
q
ae.S/ Djf .S W V
0
/ -f .S W V/j,all
S
6
Absolute error, any location
model
7
Relative error, for all
S
with
f
(
S
:
V
) >
0
rel.S/ D ae.S/=f .S W V/,all
S
mae
.
f
0
;f
/
D max fae.S/ W S; Sǝ; jSjDpg
8
Maximum absolute error
9
Error bound
eb
A number
eb
with
ae
(
S
)
eb
for all
S,
jf.S W V
0
/=f.S W V/-1j
eb
=f .S W V/;
jf.S W V/=f.S W V
0
/-1j
eb
=f .S W V
0
/ for all
S
Ratio error bounds (when
f
(
S
:
V
),
f
(
S
:
V
0
) > 0)
a measure,
diff
(
S
0
,
S*
), of the “difference”
between
p
-servers
S
0
and
S*
10
Location error
