Civil Engineering Reference
In-Depth Information
Table 4.1
Variables and notation used in sequential CART.
Parameter
Description
Approximate level set:
L
={
x
∈
i
A
(
i
)
}
where
A
(
i
)
q
,
p
is the
q
th
sub-region of the CART partition on
R
n
from iteration
i
that has
L
q
,
p
parent subregion
p
p
; also note that
A
(0
0
is the root
sub-region and has no parent sub-region, which is defined over [0, 1]
d
; when the parent sub-region is not important, we simply
refer to sub-regions with subscript only (e.g.,
A
(
i
)
A
(
i
)
q
,
p
q
th
sub-region that has a parent subregion
p
from sampling iteration
i
, and thus
A
(
i
)
q
,
p
A
(
i
−
1)
⊆
q
,
p
instead of
A
(
i
)
q
,
p
)
B
(
i
)
q
Sub-region boundaries of
A
(
i
)
q
(or
A
(
i
)
q
,
p
) which has length 2
d
and specifies the minimum and maximum value of the bounding
hyperrectangle in each of the
d
dimensions; the first
d
components (1,
...
,
d
) specify the lower bounds in each dimension, and the
second
d
components (
d
+
1,
...
,2
d
) specify the upper bounds in each dimension
X
(
i
)
q
,
p
Experimental design points scattered over
A
(
i
)
q
,
p
based on some sampling method (i.e., Latin hypercube sampling); has size
n
q
and
each design point has length
d
;
x
q
,
k
refers to the
kth
design point in
X
(
i
)
q
,
p
Y
(
i
)
q
,
p
Response (numerical value) of experimental design points
X
(
i
)
q
,
p
also of size
n
q
(i.e., responses to design points in
A
(
i
)
q
,
p
);
y
q
,
k
refers
to the
kth
response of design point
x
q
,
k
Z
(
i
)
q
,
p
Binary variable based on responses
Y
(
i
)
ˉ
L
=
ˆn
where
0
<ˆ<
1 indicates the proportion of the
n
points considered as
low
(here,
n
is the number of points sampled at any time during
sequential experimentation that fall into subregion
A
(
i
)
q
,
p
indicating which responses are low points
ˉ
L
and high points
ˉ
H
;
q
,
p
, and this may change as experimentation proceeds); thus
ˉ
H
=
n
−
ˉ
L
;
currently using
ˆ
=
0
.
8; note that
z
q
,
k
is binary indicator corresponding to design point
x
q
,
k
and response
y
q
,
k
(
i
)
q
,
p
Binary convergence values indicating if design runs in
X
(
i
)
q
,
p
converged based on empirical convergence criteria for reinforcement
learning (1
=
converged, 0
=
not converged); also of length
n
(i.e., responses to design points in
A
(
i
)
q
,
p
);
ˈ
q
,
k
corresponds to design
point
x
q
,
k
, response
y
q
,
k
, and low/high set indicator
z
q
,
k
X
q
,
p
,
Y
q
,
p
,
Z
q
,
p
,
q
,
p
Variables
X
,
Y
,
Z
, and
from any iteration that fall into subregion
q
T
(
i
)
q
,
p
CART model that models
Z
(
i
)
q
,
p
based on
X
(
i
)
q
,
p
sampled over sub-region
A
(
i
)
q
,
p
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