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s
such that
x
i
∈
X
. Then we score each variable
x
i
using the function
n
W
i
(
)
s
(
x
i
)
=
S
sZ
s
s
=
1
and each set using
T
(
X
)
=
S
(
x
i
).
x
i
∈
X
We can obtain a probability distribution on
M
j
by dividing the set scores by
X
∈
M
j
T
. The best performing minimal set is the one with the highest nor-
malized set score.
Example 3.9.
Suppose we have the following minimal sets for a given node in a
6-node system:
X
1
(
X
)
. Then
the scores are 1, 7/24, 1/8, and 7/108, respectively. In this case, the singleton set
X
1
has the highest score. Even though
X
2
and
X
3
are the same size,
X
2
has a higher
set score than
X
3
since the variables
x
2
and
x
3
have higher variable scores than
x
4
,
namely, 1/2, 7/12, and 1/4, respectively.
For other scoring strategies, see [
18
]. Note that the strategy presented here corre-
sponds to the
={
x
1
}
,
X
2
={
x
2
,
x
3
}
,
X
3
={
x
2
,
x
4
}
,
X
4
={
x
3
,
x
5
,
x
6
}
method in that paper.
Exercise 3.12.
Using the time series immediately preceding Exercise
3.7
, compute
the minimal sets for each node.
(
S
1
,
T
1
)
Exercise 3.13.
Why can the following not be minimal sets?
1.
{{
x
1
}
,
{
x
1
,
x
2
}
,
{
x
3
,
x
4
,
x
5
}}
2.
{{
x
1
,
x
2
}
,
{
x
2
,
x
3
}
,
{
x
1
,
x
2
,
x
3
}}
3.
{{
x
1
x
2
}
,
{
x
3
,
x
4
,
x
5
}}
4.
{{
x
2
}
,
{
x
3
+
}}
1
Exercise 3.14.
While sets of standardmonomials depend on the choice of monomial
order, minimal sets do not. What is the relationship between a minimal set and a set
of standard monomials?
Project 3.5.
One can sometimes extract information about a data set from the struc-
ture (regularity or irregularity in the distribution) of a given collection of minimal sets.
What can you infer from the following minimal sets for a 5-dimensional system? Each
collection of minimal sets has a different corresponding data set. Look for patterns
in occurrences of variables in the minimal sets and explore what these patterns say
about the corresponding data set.
1.
{{
.
Hint: What does it mean that x
4
is in every minimal
set?That everyminimal set has the same variable togetherwith any other variable
(except x
5
)? That x
5
is not present?
x
1
,
x
4
}
,
{
x
2
,
x
4
}
,
{
x
3
,
x
4
}}
2.
{{
x
1
,
x
2
}
,
{
x
2
,
x
3
}
,
{
x
1
,
x
3
}
,
{
x
5
}}
Hint: What does it mean that x
5
appears in a
minimal set by itself?
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