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Combined, the above three metrics allow us to perform abstract
comparisons of helixes. It should be mentioned again that
during this stage of abstract comparisons we do not make use of
information on the angular extent and magnitude of
deformations, but save this information for the more detailed
quantitative comparisons.
The above three cost metrics are combined in an integrated
index Sim, to express the similarity between a reference (H")
and a matching candidate (H^) helix as:
ay X COS Lelocity E a3 COS rotation +a 2 COS! deformation (1)
Sim’ = : :
: (number of nodes + number _of prongs)
where:
COSl,4,,;, are the cost metrics referring to table 1,
aggregated over all nodes,
COSl,,,;,,;, are the cost metrics referring to table 2,
aggregated over all nodes,
COS deformation are the cost metrics referring to table 3,
aggregated over all prongs,
ay, a, a4, are the corresponding relative weights for
each component, with a,+a,+ag=1.
In general, all types of MST cost metrics receive equal weight
(ay = a, = ag = 1/3). It is possible for certain applications to put
more emphasis on certain aspects than others (e.g. focusing on
velocity variations more than rotations), and we can easily
accommodate this by moditying the corresponding coefficients.
The combined index Sim, ranges between 0 and 2, with 0
corresponding to a perfect match and 2 reflecting the highest
possible dissimilarity. Lower values reflect better matches to a
reference helix, and this information is used to rank the
matching candidates according to their similarity to a reference
helix.
3.2 Quantitative Comparisons
The other type of comparison is quantitative, with specific
differences computed between the values of nodes and prongs.
In this case, instead of a somewhat arbitrary value of "2"
assigned to a pair of dissimilar nodes, the angle of acceleration
and the angle of deceleration are compared by taking the
absolute value of the difference between them. Similar
differences are found between angles of rotation and the
magnitudes of expansion or contraction. In this type of
comparison, the following equation is utilized:
Sim, =a,2(n! -—n? ta, 2 (p! -pl Yea $9.97)
(2)
+a, (rl —/)+ a, > la -a/ )
where (ny-nd) expresses the Euclidean spatiotemporal
distance among a reference and a corresponding
candidate node, aggregated across all nodes,
(p.-pJ) expresses the Euclidean spatiotemporal
distance among a reference and a corresponding
candidate prong, aggregated across all prongs,
(9.-q.') expresses the difference in velocity gradient
or rotation among corresponding nodes, aggregated
across all nodes,
47
(r,-rd) expresses the difference in deformation
magnitude among corresponding prongs, aggregated
across all prongs,
(aj -aj) expresses the difference in deformation angle
among corresponding prongs, aggregated across all
prongs,
An, Ap, Ag, A, A, are the corresponding relative weights
for each component, with a,taprag tarta]
We normalize all quantities by dividing their actual values by
their range, so that in this case, a value of 1 is assigned to the
most dissimilar pairs and a value of 0 is given to pairs that are
exactly the same. Once a degree of similarity has been
determined, whether by abstract or quantitative methods, we
can decide whether these helixes belong in a group or should
remain as separate entities. Papers on similarity that address
relevant issues include (Stefanidis, Agouris et al. 2002;
Vlachos, Gunopulos et al. 2002; Stefanidis, Eickhorst et al.
2003).
4. GROUPING HELIXES
If the helixes in question are sufficiently similar, then it may be
useful to group them together into a single entity and to express
the behavior of their component objects with an “aggregate
helix.” The aggregate helix that is created could then be used
for predictions about the future behavior of all polygons that
begin in a similar way to the first few nodes and prongs of the
aggregate (Figure 3). The user can sclect the level of similarity
that must be reached in order to justify this decision, with more
detailed applications needing helixes with comparison values
approaching zero.
re LE,
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Figure 3: Aggregate helix as formed from individual helixes
When sufficiently low values are found, node and prong
locations of the individual helixes can be averaged so that the
new aggregate helix is a composite of the original helixes.
Attributes that are associated with each node and prong can be
calculated in a variety of ways, including averaging all values
for each instance, looking for minimum or maximum value, or
using categorical rules in order to choose the best value for a
given application.
When dealing with the aggregate helix that has been
constructed, there may be instances when component helixes
become sufficiently different over time and should be split from
each other. "We can discover such instances by computing
deviations of node/prong values from the aggregate average and
splitting the helixes when a user specified threshold is crossed.
