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where
f
D
denotes the unknown radially symmetric PDF of the
d
ij
.Sincewe
are interested in maximizing
V
R
2
we are consequently interested in having
the highest possible value of
f
D
(
d
0
) . This will happen when vector
d
0
is
aligned in such a way that its end tip falls in a highly peaked region of
f
D
.
In this case
d
0
stands along the dominant structuring direction.
Example 6.3.
Table 6.14 shows the entropic dissimilarity matrix for the 14
points of Fig. 6.21, using a 9-neighborhood. The points of Fig. 6.21 are refer-
enced left to right and top to bottom as 1 to 14; point Q is, therefore, point 11.
Tabl e 6 . 14
Entropic dissimilarity matrix for Fig. 6.21 example.
Points 12345678901234
1
8.24 8.37 8.82 9.61
8.56 8.61 8.81 9.17 10.21
2
8.46
8.46 8.55 8.87
8.71 8.75 8.77 8.80 9.29
3
8.62 8.54
8.54 8.62
8.90 8.80 8.80 8.71 8.94
4
8.61 8.53
8.53 8.61
8.92 8.81 8.70 8.81 8.92
5
8.62 8.54
8.54 8.62
8.94 8.71 8.80 8.80 8.90
6
8.87 8.55 8.46
8.46
9.29 8.80 8.77 8.75 8.71
7
9.61 8.82 8.37 8.24
10.21 9.17 8.81 8.61 8.56
8
8.56 8.59 8.77 9.27 10.16
8.24 8.36 8.84 9.59
9
8.72 8.74 8.75 8.89 9.24
8.45
8.45 8.56 8.86
10
8.93 8.81 8.79 8.80 8.90
8.61 8.52
8.53 8.61
11
8.83 8.73 8.72 8.73 8.83
8.66 8.58
8.58 8.66
12
8.90 8.80 8.79 8.81 8.93
8.61 8.53
8.52 8.61
13
9.24 8.89 8.75 8.74 8.72
8.86 8.56 8.45
8.45
14
10.16 9.27 8.77 8.59 8.56
9.59 8.84 8.36 8.24
To compute, in this example, the dissimilarity values for, say, point 1,
one starts by determining its 9 nearest neighbors and then compute the 9
entropies for each connection from point 1 to each nearest neighbor. For this
reason, each row of the entropic dissimilarity matrix has only 9 values. Also,
for the same reason, the entropic dissimilarity matrix is not symmetric since
the 9 values are related to the nearest neighbors of each point and the nearest
neighbors from 2 points
a
and
b
are not necessarily the same.
6.4.4.2
The LEGClust Proximity Matrix
The concept of dissimilarity matrix was formalized in 6.4.3. Using a dissim-
ilarity matrix
4
one can build a proximity matrix, L,whereeach
i
th row
represents the dataset objects, each
j
th column the proximity order (1st
column=closest object, ..., last column=farthest object), and each matrix
4
H
R
2
(
D,
q
i
)
does not verify all conditions of a common distance measure since
it is not symmetric and may not verify the triangular inequality; it is, however,
sucient to build a proximity matrix.
