Graphics Reference
In-Depth Information
Figure
.
.
Concept of the relativity of a statistical graph for a continuous dataset (the Iris data)
he relativity concept does not usually hold for a matrix visualization or paral-
lel coordinates type of display, since one can easily destroy the property with a ran-
dom permutation. Itis common practice toapply various permutation algorithms to
sort the columns and rows of the designated matrix, so that similar (different) sam-
ples/variables are permuted to make them closer (more distant) rows/columns.
Global Criterion: Robinson Matrix
It is usually desirable to permute a matrix to make it resemble a Robinson matrix
(Robinson,
) as closely as possible, because of the smooth and pleasant visual
effect of permuted matrix maps. A symmetric matrix is called a Robinson matrix if
its elements satisfy r
ij
k.Iftherowsand
columns of a symmetric matrix can be permuted to those of a Robinson matrix, we
callitpre-Robinson.Foranumericalcomparison, threeanti-Robinson lossfunctions
(Streng,
) are calculated for each permuted matrix, D
r
ik
if j
k
i and r
ij
r
ik
if i
j
<
<
<
<
=
d
ij
,fortheamountof
deviation from a Robinson form with distance-type proximity:
,
p
i
=
AR
i
I
d
ij
d
ik
I
d
ij
d
ik
(
)=
j
<
k
<
i
(
<
)+
i
<
j
<
k
(
)
,
)=
p
AR
(
s
I
(
d
ij
<
d
ik
)ċ
d
ij
−
d
ik
+
i
<
j
<
k
I
(
d
ij
d
ik
)ċ
d
ij
−
d
ik
j
<
k
<
i
i
=
.
)=
p
AR
(
w
I
(
d
ij
<
d
ik
)
j
−
k
d
ij
−
d
ik
+
i
<
j
<
k
I
(
d
ij
d
ik
)
j
−
k
d
ij
−
d
ik
j
<
k
<
i
i
=
AR
i
counts only the number of anti-Robinson events in the permuted matrix;
(
)
AR
s
sumstheabsolutevaluesoftheanti-Robinsondeviations;AR
w
isaweighted
(
)
(
)
version of AR
(
s
)
penalizedbythedifferenceinthecolumnindicesofthe twoentries.
Elliptical Seriation
Chen (
)introduced a permutation algorithm called rank-two elliptical seriation
that extracts the elliptical structure of the converging sequence of iteratively formed