Graphics Reference
In-Depth Information
For categorical data, agreement scores can be used where, for example, if objects
r and s share the same category, then δ
rs
=
andδ
rs
=
iftheydonot.Other,more
elaborate, agreement scores can be devised.
When data are mixed, with binary, quantitative and categorical variables, Gower
(
) suggests using a general similarity coe
cient,
p
i
=
w
rsi
s
rsi
=
s
rs
(
.
)
p
i
=
w
rsi
where s
rsi
is the similarity between the rth and sth objects based on the ith variable
alone and w
rsi
is unity if the rth and sth objects can be compared on the ith variable
andzerootherwise. Forquantitative variables, Gowersuggests s
rsi
R
i
,
where R
i
is the range of the observations for variable i. For presence/absence data,
Gower suggests s
rsi
=
−
x
ri
−
x
si
=
ifobjectsr and s both score “presence,” and zero otherwise,
while w
rsi
ifobjectsr and s both score “absence,” and unity otherwise. For nom-
inal data Gower suggests s
rsi
=
=
ifobjectsr and s share the same categorization, and
zero otherwise.
Metric MDS
3.2
Given n objects with a set of dissimilarities
, one dissimilarity for each pair of
objects, metric MDS attempts to find a set of points in some space where each point
represents one of the objects and the distances between points
d
rs
d
rs
are such that
d
rs
=
f
(
δ
rs
)
(
.
)
where f is a continuous parametric monotonic function. he function f can either
be the identity function or a function that attempts to transform the dissimilarities
to a distance-like form. he first type of metric scaling described here is classical
scaling, which originated in the
s when Young and Householder (
) showed
that, starting with a matrix of distances between all pairs of the points in a Euclidean
space, coordinates for the points could be found such that distances are preserved.
Torgerson (
) brought the subject to popularity using the technique for scaling,
where distances are replaced by dissimilarities.
he algorithm for recovering coordinates from distances between pairs of points
is as follows:
. Form matrix
δ
rs
A=[−
]
.
n
−
n
n
T
,with
. Form matrix
B=HAH
,where
H
is the centring matrix
H=I−
n
avectorofones.
. Find the spectral decomposition of
T
, where Λ is the diagonal ma-
B
,
B=V
Λ
V
trix formed from the eigenvalues of
B
,and
V
is the matrix of corresponding
eigenvectors.
