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In-Depth Information
planted at p different sites which vary in exposure to the weather. Let x
ij
ifthe
ith species survives at the jth site, and zero if it does not. Let u
i
be a hardiness score
for the ith species and v
j
an exposure score for the jth site. It is assumed that the ex-
posurescoreatthe jth site is proportional to the mean hardiness score of the species
at that site. hus
=
v
j
i
u
i
x
ij
i
x
ij
.
(
.
)
Similarly, it is assumed that the hardiness score of species i is proportional to the
mean exposure score of the sites occupied by that species. hus
u
j
j
v
j
x
ij
j
x
ij
.
(
.
)
Let r
i
=
j
x
ij
, c
j
=
i
x
ij
. Reciprocal averaging solves the equations
ρu
i
=
j
v
j
x
ij
r
i
(
i
=
,...,n
)
(
.
)
ρv
j
=
i
u
i
x
ij
c
j
(
j
=
,...,p
)
(
.
)
where ρ is a scaling parameter, to obtain the hardiness and exposure scores.
Reciprocal averaging was used on the Shakespeare data used in Sect.
, but where
it has been turned into a binary format scoring
/
if a wordis absent/present in each
of the plays. he resulting plots are shown in Figs.
.
and
.
.
As in Fig.
.
it is the personal words (princess, bishop, clown, moor etc.) that
convey most information about the play.
When examining the plays, losing the detail (the word count for each play) has
clearly affected the detail displayed, although certain similarities within the plots can
be seen, for instance the cluster asy, cym, tam, tem, mer seen in Fig.
.
is within
alargerclusterwithinFig.
.
.
Large Data Sets
and Other Numerical Approaches
3.9
BehindmostMDStechniques thereisaneedforaccurateande
cient algorithms for
minimizing functions, but many MDS programs and algorithms cannot cope with
very large data sets, as they suffer from high computational complexity. hey cannot
feasibly be applied to data sets over a few thousand objects in size. However, meth-
ods have been proposed to overcome this problem, for example Fast Spring Model-
Visualisation (FSMvis). FSMvis adopts a novel hybrid approach based upon stochas-
tic sampling, interpolation, and spring models. Following Morrison et al. (
) the
mechanics of the spring model are described.
