Multidimensional Scaling - Data Visualization - page 319

( ),Gowerand Legendre( ),Digbyand Kempton( ),Jacksonetal.( ),
Baulieu ( ) and Snijders et al. ( ), discuss various similarity and dissimilarity
measures together with their associated problems. Table . lists the more popular
dissimilarities for quantitative data, where X=[ x ri ] denotes the data matrix ob-

tained for n objects on p variables ( r = ,...,n; i = ,...,p ) . he vector for the rth

x r

object is denoted by ( x )

r and so

X=[ ] .he w i are weights, and these and the

parameter λ are chosen as required.
When all the variables are binary, it is customary to construct a similarity coe -
cient and then totransform this into adissimilarity coe cient with atransformation
such as δ rs

s rs . he measure of similarity between objects r and s is based on
Table . .hetableshowsthenumberofvariables,a,outofthetotalpvariableswhere
both objects score “ ”, the number of variables, b,wherer scores “ ” and s scores “ ”,
etc. Table . gives a list of similarity coe cients based on the four counts a, b, c, d.
Various situations call for particular choices of coe cients. In practice, more than
one can be tried, hoping for some robustness against choice. Hubalek ( ) gives
a very comprehensive list of similarity coe cients for binary data.

= −

Table . . Dissimilarity measures for quantitative data

Dissimilarity measure Formula

Euclidean distance δ rs = ( x ri − x si )

i

Weighted Euclidean δ rs =

i w i

( x ri − x si )

′ Σ −

Mahalanobis distance δ rs =( x ri − x si ) ( x r − x s )

City block metric δ rs = x ri − x si

i

λ

λ

Minkowski metric δ rs =

i w i

x ri − x si λ

x ri − x si

Canberra metric δ rs

=

i

x ri + x si

( x ri − x si )

p

Divergence δ rs =

i

( x ri + x si )

x ri − x si

p

i

Bray-Curtis δ rs =

( x ri + x si )

i

x ri − x si

i

p

Soergel δ rs =

i max

( x ri , x si )

−

Bhattacharyya distance δ rs = x ri x si

i

min ( x ri , x si )

Wave-Hedges δ rs = −

)

i

max ( x ri , x si

i x ri x si

Angular separation δ rs = −

i x ri i x si

x r x s

( x ri − )( x si − )

i

Correlation δ rs = −

[ ( x ri − x r ) ( x si − x s ) ]

i i