Download Phipps*s Hawaiian Marriage data, reconfigured.

Minkowski (Family of) Distances
Put in eqn (for cluster and mds, or just mds?)
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Minkowski distances
i&j are stimuli, k=dims (r of them)
q=2? Euclidean distance (as the crow flies)
model of psych processing? No, but easy & useful w data
q=1 “city block” distance
model? More compensatory
q=∞ “dominance” distanc
largest diff betw 2 points on any dim is the only relevant one.
psych processing? Maybe, e.g., “diet drink vs. not” >important than other
qualities. Least compensatory.
Family of Distances
q
 r
dij   X ik  X jk 
 k 1

i&j are stimuli, k=dims (r of them)
Minkowski distances
1/ q
q=2? Euclidean distance (as the crow flies)
model of psych processing? No, but easy & useful w data
q=1 “city block” distance
model? More compensatory
q=∞ “dominance” distance
largest diff betw 2 points on any dim is the only relevant one.
psych processing? Maybe, e.g., “diet drink vs. not” >important than other
qualities. Least compensatory.
Dissimilarities // Distance, So…
Stimulus:
point A has coordinates (XA1, XA2).
point B has coordinates (XB1, XB2)
(subscripts are: first=stimulus, second=dimension)
Define
2
d AB = ( X A1 - X B1 ) + ( X A2 - X B 2)
for r > 2 dimensions:
 #dimensions
r
=
2
(
)
 X AK X BK
k=1
2
Distances
1. Non-negativity & equivalence
a.
b.
c.
dij>0
dii = djj = 0
dij = 0 only if points i&j coincide (on all r dims)
2. Symmetry
a.
b.
dij = dji
Ok in geometry, but in data,  ij  ? ji
i.
E.g., How similar is Cuba to Russia? How similar is Russia to Cuba?
ii.
E.g., How similar is store-brand to Coke? How similar is Coke to store-brand?
iii. Fix: “How similar are Coke and Pepsi”
3. Triangle Inequality
a.
b.
c.
d.
e.
j
dik<dij + djk
i
j
k
dik=dij + djk
i
dik<dij + djk
> is not possible
Will it hold in data?
Not necessarily: Pepsi & Coke 2, Pepsi & DietPepsi 2, Coke & DietPepsi 5
k
P
2
2
DP
C
5>2+2