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Minkowski (Family of) Distances

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Put in eqn (for cluster and mds, or just mds?)

- 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.

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” 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.

Stimulus: point A has coordinates (XA1, XA2).

point B has coordinates (XB1, XB2)

(subscripts are: first=stimulus, second=dimension)

Define

for r > 2 dimensions:

#dimensions

j

i

j

k

i

k

P

2

2

DP

C

- Non-negativity & equivalence
- dij>0
- dii = djj = 0
- dij = 0 only if points i&j coincide (on all r dims)

- Symmetry
- dij = dji
- Ok in geometry, but in data, ?
- E.g., How similar is Cuba to Russia? How similar is Russia to Cuba?
- E.g., How similar is store-brand to Coke? How similar is Coke to store-brand?
- Fix: “How similar are Coke and Pepsi”

- Triangle Inequality
- dik<dij + djk
- dik=dij + djk
- dik<dij + djk
- > is not possible
- Will it hold in data? Not necessarily: Pepsi & Coke 2, Pepsi & DietPepsi 2, Coke & DietPepsi 5

5>2+2