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# Minkowski (Family of) Distances PowerPoint PPT Presentation

Minkowski (Family of) Distances. 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

Minkowski (Family of) Distances

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### Minkowski (Family of) Distances

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.

### Family of Distances

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.

### Dissimilarities // Distance, So…

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

### Distances

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