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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.

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Presentation on theme: "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."— Presentation transcript:

1 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.

2 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.

3 Dissimilarities // Distance, So… Stimulus: point A has coordinates (X A1, X A2 ). point B has coordinates (X B1, X B2 ) (subscripts are: first=stimulus, second=dimension) Define for r > 2 dimensions:  #dimensions

4 Distances 1.Non-negativity & equivalence a.d ij >0 b.d ii = d jj = 0 c.d ij = 0 only if points i&j coincide (on all r dims) 2.Symmetry a.d ij = d ji b.Ok in geometry, but in data, ? 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.d ik <d ij + d jk b.d ik =d ij + d jk c.d ik <d ij + d jk d.> is not possible e.Will it hold in data? Not necessarily: Pepsi & Coke 2, Pepsi & DietPepsi 2, Coke & DietPepsi 5 ikj i k j DP C P 2 5>2+2 2

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