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Minkowski (Family of) Distances

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

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Minkowski (Family of) Distances

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  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 (XA1, XA2). point B has coordinates (XB1, XB2) (subscripts are: first=stimulus, second=dimension) Define for r > 2 dimensions:  #dimensions

  4. 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

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