Perceptual maps and multidimensional scaling - Multidimensional scaling models - Distance metrics
3 important questions on Perceptual maps and multidimensional scaling - Multidimensional scaling models - Distance metrics
What are distances defined as and what is the formula?
Distances are defined as relations between points in space.
- For 2d distances we use the euclidean distance.
- For 3, 4, 5d we call the number of dimensions r.
What are the 3 rules that determine if a metric can be called a distance?
- Non-negativity and equivalence.
- dij>0. Meaning distances can be zero, but not negative.
- Distance between a point and itself is zero. dii=djj=0.
- dij=0 only if points i and j coincide on all r dimensions.
- Rule of symmetry. dij=dji.
- They satisfy the "triangle inequality". For any 3 points i, j and k, dik</- dij + djk.
How is the problem of violation of triangle inequality solved?
By using the additive constant. delta*ij = deltaij + c, with delta being the data and c = maxijk (deltaij - deltaik - deltajk).
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