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11_distances

# Distance based models figure 83 circles and ellipses

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Unformatted text preview: .3: Circles and ellipses p.235 Sometimes it’s useful to use different scales for different coordinates. 8. Distance-based models Given an instance space X , a distance metric is a function Dis : X £ X ! R such that for any x , y , z 2 X : Therefore: use an ellipse rather than a circle to identify points p.235 t distances between a point and itself are zero: Dis(x , x ) = 0; t all other distances are larger than zero: if x 6= y then Dis(x , y ) > 0; Also consider rotating the ellipse. p t distances are symmetric: Dis( y , x ) = Dis(x , y ); t detours can not shorten the distance: Dis(x , z ) ∑ Dis(x , y ) + Dis( y , z ). August 25, 2012 p 2 2 p 2 2 ! 45 degree clockwise rotation (left) 1 Lines ! connecting points at order-p Minkowski distance 1 from the origin for (from 0 2 inside) p = 0.8; p = 1 (Manhattan distance, the rotated square in red); p = 1.5; p = 2 S= 01 (Euclidean distance, the violet circle); p = 4; p = 8; and p = 1 (Chebyshev distance, the The Minkowski distance with p<1 does not satisfy the triangle inequality. Machine Learning: Making Sense of D...
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