11_distances

# We will often refer to disp simply as the p norm what

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Unformatted text preview: refer to Disp simply as the p -norm. What happens when p goes to infinity? Dis 1 (x, y) = max |xi i We get the Chebyshev distance Peter Flach (University of Bristol) yi | Machine Learning: Making Sense of Data August 25, 2012 225 / 349 Figure 8.3 7 (left) Lines connecting points at order-p Minkowski distance 1 from the origin for (from 8 inside) p = 0.8; p = 1 (Manhattan distance, the rotated square in red); p = 1.5; p = 2 (Euclidean distance, the violet circle); p = 4; p = 8; and p = 1 (Chebyshev distance, the blue rectangle). Notice that for points on the coordinate axes all distances agree. For the other points, our reach increases with p ; however, if we require a rotation-invariant 2 distance metric then Euclidean distance is our only choice. (right) The rotated ellipse xT RT S2 Rx = 1/4; the axis-parallel ellipse xT S2 x = 1/4; and the circle xT x = 1/4. 10/21/13 8. Distance-based models p.235 Deﬁnition 8.2: Distance metric Properties of a distance The Mahalanobis distance 8. Distance-based models Figure 8...
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