11_distances

# Notice that for points on the coordinate axes all

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Unformatted text preview: xes all distances agree. For the other points, our reach increases with p ; however, if we require a rotation-invariant The Mahalanobis distance distance metric then Euclideancentroid classifier(right) The rotated ellipse The closest distance is our only choice. revisited xT RT S2 Rx = 1/4; the axis-parallel ellipse xT S2 x = 1/4; and the circle xT x = 1/4. The shape of the ellipse is estimated from data as the inverse of the covariance matrix: Dis M (x, y|⌃) = p (x y )| ⌃ 1 (x Theorem 8.1 (The arithmetic mean minimizes the squared Euclidean Machine Learning: Making Sense of Data August 25, 2012 distance). Peter Flach (University of Bristol) 226 / 349 The arithmetic mean of a set of data points D is the unique point that minimizes the sum of squared Euclidean distances to those data points. y) (The inverse of the covariance matrix has the effect of decorrelating and normalizing features) In other words: argmin y X x2D ||x y||2 = µ Proof: Find the minimum by taking the derivative and setting to 0. (page 238 in the book) 11 12 3 10/21/13 The closest cent...
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