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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