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 Feb 26, 2008, 6:18:38 AM (13 years ago)
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trunk/doc/Statistics.doxygen
r1153 r1154 302 302 and calculating the weighted median of the distances. 303 303 304 \section Distance 304 \section Distance 305 306 A Distance measures how far apart two ranges are. A Distance should 307 preferably meet some criteria: 308 309  It is symmetric, \f$ d(x,y) = d(y,x) \f$, that is distance from \f$ 310 x \f$ to \f$ y \f$ equals the distance from \f$ y \f$ to \f$ y \f$. 311  Zero selfdistance: \f$ d(x,x) = 0 \f$ 312  Triangle inequality: \f$ d(x,z) \le d(x,y) + d(y,z) \f$ 313 314 \subsection weighted_distance Weighted Distance 315 316 Weighted Distance is an extension of usual unweighted distances, in 317 which each data point is accompanied with a weight. A weighted 318 distance should meet some criteria: 319 320  Having all unity weights should yield the unweighted case. 321  Rescaling invariant  \f$ w_i = Cw_i \f$ does not change the distance. 322  Having a \f$ w_x = 0 \f$ the distance should ignore corresponding 323 \f$ x \f$, \f$ y \f$, and \f$ w_y \f$. 324  A zero weight should not result in a very different distance than a 325 small weight, in other words, modifying a weight should change the 326 distance in a continuous manner. 327  The duplicate property. If data is coming in duplicate such that 328 \f$ x_{2i}=x_{2i+1} \f$, then the case when \f$ w_{2i}=w_{2i+1} \f$ 329 should equal to if you set \f$ w_{2i}=0 \f$. 330 331 For a weighted distance, meeting these criteria, it might be difficult 332 to show that the triangle inequality is fulfilled. For most algorithms 333 the triangle inequality is not essential for the distance to work 334 properly, so if you need to choose between fulfilling triangle 335 inequality and these latter criteria it is preferable to meet the 336 latter criteria. Here follows some examples: 337 338 \subsection EuclideanDistance 339 340 \subsection PearsonDistance 305 341 306 342 \section Kernel
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