Changeset 1154

Feb 26, 2008, 6:18:38 AM (13 years ago)

some docs on weighted distance...

1 edited


  • trunk/doc/Statistics.doxygen

    r1153 r1154  
    302302and calculating the weighted median of the distances.
    304 \section Distance
     304\section Distance
     306A Distance measures how far apart two ranges are. A Distance should
     307preferably meet some criteria:
     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$.
     311 - Zero self-distance: \f$ d(x,x) = 0 \f$
     312 - Triangle inequality: \f$ d(x,z) \le d(x,y) + d(y,z) \f$
     314\subsection weighted_distance Weighted Distance
     316Weighted Distance is an extension of usual unweighted distances, in
     317which each data point is accompanied with a weight. A weighted
     318distance should meet some criteria:
     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$.
     331For a weighted distance, meeting these criteria, it might be difficult
     332to show that the triangle inequality is fulfilled. For most algorithms
     333the triangle inequality is not essential for the distance to work
     334properly, so if you need to choose between fulfilling triangle
     335inequality and these latter criteria it is preferable to meet the
     336latter criteria. Here follows some examples:
     338\subsection EuclideanDistance
     340\subsection PearsonDistance
    306342\section Kernel
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