Changeset 1154


Ignore:
Timestamp:
Feb 26, 2008, 6:18:38 AM (14 years ago)
Author:
Peter
Message:

some docs on weighted distance...

File:
1 edited

Legend:

Unmodified
Added
Removed
  • trunk/doc/Statistics.doxygen

    r1153 r1154  
    302302and calculating the weighted median of the distances.
    303303
    304 \section Distance
     304\section Distance
     305
     306A Distance measures how far apart two ranges are. A Distance should
     307preferably 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$.
     311 - Zero self-distance: \f$ 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
     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:
     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
     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:
     337
     338\subsection EuclideanDistance
     339
     340\subsection PearsonDistance
    305341
    306342\section Kernel
Note: See TracChangeset for help on using the changeset viewer.