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proxm

PROXM

Proximity mapping

   W = PROXM(A,TYPE,P,WEIGHTS)
   W = A*PROXM([],TYPE,P,WEIGHTS)
   W = A*PROXM(TYPE,P,WEIGHTS)
   D = B*W

Input
 A Dataset used for representation
 B Dataset applied to the representation set
 TYPE Type of the proximity (optional; default: 'distance')
 P Parameter of the proximity (optional; default: 1)
 WEIGHTS Weights (optional; default: all 1)

Output
 W Proximity mapping
 D Dataset, proximity matrix between B and A

Description

Computation of the [K x M] proximity mapping (or kernel) defined by  the [M x K] dataset A. Unlabeled objects in A are neglected unless A is  entirely unlabeled. If B is an [N x K] dataset, then D=B*W is the [N x M] proximity matrix between B and A. The proximities can be defined by the  following types (A and B are here 1 x K vectors)

     'polynomial'   | 'p': SIGN(A*B'+1).*(A*B'+1).^P
     'homogeneous'  | 'h': SIGN(A*B').*(A*B').^P
     'exponential'  | 'e': EXP(-(||A-B||)/P)
     'radial_basis' | 'r': EXP(-(||A-B||.^2)/(P*P))
     'sigmoid'      | 's': SIGM((SIGN(A*B').*(A*B'))/P)
     'distance'     | 'd': ||A-B||.^P
     'minkowski'    | 'm': SUM(|A-B|.^P).^(1/P)
     'city-block'   | 'c': SUM(|A-B|)
     'cosine'       | 'o': 1 - (A*B')/||A||*||B||
     'ndiff'        | 'n': SUM(A~=B)
     'hellinger'    | 'g': ||A.^0.5-B.^0.5||

In the polynomial case for a non-integer P, the proximity is computed  as D = SIGN(S+1).*ABS(S+1).^P, in order to avoid problems with negative  inner products S = A*B'. The features of the objects in A and B may be  weighted by the weights in the vector WEIGHTS.

Note that for computing (squared) Euclidean distances DISTM is much  faster than PROXM.

Example(s)

 W = A*proxm('m',1);              % define L1 distances
 W = A*proxm('m',1); D = B*W;     % L1 distances between B and A
 W = proxm('r',2)*mapex; D = A*W; % Distances between A and itself

See also

mappings, datasets, distm,

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