PRTools manual in preparation -- Mapping types

The concept of a mapping is to map objects from one vector space to another. There are several stages in the definition and application of a mapping. We will first describe these stages and then present how they are typically used. Readers who are interested to program routines that produce mappings of a particular type should study how to program a mapping.

There are four mapping types defined as described in the below table. Their typical use is given in the second column. A, B and C are datasets. U, V and W are mappings.

Mapping types
fixed	`C = A*W` `C = map(A,W)`	Fixed mapping are fully user defined by their parameters. They map data a dataset `A` object by object into a different space resulting in a dataset `C`. The exact mapping operation does not depend on the data, just on user supplied parameters. An example is the sigmoid mapping `W = sigm([],s)` which maps all features on the interval `[0,1]` after applying some scaling defined by `s`.
untrained	`W = B*U` `W = map(B,U)`	An untrained mapping `U` specifies a trainable mapping procedure without supplying the training dataset. By applying it to the training dataset `B` it results in a trained mapping `W`. An example is the principal component analysis `W = pca([],alf)` for linear feature extraction.
trained	`C = A*W` `C = map(A,W)`	A trained mapping `W` can be considered as a fixed mapping that is optimized for a given training dataset. By applying it to to a dataset `A` it maps object by object into another space resulting in a dataset `C`. An example is the application of a trained mapping as above to new dataset `A` resulting in a dataset `C` with less features.
combiner	`W = U*V` `W = map(U,V)`	A combiner `V` accept as an input another mapping `U` and combines the two into a new mapping `W`. In case `V` is not a combiner but another type of a mapping, `UV` results into a sequentially combined mapping. An example is the routine `classc` which transforms the outputs of a density based classifier, e.g. `parzenc`, into posterior probabilities: `W = parzencclassc`.