Affine mappings

Affine mappings

Many operations on datasets can be described in the mathematical sense as affine mappings: projections, (non-orthogonal) rotations, scaling, translation, linear classifiers. Although there are many routines for defining and optimizing such operations, the execution is centralized in a single command: affine. This command defines the mapping. Because of its multiple use it is rather overloaded with options. Below some details are given. They are mainly needed for programming new mappings and classifiers. For using the given PRTools commands this information is not needed.

Definition

An affine mapping is defined by

    W = affine(R,offset,lab_in,lab_out,k_in,k_out)

and applied to a dataset A with k_in features by

    B = A*W

resulting in another dataset B with k_out features.

R is the rectangular rotation matrix of size k_in * k_out. These two size parameters are optional as they can be derived automatically from the size of R. They are just needed in case for one reason or the other this does not apply or the mapping should be given a different size.

offset is an optional parameter that defines a vector of length k_out which is added to all objects after the rotation.

In lab_in the feature labels of the input space may be stored. They will be stored in W, just for annotation.

In lab_out the feature labels of the output space may be stored. They will be used to annotate the features of B. w = affine(R)

Affine mappings