# Operations on Mappings

Mappings can be combined in many ways, with each other and with scalars and with matrices of doubles. Here it will be discussed what the standard Matlab operators do with mappings. As classifiers are a special type of mapping these operations are of significant importance for combining classifiers, but the implementation is general and combining other types of mappings can be useful as well.

Combining of mappings can be understood from the definition of a mapping: it maps one space into another:

`    input space --> mapping --> output space`

Applied to a collection of objects represented in these spaces by datasets `A` and `B`, coded by` B = A*W`:

`    input dataset A --> mapping W --> output dataset B`

The PRTools rules for operations on mapping and the various ways to combine them are such that an a dataset `A` mapped by a combination of (operation on) mappings is the same as the combination (operation) applied to the resulting individual dataset.:

```    input dataset A --> C(W1,W2, ... ,Wn) --> C(B1,B2, ... ,Bn)
input dataset A --> C(W1,W2, ... ,Wn) --> C(A*W1,A*W2, ... , A*Wn)```

There are some special cases and consequences, in particular in relation with the training of mappings. They are discussed separately.

• Stacked combining  combines sets of mappings from the same input space. This is done by a horizontal concatenation of mappings:`W = [W1 W2 W3 ... Wn]`. This is especially important for combining classifiers in the same feature space.
• Parallel combining combines sets of mappings from different input spaces. This is done by a vertical concatenation of mappings: `W = [W1; W2; W3; ...; Wn]`. This is especially important for combining classifiers defined for different features.
• Sequential combining combines a first mapping with a second mapping into a new mapping using the output space of the first as input for the second mapping. The resulting mapping maps from the input space of the first into the output space of the next mapping. This is coded as `W = W1*W2*W3* ... *Wn.`This is especially important for the combination of feature reduction with classification.
• Dyadic combining combines two mappings or mappings with doubles (scalars or matrices of the proper size), e.g. W = W1+W2, or` W = s*W1` in which` s `is a scalar or` W = W1>W2`.