2 Standard n-tuple classifiers

In standard n-tuple classifiers [ 2 , 1 ] the d -dimensional input space is sampled by m n-tuples. The range of each dimension in the general case is the alphabet

but most n-tuple methods reported in the literature are defined over a binary input space where

and

Each n-tuple defines a fixed set of locations in the input space. Let the set of locations defining the j th n-tuple be:

where each

is chosen as a random integer in the specified range. This mapping is normally the same across all classes. For a given d -dimensional input pattern

an address

may be calculated for each n-tuple mapping

as shown in Equation 2 .

These addresses are used to access memory elements, where there is a memory

for each class c in the set of all classes C and n-tuple mapping

. We denote the value at location b in memory

. The set of all memory values for all the n-tuple mappings for a given class we denote

, the model for a given class. The size of the address space of each memory

. In standard n-tuple systems, each address location accesses a single bit of information. The complete algorithm for training a standard n-tuple classifier is given in Table 1 , and the recognition algorithm is given in Table 2 . X is the complete set of training patterns while the subset of patterns of class c is denoted as

and

is the i th pattern in the c th class.

**Table 1:** Algorithm for training the standard n-tuple Classifier.
	Algorithm for training standard n-tuple classifier
Step 1:	Initialise all n-tuples
	For each class
	For each n-tuple
	For each address

Step 2:	Train all n-tuples on all training patterns
	For each class
	For each pattern
	For each
	Set Current Address
	Set

**Table 2:** Algorithm for performing pattern classification with the standard n-tuple classifier
	Standard n-tuple recognition algorithm
	Classifies image in input array into class
Step 1:	Initialise Recognition Vector
	is a \| C \|-dimensional vector of real numbers
	For each class
	Set
Step 2:	Look up memory contents of each n-tuple in each class
	For j := 1 to m
	Set
	For each class
	Set
Step 3:	Classification
	Assign to class c where

Training is performed by adjusting the values stored at each address for each pattern in each class, where all values are initially set to zero. When an address

is accessed by a pattern

of class c under mapping

then

is set to one.

For recognition the total output for each class is simply the sum of the outputs for each n-tuple in that class as shown in the fourth line of Step 2 in Table 2 :