ANEXOS

Feature and colony merging are triggered by the number of ant feature encounters and colony Id recordings. Feature merging occurs if an ant keeps repeatedly seeing the same features carried by other ants. These features will be acquired by the ant, and from then on the ant will search for the combination of such features, which is called the carried features list. On the other hand colony formation occurs when the colony count of encountered ants goes over the colony threshold. The ant joins the colony with the highest ant population from the colony Ids which have been detected more than the threshold number of times.

These merging activities are linked, but they are not synonymous with each other, since ants do not need to share the same features to belong to the same colony. A cluster is expressed in terms of an ant colony and the features being searched for by each ant member of that colony.

Both feature and colony merging operators are limited by a time-window parameter. There- fore, only encounters which occur within the last number of steps represented by this time- window are taken into consideration. The MPACA is synchronised via an incremental system time process. At each time increment, each and every ant moves one step. If the ant reaches a node, the ant will update its ant feature encounters queue. When the time-stamp of the ant feature encounter exceeds the selected time-window, this is removed from the queue. The time- window keeps the size of the ant feature structure in check, and also ensures that only the most recent encounters are validated, as more time distant encounters are ignored.

3.5.4.1 Merging Features: A Learning and Forgetting Mechanism

The MPACA combines the features being sought in an asynchronous manner. It is important to note that this merging is applicable only at each distinct feature dimension. This limit inhibits the focus ant from having a combinatorial explosion of carried features. A feature vector has at most only a singular representation for each of the features such as weight, height, colour and shape.

When the ant merges features, the number of nodes matching the ant’s feature detectors de- creases, concurrently reducing the pheromone laying chances of the ant. This consequently leads to a natural check on the combination process, since specialising the detection reduces the number of matching nodes and the probability of an ant being in deposit mode. This results in ants becoming more localised to a particular area of graph space. When this happens, the chances of ants encountering the same subset of ants increases. This subsequently increases the likelihood of ants belonging to the same colony.

Both merging processes are implemented in algorithm (7). A list of features is produced from counting the frequencies for each encountered feature value within the given time-window, and where counts exceed the feature merging threshold. For each feature in this list, as long as the feature dimension is not already being carried by the ant, its carried feature list is augmented with this new feature. Thus, a feature merge occurs.

When ants become more localised, this reinforces both feature and colony merging. This is the correct functionality behind the algorithm. However, there might be instances, especially in the opening phases of the algorithm, where feature merging should not occur so rapidly. This is rectified by a forgetting mechanism, which is implemented as part of the time-window concept. The time-window is also used as a forgetting mechanism, this clears carried features within an ant which are no longer relevant. It helps prevent matching static features, enabling the ants to learn new feature combinations for different areas of space. Hence, feature merging is temporary, and this continues until an equilibrium is established, where ants stabilise the features they are carrying. A crucial aspect is that ants cannot drop the instinct feature that they were imprinted with at initialisation, since this is time independent.

The process of feature merging is distinct from colony formation for a number of reasons. Most prominently, this mechanism allows for various levels of feature merging to occur within the same colony. This means that a colony can consists of various ants which have combined over multiple but not all dimensions present. This allows various distributions of ants, having dis- tinct feature combinations to belong to the same colony. More importantly, the separate feature merging allows the MPACA to detect interactions between variables and encode them for non- linear separability such as the XOR problem, which draws on the relationship between values of separate features occurring together.

Another aspect is that the mechanism allows for colony formation that can handle missing data. Once more, as not all ants need to merge on all dimensions, missing data for particular feature dimension values can be compensated for.

3.5.4.2 Colony Merging

As the algorithm begins execution, each ant is assigned a colony Id, representative of the ini- tial node that it is born on which is consists of an increasing numerical sequence. During the MPACA execution, ants encounter other ants, and determine the colony Id which they carry. This colony Id is used for colony membership.

Algorithm 7 Feature and Colony Merging

Let α be the frequency grouping of encountered ants by feature Ids, tuple( f , n), where f and n represent the feature Id and its respective count.

Apply a filter where only values when (n > feature merging threshold parameter) are kept. Let β be the frequency grouping of encountered ants only in deposit mode by colony Ids, tuple(c, n), where c and n represent the colony Id and its respective count.

Apply a filter where only values when (n > colony merging threshold parameter) are kept. for (all features, f ∈ α) do

if ( f /∈ carriedFeatures of the ant ) then Feature carried by ant ← include feature f end if

end for

Let γ be all carried features for the focus ant for (all carried features, carriedFeature ∈ γ) do

Let δ = frequency of occurrence for the current carried feature if (carried feature == base/instinct feature) then

Do nothing else

if (δ < feature merging threshold within time-window parameter) then Drop carried feature from carried feature list for the focus ant end if

end if end for

Let ε be a repository for all ant Ids for (all colony Ids, c ∈ β ) do

Locate ant Ids within the encountered ants that are in deposit mode which correspond to c Update ε ← with value c

end for

Locate colony Id from, c ∈ β which has the highest number of ants Set ant’s colony Id ← to colony Id to this value

feature encounters. The only additional filter applied is that only ant encounters of ants in deposit mode are considered, as only such encounters represent ants which were satisfied to be in that particular area of graph space. The colony Ids grouped by their frequency, and frequency counts, which exceed the colony merging threshold are used. This will return the colony Id, amongst the colonies within which the other ants seen belong to, which has the highest population of ants. The final step adjusts the colony Id value of the focus ant. Thus, merging is done at the ant level by adjusting the colony Id.