2.6 Selected Models in Detail
2.6.2 Type II Clustering using Ant Aggregations and Ants’ Self-Aggregation
2.6.2.3 Ant Aggregation through Pheromone in a 2D Grid
The SACA based algorithms lack the benefits of cooperative ant actions, tantamount with ant colonies. The next algorithm eliminates this limitation by allowing ants to deposit pheromone traces. The pheromone traces enhance the gathering of ants within particular spatial positions
within the environment, with the behaviour produced also being a stigmergic one. Aggregation Pheromone Clustering (APC)
The APC builds on the principle that pheromone causes clumping or clustering behaviour which brings individuals of the same species closer to each other [Ghosh et al., 2008], [Tsutsui et al., 2005]. In this model, ants are representative of data elements and are allowed to move in the search space looking for points with higher pheromone density. Ant movement is controlled by the pheromone intensity around the ants’ location. The higher the accumulation of pheromone at a point, the more likely it is that ants will move towards this point. This eventually results in the formation of homogeneous groups of data. Since the number of clusters formed could exceed the number of clusters present in the problem, a second agglomerative average linkage algorithm is used.
The APC operates on a dataset of n patterns, x1, x2, x3, x4, ..., xn, and the population of n ants, a1, a2, a3, a4, .., an, where an ant ai represents the data pattern xi. Each ant emits pheromone within its neighbourhood, which intensity of pheromone by ant ai located at xi decreases with its distance from xi. The pheromone intensity at a point closer to xi, exceeds that of points far- ther away from it. The pheromone intensity deposited by ant ai at point x, is ∆τ(ai, x). This pheromone spread mechanism is modelled by a Gaussian distribution, a mechanism which also compensates for the similarity between x and xi, where the smaller the Euclidean distance be- tween the two points, the higher the pheromone concentration is, and vice-versa, as per equation (2.31):
∆τ (ai, x) = exp −d(xi,x)2
2δ 2 (2.31)
where δ denotes the spread of Gaussian function and d(xi, x) is the Euclidean distance between xi and x. The total aggregation pheromone density at x, deposited by the entire population of n ants is computed using equation (2.32):
∆τ (x) = n
∑
i=1 exp− d(xi,x)2 2δ 2 (2.32)Ant ai, initially located at xi, moves to location x0iif the total aggregation pheromone density at x0iis greater than that at xi. This movement is defined by equations (2.33, 2.34):
x0i= xi+ η. Next(ai) n (2.33) where: Next(ai) = n
∑
j=1 (xj− xi).exp− d(x j,xi)2 2δ 2 (2.34)with η being a proportionality constant at each step size.
The process of determining new locations is continued until the ant finds a location where the total aggregation of pheromone is higher than its neighbouring points. When this occurs the point x0i for ant ai is assumed to be a new potential cluster centre, Zj, where j = 1, 2, ...,C, with Cbeing the number of clusters, and the data point with which the ant was associated earlier, xi, is assigned to the cluster formed with centre Zj.
Any other data points which are within the distance δ /2 from Zjare assigned to this new cluster. If the distance between xi0and the existing cluster centre, Zj, is less than 2δ and the ratio of their pheromone densities is greater than the predefined threshold density, the data point xiis allocated to the cluster having cluster centre at Zj. Higher density values indicate that data points should belong to the same cluster.
The APC is first applied to generate a number of clusters. The next step involves determining which clusters are maintained, and which ones are too small and therefore engulfed into larger ones. In order to achieve this, the use of an agglomerative hierarchical clustering algorithm [average linkage, section (2.2.4.1)] is used. This renders the APC a two-stepped approach. Research by Ghosh et al. validate their results against a number of metrics, which are further compared against the MPACA algorithm in the evaluation section [Ghosh et al., 2008].
Evaluation Criteria
The architecture used is that of a 2D grid, where the initial spatial allocations are irrespective to the attributes of the object. Each data element (object) is represented by an ant, and placed within this 2D grid. The ants move with the aim of creating homogeneous groups of data. Ant movement is guided by pheromone intensity, and is governed by the intensity of aggregation pheromone deposited by all other ants at a point. This gradual pheromone build up and move- ment results in the formation of ant aggregations. The final agglomeration of ants, where ants are proxies for data elements, represents the clusters formed. The colony is non-influential, and only represents the collection of ants used. It returns no extra information on the formation of clusters. The mechanism is decentralised, as the ant executes actions in isolation. There is also full use of stigmergy, as pheromone traces are key elements for movement and other ant actions. The ant is primitive, as it does not have any additional functionalities. The application area is that of clustering.
Relevance to the MPACA
The importance of the APC within this literature review is that it introduces the idea of clustering by pheromone attraction and aggregation. This approach is closer to the MPACA than any of the
other methods introduced earlier. Albeit, the pheromone usage is dissimilar to the way it is used by the MPACA. The complexity of pheromone usage in the MPACA is by far superior to that in the APC. In the MPACA, there are pheromone scents which are specific to each feature or feature combination. The APC only uses a generic pheromone as an indication for ant density. There is no learning mechanism that takes place within the ants, only aggregation.
The algorithms discussed so far have introduced the use of pheromone scents and the ability to catalyse the coordination of ant movements. Pheromone scents are not solely used to transfer information about some position or locale, but they also serve to transfer other colony level information, resulting in a shared learning mechanism. An algorithm which makes use of such mechanisms is discussed next.
Unlike other pheromone driven algorithms presented later in type IV, the initial distribution of objects within space is not significant. If the learning process was to be run again, one would hope to get the same clustering of ants but not necessarily in the same areas of the 2D grid.