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The effect of the SOM training process is to cluster together similar patterns, while preserving the topology of input space. After training, all that is given is the set of trained weights with no explicit cluster boundaries. An additional step is required to find these cluster boundaries. One way to determine and visualize these cluster boundaries is to calculate the unified distance matrix (U-matrix), which contains a geometrical approximation of the codebook vector distribution in the map. The U-matrix expresses for each neuron, the distance to the neighboring codebook vectors. Large values within the U-matrix indicate the position of cluster boundaries. Using a gray-scale scheme, Figure 4.4(a) visualizes the U-matrix for the iris classification problem. For the same problem, Figure 4.4(b) visualizes the clusters on the actual map. Boundaries are usually found by using Ward clustering of the codebook vectors. Ward clustering follows a bottom-up approach where each neuron initially forms its own cluster. At consecutive iterations, two clusters which are closest to one another are merged, until the optimal, or specified number of clusters has been constructed. The end result of Ward clustering is a set of clusters with a small variance over its members, and a large variance between separate clusters. The Ward distance measure is used to decide which clusters should be merged. The distance measure is defined as , nrns 2

where with r2 = x2 + y2 + a2. To summarise then, the total exciton energy can be written as:

do fc>0 k,s,y := k-l , 2x5 + a[k-l] , y2xXa[k~l}

with A given in equation (6.35), B following analogously, C given by equation (6.36) and D by equation (6.22). Evaluation of the entities, F(a), G(a), J(a) and K(a), necessitates the introduction of a specific form for the relative motion term i/V- The exciton binding energy Exo follows simply from:

{ y = XM } .

where r and s are cluster indices, nr and ns are the number of patterns within the clusters, and wr and ws are the centroid vectors of these clusters (i.e. the average of all the codebook vectors within the cluster). The two clusters are merged if their distance, drs, is the smallest. For the newly formed cluster, q, wq = 1

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6.4 1S EXCITON In line with other work in the literature, the choice of wave function representing the electron-hole interaction that is to be employed is a hydrogenic type, given by:

Two small changes are required before our task is complete. The first is to remove the assignment to s, as forewarned. The second is to take account of the binary value of a[k-l]. When a[k-l] -0,y2xXa[k~l] simplifies to y2 and, when a[k-l] = l,y 2 xX a[k ~ 1] simplifies to y 2 xX. In this way, we obtain the algorithm's final version, shown in Figure 13.1. (The use of 'where' in the invariant is informal. Formally, the invariant is

where the Bohr radius A will be used as a parameter and systematically varied in order to minimize the total energy E of the system (which is equivalent to maximizing the exciton binding energy EXO ). However, where this work differs is in the precise form of the choice of the relative coordinate r'. Using the knowledge and experience derived from the considerations of impurities in 5, a variable symmetry-type relative motion term is chosen:

O^k^N A (35 : 5x2 k = ( Z i : k ^ i < N : a [ i ] x 2 i ) : y = Xs) ,

Note that, in order to preserve topological structure, two clusters can only be merged if they are adjacent. Furthermore, only clusters that have a nonzero number of patterns associated with them are merged.

or by using a = \ze Zh\ as above, then:

We discuss the use of auxiliary variables, and their eMmination using existential quantifications, in detail in Section 15.2.) Exercise 13.15. The previous algorithm made use of the bits in the binary representation of M starting with the most significant bit and ending with the least significant bit. There is an advantage in processing them in the opposite order, because they can be computed as the remainders resulting from successive divisions by 2. (Thus a[0] = M mod 2, a[l] = (M-=-2) mod 2, a[2] = (M~4) mod 2, etc.) An algorithm that uses this approach is based on the invariant (Kfc A yxzk = XM . Develop such an algorithm.

The second variational parameter, , allows the exciton to assume any shape of wave function that is energetically favourable. Traditionally, the case with C = 0 has become known as the two-dimensional exciton [134-138], and C = 1, as the threedimensional exciton [134,139]. Cases where C is allowed to take values other than 0 and 1 are rarely found in the literature because of the increased complexity in the resulting mathematics of the problem [140-143]. In the present approach, C will be allowed to take all values from 0 upwards; however, due to complications in the mathematics, in particular deriving the limits of integration for the evaluation of K(a) (see later), it is necessary to introduce the following transformations:

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