Sammon’s mapping

A shght change to the objective function for MDS will make the objective function for the Sammon s mapping, where the squared term in the objective function... 

Another nonlinear mapping method, the Sammon s mapping, is closely related to the metric MDS. The only difference is that the errors in distance preservation are normalized with the distance in the original space. Thus, preservation of small distances is emphasized. The error function is defined as... 

The aim of non-linear Sammon s mapping (NLM) is to represent a p-dimensional space, containing n objects, by a two-dimensional map that preserves the distances between the objects optimally. This optimization problem is not easy and requires extensive computing time because of the large number (2n) of parameters. The starting point is a distance matrix for the p-dimensional space with the Euclidean distance most widely applied. The initial 2n map co-ordinates can be chosen randomly but the scores of the first and second principal component may be used favourably. A mapping error E that reflects the differences of the distances between two objects i and J in p-space (c/,/) and in the 2-dimensional map (dij) has to be minimized (equation 35) ... 

When a non-parametric, monotonic relationship is searched between the distance matrix and the distance between objects in the projection space, non-metric MDS is introduced. Usually the approaches differ in the stress criterion which is chosen to be minimized, and no analytical solution is available, so that other methods must be considered, such as an iterative, gradient descent optimization in Sammon s mapping, repeating the mapping several times starting from different sets and with different parameters to avoid local... 

Sammon s NLM is one form of multidimensional scaling (MDS). There exist a number of other MDS methods with the common aim of mapping the similarities or dissimilarities of the data. The different methods use different distance measures and loss functions (see Cox and Cox 2001). 

For comparison the same fatty acid data have been visualized by Sammon s NLM with parameter p set to 1 which makes a local mapping preserving well the small distances between objects (Figure 3.25, left). The NLM result shows a compact cluster of the fresh reference samples (class 6), and somewhat separated the desert sample (5). The horizontal map coordinate separates fresh samples from the high mountain samples (4) and the other samples (glacier samples, 2 lake samples, 3 and iceman samples, 1). The clustering has been indicated by manually drawn ellipses it is very similar as obtained with the scores of the PCI and PC2. Furthermore, Figure 3.25... 

FIGURE 3.25 Sammon s NLM with p— 1 (left) and Kohonen map (right) of fatty acid concentration data from mummies and reference samples. 

Agrafiotis [60] has also developed a simulated annealing method for maximising diversity. The method employs a user-defined objective function and can therefore be tailored to encode different selection criteria. The results of subset selection can be visualized using Sammon s nonlinear mapping algorithm [61]. 

Ekins, S., Balakin, K.V., Savchuk, N. and Ivanenkov, Y. (2006) Insights for human efher-a-go-go-related gene potassium channel inhibition using recursive partitioning and Kohonen and Sammon mapping techniques. Journal of Medicinal Chemistry, 49,... 

Frickenhaus, S., Kannan, S., Zacharias, M. Efficient evaluation of sampling quality of molecular dynamics simulations by clustering of dihedral torsion angles and sammon mapping. J. Comput. Chem. 2009, 30, 479-92. 

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