Nonlinear mapping

Nonlinear maps (NLMs) represent all relative distances between all pairs of compounds in descriptor space in a two-dimensional map (see Fig. 11). The distance of two points on the map directly reflects the similarity of the compounds [111-113]. 

Nonlinear maps are preferably calculated as two-dimensional maps. Due to the enormous demand for computer power, the method is limited to a maximum of a few hundred compounds. 

To compute a nonlinear map, the distances between all pairs of descriptors are calculated. The initial positions of the compounds on the map are chosen randomly and then modified in an iterative algorithm until all distances are represented as well as possible. The core algorithm of NLM is a partial least-squares error minimization (PLS). The total error of mapping must be smaller than the distances between the molecules and is therefore given on the NLM, e.g., as sum of error squares, E2. 

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The neurons in both the hidden and output layers perform summing and nonlinear mapping functions. The functions carried out by each neuron are illustrated in Fig. 2. Each neuron occupies a particular position in a feed-forward network and accepts inputs only from the neurons in the preceding layer and sends its outputs to other neurons in the succeeding layer. The inputs from other nodes are first weighted and then summed. This summing of the weighted inputs is carried out by a processor within the neuron. The sum that is obtained is called the activation of the neuron. Each activated neu-... 

A network that is too large may require a large number of training patterns in order to avoid memorization and training time, while one that is too small may not train to an acceptable tolerance. Cybenko [30] has shown that one hidden layer with homogenous sigmoidal output functions is sufficient to form an arbitrary close approximation to any decisions boundaries for the outputs. They are also shown to be sufficient for any continuous nonlinear mappings. In practice, one hidden layer was found to be sufficient to solve most problems for the cases considered in this chapter. If discontinuities in the approximated functions are encountered, then more than one hidden layer is necessary. 

Sammon JE. A nonlinear mapping for data structure analysis. IEEE Trans Comput 1969 C-18 401-9. 

Agrafiotis DK, Lobanov VS. Nonlinear mapping networks. / Chem Inf Comput Sci 1997 40 1356-62. 

For an introduction to NNs and their functionality, the reader is referred to the rich literature on the subject (e.g., Rumelhart et al, 1986 Barron and Barron, 1988). For our purposes it suffices to say that NNs represent nonlinear mappings formulated inductively from the data. In doing so, they offer potential solutions to the functional estimation problem and will be studied as such. 

In addition to looking for data trends in physical property space using PCA and PLS, trends in chemical structure space can be delineated by viewing nonlinear maps (NLM) of two-dimensional structure descriptors such as Unity Fingerprints or topological atom pairs using tools such as Benchware DataMiner [42]. Two-dimensional NLM plots provide an overview of chemical structure space and biological activity/molecular properties are mapped in a 3rd and/or 4th dimension to look for trends in the dataset. 

In general, the evaluation of interlaboratory studies can be carried out in various ways (Danzer et al. [1991]). Apart from z-scores, multivariate data analysis (nonlinear mapping, principal component analysis) and information theory (see Sect. 9.2) have been applied. 

Cluster analysis will be discussed in Chapter 6 in detail. Here we introduce cluster analysis as an alternative nonlinear mapping technique for exploratory data analysis. The method allows gaining more insight into the relations between the objects if a... 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

For comparison also a dendrogram (Figure 3.28) and a nonlinear mapping (NLM) (Figure 3.29) have been performed on the PAH data. Results from these methods show a clear separation of the samples from Linz and Vienna, but not much more details. The clusters in the NLM plots are very similar to the clusters in the PCA score plots. Thus, preserving the distances using two dimensions—the goal of... 

Rassokhin, D., Lobanov, V. S., and Agrafiotis, D. K. (2000) Nonlinear mapping of massive data sets by fuzzy clustering and neural networks../. Comput. Chem. 21, 1-14. 

Here E is the solute excess molar refractivity, S is the solute dipolarity/ polarizability A and B are the overall or summation hydrogen-bond acidity and basicity, respectively and V is the McGowan characteristic volume lower-case letters stand for respective coefficients which are characteristic of the solvent, c is the constant. By help of sfafisfical methods like the principal component analysis and nonlinear mapping, the authors determined the mathematical distance (i.e., measure of dissimilarify) from an IL fo seven conventional solvents immiscible with water. It appears that the closest to the IL conventional solvent is 1-octanol. Even more close to IL is an aqueous biphasic system based on PEG-200 and ammonium sulfate (and even closer are ethylene glycol and trifluoroethanol, as calculated for hypofhefical water-solvenf sysfems involving fhese solvenfs). 

An overall picture of similarity between [C4CiIm][PFg] and conventional solvents is given by the so-called nonlinear map, or Sammon map [20] in Figure 9.5. The closer the points representing solvents on the map, the more similar the solvents are. As is seen, the position of the IL on the map is between esters and aromatic hydrocarbons. 

When the significant eigenvectors are more than 2 or 3, the information cannot be easily visualized by few eigenvector plots. In these cases the use of nonlinear mapping (NLM) can give a planar representation of the objects with greater fidelity to the structure of the information in the hyperspace of the variables... 

Nonlinear mapping has not been widely applied in food chemistry because ... 

An example of the use of NLM is shown in Fig. 6. Nonlinear mapping was also used in the representation of the amino acid sj ctrum of French red wines 110 objects (Bordeaux, Beaujolais and non-Beaujolais Bourgogne wines) characterized by 20 amino acids were represented. 

To reduce computing time, a two-step method has been suggested , the simplified nonlinear mapping (SNLM), and it was first used to present samples of olive oil of several Italian regions In the first step, the NLM coordinates of a... 

Fig. 6. Nonlinear mapping of Bordeaux and Rhone wines. (Adapted from Ref.

Fig. 7. Simplified nonlinear mapping of Italian red wines (category centroids as basepoints). 8 variables...

For many problems, rather different solutions of similar fitness exist. To obtain a quick overview of the structure of the underlying high-dimensional fitness landscape, we used a self-organizing map (SOM) that was developed originally by Kohonen [14] in this study. Figure 10.1 shows a nonlinear mapping of a seven-... 

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