Clustering multidimensional objects

This technique makes use of the innate human cognition to perform clustering in multidimensional space (Osbourn and Martinez, 1995). It is unsupervised and model-free, therefore requires from the user only the data input. A special mask mathematically defines the visual region of influence. Its shape is based on human visual perception, taking advantage of the human brain to recognize and cluster objects (Fig. 10.10a). Its properties are as follows. Two points in space are clustered only if no other point lies within area of the mask which thus defines the exclusion region. In this way, an n-dimensional problem is reduced to set of n two-dimensional problems. 

Preference mapping can be accomplished with projection techniques such as multidimensional scaling and cluster analysis, but the following discussion focuses on principal components analysis (PCA) [69] because of the interpretability of the results. A PCA represents a multivariate data table, e.g., N rows ( molecules ) and K columns ( properties ), as a projection onto a low-dimensional table so that the original information is condensed into usually 2-5 dimensions. The principal components scores are calculated by forming linear combinations of the original variables (i.e., properties ). These are the coordinates of the objects ( molecules ) in the new low-dimensional model plane (or hyperplane) and reveal groups of similar... 

An object is characterized by a set of measures, and it may be represented as a point in multidimensional space defined by the axes, each of which corresponds to a variate. In Figure 5, a data matrix X defines measures of two variables on two objects A and B. Object A is characterized by the pattern vector, a = Xu, and B by the pattern vector, b = X21, X22. Using a distance measure, objects or points closest together are assigned to the same cluster. Numerous distance metrics have been proposed and applied in the scientific literature. 

Like PCA, non-linear mapping (NLM), or multidimensional scaling, is a method for visualizing relationships between objects, which in the medicinal chemistry context often are compounds, but could equally be a number of measured activities. It is an iterative minimization procedure which attempts to preserve interpoint distances in multidimensional space in a 2D or 3D representation. Unlike PCA however, the axes are not orthogonal and are not clearly interpretable with respect to the original variables. Nonlinear mapping has been used to cluster aromatic and aliphatic substituents. ... 

More objective procedures in series design are clustering methods in multidimensional parameter space substituents from different clusters are selected for synthesis (chapter 3) [50, 154, 403]. As this approach cannot automatically avoid collinearity or multicollinearity, several different standard sets of aromatic substituents have been proposed e.g. [652, 653]). A distance mapping technique may be used to select further substituents on the basis of a nj ximum distance to the substituents which already are included [652]. A modification [654] of this approach uses the determinant of the parameter correlation matrix as the criterion for substituent selection. 

One-Way versus Two-Way Clustering Clustering methods that were mentioned thus far can be referred to as one-way clustering. When processing multidimensional data objects, one-way clustering can be applied to either the rows or the columns of the data matrix, separately. For a given dataset, one-way methods perform... 

An analytical method can be represented by a point or by a region in a multidimensional "space of procedures". The coordinates correspond to the parameters of the method, like accuracy, time, cost, etc.. Kaiser C4213 applied the information theory.to the estimation of the "informing power" of analytical procedures. Pattern recognition methods have been proposed by Wold et. al. C36, 341, 3433 for an objective evaluation of analytical methods. A data matrix is obtained by the application of me-thods to a number of real samples. Mathematical models were constructed for the clusters describing the methods under consideration. 

Cluster analysis (Everitt et al. 2001) is a tool for grouping various objects on the basis of their distance in a multidimensional space. In chemistry, cluster analysis is used for the interpretation of analytical results. For example, in food or drink samples, the concentrations of many chemicals are measured, and the question is which of the samples are similar on the basis of the analytical results. The first step is always the transformation of the raw measurement data into a distance matrix. The general features of a distance matrix are that the diagonal elements are zero (everything is at zero distance from itself), all matrix elements are non-negative (negative distance cannot be interpreted) and the matrix is symmetrical (to and from distances are identical). It is clear that the distance matrix defined by Eq. (8.29) fulfils these requirements. 

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