Multi-dimensional scaling

S. S. Schiffman, M.L. Reynolds and F.W. Young, Introduction to Multi-dimensional Scaling Theory, Methods and Applications. Academic Press, New York, 1981. 

Agrafiotis, D. K., Rassokhin, D. N., and Lobanov, V. S. (2001) Multi-dimensional scaling and visualization of large molecular similarity tables. J. Comput. Chem. 22,... 

Multi-dimensional scaling with ideal point... 

Like PCA, NLM or multi-dimensional scaling, is a method for visualizing relationships between objects, which in 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 multi-dimensional 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. However, it can be valuable in cases where the first two or three PCs are influenced by outliers (extreme data points) or only explain a small percentage of the original data. NLM has been used to cluster aromatic and aliphatic substituents," " for example. 

Torgerson, W. S. (1952) Multi-dimensional scaling I. Theory and method. Psy-chometrika 17, 401 19. 

Fig. 2.1. Outline of the hybrid algorithm. The unstructured array of sensors is clustered using multi-dimensional scaling (MDS) with a mutual information (MI) based distance measure. Then Vector Quantization (VQ) is used to partition the sensor into correlated groups. Each such group provides input to one module of an associative memory layer. VQ is used again to provide each module unit with a specific receptive field, i.e. to become a feature detector. Finally, classification is done by means of BCPNN.

Fig, 23.10 Flavour terms by multi-dimensional scaling. (After Brown and Clapperton [52].)... 

Heiser, W. J. (1990). A generalized majorization method for least squares multi-dimensional scaling of pseudodistances that may be negative. Psychometrika, 56, 7-27. 

The first method is to develop a cognitive or semantic map of the expert s knowledge using quantitative methods. This method requires the expert to complete word associations of all of die related concepts in the content domain. The intercorrelations are multi-dimensionally scaled to generate a structural map (see Jonassen, [7] for a description of this technique). This structural map would then be used as a graphical browser or concept map for accessing information in the hypertext. 

Borg I, Groenen P (1997) Modem multi-dimensional scaling. Springer, New York... 

This discussion will focus on two main techniques to perform the reduction (1) principal component analysis and (2) factor analysis. Both of these techniques attempt to find an appropriate low-dimensional representation of the covariance matrix. Other approaches such as multi-dimensional scaling, non-linear mapping, and Kohonen networks are reviewed briefly in this section, and discussed in greater detail in Section 5. 

Multi-dimensional Scaling, Non-linear Mapping, and Kohonen Networks... 

Multi-dimensional scaling (MDS) emerged from the need to visualize a set of objects described by means of a similarity or dissimilarity matrix, The technique originated in the field of psychology and can be traced back to the work of Torgerson and Kruskal. The problem is to construct a configuration of points in a low-dimensional space from information about the distances between these points. In particular, given a set of k data points in the input space xi,i= 1,2, a... 

Non-linear mapping is a multivariate statistical technique that is closely related to multi-dimensional scaling. Just like MDS, the objective is to approximate local geometric relationships on a two- or three-dimensional plot. 

Kruskal, J. B., Non-metric multi-dimensional scaling a numerical method, Phychometrika, 29 (1964) 115-129. 

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