Scaling, multidimensional

Multidimensional scaling [70] is a method for obtaining the best low-dimensional representation of a high-dimensional data set. Normally, a two- or three-dimensional representation is required, since it can then be plotted and inspected visually for clusters. In the classical scaling technique, the low-dimensional representation is obtained by extracting the eigenvectors of the (Af xAf) dissimilarity matrix. However, it can be shown that this operation is equivalent to a PC A of the (ApXAp) covariance matrix, provided that the distances in the dissimilarity matrix are Euclidian or near-Euclidian [53]. Since the covariance matrix is invariably of smaller order than the dissimilarity matrix, PCA is to be preferred on computational grounds. The only exception is if the dissimilarity matrix is available but the covariance matrix is not, a circumstance that rarely arises in structural chemistry work. 

Ordinal multidimensional scaling is an empirical technique for iteratively improving a low-dimensional representation of a high-dimensional dataset [53, 70]. The technique is non-parametric, because it is based on the ranks of the dissimilarities rather than their actual values. The iterative procedure aims to minimize a quantity called stress . Small values of stress are obtained if the ranks of the Euclidian distances between points on the low-dimensional plot are similar to the ranks of the dissimilarities of the corresponding observations in the original multidimensional space. 

The first application using MDS in molecular diversity analysis was introduced by a group at Chiron as a means of reducing the enormous dimensionality of binary chemical descriptors They found that 2048-bit Daylight fingerprints associated with 721 commercially available primary amines could be reduced to only five dimensions that reproduced all 260,000 original dissimilarities with a standard deviation of only 10%. Similarly, only seven dimensions were required to reduce the 642,000 pairwise similarities among a set of 1133 carboxylic acids and acid chlorides to the same standard deviation. 

A simple example may serve to introduce the subject. Table 38.4 gives the distances between a few major European cities measured by the approximate flight 

Even when the input data are much less accurate a reasonable configuration can be recovered by MDS. For example, replacing the distances by their rank numbers and then applying non-metric MDS gives the configuration displayed in Fig. 38.5, which still represents the actual distance configuration quite well. 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized  

Minimizing this function is equivalent to finding a low-dimensional configuration of points which has Euclidean object-to-object distances by as close as possible to some transformation/(.) of the original distances or dissimilarities, dy. Thus, the model distances by are not necessarily fitted to the original dy, as in classical MDS, but to some admissible transformation of the measured distances. For example, when the transformation is a general monotonic transformation it preserves the 

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Garrido L, Gomez S, Roca J. Improved multidimensional scaling analysis using neural networks with distance-error backpropagation. Neural Comput 1999 11 595-600. 

A close analogy exists between PCoA and PCA, the difference lying in the source of the data. In the former they appear as a square distance table, while in the latter they are defined as a rectangular measurement table. The result of PCoA also serves as a starting point for multidimensional scaling (MDS) which attempts to reproduce distances as closely as possible in a low-dimensional space. In this context PCoA is also referred to as classical metric scaling. In MDS, one minimizes the stress between observed and reconstructed distances, while in PCA one maximizes the variance reproduced by successive factors. 

Fig. 38.6. Three-dimensional configuration according to a non-metric multidimensional scaling applied to 24 types of bread differing in appearance as assessed by 12 panellists.

J.B. Kruskal, Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29 (1964) 1-27,115-129. 

J.B. Kruskal and M. Wish, Multidimensional Scaling. SAGE, Newberry Park, CA, 1991. 

Fig. 16.2 A multidimensional scaling (MDS) plot in three dimensions for scent profiles of individuals from three spotted hyena clans (o Isiaka clan, N = 8 Pool clan, N = 14 A Mamba clan, N = 23). Similarities between samples were calculated with the Bray-Curtis coefficient. MDS Stress 0.08... 

Multidimensional scaling (MDS) is a collection of analysis methods for data sets which have three or more variables making up each data point. MDS displays the relationships of three or more dimensional extensions of the methods of statistical graphics. 

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 an excellent review of multidimensional scaling, see http //forrest.psych.unc.edu/teaching/p208a/ mds/mds.html. 

Figure 5. Multidimensional scaling plotof GPCR SAR space as defined by the presence or absence of Themes. FamilyAGPCRs are indicated by filled squares Family B by triangles and Family C by diamonds.

Borg, I. and Groenen, P. (1997) Modern multidimensional scaling. Springer, New York. 

Kruskal, J. (1977) The relationship between multidimensional scaling and clustering. In Classification and Clustering, Van Ryzin, J. (ed.), Academic Press, New York. 

How is dimension reduction of chemical spaces achieved There are a number of different concepts and mathematical procedures to reduce the dimensionality of descriptor spaces with respect to a molecular dataset under investigation. These techniques include, for example, linear mapping, multidimensional scaling, factor analysis, or principal component analysis (PCA), as reviewed in ref. 8. Essentially, these techniques either try to identify those descriptors among the initially chosen ones that are most important to capture the chemical information encoded in a molecular dataset or, alternatively, attempt to construct new variables from original descriptor contributions. A representative example will be discussed below in more detail. 

Heath, A.C., Meyer, J., Eaves, L.J., and Martin, N.G. (1991a) The inheritance of alcohol consumption patterns in a general population twin sample I. Multidimensional scaling of quantity/fre-quency data. J Stud Alcohol 52 345-352. 

Odor and taste quality can be mapped by multidimensional scaling (MDS) techniques. Physicochemical parameters can be related to these maps by a variety of mathematical methods including multiple regression, canonical correlation, and partial least squares. These approaches to studying QSAR (quantitative structure-activity relationships) in the chemical senses, along with procedures developed by the pharmaceutical industry, may ultimately be useful in designing flavor compounds by computer. 

Four studies are described here that relate physicochemical properties to odor quality as defined by maps derived by multidimensional scaling procedures. The mathematical procedures used to relate the physicochemical properties to the maps are discussed as well. 

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