Multidimensional scaling methods

Other methods exist besides PCA-based approaches that are able to extract a pattern from computed statistics. One of them is the so-called classical multidimensional scaling method in which a fictitious, low-dimensional distribution of points is calculated to reproduce the distance relations among the points in the intermediate dimensionality. An example is the map of the distances in RMSD for configurations of dialanine. There, the dihedral angle 4> rotation is very similar to the distances that one would obtain from a suitable distribution of points along a ring. This implies that not all the coordinates of the intermediate representation are useful, because the landscape in this case is effectively two-dimensional. 

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). 

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. 

While some researchers feel that individual components making up the sensory response need to be characterized (9), others feel that methods dealing with the composite sensation are more appropriate. For this reason methods which do not rely on internal representation of terms have been applied. One such method is multidimensional scaling (MBS), which treats data based on a persons total perception of the dissimilarity between objects.(10)... 

Relationships between the individual LOE can be examined via principal components analysis (PCA). Correlations among principal components for individual LOE indicate concordance or agreement. Relationships between different SQT LOE can also be assessed using other methods including Mantel s test (Legendre and Fortin, 1989) coupled with a measure of similarity or ordination canonical discriminant (or correspondence) analyses multidimensional scaling (MDS). 

The simplest experiments are those that focus just on the ligand. These are typically used to determine solution conformations or 3D structures of ligands. Homonuclear1H II ) or 21) NMR experiments are used mainly here. At the other end of the scale, experiments to study the macromolecular binding partner often require labeled protein and multidimensional NMR methods, as indicated on the right-hand side of Fig. 1. Finally, many NMR experiments provide information... 

Harchman, R.A. and Lundy, M.E., The PARAFAC model for three-way factor analysis and multidimensional scaling, in Research Methods for Multimode Data Analysis, Law, H.G. et al., Eds., Praeger, New York, 1984. 

Figure 7.3 Authentication of monovarietal virgin olive oils results of applying multidimensional scaling to volatile compounds. The distance was Manhattan (city block) and the amalgamation was Ward s method. Note A, cv. Arbequina C, cv. Coratina K, cv. Koroneiki P, cv. Picual (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

Schiffman, S., Reynolds, M.L. and Young, F.W. (1981) Introduction to Multidimensional Scaling. Theory, Methods and Applications, Academic Press, Orlando, FL, USA. 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

In order to provide an easy way to visualize how the methods differ from each other the accuracy values (Q3, Ca, Cp and Cc) for each method where subjected to nonmetric multidimensional scaling analysis (Cox and Cox, 1994). The resulting graphs provide a representation in a two-dimensional space of the methods and those are shown in figure 3. As can be observed, the interrelationship between the methods... 

Figure 3. Multidimensional scaling analysis of the dissimilarities between accuracies of different protein secondary structure prediction methods. The method codes can be found in Table I.

In this chapter, we would like to demonstrate that one of the very old MVA tools, nonmetric multidimensional scaling (nMDS) [3], can work well as an unsupervised truly data-driven method for data reduction. We first explain an efficient maximally nonmetric algorithm [4] and then demonstrate its superiority to linear MVA methods. We also demonstrate that the subsequent application of linear MVA after data reduction by nMDS can often be a powerful data mining technique. 

There have been attempts to deal with the issue of nonlinearity in data sets. Detrended principal components (DPC) use a polynomial expression to remove the nonlinear relationships from the PCA axes. DPC are useful for data sets of moderate nonlinearity. Detrended correspondence analysis uses a more complex algorithm to eliminate the nonlinearity but requires a more complex computation. Nonmetric multidimensional scaling (NMDS) is a robust method that deals with nonlinearities by using ranks. 

Kruskal, J. B. (1964) Non-metric multidimensional scaling A numerical method. Psychometrika 29, 115-129. 

Two statistical methods used in behavioral science, multidimensional scaling (9) and hierarchical clustering (10), are used to analyze the error patterns represented in confusion matrices. Together, they can be the basis for a technique which examines the nature of a neural network s decision process. 

A whole spectrum of statistical techniques have been applied to the analysis of DNA microarray data [26-28]. These include clustering analysis (hierarchical, K-means, self-organizing maps), dimension reduction (singular value decomposition, principal component analysis, multidimensional scaling, or correspondence analysis), and supervised classification (support vector machines, artificial neural networks, discriminant methods, or between-group analysis) methods. More recently, a number of Bayesian and other probabilistic approaches have been employed in the analysis of DNA microarray data [11], Generally, the first phase of microarray data analysis is exploratory data analysis. 

Corresponding to the dimension d = 2, the poset shown in Fig. 19 can alternatively be visualized by a two-dimensional grid as is shown in Fig. 22. Both visualizations have their advantages. Structures within a Hasse diagram, e.g., successor sets, or sets of objects separated from others by incomparabilities, can be more easily disclosed by a representation like that of Fig. 19. In multivariate statistics reduction of data is typically performed by principal components analysis or by multidimensional scaling. These methods minimize the variance or preserve the distance between objects optimally. When order relations are the essential aspect to be preserved in the data analysis, the optimal result is a visualization of the sediment sites within a two-dimensional grid. 

While PCA is a linear projection method, there also exist nonlinear projection methods, e.g. multidimensional scaling [Mardia et al. 1979] and nonlinear PCA [Dong McAvoy 1996], A good overview of nonlinear multivariate analysis tools is given by [Gift 1990],... 

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