Multidimensional ranking

Sorensen PB, Bruggemann R, Thomsen M, Lerche DB (2005) Application of multidimensional rank-correlation, MATCH Communications in Mathematical and in Computer Chemistry 54 643-670... 

Sorensen PB, Briiggemann R, Thomsen M and Lerche D (2005) Applications of multidimensional rank-correlation. Match - Comrnun Math Comput Chem 54 643-670... 

We now consider a subspace of S which is orthogonal to v, and we repeat the argument. This leads to V2, and in the multidimensional case to all r columns in V. By the geometrical construction, all r latent vectors are mutually orthogonal, and r is equal to the number of dimensions of the pattern of points represented by X. This number r is the rank of X and cannot exceed the number of columns p in X and, in our case, is smaller than the number of rows in X (because we assume that n is larger than p). 

To construct the multidimensional boxes, a training set of samples with known class ideniit) is obtained. The training set is divided into separate sets, one for each class, and principal components are calculated separately for each of the classes. The number of relevant principal components (rank) is determined for each class and the SIMCA models are completed by defining boundary regions for each of the PCA models. 

Since aromaticity is a central issue in the present report, it is of importance to summarize briefly the measures and attributes of aromaticity, while for more detailed discussion the reader is recommended recent reviews [47-49], Although the use of quantitative measures for the description of aromaticity is questioned from time to time [52], and their direct use to rank aromaticity in different compounds might be misleading due to the multidimensional nature [53-56] of this phenomenon, the usefulness of different quantitative measures cannot be questioned. Four main criteria can be selected in the discussion of aromaticity. These are the energetic, geometric, magnetic, and reactivity criteria. Among these the latter is usually more qualitative, therefore only the first three will be mentioned briefly here. 

The resulting semi-quantitative model was used in conjunction with structure-based docking and scoring, 3D-QS AR based affinity and selectivity predictions and in silico ADME models to estimate membrane permeability, solubility, and other key properties for the optimization process in this series. Hence, in this as well as in other series, multiple models can be collectively applied for ranking and prioritizing synthesis candidates and focused virtual libraries during advanced stages of multidimensional compound optimization. 

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). 

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. 

In partial order ranking - in contrast to standard multidimensional statistical analysis - neither assumptions about linearity nor any assumptions about distribution properties are made. In this way the partial order ranking can be considered as a non-parametric method. Thus, there is no preference among the descriptors. However, due to the simple mathematics outlined above, it is obvious that the method a priori is rather sensitive to noise, since even minor fluctuations in the descriptor values may lead to non-comparability or reversed ordering. An approach how to handle loss of information by using an ordinal in stead of a matrix can also be found in the chapter by Pavan et al., see p. 181). 

In partial order ranking - in contrast to standard multidimensional statistical analysis - neither assumptions about linearity nor any assumptions about distribution properties are made. Partial order ranking may be considered as a parameter-free method. Thus, there is no preference among the... 

Liu XQ, Sidiropoulos ND, Cramer-Rao lower bounds for low-rank decomposition of multidimensional arrays, IEEE Transactions on Signal Processing, 2001,49, 2074—2086. 

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. 

Sometimes, several feature selection methods are used for a given analysis. For example, an analyst might reduce chromatogram to a peak table, selecting a series of candidate variables of interest and then perform further variable ranking and optimization on the integrated peak table, especially in the case of multidimensional separations where hundreds, if not thousands of compounds can be resolved (Felkel et al., 2010). 

The rank-by-feature framework was shown effective in facUitating users exploration of large multidimensional datasets fi om several different research fields including microarray data analysis (Seo and Shneiderman, 2006). We believe that it can be successfully apphed to biomedical datasets that are very often large multidimensional datasets. In the following sections, we introduce visual interface frameworks and ranking criteria for ID and 2D projections for multidimensional datasets. 

We believe that the proposed strategy for multidimensional data exploration with room for iteration and rapid shifts of attention enables noviees and experts to make discoveries more reliably. The graphics, ranking and interaction for diseovery (GRID) principles are as follows ... 

The rank-by-feature framework enables users to apply a systematic approach to understanding the dimensions and finding important features using axis-parallel ID and 2D projections of multidimensional datasets. Users begin by seleeting a ranking criterion and then can see the ranking for all ID or 2D projections. They can select... 

Seo, J., and Shneiderman, B. (2004). A rank-by-feature framework for unsupervised nniltidimensional data exploration using low dimensional projections. In Proceedings of IEEE Symposium on Information Visualization, M. Ward and T. Munzner (Eds.). Austin, TX lEER Computer Society Press, pp. 65-72. Seo, J., and Shneiderman, B. (2005). A rank-by-feature framework for interactive exploration of multidimensional data. Informat Visualiz, 4(2) 99-113. 

Intelligent computer assisted interpretation of spectroscopic data should be based on the knowledge from large structure oriented data collections. Both the inspection of spectral features and the statistical evaluation of similar structures (from library searches) can provide a set of probability ranked substructures which are readily assembled to target structures. The idea of substructure analysis allows the chemist to combine the results of different interpretation strategies, different databases and different spectroscopic methods to yield the structural information desired. Thus in a multidimensional data system hke SPECINFO structural noise can be effectively suppressed, if all information available in the spectroscopic laboratory is combined in a central intelligent computer system. 

For risk assessment, a DSS (Lopes et al, 2009) is used that incorporates a decision model proposed by Brito Almeida (2009). This DSS is a system designed to assess risk levels of each pipeline section with its characteristics, ranking all sections in a multidimensional hierarchy of risk, based on Multi Attribute Utility Theory (MAUT). 

Through the approach used, it was possible to present a rich analysis which considers all risk dimensions and so instead of analyzing only individual risk measures or frequencies related to a specific consequence, it provides a ranking based on the multidimensional risk of each section of this gas pipeline. 

In addition to the well-known scatter-plots, two other graphic tools have been considered in this study Cobweb plots and contribution to the sample mean plots (CSM plots). Cobweb plots have been designed to show multidimensional samples in a two-dimensional graph. Vertical parallel lines separated by equal distances are used to represent the sampled values of several inputs/outputs. Each vertical line is used for a different input/output. Either the raw values or the ranks may be represented. Sampled values are marked in each vertical line and jagged hues connect the values corresponding to the same run. The analysis of cobweb plots enables the characterisation of dependence and conditional dependence. 

Falahee, M. and Macrae, A. W. (1997). Perceptual variation among drinking waters The reliability of sorting and ranking data for multidimensional scaling. Food Quality and Preference, 8, 389-394. 

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