Principal Component Analysis variable contribution

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. 

Principal components analysis Computational approach to reduce the dimensionality (Le.. the number of variables) in a data analysis, by weighting variables according to their contribution to the overall variation. A plot of points in the first two principal components is like a two-dimensional (2D) shadow" of the multidimensional data (hat can be used to find clusters and relationships among the points (i.c.. compounds). 

By analyzing the background information and by scrutinizing the constituents of the reaction system, it is possible to select pertinent descriptors of the system. This allows all aspects on the reaction to be taken into account prior to any experiment. Principal components analysis of the variation of the descriptors over the whole set of possible constituents of the system will reveal the principal properties. Sometimes a descriptor variable which does not contribute at all to any systematic variation over the set of compounds has been included among the descriptors. In such cases, it can reasonably be assumed that the descriptor has little relevance to the problem under study. Irrelevant descriptors are easily detected by principal components analysis. 

Separating measured data vectors or matrices into independent lower order approximations and residual terms is useful both in process performance evaluation, as variance contributions can be clearly separated, and in feedback process control, as the number of decision variables can be significantly reduced while the adverse effects of autocorrelation are eliminated. In the following two sections orthogonal decomposition approaches using Gram polynomials and principal components analysis (PCA) will be introduced. 

Principal component analysis is used to reduce the information in many variables into a set of weighted linear combinations of those variables it does not differentiate between common and unique variance. If latent variables have to be determined, which contribute to the common variance in a set of measured variables, factor analysis (FA) is a valuable statistical method, since it attempts to exclude unique variance from the analysis. 

Principal components analysis (PCA) is based on a mathematic procedure whieh reduces the number of original variables (1024 intensities of a spectrum in our case) into some principal components (generally less than 10) whieh are linearly uncorrelated. Each individual (spectrum) has a weight (seores) associated with each of the components (loadings) which represents the importance of the contribution of a component for this... 

Fig. 4.7. Illustration of the process of principal components analysis to produce a new data matrix of Q scores for P samples where Q is equal to (or less than) the smaller of /V (variables) or P (samples). The loadings matrix contains the contribution (loading) of each of the Af variables to each of the O principal components.

There are a few approaches to fadUtate the interpretation of data on the complex and variable mixtures of chlorobiphenlys in marine environmental samples. The composition of mixtures can be represented as mole percent contributions of individual CBs to their sum Duinker et al., 1980) or as molar ratios, e.g., CBx/CB153 Boon et al., 1992). This allows visual, qualitative and quantitative comparisons between samples with widely different overall compositions. Quantitative and less arbitrary comparison is possible with statistical methods such as principal component analysis Jackson, 1991). Finally, distribution patterns between the CB mixtures in solution and in suspended particles can be studied by plotting concentration ratios in these compartments Le., distribution coefficients) against known molecular properties, e.g., octanol-water distribution coefficients Schulz-BuU et al., 1998). 

Principal component analysis (PCA) is a statistical technique with a long history in multivariate data analysis (see Chemometrics Multivariate View on Chemical Problems) PCA reduces a set of partially cross-correlated data into a smaller set of orthogonal variables (principal components) without a significant loss in the contribution to variation. In effect, the method detects and combines descriptors which behave in a similar way into a new set of variables that are non-correlated, i.e., they are orthogonal. 

The second step in factor analysis is interpretation of the principal components or factors. This is accomplished by examining the contribution that each of the original measured variables makes to the linear combination describing the factor axis. These contributions are called the factor loadings. When several variables have large loadings on a factor they may be identified as being associated. From this association one may infer chemical or physical interactions that may then be interpreted in a mechanistic sense. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

A principal components (PC) analysis was also performed to evaluate the elements in the data set that contributed to the variance. PC plots graphically indicate a linear combination of original variables, oriented in the direction of greatest variance. PC space also displays the elements with the greatest variation by graphically displaying them with the longest vectors. 

Lipid A constitutes the covalently bound lipid component and the least variable component of LPS (25). It anchors LPS to the bacterial cell by hydrophobic and electrostatic forces and mediates or contributes to many of the functions and activities that LPS exerts in prokaryotic and eukaryotic organisms. In the following sections, the primary structure of lipid A of different Gram-negative bacteria is described, together with some of its characteristic biological properties. Furthermore, this article describes some of the principal methods that have been used for the structural analysis of lipid A and discusses their merits and limitations. 

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