Principal Components Analysis , essential

The essential degrees of freedom are found by a principal component analysis of the position correlation matrix Cy of the cartesian coordinate displacements Xi with respect to their averages xi), as gathered during a long MD run ... 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

The goal of factor analysis (FA) and their essential variant principal component analysis (PCA) is to describe the structure of a data set by means of new uncorrelated variables, so-called common factors or principal components. These factors characterize frequently underlying real effects which can be interpreted in a meaningful way. 

An important application field of factor and principal component analysis is environmental analysis. Einax and Danzer [1989] used FA to characterize the emission sources of airborne particulates which have been sampled in urban screening networks in two cities and one single place. The result of factor analysis basing on the contents of 16 elements (Al, B, Ba, Cr, Cu, Fe, Mg, Mn, Mo, Ni, Pb, Si, Sn, Ti, V, Zn) determined by Optical Atomic Emission Spectrography can be seen in Fig. 8.17. In Table 8.3 the common factors, their essential loadings, and the sources derived from them are given. 

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. 

A more detailed decomposition of macromolecular dynamics that can be used not only for assessing convergence but also for other purposes is principal components analysis (PCA), sometimes also called essential dynamics (Wlodek et al. 1997). In PCA the positional covariance matrix C is calculated for a given trajectory after removal of rotational and translational motion, i.e., after best overlaying all structures. Given M snapshots of an N atom macromolecule, C is a 3N X 3A matrix with elements... 

The data processing of the multivariate output data generated by the gas sensor array signals represents another essential part of the electronic nose concept. The statistical techniques used are based on commercial or specially designed software using pattern recognition routines like principal component analysis (PCA), cluster analysis (CA), partial least squares (PLSs) and linear discriminant analysis (LDA). 

However, since failures may involve a large number of parameters, often not independent from each other, the univariate techniques may be not so efficient therefore, they have been replaced by multivariate techniques, which are powerful tools able to compress data and reduce the problem dimensionality while retaining the essential information. In detail, Principal Component Analysis (PCA) [12, 47] is a standard multivariate technique, whose main goal is to transform a number of... 

Self-organizing maps in conjunction with principal component analysis constitute a powerful approach for display and classification of multivariate data. However, this does not mean that feature selection should not be used to strengthen the classification of the data. Deletion of irrelevant features can improve the reliability of the classification because noisy variables increase the chances of false classification and decrease classification success rates on new data. Furthermore, feature selection can lead to an understanding of the essential features that play an important role in governing the behavior of the system or process under investigation. It can identify those measurements that are informative and those measurements that are uninformative. However, any approach used for feature selection should take into account the existence of redundancies in the data and be multivariate in nature to ensure identification of all relevant features. 

Another important class of MVA is represented by cluster analysis methods and principal component analysis (PCA). The latter is a representative of data reduction methods that exploit linear algebra. We do not, however, believe all the important patterns can be captured by linear algebraic methods. Finear mathematical methods are ideal for data compression, because to recover the original data distortion is undesirable. Thus, data compression is essentially applied Fourier analysis [2], In contrast, data mining is a kind of pattern... 

Figure 1. Principal Components Analysis (PCA) and Factorial Discriminating Analysis (FDA) ofRavensara aromatica Essential oils (26).

Principal components analysis has also been applied to array time series data (83-85) and a limited number of principal components usually accounts for the essential features of the data set, allowing considerably reduced complexity for example, the sporulation data was modeled using as few as two principal components (83). 

An alternative approach to that of picking out the essential tests in a specification using regression analysis is to take a look at the specification as a whole and extract the essential features (termed principal components 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. 

Fig. 9.12. Scattergram of the scores on the first two factors from a principal component analysis of the torsion angles of 181 furanose rings. The two factors include essentially 100% of the variance of the ring torsion angles. The numbers show the position of some selected ring conformations. 1 envelope (e) with mirror plane through C(2 ) endo 2 e C(3 ) endo 3 e C(3 ) exo 4 e C(2 ) exo 5 twist with Ci axis through C(T) 6 e 0(T) endo...

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