Correlative component analysis

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 ... 

To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

I Principal Component Analysis (PCA) transforms a number of correlated variables into a smaller number of uncorrelated variables, the so-called principal components. 

The dimensionality of a data set is the number of variables that are used to describe eac object. For example, a conformation of a cyclohexane ring might be described in terms c the six torsion angles in the ring. However, it is often found that there are significai correlations between these variables. Under such circumstances, a cluster analysis is ofte facilitated by reducing the dimensionality of a data set to eliminate these correlation Principal components analysis (PCA) is a commonly used method for reducing the dimensior ality of a data set. 

An alternative to principal components analysis is factor analysis. This is a technique which can identify multicollinearities in the set - these are descriptors which are correlated with a linear combination of two or more other descriptors. Factor analysis is related to (and... 

The data from sensory evaluation and texture profile analysis of the jellies made with amidated pectin and sunflower pectin were subjected to Principal component analysis (PC) using the statistical software based on Jacobi method (Univac, 1973). The results of PC analysis are shown in figure 7. The plane of two principal components (F1,F2) explain 89,75 % of the variance contained in the original data. The attributes related with textural evaluation are highly correlated with the first principal component (Had.=0.95, Spr.=0.97, Che.=0.98, Gum.=0.95, Coe=0.98, HS=0.82 and SP=-0.93). As it could be expected, spreadability increases along the negative side of the axis unlike other textural parameters. 

Reduced rank regression (RRR), also known as redundancy analysis (or PCA on Instrumental Variables), is the combination of multivariate least squares regression and dimension reduction [7]. The idea is that more often than not the dependent K-variables will be correlated. A principal component analysis of Y might indicate that A (A m) PCs may explain Y adequately. Thus, a full set of m... 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

Ab initio electron correlated calculations of the equilibrium geometries, dipole moments, and static dipole polarizabilities were reported for oxadiazoles <1996JPC8752>. The various measures of delocalization in the five-membered heteroaromatic compounds were obtained from MO calculations at the HF/6-31G level and the application of natural bond orbital analysis and natural resonance theory. The hydrogen transfer and aromatic energies of these compounds were also calculated. These were compared to the relative ranking of aromaticity reported by J. P. Bean from a principal component analysis of other measures of aromaticity <1998JOC2497>. 

To understand these processes and correlate residue profiles with specific toxic responses required congener-specific methods of analysis and complex statistical techniques (principal component analysis). Using these techniques, it was established that eggs of Forster s terns of two colonies differed significantly in PCB composition (Schwartz and Stalling 1991). Similar techniques were used to identify various PCB-contaminated populations of harbor seals (Phoca vitulina) in Denmark (Storr-Hansen and Spliid 1993). 

Harris et al. also tested the taxonic conjecture by examining one of the logical consequences of taxonicity, a prediction that taxon membership should emerge as the first unrotated component in principal component analysis (Meehl, 1992). The idea behind this corollary is that in the absence of nuisance correlations, all observed correlations between indicators are due to latent taxon membership. Furthermore, the purpose of a principal component analysis is to reduce the observed correlations to a smaller set of factors... 

The authors wanted to select indicators that specifically tap melancholic depression. To evaluate this construct, a principal components analysis of the joint pool of K-SADS and BDI items was performed. Two independent statistical tests suggested a two-component solution, but the resulting components appeared to reflect method factors, rather than substantive factors. Specifically, all of the BDI items loaded on the first component (except for three items that did not load on either component) and nearly all of the K-SADS items loaded on the second component. In fact, the first component correlated. 98 with the BDI and the second component correlated. 93 with the K-SADS. Ambrosini et al., however, concluded that the first component reflected depression severity and the second component reflected melancholic depression. This interpretation was somewhat at odds with the data. Specifically, the second component included some K-SADS items that did not tap symptoms of melancholia (e.g., irritability and anger) and did not include some BDI items that measure symptoms of melancholia (e.g., loss of appetite). 

When applied to electronic nose data the presence of various sources of correlated disturbances has to be considered. As an example, sample temperature fluctuations induce correlated disturbances, which may be described by principal components of highest order. When these disturbances are important the first principal component has to be eliminated in order to emphasize the relevant data properties. A set of algorithms called Minor Component Analysis (MCA) was introduced to take into account these phenomena mainly in image analysis [17]. 

The hypothesis of a normal distribution is a strong limitation that should be always kept in mind when PCA is used. In electronic nose experiments, samples are usually extracted from more than one class, and it is not always that the totality of measurements results in a normally distributed data set. Nonetheless, PCA is frequently used to analyze electronic nose data. Due to the high correlation normally shown by electronic nose sensors, PCA allows a visual display of electronic nose data in either 2D or 3D plots. Higher order methods were proposed and studied to solve pattern recognition problems in other application fields. It is worth mentioning here the Independent Component Analysis (ICA) that has been applied successfully in image and sound analysis problems [18]. Recently ICA was also applied to process electronic nose data results as a powerful pre-processor of data [19]. 

Correlations are inherent in chemical processes even where it can be assumed that there is no correlation among the data. Principal component analysis (PCA) transforms a set of correlated variables into a new set of uncorrelated ones, known as principal components, and is an effective tool in multivariate data analysis. In the last section we describe a method that combines PCA and the steady-state data reconciliation model to provide sharper, and less confounding, statistical tests for gross errors. 

Finally, a method for dealing with the inherent correlation existing in chemical processes was discussed. This method combines principal component analysis (PCA) and the steady-state data reconciliation model to provide sharper and less confounding statistical tests for gross errors. 

The reason for the correlation between the localization and the amino acid composition was sought by Andrade et al. (1998). They examined the amino acid composition of proteins with known localization and three-dimensional structure in three ways total composition, surface composition, and interior composition. The principal component analysis showed the best correlation between the surface composition and the localization. Therefore, they concluded that the correlation is the result of evolutionary adaptation of proteins to the surrounding environment. 

N. Delaunay, V. Pichon and M.C. Hennion, Experimental comparison of three monoclonal antibodies for the class-selective immunoextraction of triazines. Correlation with molecular modeling and principal component analysis studies. J. Chromatogr.A 999 (2003) 3-15. 

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