Principal component analysis covariance

The purpose of the analysis at first is to identify significant covariates among demographic data and the other phenotypes and delineate correlated phenotypes by principal component analysis. Covariates are determined by generating a covariance matrix for all markers and selecting each significantly correlated markers for use as a covariate in the association test of each marker. Serological markers and baseline... 

The important underlying components of protein motion during a simulation can be extracted by a Principal Component Analysis (PGA). It stands for a diagonalization of the variance-covariance matrix R of the mass-weighted internal displacements during a molecular dynamics simulation. 

Step 2 This ensemble is subjected to a principal component analysis (PCA) [61] by diagonalizing the covariance matrix C G x 7Z, ... 

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m X m matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C, is defined as... 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

Principal Component Analysis (PCA) PCA is used to recognize patterns in data and reduce the dimensionality of the problem. Let the matrix A now represent data with the columns of A representing different samples and the rows representing different variables. The covariance matrix is defined as... 

Grain-size and geochemical trends in this reanalysis were calculated from the first component derived from a Principal Component Analysis (PCA) of covariance matrices resulting from centered log-ratio transformation (van Eynatten 2004 Szava-Kovats 2008). Given a d-part composition, x=[xi, xd], the log-ratio transformation to composition y is... 

Principal component analysis (PCA) is aimed at explaining the covariance structure of multivariate data through a reduction of the whole data set to a smaller number of independent variables. We assume that an m-point sample is represented by the nxm matrix X which collects i=l,...,m observations (measurements) xt of a column-vector x with j=, ...,n elements (e.g., the measurements of n=10 oxide weight percents in m = 50 rocks). Let x be the mean vector and Sx the nxn covariance matrix of this sample... 

Principal Component Analysis (PCA) is the most popular technique of multivariate analysis used in environmental chemistry and toxicology [313-316]. Both PCA and factor analysis (FA) aim to reduce the dimensionality of a set of data but the approaches to do so are different for the two techniques. Each provides a different insight into the data structure, with PCA concentrating on explaining the diagonal elements of the covariance matrix, while FA the off-diagonal elements [313, 316-319]. Theoretically, PCA corresponds to a mathematical decomposition of the descriptor matrix,X, into means (xk), scores (fia), loadings (pak), and residuals (eik), which can be expressed as... 

Evaluation of the statistical properties is a fundamental part of any statistical analysis and here we concentrated on the distribution of each variable. To reduce the dimensionality of this data set we used principal component analysis (PCA) to explore the covariance structure of these data and to reduce the variables to a more manageable number (PAl method with no rotation, 21). 

Technique 2 Elgenanalysls. It Is well known that the structure of a data set can be uncovered by performing an elgenanalysls of Its covariance matrix.(14) This Is often called principal component analysis. That Is, we arrange the M measurement made on each of N objects as a column vector and combine them to form an M x N matrix, A. A matrix B, resembling the covariance matrix of this data set, Is an M x M matrix AA whose elements are given by... 

Principal component analysis is based on the eigenvalue-eigenvector decomposition of the n h empirical covariance matrix Cy = X X (ref. 22-24). The eigenvalues are denoted by > 2 — Vi > where the last inequality follows from the presence of same random error in the data. Using the eigenvectors u, U2,. . ., un, define the new variables... 

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... 

With these patterns in mind, we conducted a principal components analysis of the two datasets using a covariance matrix, since some of the elements have especially high concentrations that could swamp those with lower concentrations in the analysis (Figure S). Here we can see that, in the plaza, samples from the north half vary by P concentration, while those of the south vary according to levels of Ba and Mg the reverse is true for western versus eastern samples (not pictured). In the patio, all samples tend to vary along Factor 1, in which Al, Ba, Fe, and Mn account for most of the variance in the data. This suggests that activity loci in the plaza and patio vary by comer or quadrant. 

The method of PLS bears some relation to principal component analysis instead of Lnding the hyperplanes of maximum variance, it Lnds a linear model describing some predicted variables in terms of other observable variables. It is used to Lnd the fundamental relations between two matrices (X andY), that is, a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to Lnd the multidimensional direction irMIspace that explains the maximum multidimensional variance direction in flrfspace. 

Principal component analysis is a popular statistical method that tries to explain the covariance structure of data by means of a small number of components. These components are linear combinations of the original variables, and often allow for an interpretation and a better understanding of the different sources of variation. Because PCA is concerned with data reduction, it is widely used for the analysis of high-dimensional data, which are frequently encountered in chemometrics. PCA is then often the first step of the data analysis, followed by classification, cluster analysis, or other multivariate techniques [44], It is thus important to find those principal components that contain most of the information. 

Croux, C. and Haesbroeck, G., Principal components analysis based on robust estimators of the covariance or correlation matrix influence functions and efficiencies, Biometrika, 87, 603-618, 2000. 

The program used in this study was a modified version of Dixon s BMD08M factor analysis with varimax rotation (12), The principal components analysis was conducted using covariance matrices. Five factors were created from the data set. Examination of the individual proportion of the total variance contributed by each of the factors demonstrated that 96.3% of the total variance could be accounted for by the first three factors. These three factors were used in the following cluster analysis. 

The vectors of means = (xi, I2,..., x ) and deviations = (ii, S2,. ..,Sp), and matrices of covariances S = (Sij) and correlations R = (tij) can be calculated. For this data matrix, the most used non-supervised methods are Principal Components Analysis (PCA), and/or Factorial Analysis (FA) in an attempt to reduce the dimensions of the data and study the interrelation between variables and observations, and Cluster Analysis (CA) to search for clusters of observations or variables (Krzanowski 1988 Cela 1994 Afifi and Clark 1996). Before applying these techniques, variables are usually first standardised (X, X ) to achieve a mean of 0 and unit variance. 

The extraction of the eigenvectors from a symmetric data matrix forms the basis and starting point of many multivariate chemometric procedures. The way in which the data are preprocessed and scaled, and how the resulting vectors are treated, has produced a wide range of related and similar techniques. By far the most common is principal components analysis. As we have seen, PCA provides n eigenvectors derived from a. nx n dispersion matrix of variances and covariances, or correlations. If the data are standardized prior to eigenvector analysis, then the variance-covariance matrix becomes the correlation matrix [see Equation (25) in Chapter 1, with Ji = 52]. Another technique, strongly related to PCA, is factor analysis. ... 

The WHIM algorithm performs a principal component analysis (PCA) on the mean centered Cartesian coordinates of the molecule from a weighted covariance matrix of the atomic coordinates. The weights of this matrix are such properties as atomic mass, van der Waals volume, Sanderson atomic electronegativity, atomic... 

A first PLS model was established from 124 reaction systems. To ensure that this set of reaction systems was not selected in such a way that the descriptor variables were correlated, a principal component analysis was made of the variation of the eight descriptors over the set. This analysis afforded eight significant principal components according to cross validation. This showed that the variance-covariance matrix of the descriptors was a full rank matrix and that there were no severe colinearities among the descriptors. 

These are autocovariances and cross-covariances calculated from sequential data with the aim of transforming them into uniform-length descriptors suitable for QSAR modeling. ACC transforms were originally proposed to describe peptide sequences [Wold, Jonsson et al, 1993 Sjbstrbm, Rannar et al., 1995 Andersson, Sjostrom et al., 1998 Nystrom, Andersson et al., 2000]. To calculate ACC transforms, each amino acid position in the peptide sequence is defined in terms of three orthogonal z-scores, derived from a Principal Component Analysis (PC A) of 29 physico-chemical properties of the 20 coded amino acids. 

An anisometry descriptor defined as a function of the eigenvalues, obtained by —> Principal Component Analysis applied to the covariance matrix calculated from the —> molecular matrix M ... 

WHIM descriptors are built in such a way as to capture relevant molecular 3D information regarding molecular size, shape, symmetry, and atom distribution with respect to invariant reference frames. The algorithm consists in performing a Principal Components Analysis on the centered Cartesian coordinates of a molecule (centered molecular matrix) by using a weighted covariance matrix obtained from different weighting schemes for the atoms ... 

Performing principal component analysis on the ranks can help to assess the dimensionality of the ordering context. Since the marginal distributions of the ranks are the same except for ties, the difference between covariance matrix and correlation matrix is not critical. If there are subsets of indicators that segregate strongly in their loadings, then complexity is confirmed and it may be prudent to consider partitioning of the prioritization process. 

Another recommendation was to examine the correlation matrix of the covariates prior to the analysis and determine whether any two covariates were correlated. If any two correlated covariates were found to be important predictor variables, one could possibly transform the variables into a composite variable, such as the transformation of height and weight into body surface area or body mass index, or to use only the covariate with the greatest predictive value in the model and not to include the other covariate. An untested approach would be to use principal component analysis and then use one or more of the principal components as the covariate in a model. 

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