Column-centered matrix

A special form of cross-product matrix is the variance-covariance matrix (or covariance matrix for short) Cp, which is based on the column-centered matrix Yp derived from an original matrix X ... 

Finally, we can center the matrix X simultaneously by rows and by columns, which yields the matrix of deviations from row- and column-means or the double-centered matrix Y ... 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

The theory of the non-linear PCA biplot has been developed by Gower [49] and can be described as follows. We first assume that a column-centered measurement table X is decomposed by means of classical (or linear) PCA into a matrix of factor scores S and a matrix of factor loadings L ... 

Chemometric techniques, which can easily cope with this type of data by the use of matrices, will maximize the benefit of the multivariate character. These calculation techniques require that corresponding data points (for instance the top of a peak) in different electropherograms are located in the same column of the matrix. As a consequence, preprocessing the CE data is recommended. Peak shifts are commonly corrected with warping techniques, for example, COW, while column centering, normalization, baseline correction, and MSC are also frequently performed preprocessing techniques. 

After column centering of XM in Equation (2.22), the obtained matrix Xm has pseudorank two and becomes rank-deficient. This holds in certain cases for the combination of closure and column centering (see Appendix 2.A). Discussions on closure, centering, rank-deficiency and its relationships can be found in the literature [Amrhein et al. 1996, Pell etal. 1992],... 

This is a rank one matrix and will remain so, even after centering across the first mode. The averages of the two columns are 4.1 and 8.2 respectively and the centered matrix reads as Z(1) which is also a rank one matrix. 

The matrix X represents the collected spectral data after optional procedures for preprocessing, such as normalization and column-centering. Each row of X corresponds to an object (sample), and each column represents a variable. Superscript t is used to imply transposition. Vectors are defined as columns so that transposition defines row vectors. 

Standardization means to divide each centered matrix element with the column standard deviations ... 

Here, Xy is the ith entry of the jth column vector and n is the number of objects (rows in the matrix). The essence of mean-centering is to subtract this average from the entries of the vector (Eq. (6)). 

Each element M/j of this matrix corresponds to the probability that the amino acid in column / will mutate to 1 amino acid in row j after a period of 250 PAM. The values have been multiplied by 100. (Based on Dayhoff IW 1978. Atlas oi Protein Sequence and Structure Volume 5 Supplement 3. Dayhoff M 0 (Editor) Georgetown University Medical Center, National Biomedical Research Foundation Figure 83.)... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

Usually, the raw data in a matrix are preprocessed before being submitted to multivariate analysis. A common operation is reduction by the mean or centering. Centering is a standard transformation of the data which is applied in principal components analysis (Section 31.3). Subtraction of the column-means from the elements in the corresponding columns of an nxp matrix X produces the matrix of... 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

If the data include outliers, it is advisable to use robust versions of centering and scaling. The simplest possibility is to replace the arithmetic means of the columns by the column medians, and the standard deviations of the columns by the median absolute deviations (MAD), see Sections 1.6.3 and 1.6.4, as shown in the following M-code for a matrix X. 

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