Mean centered data matrix

To use mean centering, it is necessary to substitute the mean-centered data matrix A+ into the SVD and in all subsequent calculations where A would normally be used in conjunction with the U, S, or V from the principal component model. 

Recall from Chapter 4, Principal Component Analysis, that a mean-centered data matrix with n rows of mixture spectra recorded at m wavelengths, where each mixture contains up to k constituents, can be expressed as a product of k vectors representing concentrations and k vectors representing spectra for the pure constituents in the mixtures, as shown in Equation 5.20. 

With principal component analysis, it is possible to build an empirical mathematical model for the mean-centered data matrix X, as shown by... 

Decomposition of each measured variable on the selected wavelet results in decomposition of the variance of the data matrix into its contributions at multiple scales. Thus, for a mean centered data matrix. 

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. 

COA, PCA, and many other ordinations can be viewed as matrix decomposition (SVD) following transformation of the data matrix (fig. 5.4). Transformations can include centering with respect to variable means, normalization of variables, square root, and logarithmic transforms. In each case, the transformation modifies the view of the data, and thus different questions are posed. PCA is typically a decomposition of a column mean centered (covariance matrix). That is, the mean of each column (array) is subtracted from each individual gene-expression value before SVD. For more information, see Wall [46], where the mathematical relation between PCA and... 

The purpose of translation is to change the position of the data with respect to the coordinate axes. Usually, the data are translated such that the origin coincides with the mean of the data set. Thus, to mean-center the data, let be the datum associated with the kth measurement on the /th sample. The mean-centered value is computed as = x.f — X/ where xl is the mean for variable k. This procedure is performed on all of the data to produce a new data matrix the variables of which are now referred to as features. 

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 ... 

Mean-centering consists in extracting the mean value of the variable to each one of the values of the original variable. In this way, each variable in the new data matrix (centered matrix) presents a mean equal to zero. 

Zero-centered data means that each sensor is shifted across the zero value, so that the mean of the responses is zero. Zero-centered scaling may be important when the assumption of a known statistical distribution of the data is used. For instance, in case of a normal distribution, zero-centered data are completely described only by the covariance matrix. 

FIGURE 2.9 Basic statistics of multivariate data and covariance matrix. xT, transposed mean vector vT, transposed variance vector vXOtal. total variance (sum of variances vb. .., vm). C is the sample covariance matrix calculated from mean-centered X. 

To illustmie the importance of mean centering, PCA is performed on a matrix of dataSefore and after mean centering. Wlien the data are not mean centered (see Egure 4.27a), the first PC must describe the direction from the ori-... 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

Consequently, classical PCR (CPCR) starts by mean-centering the data. Then, in order to cope with the multicollinearity in the x-variables, the first k principal components of Xnp are computed. As outlined in Section 6.5.1, these loading vectors ppJl = (p,... ) are the k eigenvectors that correspond to the k dominant eigenvalues of the empirical covariance matrix Sj = xrX. Next, the -dimensional scores of each data point t are computed as j. = In the final step, the centered response variables y. are... 

FIGURE 11.17 PCA loadings for raw mean-centered augmented data matrix from top to bottom, first to fifth (PCI to PC5) principal components. Compound names and abbreviations are as follows alachlor (ALA), atrazine (ATR), bentazone (BEN), biphenyl (BIF), 3-chlorophenol (3-CP), 4-chlorophenol (4-CP), (2,4-dichlorophenosy)acetic acid (2,4-D), dichloroprop (DCP), dimethoate (DIM), linuron (LIN), h-chloro-z-methyphenoxyacetic acid (MCPA), mecoprop (MEC), 4-chloro-3-methylphenol (MEP), metholachlor (MET), pen-tachlorophenol (PCP), simazine (SIM), (2,4,5-trichlorophenoxy)acetic acid (2,4,5-T), tribu-tylphosphate (TBP), 2,4,6-trichlorophenol (TCP). 

The procedure by which the factorization described above is carried out on a data matrix obtained after centering and scaling each descriptor to unit variance is often called Factor Analysis, (The term "Factor analysis" in mathematical statistics has a slightly different meaning which will not be discussed here. For details of this, see [9]). It can be effected by computing the eigenvectors to (X - X) (X - X). Another, and more efficient method is to use a procedure called Singular value descomposition (SVD), see Appendix 15B. 

Let Y he N X K dimensional data N < K) with mean-centered rows and columns as it is for Yr. Although the only requirement for PCA application is mean-centering of columns, having the rows mean-centered as well due to CD profile removal provides better scaling to remaining data. Define a covariance or scatter matrix Z = YY and let U = [uiU2...ujv] with Uj = be the orthonormal eigenvectors of Z such that... 

Figure 9.16. Hypothesized structure (top) of a data matrix, X, as a bilinear model plus offsets constant over all elements. Centering by subtracting the grand mean of X (bottom) will not remove the offsets from the data. The scalar m holds the grand average of X.

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