Data matrices centering

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. 

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. 

Whether or not we scale, weight, and/or center our data, a mandatory pretreatment is required by most of the algorithms used to calculate the eigenvectors. Most algorithms require that we square our data matrix, A, by either pre- or post-multiplying it by its transpose ... 

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. 

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 ... 

The PCA model gives a representation of the centered (and scaled) data matrix... 

PCA decomposes a (centered) data matrix X into scores T and loadings P, see Chapter 3. For a certain number a of PCs which is usually less than the rank of the data matrix, this decomposition is... 

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... 

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). 

ID ll spectrum, and crosspeaks are arranged symmetrically around the diagonal. There is only one radio frequency channel in a homonuclear experiment, the H channel, so the center of the spectral window (set by the exact frequency of pulses and of the reference frequency in the receiver) is the same in If and F (Varian tof, Bruker ol). The spectral widths should be set to the same value in both dimensions, leading to a square data matrix. Heteronuclear experiments have no diagonal, and two separate radio frequency channels are used (transmitter for F2, decoupler for F ) with two independently set spectral windows (Varian tof and dof, sw, and swl, Bruker ol and o2, sw(If), and sw(I )). Heteronuclear experiments can be further subdivided into direct (HETCOR) and inverse (HSQC, HMQC, HMBC) experiments. Direct experiments detect the X nucleus (e.g., 13C) in the directly detected dimension (Ff) using a direct probe (13C coil on the inside, closest to the sample, H coil on the outside), and inverse experiments detect XH in the To dimension using an inverse probe (XH coil on the inside, 13C coil outside). 

Fig. 30. Schematic design of a simple but very useful and efficient data reduction algorithm. Data representing the time trajectory of an individual variable are only kept (= recorded, stored) when the value leaves a permissive window which is centered around the last stored value. If this happens, the new value is appended to the data matrix and the window is re-centered around this value. This creates a two-column matrix for each individual variable with the typical time stamps in the first column and the measured (or calculated) values in the second column. In addition, the window width must be stored since it is typical for an individual variable. This algorithm assures that no storage space is wasted whenever the variable behaves as a parameter (i.e. does not change significantly with time, is almost constant) but also assures that any rapid and/or singular dynamic behavior is fully documented. No important information is then lost...

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. 

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... 

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.

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. 

In the present notation, it is assumed that the absorbance data are centered and that, therefore, there is no intercept at the absorbance axis. If uncentered data are used, the first column in the concentration matrix should consist of Is, and in the K matrix, the intercept coefficients would have to be introduced as the first row. 

Initially, the n experimental spectra, each comprising p data points, are collected into a n,p-dimensional data matrix A. Any row of the matrix A comprises all p absorbance values of a particular spectrum. Any column consists of all n absorbance values at a particular wavelength. As a first step the data matrix is either centered (Az) or standardized (Ag) (cf Section 22.2). In order to achieve the above stated aims, this pretreated matrix is afterwards split into two matrices by the chosen algorithm ... 

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