Centering the Data

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. 

We first mean center each data point, ay, and then divide it by the scale factor. If we do not wish to mean-center the data, we finish by adding the mean value back to the scaled data point... 

Preprocessing by log double-centering consists of first taking logarithms, and then to center the data both by rows and by columns ... 

This new analytical method determines the rate constant and activation energy of Kevlar s photooxidative processes. The 0.2 atm of oxygen-18-labelled environment in a solar chamber simulates the air-exposure under sunlight conditions. The technique also allows the radial 0-distribution measurement from the fiber surface toward the fiber center. The data from the accelerated experimental conditions in the solar chamber in an 02-atmosphere are differentiated from the usual daylight exposure effects. 

Branching at the Electrophilic Center. The data indicate that branching at the a-carbon on the substrate tends to reduce differences in reactivity between sulfur nucleophiles and H20. This conclusion is based upon the following trends among the tabulated data. 

The first attempt to establish the mechanisms of the anomerizations was published by Bonner.79 An extensive study was made of the anomerizations of the D-glucopyranose pentaacetates in mixtures of acetic anhydride and acetic acid in the presence of sulfuric acid. The rate of reaction was found to be greatest in pure acetic anhydride. The anomerizations were shown to be inversions specific for the anomeric center. The data did not allow definite conclusions regarding the reaction mechanisms. Nevertheless, a mechanism was proposed, for both the forward and reverse reactions, which appeared the most attractive of those which could be postulated to account for the experimental facts that the anomerization... 

Near-infrared (NIR) spectra of water-methanol mixtures are examined to demonstrate the fundamental aspects of calibration. These spectra are used because they present unique challenges to calibration. Another reference NIR data set is also briefly evaluated. The reader should remember that the information presented is generic and applies to all calibration situations, not just spectroscopic data. Additionally, for discussion purposes, the quantitative information for the target analyte will be concentration. However, other chemical or physical properties can also be modeled. Throughout this chapter, unless noted otherwise as in Sections 5.3 and 5.4, it will be assumed that the described models have had the intercept term eliminated. The easiest way to accomplish this is to mean-center the data. 

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 5a. Temperature data (°C) compiled from transects across warm core Ring 81-G. The upper scale represents kilometers from closest point of approach to ring center. The data were collected using an in situ temperature sensor, XBTs, and CTDs. Note the intrusion of cold water at depth in the ring boundary zone.

Data analysis and sensibly applied statistical tools are of crucial importance for metabolomics. Good experimental design is of course a fundamental first requirement. There have been a number of books and research papers written recently discussing statistics use and models for data analysis of metabolomics.100-104 Statistical and experimental robustness have been the focus of metabolomics and demonstrated in a study of NMR protocols and multivariate statistical batch processing, which were examined for consistency over six different centers. The data were shown to be sufficiently robust to generate comparable results across each center.105... 

To make the principal components comparable, we center the data to find the first principal component as a direction from the origin of our coordinate system (Figure 4.4). We achieve this by calculating the mean of the variations in distance in each row to get m mean values (one for each descriptor), and subtract them from the descriptor matrix to get... 

The first step of Croux and Ruiz-Gazen making PCA more robust is centering the data with a robust criterion, the LI-median, that is, the point which minimizes the sum of Euclidean distances to all points of the data. In a next step, directions in the data space, which are not influenced by outliers, are determined by maximizing a robust parameter, the estimator. To calculate this estimator, first all objects are projected onto normalized vectors passing through each point and the LI-median center. Then for each projection, the Qn, that is, the first quartile of all pairwise differences, is calculated as follows ... 

Figure 9.5. Offset across the second mode. Raw data (shown upper left) are simulated UV-Vis spectra of samples containing one analyte. To the right, the same spectral data are shown using centering across the second mode (i.e. row-wise - the opposite direction of Figure 9.4). As shown in the lower plots the linear relation to the concentration of the analyte is not present in the raw data but is obtained upon centering the data.

Offsets are often handled by first centering the data and subsequently fitting the bilinear model to the centered data as also shown in the previous plots. If the data are centered by subtracting the column-average from every element in the column, this is referred to as centering across the first mode. Mathematically it can be expressed as... 

As shown above the parameters can be estimated in two steps. Centering the data across the first mode will remove the offsets p, and the bilinear model is subsequently fitted to the centered data Z, thus minimizing the loss function... 

Fit the model including offsets to the (now complete) data set. For PCA, this amounts to centering the data and fitting the PCA model. 

Figure 9.11 Reducing parameter correlation by centering the data.

An adjustable choke is located at each well for adjusting the well outlet pressure. The chokes are controlled at a discharge pressure of 2000 psi (136 barr). A wellhead control panel is located at each well. All critical controls for each well are included in the local panel. All well operation is carried out locally, but key parameters are monitored by operators from the treatment plant main control center. The data from each well to the main control center is sent via a radio communication system. 

If analyzing multiple slides, scale or center the data sets across slides. 

The data in Fignre 7.46 show how one can use the indenter-caused defects to estimate the distribution of the stresses in a sample, and for the assessment of the risk, they pose to the stability of the glass. In this case the indentation was not in the center bnt was imposed at various distances, r, from the center. The data show that the sample strength increased with an increase in r. However, even in the case of indentations close to the sample edge, the strength was still lowered considerably, which corresponds to a particnlar level of stresses applied at a given point. 

The time-resolved absorption measurements were performed on Q -reconstituted reaction centers. The data obtained at the probing wavelength of 665 nm reveal the consequences of the mutations for the binding of the quinone and for the electron transfer kinetics in the reaction centers. 

If all features are mean centered the data are shifted so that the origin of the multivariate feature space becomes the centroid of the data points. 

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