Mean centering data

Centering and normalization are often useful in chemometrics analysis of NIR data. Mean centering is simply an adjustment to a data set to reposition the centroid of the... 

The two main ways of data pre-processing are mean-centering and scaling. Mean-centering is a procedure by which one computes the means for each column (variable), and then subtracts them from each element of the column. One can do the same with the rows (i.e., for each object). ScaUng is a a slightly more sophisticated procedure. Let us consider unit-variance scaling. First we calculate the standard deviation of each column, and then we divide each element of the column by the deviation. 

Mean-centering, as is shown by experience, can be successfully employed in combination with another data pre-processing technique, namely scaling, which is discussed later. 

It is necessary to pre-process data by mean-centering, scaling or autoscaling... 

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. 

Figure Cl shows a hypothetical set of data before mean centering. Figure C2 shows the same data set after mean centering. We can imagine that this is a plot of the y data (let s call them concentration values) for a two component system. For each of the 15 samples in the data set, we plot the concentration of the first component along the x-axis and the concentration of the second...

Figure C1. Hypothetical data set before mean-centering.

Figure C2. Hypothetical data set after mean-centering.

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. 

Figure C3 shows the same data from figure Cl after variance scaling. Figure C4 shows the mean centered data from figure C2 after variance scaling. Variance scaling does change the positions of the data points from one another, but does not change the location of the centroid of the data set.

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... 

Figure C5 shows the data from Figure Cl after this type of scaling to uniform variance. Figure C6 shows the mean-centered data from Figure C2 after the same treatment.

Normalization is performed on a sample by sample basis. For example, to normalize a spectrum in a data set, we first sum the squares of all of the absorbance values for all of the wavelengths in that spectrum. Then, we divide the absorbance value at each wavelength in the spectrum by the square root of this sum of squares. Figure C7 shows the same data from Figure Cl after variance scaling Figure C8 shows the mean centered data from Figure C2 after variance... 

Figure E3. Hypothetical mean-centered data set containing points with, A, atypically high and, B, atypically low leverage.

Computationally, canonical correlation analysis can be implemented using the following steps, where it is assumed that the data X and Y are mean-centered. 

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. 

This measure is equivalent to the correlation coefficient between two sets of mean-centered data—corresponding here to the vector components of xA and xB. It is frequently used for the comparison of spectra in IR and MS. 

Data transformations can be applied to change the distributions of the values of the variables, for instance to bring them closer to a normal distribution. Usually, the data are mean centered (column-wise), often they are autoscaled (means of all... 

Data for a demo example with 10 objects and two mean-centered variables x and x2 are given in Table 3.1 the feature scatter plot in Figure 3.1. The loading vector for PCl,/>i, has the components 0.839 and 0.544 (in Section 3.6 we describe methods to calculate such values). Note that a vector in the opposite direction (—0.839, —0.544) would be equivalent. The scores C of PCI cover more than 85% of the total variance. 

FIGURE 3.6 Effect of mean-centering on PCA. In the left plot the data are not centered at the origin therefore, the scores are also not centered. The right plot shows centered data which also result in centered scores. 

For PCA, it is generally recommended to use mean-centered data. Note that there are different possibilities for mean-centering. One could subtract arithmetic column-means from each data column, but also more robust mean-centering methods can be applied (see Section 2.2.2). 

Another important aspect of data preparation for PCA is scaling. The PCA results will change if the original (mean-centered) data are taken or if the data were, for instance, autoscaled first. Figure 3.7 (left) shows mean-centered data... 

---