Mean centering

To know how to carry out scaling, mean-centering, and auto-scaling... 

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. 

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)). 

As mentioned above, one can use UV-scaling together with mean-centering. This is called autoscaling (Eq. (9)). 

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.

Autoscaling is another term that has been used in different ways by diffemt people. It is often used to indicate "mean centering followed by variance scaling." Others use it to indicate normalization (see below). 

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.

A nearly perfect diverging wavefront would exit the test plate appearing as though it came from a source 100 m away. A segment would be positioned so that its mean center of curvature was coincident with that virtual source 100 m away. In the worst case in our example, the un-equal air path would be about 4 m rather than 204 m. Interference would take place between the wavefront reflected off the 100 m radius side of the test plate and the segment. The roughly 3 m back to the source and beamsplitter is common path and will not affect the interference pattern. 

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. 

---