Mean centered value

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 next step in the calculation consists of generating the matrix of within-group deviations (the matrix of mean centered values for each group). A mean centered (or corrected) value for an element Xy from the group U (j e U) (first four rows of elements in this example) is calculated as the difference ... 

Consider a one-dimensional random walk, with a probability p of moving to the right and probability q = 1 — p of moving to the left. If p = g = 1/2, the distribution has mean p = 0 and spreads in time with a standard deviation a = sJijA. In general, though, p = (p — g)t and a = y pgt. In particular, as p moves away from the center value 1/2, the center of mass of the system Itself moves with velocity P = p — q. 

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...

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. 

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... 

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... 

Table 17.1 Crystallographic data of the hexagonal and cubic closest-packings of spheres. +F means +(j,0), +(j,0, j), +(0, j, j) (face centering). Values given as 0 or fractional numbers are fixed by space-group symmetry (special positions)...

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. 

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. 

For mean-centered X the matrix To has size nxm and contains the PCA scores normalized to a length of 1. S is a diagonal matrix of size mxm containing the so-called singular values in its diagonal which are equal to the standard deviations of the scores. PT is the transposed PCA loading matrix with size mxm. The PCA scores, T. as defined above are calculated by... 

The mean-centering operation effectively removes the absolute intensity information from each of the variables, thus enabling snbsequent modeling methods to focus on the response variations about the mean. In PAT instrument calibration applications, mean-centering is almost always nsefnl, because it is almost always the case that relevant analyzer signal is represented by variation in responses at different variables, and that the absolute values of the responses at those variables are not relevant to the problem at hand. 

Verify that the d0 values thus calculated show a relatively narrow distribution around a mean value close to 0.2 nm. Criticize or defend the following proposition As a mean center-to-center intermolecular spacing, this value is on the low side as a back-calculated parameter, however, it probably compensates for deviations from the assumed geometry, breakdown of Equation (33) at short distances, or other shortcomings of the molecular additivity principle. 

Reflectance readings of the polished pellets were made on all fractions across the xenolith, utilizing a Leitz Ortholux microscope and a Photovolt photometer. Figure 12 shows the mean reflectance values of each fraction plotted vs. the distance across the xenolith. These values generally decrease toward the center of the xenolith as the distance from the sill increases. The mean reflectance values range from 5.7 to 8.1%. 

One common microarray data normalization method is to calculate a normalization factor on a per array basis or across an entire experiment. The primary assumption for using a singular normalization factor is that the volume of labeled sample is comparable across the two channels. Thus, due to the large population of labeled cDNA within the uniform volume it is assumed that the same number of labeled cDNAs exist in both samples. Ideally, the overall intensity in the two channels will be the same. Furthermore, any increases in labeled cDNAs, due to increases in mRNA, must result in decreases of some other labeled cDNAs. Typical methods include mean- or median-centering, where the mean/median values are centered within the data distribution, and z-score normalization which adds a scaling factor to mean-centering. 

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