Column-standard deviation

A vector of column-standard deviations provides for each column of X a measure of the spread of the elements around the corresponding column-mean ... 

The geometrical interpretation of column-standard deviations is in terms of distances of the points representing the columns of Y from the origin of 5" (multi-... 

In this case the column-norms dp are called column-standard deviations. The square of these numbers are the column-variances, whose sum represents the global variance in the data. Note that the column-variances are heterogeneous which means that they are very different from each other. 

Column-standardization is the most widely used transformation. It is performed by division of each element of a column-centered table by its corresponding column-standard deviation (i.e. the square root of the column-variance) ... 

In this case it is required that the original data in X are strictly positive. The effect of the transformation appears from Table 31.6. Column-means are zero, while column-standard deviations tend to be more homogeneous than in the case of simple column-centering in Table 31.4 as can be seen by inspecting the corresponding values for Na and Cl. 

In PC A, components are associated with maximal variance directions. A fundamental assumption of PCA is that the variables are linearly related and variables are measured on the same scale. In the case where variables are measured on different scales, normalized PCA, where values are column mean centered and also divided by the column standard deviation prior to decomposition, must be used. Although most software packages only provide these two PCA options (same scale centered PCA different scales normalized), there are in fact several other options with PCA, and confusion can frequently arise from the use for the same terminology (PCA) for each option. PCA has problems with data with many zeros in them. Interpretation of PCA of microarray data is sometimes difficult, because much of the variance may not be associated with covariates or sample classes of interest. Thus, from a biological point of view, it is worth examining the variance associated with each axis carefully (fig. 5.5). 

After scaling, each observation has unit variance but different means. Standardization involves both centering and scaling. Each observations has the mean subtracted and is then divided by the column standard deviation,... 

Figure 12. Unnotched Charpy impact strength of kenaf, lyocell and mixed kenaf/lyocell reinforced PLA composites (mean value as column, standard deviation as error bars).

Standardization means to divide each centered matrix element with the column standard deviations ... 

Table 2. Reading statistics reveal that on task pictures slow the reader s speed, and advertisements increase the regression rate (shown as reg. rate). All but two rows have significant differences between the on task and advert conditions the p-value is given in the right column. Standard deviations are shown in parentheses.

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. 

The result is that each column vector has unit standard deviation. 

Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoretical plate is defined as the variance per unit length of the column... 

Since the t distribution relies on the sample standard deviation. s, the resultant distribution will differ according to the sample size n. To designate this difference, the respec tive distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus I. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). 

Figure 7 The production and emission of NO during denitrification in agricultural soil treated with NO3 fertilizer (KNO3) and the nitrification inhibitor Dyciandiamide (10%) under aerobic (air) and anerobic conditions (N,). Fluxes are means from three soil columns, error bars represent standard deviations from the mean. V = vertical flow through the column H = Horizontal flow over the soil surface.

The peak width at the points of inflexion of the elution curve is twice the standard deviation of the Poisson or Gaussian curve and thus, from equation (8), the variance (the square of the standard deviation) will be equal to (n), the total number of plates in the column. 

The curves show that the peak capacity increases with the column efficiency, which is much as one would expect, however the major factor that influences peak capacity is clearly the capacity ratio of the last eluted peak. It follows that any aspect of the chromatographic system that might limit the value of (k ) for the last peak will also limit the peak capacity. Davis and Giddings [15] have pointed out that the theoretical peak capacity is an exaggerated value of the true peak capacity. They claim that the individual (k ) values for each solute in a realistic multi-component mixture will have a statistically irregular distribution. As they very adroitly point out, the solutes in a real sample do not array themselves conveniently along the chromatogram four standard deviations apart to provide the maximum peak capacity. 

Equation (3) allows the calculation of the distance traveled axially by a solute band before the radial standard deviation of the sample is numerically equal to the column radius. Consider a sample injected precisely at the center of a 4 mm diameter LC column. Now, radial equilibrium will be achieved when (o), the radial standard deviation of the band, is numerically equal to the radius, i.e., o = 0.2 cm. 

The standard deviation of the extra-column dispersion is given as opposed to the variance because, as it represents one-quarter of the peak width, it is easier to visualize from a practical point of view. It is seen the values vary widely with the type of column that is used, (ag) values for GC capillary columns range from about 12 pi for a relatively short, wide, macrobore column to 1.1 pi for a long, narrow, high efficiency column. 

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. 

Table 11 shows the precision obtained with the Eagle-Picher Turbidimeter. Column 4 is the standard deviation of the specific surface values, and column 5 gives these as percentage of the mean specific surface values... 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

It is now necessary to attend to the second important function of the column. It has already been stated that, in order to achieve the separation of two substances during their passage through a chromatographic column, the two solute bands must be moved apart and, at the same time, must be kept sufficiently narrow so that they are eluted discretely. It follows, that the extent to which a column can constrain the peaks from spreading will give a measure of its quality. It is, therefore, desirable to be able to measure the peak width and obtain from it, some value that can describe the column performance. Because the peak will be close to Gaussian in form, the peak width at the points of inflexion of the curve (which corresponds to twice the standard deviation of the curve) will be determined. At the points of inflexion... 

It is clearly seen that the connecting tube dispersion of 3 p.1 will still be equivalent to 75% of the column dispersion. Ideally, the extra column dispersion, in terms of standard deviation, should only be about 1 1 but this is extremely difficult to achieve in practice. Some... 

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