Variance 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. 

Variance scaling is performed on a variable by variable basis. In other words, we would variance scale a set the concentration values of a data set on a component by component basis. Starting with the first component, we compute the total variance of the concentrations of that component. There are several variations on variance scaling. First, we will consider the most the method which adjusts all the variables to exactly unit variance. To do this we compute the variance of the variable, and then use the variance to scale all the concentrations of all the samples so that the new variance for the component is equal to unity. 

The decision whether or not to variance scale the x-block data is independent from the decision about scaling the y-block data. We can decide to scale either, both, or neither. 

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.

An alternative method of variance scaling is to scale each variable to a uniform variance that is not equal to unity. Instead we scale each data point by the root mean squared variance of all the variables in the data set This is, perhaps, the most commonly employed type of variance scaling because it is a bit simpler and faster to compute. A data set scaled in this way will have a total variance equal to the number of variables in the data set divided by the number of data points minus one. To use this method of variance scaling, we compute a scale factor, sr, over all of the variables in the data matrix, 8g,... 

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... 

Variance (cont) of prediction, 167 Variance scaling, 100, 174 Vectors basis, 94 Weighting of data, 100 Whole spectrum method, 71 x-block data, 7 x-data, 7 XE, 94 y-block data, 7 y-data, 7... 

At high Reynolds number and for Schmidt numbers near unity or larger, we are justified in assuming that Tt is nearly independent of Schmidt number. We will also need a closure for in (3.175). In general, the dissipation-range variance scales as Re, 1 = Rc, 1/2 (Fox and Yeung 1999 Vedula 2001). We will thus model the covariances by... 

Variance scaling standardizes each variable j by its standard deviation s/, usually, it is combined with mean-centering and is then called autoscaling (or z-transformation). 

VARIANCE SCALING Variance. scaling is achieved by dividing each element in a variable vector by the standard deviation of that variable. 11k prim ir rca.son for variance scaling is to remove weigliting that is artificially imposed by the scale of the variables. This is useful because many data... 

The mean of the variable x then is the zero point on the new xce raxis. Mean centering is only possible, if the variances of the different features are similar. Otherwise, both mean centering and variance scaling are necessary. 

Prior to using the variance-scaling preprocessing option, mean centering must first be used. The combination of these two preprocessing options is often called... 

Photomultipliers, charge coupled devices (CCDs) and avalanche diode detectors are able to detect single photons over the visible to near-IR range with efficiencies approaching unity. The arrival of photons at a detector is not correlated, due to the quantum nature of electromagnetic radiation. Measurements of intensity as the averaged sum of photon events has a well-defined stochastic variance associated with a Poisson distribution. This variance scales as the square root of the number of photons. 

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