Scaling/ scaled autoscaling

Autoscaling is a combination of mean centering and variance scaling. Autoscaled features have a mean of zero and a variance of one and therefore possess equal statistical influence on data interpretation (equation 2) ... 

Figure 4-6. Autoscaling. The variables are represented by variance bars, a) Raw data b) the data after UV-scaling only c) the autoscaled data [8].

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... 

If the descriptors are on different scales then those which naturally occupy a larger scale may be given more weight in the subsequent analysis, simply because of their natura units. In autoscaling the descriptors are scaled to zero mean and a standard deviatior of 1. 

An alternative to autoscaling is range scaling, where the denominator in Equation (12.20] equals the range (the difference between the maximum and minimum values). Range scaling gives a set of new values between —0.5 and +0.5. 

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). 

In contrast to correlation matrix the covariance matrix is scale-dependent. In case of autoscaled variables the covariance matrix equals the correlation matrix. 

A general comment that affects all statistical multivariate data analysis techniques, namely that each of the variables should be given equal chance to influence the outcome of the analysis. This can be achieved by scaling the variables in appropriative way. One popular method for scaling variables is autoscaling, whereby the variance of each variable is adjusted to 1. 

In data analysis, data are seldom used without some preprocessing. Such preprocessing is typically concerned with the scale of data. In this regard two main scaling procedures are widely used zero-centered and autoscaling. 

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). 

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... 

FIGURE 3.7 Effect of autoscaling on PCA. In the left plot the data are not scaled but only centered, in the right plot the data are autoscaled. The results of PCA change. 

From the definition of the objective functions (Equations 4.88 and 4.89), it can be seen that different scalings of the x-variables would result in different penalization, because only the coefficients themselves but no information about the scale of the x-variables is included in the term for penalization. Therefore the x-variables are usually autoscaled. Note that the intercept b0 is not included in the penalization term in order to make the result not be depending on the origin of the y-variable. 

Regularization. Regularization, the autoscaling of Kowalski, (35) and scaling of Massart, (36) transforms the data so that the data set has a zero mean and a variance of one for each variable. This method equalizes the influence of peaks or measurements. 

GLS preprocessing can be considered a more elaborate form of variable scaling, where, instead of each variable having its own scaling factor (as in autoscaling and variable-specific scaling), the variables are scaled to de-emphasize multivariate directions that are known to correspond to irrelevant spectral effects. Of course, the effectiveness of GLS depends on the ability to collect data that can be used to determine the difference effects, the accuracy of the measured difference effects, and whether the irrelevant spectral information can be accurately expressed as linear combinations of the original x variables. 

If the original data contain Information on the uncertainties associated with each measurement the sensitivity of the variance of the results to these errors can be studied. Approaches Include uncertainty weighting during the autoscaling procedure which Is provided for In ARTHUR, uncertainty scaling (the data standard deviation used for autoscaling Is replaced by the measurement absolute error such as presented In Table VII), and Monte Carlo simulation for estimating the variance of the statistics based on the error perturbed data ( ). 

Cd, K and Zn are not precisely determined. Previously reported (13) results for Identical split samples Indicates that most of this experimental error was due to analytical Imprecision rather than collection and handling. Many of the samples were near the detection limit for the five trace metals (As, Cd, Cu, Pb, Zn), To determine the effect of these measurement errors the PCA was repeated with uncertainty scaled data. (The data standard deviation used In autoscaling was replaced with the measurement absolute error.)... 

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