Autoscaling

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). 

No autoscaling is available that, while convenient, exposes the individual plot limits and bin boundaries to the vagaries of measurement and sampling noise the user is forced to actively select lower and upper bounds on the subdivided x-range, and the number of bins, to come up with bin boundaries that make sense. 

To investigate the variance structure in the raw physical/chemical data material a PCA was performed on the autoscaled Y-data. Figure 3 shows a loading plot of the Y-data as a function of the two first PC s describing together 57 % of the total variance. 

It can be shown that the standardized Euclidean distance is the Euclidean distance of the autoscaled values of X (see further Section 30.2.2.3). One should also note that in this context the standard deviation is obtained by dividing by n, instead of... 

In the context of data analysis we divide by n rather than by (n - 1) in the calculation of the variance. This procedure is also called autoscaling. It can be verified in Table 31.5 how these transformed data are derived from those of Table 31.4. 

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

The use of standardized data (variable standardization or column autoscaling, see Frank and Todeschini [1994]) results in data which are independent of the unit of measurement. Other types of standardization like object standardization, row autoscaling, or global standardization (global autoscaling, (xij — x)/s) do not play a large role in data analysis. 

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). 

Autoscaled data have a mean of zero and a variance (or standard deviation) of one, thereby giving all variables an equal statistical weight. Autoscaling shifts the centroid... 

FIGURE 2.5 Graphical representation of mean-centering and autoscaling for an A -matrix with two variables. 

The correlation coefficients can be arranged in a matrix like the covariances. The resulting correlation matrix (R, with l s in the main diagonal) is for autoscaled x-data identical to C. 

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... 

For autoscaled variables, each variable has a variance of 1, and the total variance is in, the number of variables. For such data, a mle of thumb uses only PCA components with a variance >1, the mean variance of the scores. The number of PCA components with variances larger than 0 is equal to the rank of the covariance matrix of the data. 

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. 

The goal of dimension reduction can be best met with PCA if the data distribution is elliptically symmetric around the center. It will not work well as a dimension reduction tool for highly skewed data. Figure 3.9 (left) shows skewed autoscaled... 

FIGURE 3.9 PCA for skewed autoscaled data In the left plot PCI explains 79% of the total variance but fails in explaining the data structure. In the right plot x2 was log-transformed and then autoscaled PCI now explains 95% of the total variance and well follows the data structure. 

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