Standardization Autoscaling

For demonstration we use the data of a cooperative test [DOERFFEL and ZWANZIGER, 1987], In this interlaboratory comparison five laboratories were involved in the analysis of slag samples three times for seven chemical elements. So the (15, 7)-data matrix consists of 5 times 3 rows and 7 columns. The raw data in % are given in Tab. 5-2. The data have been preprocessed by standardization (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. 

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... 

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. 

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... 

Standard normal variate (SNV) transformation is closely related to MSC (Barnes et al. 1989, 1993 Helland et al. 1995). SNV treats each spectmm separately by autoscaling the absorbance values (row-wise) by... 

Autoscaling is a technique for coding data so that the mean is zero and the standard deviation is unity. What should Cy, and Jy, be to autoscale the nine responses in Section 3.1 (Hint see Figure 3.3.)... 

The mean-centering part of autoscaling removes absolnte intensity information in the jc variables. However, the additional division by the standard deviation also removes total variance information in each of the vari-... 

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.)... 

Autoscaling is performed in order to remove the scale difference between the sensor intensities and slopes. Tliis involves subtracting the mean and dividing by the standard deviation for each column (see Chapter 3). 

Autoscaled data have the unique characteristic that each of the variables has a zero mean and a standard deviation of one. Like mean-centering, autoscaling removes absolute intensity information. However, unlike mean-centering, it also removes total variance information in each of the variables. It effectively puts each of the variables on equal footing before modeling is done. 

The most common possibility is autoscaling (see also Section 5.2.2, Eq. 5-3). Auto means that the data are related to measures of their own distribution, namely the mean x and the standard deviation s of an assumed normal distribution. This is achieved by subtracting the mean xj from each individual value x / (i = 1,. .., n) and dividing by sj of the respective feature xj (j = 1,. .., m) ... 

One definition of the autocorrelation function, rxx, using autoscaled values (mean = 0, standard deviation = 1) is ... 

AUTOSCAL - Mean center and standardize columns of a matrix... 

Autoscaling involves standardizing the measurement variables so that each descriptor or measurement has a mean of zero and a standard deviation of unity, that is,... 

A careful pretreatment is necessary to focus on the relevant variables. Often variables with low absolute values (<0.01 kcal moT ) and those with low standard deviations (<0.02-0.03kcal mol" ) are removed in order to eliminate noise. Autoscaling is not recommended, since all the data comes from the same source (GRID probe-target interaction energies) and all the data are expressed in the same units (kcalmof ). Thus, autoscaling might introduce noise in the model. 

Multivariate analysis techniques were applied to peak areas obtained by CE to evaluate the ripening time of the cheese. Data were autoscaled prior to model calculations. This normalization involved the subtraction of the mean and then the division of each value of a given variable by the standard deviation of all the values for this variable over the entire sample collection period (48). After normalization, all variables had the same weight because they had a mean of zero and unitary variance. 

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