Transformation variance scaling

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). 

Standardization is usually always necessary when the variables measured are recorded in different units, e.g. concentration, pH, particle size, conductivity etc. The transformed variable has no units, has a mean of zero and a standard deviation of unity. The transformation is achieved by mean centring and variance scaling the original data. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

Linear regression assumes that variance across doses is constant and that the dose response is linear. If the variance is not approximately constant, then a transformation may be applied or a weighted analysis may be carried out. If the dose scale tends to a plateau, then the dose scale may be transformed. If counts decline markedly at high doses, then linear regression is inappropriate. 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

FIGURE 6.2 Representation of multivariate data by icons, faces, and music for human cluster analysis and classification in a demo example with mass spectra. Mass spectra have first been transformed by modulo-14 summation (see Section 7.4.4) and from the resulting 14 variables, 8 variables with maximum variance have been selected and scaled to integer values between 1 and 5. A, typical pattern for aromatic hydrocarbons B, typical pattern for alkanes C, typical pattern for alkenes 1 and 2, unknowns (2-methyl-heptane and meta-xylene). The 5x8 data matrix has been used to draw faces (by function faces in the R-library Tea-chingDemos ), segment icons (by R-function stars ), and to create small melodies (Varmuza 1986). Both unknowns can be easily assigned to the correct class by all three representations. 

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. 

Major steps In this type of analysis Include Initial data scaling and transformation, outlier detection, determination of the underlying factors, and evaluation of the effect that experimental procedures may have on the variance of the results. Most of the calculations were performed with the ARTHUR software package (O. 

The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

They change Px into Py, but the difference is so minor that they are often considered the same distribution, and are denoted by the same name. The transformation can be used to transform the distribution to a standard form, for instance one with zero average and unit variance. In the case of lattice distributions one employs (5.4) to make the lattice points coincide with integers. In fact, the use of (5.4) to reduce the distribution to a simple form is often done tacitly, or in the guise of choosing the zero and the unit on the scale. 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

Cmax/AUC Ratio of Cmax to AUC Scaled parameter Variance estimate complicated due to transformation (ratio) of correlated parameters... 

Once a scaling model has been found the scaled data should be examined carefully to ascertain that the variance is equal over the domain of the data. If not then a suitable transform must be found to equalize the variance. Otherwise, no single stochastic model will accurately reflect the probability of an occurrence of the "event" in question over the data domain, much less for an extrapolated prediction. For example, if the standard deviation is proportional to the mean, a very common situation in nature, the variance is equalized by taking the log of the model variable. This is the case for both of the above examples, where the probability model was fitting to In x rather than x itself. Suitable transformations for other common situations, as well as a general method for finding transforms is given by Johnson Leone (7). 

To lighten the arithmetic in calculating variances, it is frequently worth while to take an arbitrary zero and transform all the numbers on to the new scale. Thus with the above set of numbers we might shift the axis by 9 units so that they now become... 

One additional step in factor analysis that helps the interpretation of the results is factor scaling. One scaling that is appropriate for Py-MS results interpretation is the adjustment of the principal components so that their variances are equal to unity. This is accomplished by means of the transformation ... 

Furthermore, the recognition of variations across the different scales of spatial and temporal dimensions would enable the identification of shifting therapeutic targets to address both of the individual and the time variances in personalized medicine (see Fig. 1). Accurate and robust biomarkers can also be useful for the stratification of diseases and classification of patient subgroups for more effective prevention and therapy. The prediction of drug responses would in turn help avoid adverse events for better clinical outcomes. In addition, the construction of dynamic disease predictive networks derived from the analyses of omics data would allow for the transition from reactive treatments to holistic and proactive care. With the transformation from disease-centered to human-based care, the systems and dynamical models would provide patient-centric information to enhance the participation of individuals, the goal of participatory medicine. 

If the values of some descriptor vary in magnitudes over the set of compounds it is difficult to assume that a linear model will be a good approximation to account for such large variations. In these cases, a better model can often be obtained after a logarithmic transformation of this variable prior to scaling to unit variance. 

One helpful method possible with a random variable that has a normal distribution with mean, p, and variance, is to transform values of the random variable so that they have the scale of the standard normal distribution. This... 

The constant variance assumption can be relaxed via either a rescaling of the response or a weighted fit (4). Similarly, if an appropriate model is used, the normality assumption may be relaxed (4). For example, with a dichotomous response, a logit-log model may be appropriate (5). Other response patterns (e.g., Poisson) may be fit via a generalized linear model (6). For quantitative responses, it is often most practical to find a rescaling or transformation of the response scale to achieve nearly constant variance and nearly normal responses. Finally, if samples are grouped, then blocks or other experiment design structures must be included in the model (7-12). 

The spectrum denotes the variance of the process at a certain time b and scale a. With the chosen normalization of the wavelet transformation (Eq. (12.1)), white noise is given by Sg b,a) = m(6,a)p = const.. 

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