Data variance

Fig. 8. Plot of data from patients having Hver diseases A or B or unknown X (a) on two blood enzymes (b) scores of points on the first two eigenvectors obtained from an eight-dimensional enzyme space and (c) eigenvector plot of the variance weighted data. Variance weights ranged from 3.5 to 1.2 for the eight blood enzymes measured. A weight of 1.0 indicates no discrimination information (22).

Although this is a complicated expression, the results can be given a simple interpretation. The data sample size is n, whereas the prior sample size is Ko, and therefore J. is the weighted average of the prior data and the actual data. al is the weighted average of the prior variance (Go), the data variance (5-), and a tenn from the difference in the prior... 

More recent publications on sulfosuccinates have confirmed the minimal or close to zero skin and eye irritation caused by these products. In a general screening of product safety evaluation methods the authors [16] rejected the sulfosuccinate from further consideration in the statistical analysis of experimental data (variance analysis) because the product had not shown any irritation in the Duhring-Chamber test. The sulfosuccinate (based on fatty alcohol ethoxy late) was tested in a screening with 14 other surfactants, namely, alkyl sulfates, sulfonates, ether sulfates, and a protein fatty acid condensation product. 

PLS was originally proposed by Herman Wold (Wold, 1982 Wold et al., 1984) to address situations involving a modest number of observations, highly collinear variables, and data with noise in both the X- and Y-data sets. It is therefore designed to analyze the variations between two data sets, X, Y). Although PLS is similar to PCA in that they both model the A -data variance, the resulting X space model in PLS is a rotated version of the PCA model. The rotation is defined so that the scores of X data maximize the covariance of X to predict the Y-data. 

PCA [12, 16] is a multivariate statistics method frequently applied for the analysis of data tables obtained from environmental monitoring studies. It starts from the hypothesis that in the group of original data, there is a set of reduced factors or dominant components (sources of variation) which influence the observed data variance in an important way, and that these factors or components cannot be directly measured (they are hidden factors), since no specific sensors exist for them or, in other words, they cannot be experimentally observed. 

In the SE compartment, PCI (explaining 52.3% of data variance) describes a contamination pattern of PAHs, except for the naphthalene compound. This compound is the most volatile within this group, and presents a slightly different chemical behavior. The pattern of PAHs is detected at high levels in the upper... 

Three components were identified from the analysis of the scaled Daug (44 x 15). The selection of two and four components was discarded since the total explained data variance was much lower with two components, and no additional environmentally relevant information was added with four components [30]. 

FIGURE 1.4 Projection of the glass vessels data set on the first two PCs. Both PCs together explain 67.5% of the total data variance. The four different groups corresponding to different types of glass are clearly visible. 

The eigenvectors of this matrix are linear combinations of the measurements, and the eigenvalues are a direct measure of the fraction of total variance accounted for by the corresponding eigenvector. This analysis Is the basis for the Karhunen-Loeve transformation, In which the data are projected onto the plane of the two eigenvectors with largest eigenvalue. This choice of axes displays more of the data variance than any other. 

Calculate the variance for each group of data. Variance is standard deviation squared, so the final step of taking square root i the calculation is omitted. 

If we calculate the four nonelementary discriminant functions df we find the following fractions of data variance explained 77.3% (by one function), 98.8% (by two functions), and 99.9% (by three discriminant functions). Hence we do not expect severe biased projections of our data on to the plane. In Fig. 5-23 we find some overlapping laboratories, however. In the 3D-plot of Fig. 5-24 a good, separated display of all laboratories data is indicated. So far, the data projection is satisfactory. 

The variance in the data, a measure of the spread of a set of data, is related to the precision of the data. For example, the larger the variance, the larger the spread of data and the lower the precision of the data. Variance is usually given the symbol s2 and is defined by the formula ... 

Principal component analysis (PCA) and multivariate curve resolution-alternating least squares (MCR-ALS) were applied to the augmented columnwise data matrix D1"1", as shown in Figure 11.16. In both cases, a linear mixture model was assumed to explain the observed data variance using a reduced number of contamination sources. The bilinear data matrix decomposition used in both cases can be written by Equation 11.19 ... 

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