Within-cluster variance

Note that the centroid is the simple arithmetic mean of the vectors of the cluster members, and this mean is frequently used to represent the cluster as a whole. In situations where a mean is not applicable or appropriate, the median can be used to define the cluster medoid (see Kaufman and Rousseeuw for details). The square-error (also called the within-cluster variance), for a cluster is the sum of squared Euclidean distances to the centroid or medoid for all s items in that cluster ... 

In general, one maximizes between-cluster Euclidean distance or minimizes within-cluster Euclidean distance or variance. This really amounts to the same. As described by Bratchell [6], one can partition total variation, represented by T, into between-group (B) and within-group components (W). 

When we consider the multivariate situation, it is again evident that the discriminating power of the combined variables will be good when the centroids of the two sets of objects are sufficiently distant from each other and when the clusters are tight or dense. In mathematical terms this means that the between-class variance is large compared with the within-class variances. 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

As mentioned, hierarchical cluster analysis usually offers a series of possible cluster solutions which differ in the number of clusters. A measure of the total within-groups variance can then be utilized to decide the probable number of clusters. The procedure is very similar to that described in Section 5.4 under the name scree plot. If one plots the variance sum for each cluster solution against the number of clusters in the respective solution a decay pattern (curve) will result, hopefully tailing in a plateau level this indicates that further increasing the number of clusters in a solution will have no effect. 

Figure 7.6 Principle components analysis (PCA) of PCB congener concentrations in technical Aroclor mixtures, contaminated water, caged brown trout, SPMDs, and hexane filled dialysis bags. The plot shows that 77% of the variance of samples within the 95% confidence ellipse is explained by PCI and PC2 and that caged fish and SPMDs are clustered together (PCA plot courtesy of Kathy Echols, USGS-CERC, Columbia, MO, USA).

The bottleneck in utilizing Raman shifted rapidly from data acquisition to data interpretation. Visual differentiation works well when polymorph spectra are dramatically different or when reference samples are available for comparison, but is poorly suited for automation, for spectrally similar polymorphs, or when the form was previously unknown [231]. Spectral match techniques, such as are used in spectral libraries, help with automation, but can have trouble when the reference library is too small. Easily automated clustering techniques, such as hierarchical cluster analysis (HCA) or PCA, group similar spectra and provide information on the degree of similarity within each group [223,230]. The techniques operate best on large data sets. As an alternative, researchers at Pfizer tested several different analysis of variance (ANOVA) techniques, along with descriptive statistics, to identify different polymorphs from measurements of Raman... 

The Wodicka efal. (1997) paper also defined the performance of fhe Affymetrix chip. Semiquantitative measurement of the absolute abundance of mRNA species was possible. Flybridization of total yeast-genomic DNA to the chips revealed the mean hybridization signal across 6049 probe sets to vary by 25% coefficient of variance (CV). The use of gDNA serves to normalize because most genes are represented only once in the population. In fact, the majority (98%) of the intensities were found to cluster well within two standard deviations. Thus, the concentration of a given mRNA could be estimated af >95% probability to reside within twofold of its actual concentration. Measurement at widely different total gDNA concentrations did not appreciably affect this outcome. 

Of the four different methods of cluster analysis applied, the method of Ward described in the Clustan User Manual (10), worked best when compared to the single-, complete-, or average-linkage methods. Using Ward s method, two clusters, Gn and Gm, are fused when by pooling the variance within two existing clusters the variance of the so formed clusters increases minimally. The variance or the sum of squares within the classes will be chosen as the index h of a partition. 

The result from cluster analysis presented in Fig. 9-2 is subjected to MVDA (for mathematical fundamentals see Section 5.6 or [AHRENS and LAUTER, 1981]). The principle of MVDA is the separation of predicted classes of objects (sampling points). In simultaneous consideration of all the features observed (heavy metal content), the variance of the discriminant functions is maximized between the classes and minimized within them. The classification of new objects into a priori classes or the reclassification of the learning data set is carried out using the values of the discriminant function. These values represent linear combinations of the optimum separation set of the original features. The result of the reclassification is presented as follows ... 

The application of methods of multivariate statistics (here demonstrated with examples of cluster analysis, multivariate analysis of variance and discriminant analysis, and principal components analysis) enables clarification of the lateral structure of the types of feature change within a test area. 

The expected residual class variance for class q is calculated by using the residual data vectors for all samples in the training set. The resulting residual matrix is used to calculate the residual variance within class q. This value is an indication of how tight a class cluster is in multidimensional space. It is calculated according to Equation 4.46, where s02 is the residual variance in class q and n is the number of samples in class q. 

Principal components analysis (PCA) reduces the volume of large data sets by combining correlated variables and maximizing variances to show patterns in the data. Usually, analysis of the variance (ANOVA) is used to prove that the null hypothesis, that there is no difference between the data sets, is not valid. Test results are compared with table values at a probability (normally 95%) that they will conform to that value. Data are plotted in such ways that different populations are visibly separate and the clustering within each set illustrates the degree of repeatability. 

There are many other statistical models which can be used for the evaluation of DICE studies. Inclusion of not only a group factor, but also a time factor in the experiment methods of the analysis of variance (ANOVA) can be applied to find expression changes within the temporal course of the protein expression or to find interactions between the group and time factor. Several multivariate statistical methods are of use, too. Spots with similar expression profiles can be grouped by cluster analysis or, on the other hand, new spots can be assigned to existing groups by the methods of discriminant analysis. 

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