Cluster analysis continued

Fig. 3.2 Typical applications using chemical multivariate data (schematically shown for 2-dimensional data) cluster analysis (a) separation of categories (b), discrimination by a decision plane and classification of unknowns (c) modelling categories and principal component analysis (d), feature selection (X2 is not relevant for category separation) (eY relationship between a continuous property Y and the features Xi and X2 (f)...

The methods of data analysis depend on the nature of the final output. If the problem is one of classification, a number of multivariate classifiers are available such as those based on principal components analysis (SIMCA), cluster analysis and discriminant analysis, or non-linear artificial neural networks. If the required output is a continuous variable, such as a concentration, then partial least squares regression or principal component regression are often used [20]. 

Results from Fourier clustering. Fourier clustering produces similar, but not identical results. The Fourier cluster analysis method is really a continuous method since the threshold for a frequency cut-off can in theory be made to vary continuously. 

It should be stressed at this point that both approaches must be used simultaneously (2,3). Both contain valuable information, however, their relative importance changes due to the particular situation, i.e., from point to point in the state space. In order to adapt the weights of both accordingly, the evidence of the network component has been monitored continuously. In order to determine the areas of experience a cluster analysis has been performed. With a relative importance I of the network, the importance of the Monod correlation is (1 - ). It is straightforv ard to couple la to the amount of data available to train the network in the area immediately around the actual state space position. This can be performed with the evidence measure (1),... 

A single printed sensing layer was exposed to methanol, ethanol, acetone and isopropanol vapor, respectively. During the exposure, the absorbance spectrum was continually measured and split into several wavelength intervals. The spectra have been analyzed by principal component analysis and cluster analysis. Actually, it is possible to use a single sensing film to emulate sensor arrays (29). 

By means of a cluster analysis, it is to be shown how the heat transfer conditions within clusters differ from the heat transfer between droplets and gas outside of a cluster. Therefore, at consecutive time steps the properties of the continuous and the dispersed phase are correlated in two regions of the spray. The regions are defined by two volumes given in Table 19.5. 

Figure 11 Dendrogram as the result of a hierarchical cluster analysis. A set of ten objects is characterized by the two features xi and X2 (scatter plot at the left hand side), d is the Euclidean distance between two merging points or clusters. The process starts with the two nearest neighbors (objects 5 and 6) and continues until all objects are fused into a single cluster...

One of the cluster analysis methods is the agglomerative method (Everitt et al. 2001). Using this approach, a distance threshold parameter is continuously increased. At the first step, the two nearest objects are identified, and at this stage, only their distance is below the threshold. These two objects are xmited and the location of the unified object is the arithmetic mean of the coordinates. By increasing the threshold further and further, objects are united until only one object remains. The similarity of the objects is indicated by the order of the aggregations. 

One of the most intuitive ways to describe how cluster analysis works in practice is by referring to the agglomerative hierarchical cluster analysis (HCA) method. Beside the common preliminary steps already discussed, that is definition of the metric (Euclidean, Mahalanobis, Manhattan distance, etc.) and calculation of the distance matrix and the corresponding similarity matrix, the analysis continues according to a recursive procedure such as... 

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