Hierarchical dendrograms

The determination of which features the underlying factors are composed of provides a basis for attaching a physical Interpretation to the factors. Varlmax rotation of the PGA may be utilized to aid In the Interpretation of the factors. Hierarchical dendrograms Indicate feature clusters whose composition are analogous to PC factors. The physical Interpretation of the clusters and principal components Indicates the Influence of pollution emission sources or meteorological processes on the rainwater composition at an Individual monitoring site. 

Interpretation of the hierarchical dendrogram for the clustering of chemical variables shows clearly that three major clusters are formed. One of them shows up the close relation between nitrite content and oxygen content in the river water. This is an important indicator regarding the processes involving oxygen demand and oxidation, which are important in biological transformations. The... 

Figure 8.5 Example dendrogram representing an hierarchical clustering of a set of seven compounds.

In hierarchical clustering one can obtain any number of clusters K,lhierarchical clustering, with the difference that there K is defined a priori by the user. The question then arises which A -clustering is significant. To introduce the problem let us first consider a technique that was proposed for the non-hierarchical method MASLOC [27], which selects so-called robust clusters. 

X clust <- hclust (X dist) hierarchical clustering plot (X clust) plots a dendrogram... 

FIGURE 3.24 Dendrogram of fatty acid concentration data from mummies and reference samples. Hierarchical cluster analysis (complete linkage) with Euclidean distances has been applied. 

FIGURE 3.28 Dendrogram resulting from hierarchical cluster analysis of the nri i... 

An appropriate graphical representation of the result is a tree-like dendrogram (Section 6.4). It allows to determine manually the optimal number of clusters as well as to see the hierarchical relations between different groups of objects. 

Hierarchical cluster analysis (Section 6.4)—with the result represented by a dendrogram—is a complementary, nonlinear, and widely used method for cluster analysis. The distance measure used was dTANi (Equation 6.5), and the cluster mode was average linkage. The dendrogram in Figure 6.6 (without the chemical structures) was obtained from the descriptor matrix X by... 

FIGURE 6.6 Dendrogram from hierarchical cluster analysis (average linkage) of n = 20 standard amino acids. Distance measure used was dTANi (Equation 6.5) calculated from eight binary substructure descriptors. Four structure pairs with identical descriptors merge at a distance of zero. Clustering widely corresponds to the chemist s point of view. 

Another very useful exploration technique is cluster analyis, which quantifies similarities by calculating mathematic distances. The typical graphic output is a dendrogram. A common method of cluster analysis is Hierarchical cluster analysis (HCA). 

Figure 12.24 Dendrograms obtained from hierarchical cluster analysis (HCA) of the NIR. spectra of the poly(urethane) foam samples (shown in Figure 12.16), (A) using the first two PCA scores as input, (B) using the first five PCA scores as input. In both cases, the Mahalanobis distance measure and the nearest-neighbor linkage rule were used.

Hierarchical cluster analysis (HCA) is an unsupervised technique that examines the inteipoint distances between all of the samples and represents that information in the form of a twcKlimensional plot called a dendrogram. These dendrograms present the data from high-dimensional row spaces in a form that facilitates the use of human pattern-recognition abilities. 

So, clustering techniques have been used for classification. Piepponen et al. applied a hierarchical cluster analysis (CLUSTAN) to the classification of food oils (groundnut, soya, sunflower and maize) by their fatty acid composition. The dendrogram of the distances shows four weU-separated clusters. Some suspect commercial samples of sunflower oil fall near the cluster of soya oils, so far from the clainud class that they cannot be consider i genuine. 

Aishima used hierarchical cluster analysis on gas chromatographic profiles [10 peaks out of 93 measured peaks, 48 samples seleeted out of 200 samples of eight brands (categories) of soy sauce]. The obtained dendrograms were mainly discussed in connection with the results of linear discriminant analysis and the ten peaks selected for clustering. 

Cluster analysis (CA) performs agglomerative hierarchical clustering of objects based on distance measures of dissimilarity or similarity. The hierarchy of clusters can be represented by a binary tree, called a dendrogram. A final partition, i.e. the cluster assignment of each object, is obtained by cutting the tree at a specified level [24],... 

On the basis of the binary codification of the responses to the previously described tests, a hierarchical classification was done for each water sample. A partial dendrogram was established for each sample, and the totality of the strains isolated during the experiment were compared in a general dendrogram. 

The typical output of hierarchical cluster methods is a so-called dendrogram, a treelike diagram which is very useful for discussing several possible results of the clustering process. For an illustration see Fig. 5-13 the underlying example will be explained in Section 5.3.4. 

Results from hierarchical agglomerative cluster analysis according to the algorithm of WARD (see Section 5.3) are illustrated as a dendrogram in Fig. 7-14. Distinction of the months in which the heating of buildings has a large influence from the summer months is clearly demonstrated. November and December of the second year of the in-... 

Fig. 7-14. Dendrogram of the hierarchical agglomerative cluster analysis according to WARD...

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