Centroid-based clustering method

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

To generate the dendrogram, HCA methods form clusters of samples based on their nearness in row space. A common approach is to initially treat every sample as a cluster and join closest clusters together. This process is repeated until only one cluster remains. Variations of HCA use different approaches to measure distances between clusters (e.g., single vs. centroid linking, Euclidean vs. Mahalanobis distance), fhe two methods discussed below use single and centroid linking with Euclidean distances. 

In this work, classification through HCA was based on the Euclidean distance and the average group method. This method established links between samples/cluster. The distance between two clusters was computed as the distance between the average values (the mean vector or centroids) of the two clusters. The descriptors employed in HCA were the same selected in... 

A more general model would be a hyperellipsoid as defined by the Mahalanobis distance. SIMCA uses one or several principal components as the axes of the model. If the shape of a group of objects considered is like a cigar then the first principal component would be sufficient to define a suitable model (Figure 8). If the cluster shape is similar to a brick then the second or even the third principal component are also required. If the cluster shape is similar to a ball then only the centroid is necessary for the model. Because the method is based on principal component analysis it is not very sensitive to a large numbo- of features or to collinear features. 

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