Mahalanobis/Manhattan distance

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... 

In this matrix the most similar pair of (different) objects is (1.4), while the most divergent pair is (3.4). Apart from the classical Euclidean distance defined by Equation 8.2, some further relevant measures exist such as Mahalanobis or Manhattan distance. The Mahalanobis distance, for instance, which is important in classification (see Chapter 3.10). is computed according to... 

Different approaches to estimate interpolation regions in a multivariate space were evaluated by Jaworska [Jaworska, Nikolova-Jeliazkova et al, 2005], based on (a) ranges of the descriptor space (b) distance-based methods, using Euclidean, Manhattan, and Mahalanobis distances. Hotelling T method and leverage values and (c) probability density distribution methods based on parametric and nonparametric approaches. Both ranges and distance-based methods were also evaluated in the principal component space by Principal Component Armlysis. 

In the first step of HCA, a distance matrix is calculated that contains the complete set of interspectral distances. The distance matrix is symmetric along its diagonal and has the dimension nxn, with n as the number of patterns. Spectral distance can be obtained in different ways depending on how the similarity of two patterns is calculated. Popular distance measures are Euclidean distances, including the city-block distance (Manhattan block distance), Mahalanobis distance, and so-called differentiation indices (D-values, see also Appendix B) . 

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