Manhattan distance

FIGURE 2.10 Euclidean distance and city block distance (Manhattan distance) between objects represented by vectors or points xA and xB. The cosine of the angle between the object vectors is a similarity measure and corresponds to the correlation coefficient of the vector... 

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) . 

To construct dissimilarity measures, one uses mismatches Here a + b is the Hamming (Manhattan, taxi-cab, city-block) distance, and a + h) is the Euclidean distance. 

Hamming, Manhattan, taxi-cab, city-block distance (a + fc) ... 

For those variables that are measured on a scale of integer values consisting of more than two levels, one uses the Manhattan or city-block distance. This is also referred to as the L,-norm. It is given for variable j by ... 

Manhattan distances can be used also for continuous variables, but this is rarely done, because one prefers Euclidean distances in that case. Figure 30.6 compares the Euclidean and Maiihattan distances for two variables. While the Euclidean distance between i and i is measured along a straight line connecting the two points, the Manhattan distance is the sum of the distances parallel to the axes. The equations for both types of distances are very similar in appearance. In fact, they both belong to the Minkowski distances given by ... 

The Manhattan distance is obtained for r = 1 and the Euclidean distance for r = 2. In this context the Euclidean distance is also referred to as the L2-norm. 

In the case of r = 2 we obtain the ordinary Euclidean distance of eq. (31.75), which is also called the L2-norm. In the case of r = 1 we derive the city-block distance (also called Hamming-, taxi- or Manhattan-distance), which is also referred to as... 

Hot water can be transported over longer distances with little heat loss while steam heat distribution systems can only serve high-density regions. The largest steam system in the United States is a part of New York s Consolidated Edison Company and serves a small part of Manhattan Island. The larger pipes or mains carry 200 to 250°F water under pressure. Re-... 

Euclidean distance City block (Manhattan) distance Minkowski distance Correlation coefficient (cos a), similarity... 

M d city <-dist (X,method= "manhattan" ) In general, the Minkowski distance is defined by... 

Distance measures were already discussed in Section 2.4. The most widely used distance measure for cluster analysis is the Euclidean distance. The Manhattan distance would be less dominated by far outlying objects since it is based on absolute rather than squared differences. The Minkowski distance is a generalization of both measures, and it allows adjusting the power of the distances along the coordinates. All these distance measures are not scale invariant. This means that variables with higher scale will have more influence to the distance measure than variables with smaller scale. If this effect is not wanted, the variables need to be scaled to equal variance. 

The USR (Ultrafast Shape Recognition) Method. This method was reported by Ballester and Richards (53) for compound database search on the basis of molecular shape similarity. It was reportedly capable of screening billions of compounds for similar shapes on a single computer. The method is based on the notion that the relative position of the atoms in a molecule is completely determined by inter-atomic distances. Instead of using all inter-atomic distances, USR uses a subset of distances, reducing the computational costs. Specifically, the distances between all atoms of a molecule to each of four strategic points are calculated. Each set of distances forms a distribution, and the three moments (mean, variance, and skewness) of the four distributions are calculated. Thus, for each molecule, 12 USR descriptors are calculated. The inverse of the translated and scaled Manhattan distance between two shape descriptors is used to measure the similarity between the two molecules. A value of 1 corresponds to maximum similarity and a value of 0 corresponds to minimum similarity. 

Fig. 3.3 Elements of the Euler-Venn diagrams represent compounds that were found among the first 5% of the similarity-ranked list that results from retrospective screening with the (a) COX2, (b) HIV protease and (c) MMP datasets of the COBRA dataset. The Manhattan distance...

In a subsequent study, we examined the influence of seven similarity indices on the enrichment of actives using the topological CATS descriptor and the 12 COBRA datasets [31]. In particular, we evaluated to what extent different similarity measures complement each other in terms of the retrieved active compounds. Retrospective screening experiments were carried out with seven similarity measures Manhattan distance, Euclidian distance, Tanimoto coefficient, Soergel distance, Dice coefficient, cosine coefficient, and spherical distance. Apart from the GPCR dataset, considerable enrichments were achieved. Enrichment factors for the same datasets but different similarity measures differed only slightly. For most of the datasets the Manhattan and the Soergel distance... 

Distances with C = 1 are especially useful in the classification of local data as simple as in Fig. 5-12, where simply d( 1, 2) = a + b. They are also known as Manhattan, city block, or taxi driver metrics. These distances describe an absolute distance and may be easily understood. With C = 2 the distance of Eq. 5-7, the EUCLIDean distance, is obtained. If one approaches infinity, C = oo, in the maximum metric the measurement pairs with the greatest difference will have the greatest weight. This metric is, therefore, suitable in outlier recognition. 

The distance between two points Ri and Rj in the representation space can be any nonnegative, real, commutative function that satisfies the triangle inequality (ref. 8). Usually, when comparing spectra Euclidean or Manhattan distances are employed. The generalized form of both, the Minkowski distance, can be written as follows ... 

Figure 7.3 Authentication of monovarietal virgin olive oils results of applying multidimensional scaling to volatile compounds. The distance was Manhattan (city block) and the amalgamation was Ward s method. Note A, cv. Arbequina C, cv. Coratina K, cv. Koroneiki P, cv. Picual (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

Manhattan distance. This is defined slightly differently to the Euclidean distance and is given by... 

The difference between the Euclidean and Manhattan distances is illustrated in Figure 4.23. The values are given in Table 4.19 note the Manhattan distance will always be greater than (or in exceptional cases equal to) the Euclidean distance. 

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