City block distance

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) ... 

It can be shown [5] that the Hamming distance is a binary version of the city block distance (Section 30.2.3.2). 

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 ... 

Fig. 30.6. D - is the Euclidean distance between i and i dm and dm are the city-block distances between i and i for variables Xi and X2 respectively. The city-block distance is 4ri + dm-...

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... 

Non-linear PCA can be obtained in many different ways. Some methods make use of higher order terms of the data (e.g. squares, cross-products), non-linear transformations (e.g. logarithms), metrics that differ from the usual Euclidean one (e.g. city-block distance) or specialized applications of neural networks [50]. The objective of these methods is to increase the amount of variance in the data that is explained by the first two or three components of the analysis. We only provide a brief outline of the various approaches, with the exception of neural networks for which the reader is referred to Chapter 44. 

The chemical constitution of a molecule or an ensemble of molecules (EM) of n atoms is representable by a symmetric n X n BE-matrix and corresponds accordingly to a point P in TR ( +D/a an n(n +1)/2 dimensional Euclidean space, the Dugundji space of the FIEM(A). The "city block distance of two points P i and P 2 is twice the number of electrons that are involved in the interconversion EMi EM2 of those EM that belong to the points Pi and P2. This chemical metric on the EM of an FIEM provides not only a formalism for constitutional chemistry, but also allows us to use the properties of Euclidean spaces in expressing the logical structure of the FIEM, and thus of constitutional chemistry 3e>32c>. 

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... 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

Euclidean distance, (Euclid), city block distance, d(city), or Mahalanobis distance, (Mahalanobis). 

Similarity and Distance. Two sequences of subgraphs m and n such as those in Table 1 have the property that there is a built-in one-to-one correspondence between the elements of one sequence (m,) and those of the other (/i,). Accordingly, it is straightforward to calculate various well-known (17) measures of the distance d between the sequences, e.g. Euclidean distance [2,( Wi city block distance... 

Figure 7.1 Authentication of monovarietal virgin olive oils results of applying clustering analysis to volatile compounds. The Mahattan (city block) distance metric and Ward s amalgamation methods were used in (a) the Squared Euclidean distance and (b) complete linkage amalgamation methods. Note A, cv. Arbequina (6) C, cv. Coratina (6) K, cv. Koroneiki (6) P, cv. Picual (6) 1, harvest 1991 2, harvest 1992. Olives were harvested at three levels of maturity (unripe, normal, overripe) (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

In addition to the Euclidean distance, a city block distance can be calculated. The formula giving a city block distance between spectra j and h is... 

In the casep=, the so-called Manhattan or city-block distance is obtained. This distance refers to the passage around a corner, that is,... 

Figure 5 0 City-block distance (a) and Euclidean distance (b) for two features.

In the case of k = 2, Eq. [8] corresponds to the well-known Euclidean distance of which Eq. [7] is an integral version. In the case of k = 1, we find the Manhattan or city-block distance. The choice of Euclidean distance is a computationally interesting choice, but it is by no means the only one possible. In this context, it is appropriate to mention the four requirements that should be associated with a true distance measure ... 

FIGURE 15. Euclidean distance and city block distance-... 

City Block The city block distance between the output vector and the weight vector. It is defined as follows ... 

We have used the phrase winning PE rather glibly, without explaining what it means. Usually the PE with the largest value of T in a layer is the winning PE. However, if the Euclidean distance or city block distance summation function is used, the opposite holds true, that is, the PE with the smallest value of T, is the winnen These summation functions are usually used with special-purpose transfer functions (an example is a radial basis transfer function), which perform a monotonic inversion of the effective input. ... 

Fragrance Analogues in Chemical Space 2.4.1 Similarity Searching by City-Block Distance... 

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) . 

Figure 5 Mass spectra can be considered as points or vectors in a multidimensional spectral space. For simplicity only two mass numbers (43, 58) have been selected in this example. A , abundance (peak height in %B) at mass m U, spectrum from unknown R1, reference spectrum of propanal R2, reference spectrum of acetone. Measures for spectral similarity are the Euclidean distance (d), the city block distance (A43 + Ajs) or the cosine of angle a (equivalent to S, in Eqn [2]).

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