Hamming distances

It follows all the four metric properties and is monotonic with Hamming distance. For dichotomous variables, (Euclidean distance) =Hamming 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) ... 

If the binary descriptors for the objects s and t are substructure keys the Hamming distance Eq. (6)) gives the number of different substructures in s and t (components that are 1 in either s or but not in both). On the other hand, the Tanimoto coefficient (Eq. (7)) is a measure of the number of substructures that s and t have in common (i.e., the frequency a) relative to the total number of substructures they could share (given by the number of components that are 1 in either s or t). 

Euclidean and Hamming distance measures of torsional similarity. 

The differences between the Soergel and Hamming distance nieasures for various molecules (see text). 

A very useful distance measure between ordered sets of discrete valued elements is the Hamming distance, Bn- Given two sets, x = (xi,...,Xn) and y= yi, .., yn),... 

Fig. 3.23 Hamming distance as a function of time for elementary rule R90 see text.

While the solution to this problem can be trivially solved by a conventional serial computer program, of course by say computing the Hamming distance ... 

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

Some authors use the Hamming distance as the equivalent of Euclidean distance of binary data. In that case ... 

The literature also mentions a normalized Hamming distance, which is then equal to either ... 

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... 

Bit vectors live in an -dimensional, discrete hypercubic space, where each vertex of the hypercube corresponds to a set. Figure 2 provides an example of sets with three elements. Distances between two bit vectors, vA and vB, measured in this space correspond to Hamming distances, which are based on the city-block Zj metric... 

Because these vectors live in an -dimensional hypercubic space, the use of non-integer distance measures is inappropriate, although in this special case the square of the Euclidean distance is equal to the Hamming distance. 

Fig. 2. Distance between two binary-valued feature vectors vA and vB is not given by the Euclidean distance but the Hamming distance between the two.

Although the Hamming distance (see Eq. 2.18) also applies for continuous vectors, Euclidean distances are typically used... 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

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