Similarity measures Euclidean distance

Perhaps a more useful means of quantifying structural data is to use a similarity measurement. These are reviewed by Ludwig and Reynolds (1988) and form the basis of multivariate clustering and ordination. Similarity measures can compare the presence of species in two sites or compare a site to a predetermined set of species derived from historical data or as an artificial set comprised of measurement endpoints from the problem formulation of an ecological risk assessment. The simplest similarity measures are binary in nature, but others can accommodate the number of individuals in each set. Related to similarity measurements are distance metrics. Distance measurements, such as Euclidean distance, have the drawbacks of being sensitive to outliers, scale, transformations, and magnitudes. Distance measures form the basis of many classification and clustering techniques. 

So far we have been considering leverage with respect to a point s Euclidean distance from an origin. But this is not the only measure of distance, nor is it necessarily the optimum measure of distance in this context. Consider the data set shown in Figure E4. Points C and D are located at approximately equal Euclidean distances from the centroid of the data set. However, while point C is clearly a typical member of the data set, point D may well be an outlier. It would be useful to have a measure of distance which relates more closely to the similarity/difference of a data point to/from a set of data points than simple Euclidean distance.The various Mahalanobis distances are one such family of such measures of distance. Thus, while the Euclidean distances of points C and D from the centroid of the data set are equal, the various Mahalanobis distances from the centroid of the data set are larger for point D than for point C. 

Fig. 30.5. The point i is equidistant to i and i" according to the Euclidean distances (Z) - and Z) ) but much closer to i (cos 0, ) than to i" (cos 0 ), when a correlation-based similarity measure is applied.

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 similarities between all pairs of objects are measured using one of the measures described earlier. This yields the similarity matrix or, if the distance is used as measure of (dis)similarity, the distance matrix. It is a symmetrical nx matrix containing the similarities between each pair of objects. Let us suppose, for example, that the meteorites A, B, C, D, and E in Table 30.3 have to be classified and that the distance measure selected is Euclidean distance. Using eq. (30.4), one obtains the similarity matrix in Table 30.4. Because the matrix is symmetrical, only half of this matrix needs to be used. 

Once a dimensionality for the map and the type of local measure to be used have been chosen, training can start. A sample pattern is drawn at random from the database and the sample pattern and the weights vector at each unit are compared. As in a conventional SOM, the winning node or BMU is the unit whose weights vector is most similar to the sample pattern, as measured by the squared Euclidean distance between the two. 

In 1980, Carbo et al. were the first to express molecular similarity using the electron density [17]. They introduced a distance measure between two molecules A and B in the sense of a Euclidean distance in the following way ... 

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... 

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). 

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. 

Spectral similarity search is a routine method for identification of compounds, and is similar to fc-NN classification. For molecular spectra (IR, MS, NMR), more complicated, problem-specific similarity measures are used than criteria based on the Euclidean distance (Davies 2003 Robien 2003 Thiele and Salzer 2003). If the unknown is contained in the used data base (spectral library), identification is often possible for compounds not present in the data base, k-NN classification may give hints to which compound classes the unknown belongs. 

Instead of using the Euclidean distance, also other distance measures can be considered. Moreover, another power than 2 could be used for the membership coefficients, which will change the characteristics of the procedure (degree of fuzzification). Similar to fc-means, the number of clusters k has to be provided as an input, and the algorithm also uses cluster centroids Cj which are now computed by... 

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... 

Distances in these spaces should be based upon an Zj or city-block metric (see Eq. 2.18) and not the Z2 or Euclidean metric typically used in many applications. The reasons for this are the same as those discussed in Subheading 2.2.1. for binary vectors. Set-based similarity measures can be adapted from those based on bit vectors using an ansatz borrowed from fuzzy set theory (41,42). For example, the Tanimoto similarity coefficient becomes... 

Initially cluster analysis defines a measure of simUarity given by a distance or a correlation or the information content Distance can be measured as euclidean distance or Mahalanobis distance or Minkowski distance. Objects separated by a short distance are recognized as very similar, while objects separated by a great distance are dissimilar. The overall result of cluster analysis is reported as a dendrogram of the similarities obtained by many procedures. 

To find the structures of the objects in the data set, we need a measure of similarity. Although many types of measures can be applied, the Euclidean distance is the most frequently used similarity measure. According to the law of Pythagoras, the distance between two points Oj and 02 characterized by variables x and y can be presented as follows (Figure 15.1) ... 

Mahalanobis distance. This method is popular with many chemometricians and, whilst superficially similar to the Euclidean distance, it takes into account that some variables may be correlated and so measure more or less the same properties. The distance between objects k and l is best defined in matrix terms by... 

Other similarity coefficients used in similarity studies include the cosine coefficient, and the Hamming and Euclidean distance measures [7], Similarity coefficients can also be applied to vectors of attributes where the attributes are real numbers, for example, topological indices or physiochemical properties. 

The choice of representation, of similarity measure and of selection method are not independent of each other. For example, some types of similarity measure (specifically the association coefficients as exemplified by the well-known Tanimoto coefficient) seem better suited than others (such as Euclidean distance) to the processing of fingerprint data [12]. Again, the partition-based methods for compound selection that are discussed below can only be used with low-dimensionality representations, thus precluding the use of fingerprint representations (unless some drastic form of dimensionality reduction is performed, as advocated by Agrafiotis [13]). Thus, while this chapter focuses upon selection methods, the reader should keep in mind the representations and the similarity measures that are being used recent, extended reviews of these two important components of diversity analysis are provided by Brown [14] and by Willett et al. [15]. 

The manner in which sample-to-sample resemblance is defined is a key difference between the various hierarchical clustering techniques. Sample analyses may be similar to one another in a variety of ways and reflect interest in drawing attention to different underlying processes or properties. The selection of an appropriate measure of similarity is dependent, therefore, on the objectives of the research as set forth in the problem definition. Examples of different similarity measures or coefficients that have been used in compositional studies are average Euclidean distance, correlation, and cosine. Many others that could be applied are discussed in the literature dealing with cluster analysis (15, 18, 19, 36, 37). 

The Euclidean distance is a good measure of similarity when the binary sets are relatively rich, and is mostly used in situations in which similarity is measured in a relative sense (the distance of two compounds to the same target). James and co-workers" prefer the Tanimoto coefficient when absolute comparisons (between two independent pairs of molecules) are made. 

To apply these multivariate techniques, we require a data matrix with the information corresponding to n observations of p quantitative variables X, X2,..Xp). We could, also, have some qualitative variables, coded numerically, to classify the observations into groups. From a geometric perspective, the n observations of the data matrix would correspond to n points of the Euclidean space of the p variables, and the Euclidean distance between observations would correspond to a measure of proximity (similarity). 

---