Euclidian distances

Here, the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

The //-space Euclidian distances for every object relative to the group averages A and B are calculated and plotted. 

Note. If the N dimensions yield very different numerical values, such as 105 3 mmol/L, 0.0034 0.02 meter, and 13200 600 pg/ml, the Euclidian distances are dominated by the contributions due to those dimensions for which the differences A-B, AS, or BS are numerically large. In such cases it is recommended that the individual results are first normalized, i.e., x = (x - Xn,ean)/ 5 t, where Xmean and Sx are the mean and standard deviation over all objects for that particular dimension X, by using option (Transform)/(Normalize) in program DATA. Use option (Transpose) to exchange columns and rows beforehand and afterwards The case presented in sample file SIEVEl.dat is different the individual results are wt-% material in a given size class, so that the physical dimension is the same for all rows. Since the question asked is are there differences in size distribution , normalization as suggested above would distort tbe information and statistics-of-small-numbers artifacts in the poorly populated size classes would become overemphasized. 

Table 8.2. Euclidian distances dE and distance s ranks of an unknown wine sample to 16 samples with known growing region (GR), according to Danzer et al. [2001]... 

The multivariate tools typically used for the NIR-CI analysis of pharmaceutical products fall into two main categories pattern recognition techniques and factor-based chemometric analysis methods. Pattern recognition algorithms such as spectral correlation or Euclidian distance calculations basically determine the similarity of a sample spectrum to a reference spectrum. These tools are especially useful for images where the individual pixels yield relatively unmixed spectra. These techniques can be used to quickly define spatial distributions of known materials based on external reference spectra. Alternatively, they can be used with internal references, to locate and classify regions with similar spectral response. 

Cluster Analysis. Cluster analysis using BMDP s PKM method was performed on the data with several methods of data transformation, normalization, and variable standardization. Qualitative clustering results for these different procedures of data manipulation were similar. The method finally selected Is that discussed above, l.e., normalization of each sample, so the concentrations sum to unity and use of Euclidian distances with no standardization of variables as a measure of sample similarity. 

The term djic is the Euclidian distance between observation i and centroid k. The test statistic is then... 

The distance methods operate differently. The classification of a test set member is based on the class assignment of the samples in the training set nearest to the unknowns. The type of distance used can differ but is usually the Euclidian distance, and the number of nearest neighbors is selected in advance. Usually the 3 to 5 nearest neighbors are selected and the possibility that the unknown may not be represented in the training sets is allowed. 

The euclidian distance (ED) of the selected phases relative to the most non-polar... 

Fig. 10.9 Example of dendrogram. The dissimilarity between compounds is shown as Euclidian distance on the horizontal axis (Suslick, 2004)...

KNN)13 14 and potential function methods (PFMs).15,16 Modeling methods establish volumes in the pattern space with different bounds for each class. The bounds can be based on correlation coefficients, distances (e.g. the Euclidian distance in the Pattern Recognition by Independent Multicategory Analysis methods [PRIMA]17 or the Mahalanobis distance in the Unequal [UNEQ] method18), the residual variance19,20 or supervised artificial neural networks (e.g. in the Multi-layer Perception21). 

In the NN method, the property F of the target compound is calculated as an average (or weighted average) of that for its NN in the space of descriptors selected for the model. Different metrics (Euclidian distances, Tanimoto similarity coefficients, etc.), can be used to identify the neighbors. Their number k is optimized using a cross-validation procedure for the training set. 

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... 

Figure 14,2.1. Dendrogram obtained from the Euclidian distance of the first three principal components calculated from results of conventional coal analysis.

The sum of the squared Euclidian distances is used as the objective function to be minimized ... 

---