Euclidean distance method

The distribution of chondroihn sulfate was calculated using the partial least squares and Euclidean distance method, as introduced by Potter et al. [103]. The FT-IR images revealed a heterogeneous chondroihn sulfate distribuhon, not only between cartilage zones but also within the territorial and inter-territorial mahices. 

There are a variety of algorithms available for this type of spectral matching. One of the most popular is the Euclidean distance method ... 

Another often used algorithm for simple spectral matching is the vector correlation method. It is similar to the Euclidean distance method but does not require that the spectra be normalized. In this method, each spectrum is centered around the mean response value to calculate the hit quality index. 

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

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. 

S.J. Dixon and R.G. Brereton, Comparison of performance of five common classifiers represented as boundary methods Euclidean distance to centroids, linear discriminant analysis, quadratic discriminant analysis, learning vector quantization and support vector machines, as dependent on data structure, Chemom. Intell. Lab. Syst, 95, 1-17 (2009). 

The infomsdon submitted to the computer program includes the pre-processed daa, the metric for classification (e.g.. Euclidean distance), and the choice of cisatering method (e.g., single link). Intercluster distances are calculated and used to construct a dendrogram. 

To generate the dendrogram, HCA methods form clusters of samples based on their nearness in row space. A common approach is to initially treat every sample as a cluster and join closest clusters together. This process is repeated until only one cluster remains. Variations of HCA use different approaches to measure distances between clusters (e.g., single vs. centroid linking, Euclidean vs. Mahalanobis distance), fhe two methods discussed below use single and centroid linking with Euclidean distances. 

To classify a new sample, fc-NN computes its distances (usually, the multivariate Euclidean distances, see Eq. 7) from each of the samples of a training set, whose class membership is known. The k nearest samples are then taken into consideration to perform the classification generally, a majority vote is employed, meaning that the new object is classified into the class mostly represented within the k selected objects. Being a distance-based method, it is sensitive to the measurement units and to the scaling procedures applied. 

Figure 4. Dendrogram showing sub-division of main chemical group according to cluster analysis (Euclidean distance Ward s method).

In most applications easily interpretable results are obtained by the method of WARD where the parameters are defined by relationships between the numbers of objects (see also Section 7.2.1.2.2). If squared EUCLIDean distances are used, this strategy can also... 

In collocation methods, different sets of basis functions can also be used, for example radial basis functions, which are functions that depend only on the euclidean distance between collocation points i and j, ri j. 

Figure 19 shows the main steps of a method of characterization of spores and generated filaments which is based on the Euclidean Distance Map. The central zone of the EDM (i.e. the zone with the highest distance value Rm) is dilated Rm times. The size of the object in Fig. 20 e gives the size of the spore. A logical subtraction of Fig. 19 e from Fig. 19 c leaves only the branches visible in Fig. 19f. 

Search algorithms have advanced over the years to the point that most of the spectral data are used in the search. The methods are referred to as full-spectra searches because the entire spectral pattern is used in the matching procedure. Again, a number of similarity metrics are used, but most produce similar results. Typically, the spectral range for the search is selectable, and the library and target spectra are all normalized so that the total spectral area is 1.0. Either the Euclidean distance or the dot product between the target and library entries is calculated. The Euclidean distance is defined as... 

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).

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... 

---