Similarity measures correlation coefficient

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). 

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

On the other hand, the analyst might not be interested in global retention indices. Indeed, by increasing the temperature for SF3, he would obtain similar retention indices as for the other two. He will then observe that the relative retention time, i.e. the retention times of the substances compared with each other, are the same for SF, and SF3 and different from SFj. Chemically, this means that SF3 has different polarity from SFj, but the same specific interactions. This is best expressed by using the correlation coefficient as the similarity measure. Indeed, rj3 = 1, indicating complete similarity, while r 2 23 much lower. Since both... 

Because x, as well as w are normalized, represents the cosine or correlation coefficient between the two vectors. In a variant of ART, Fuzzy ART, a fuzzy similarity measure is used instead of the cosine similarity measure [14]. 

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. 

A number of performance criteria are not primarily dedicated to the users of a model but are applied in model generation and optimization. For instance, the mean squared error (MSE) or similar measures are considered for optimization of the number of components in PLS or PC A. For variable selection, the models to be compared have different numbers of variables in this case—and especially if a fit criterion is used—the performance measure must consider the number of variables appropriate measures are the adjusted squared correlation coefficient, adjR, or the Akaike S information criterion (AIC) see Section 4.2.3. 

Fields can be utilized in virtual screening applications for assessing the similarity (alignment) or complementarity (docking) of molecules. Two similarity measures have achieved the most attention. These are the so-called Garbo- [195] and Hodgkin indexes [196] respectively. Others are Pearson s product moment correlation coefficient [169] and Spearman s rank correlation coefficient [169]. 

Identification involves the confirmation of a certain chemical entity from its spectrum by matching against the components of a spectral library using an appropriate measure of similarity such as the correlation coefficient, also known as the spectral match value (SMV). SMV is the cosine of the angle formed by the vectors of the spectram for the sample and the average spectrum for each product included in the library. 

This work was extended by De Michelis and Calvelo (1994) who measured dispersion coefficients for 1cm cubes and for 1cm x 1cm x 1.5 cm cuboids. The data were again correlated by an expression taking the form of equation 3.30, with similar exponents on bed height (2.46 and 2.58 for cubes and cuboids respectively) and gas velocity (3.13 and 3.34 respectively) but coefficients of 0.110 and 0.256 respecfively fhe coefficient in equation 3.30 increased with decreasing particle sphericity. 

Fewer intercomparison studies have been carried out for peroxy radicals than for OH. Two chemical amplification methods were compared during a measurement campaign in Brittany, France (Cantrell et al., 1996). Although the measurements tended to track one another, there is more scatter than might be expected, given the similar nature of the instruments. For example, a plot of the data from one instrument against those from the second had a slope of 0.71 but a correlation coefficient of only r = 0.36. In another study (Zenker et al., 1998), comparison of three chemical amplifier techniques to matrix isolation-ESR gave... 

We compiled literature and our own extraction data to compare the distribution of the same solutes from water to [C4CiIm][PFg] and 48 various conventional solvents. As a measure of similarity of the extraction properties of any two solvents, we used the Pearson correlation coefficient between IgD for the same solutes. Note that a high correlation coefficient does not mean that the distribution ratios determined with the two solvents are close by absolute value rather, it means that the distribution ratios change in the same marmer from one solute to another. 

This NMR titration method was subsequently applied to equilibrium IEs on acidity.30 33 Like the previous methods, it too benefits from the high sensitivity of 13C and 19F chemical shifts, and even 111 chemical shifts, to both isotopic substitution and state of protonation. Figure 1 shows the NMR titration of a mixture of tri(methyl-d)amine and tri(methyl-t/2)amine in D20, plotted according to Equation (19). The slope is 1.1618 0.0004. The intercept is -0.0061 0.0046, properly zero. The correlation coefficient is an impressive 0.999999, which is an indication of the accuracy achievable. Another remarkable result was the measurement of the relative basicity of the two exceedingly similar isotopomers of 1 -benzyl-4-methylpiperidine-2,2,6-t/3 (6). These are truly isotopomers (here stereoisomers), which bear the same number of isotopic substitutions and differ only in the position of the isotope, which is either axial or equatorial. 

In principle, the features describing the objects can also be subjected to cluster analysis. In this case one may think immediately of the correlation coefficient, r, or the coefficient of determination, COD, as a measure of the similarity of each pair of features. Accordingly, 1 - r or 1 - COD is useful as a measure of distance. 

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