Dissimilarity plots

A dissimilarity plot is then obtained by plotting the dissimilarity values, dj, as a function of the retention time i. Initially, each p 2 matrix Y, consists of two columns the reference spectrum, which is the mean (average) spectram (normalised to unit length) of matrix X, and the spectrum at the /th retention time. The spectrum with the highest dissimilarity value is the least correlated with the mean spectrum, and it is the first spectrum selected, x, . Then, the mean spectrum is replaced by x, as reference in matrices Y, (Y, = [x j x,]), and a second dissimilarity plot is obtained by applying eq. (34.14). The spectrum most dissimilar with x, is selected (x 2) and added to matrix Y,-. Therefore, for the determination of the third dissimilarity plot Y, contains three columns [x, x 2 /]> wo reference spectra and the spectmm at the /th retention time. 

In summary, the selection procedure consists of three steps (1) compare each spectrum in X with all spectra already selected by applying eq. (34.14). Initially, when no spectrum has been selected, the spectra are compared with the average spectrum of matrix X (2) plot of the dissimilarity values as a function of the retention time (dissimilarity plot) and (3) select the spectrum with the highest dissimilarity value by including it as a reference in matrix Y,-. The selection of the spectra is finished when the dissimilarity plot shows a random pattern. It is considered that there are as many compounds as there are spectra. Once the purest spectra are available, the data matrix X can be resolved into its spectra and elution profiles by Alternating Regression explained in Section 34.3.1. 

Initially, the first two principal components were calculated. This yielded the principal components which are given in Figure 9-9 (left) and plotted in Figure 9-9 (right). The score plot shows which mineral water samples have similar mineral concentrations and which are quite different. For e3oimple, the mineral waters 6 and 7 are similar whUe 4 and 7 are rather dissimilar. 

A comparison of the Rh/S102 and Rh/T102 plots of 3 vs p and the derived rate constants serves to further show the similarity between normal- and SMSI-Rh/TlO, and their dissimilarity with Rh/SlOo. The Rh/S102 6 vs p plots at 48 and 158 C are super-Imposable within our experimental error, while they are significantly different from those for Rh/T102 normal or SMSI. 

By way of illustration, let us consider the separation of 0.2% prednisone in etrocortysone eluting with a chromatographic resolution equal to 0.8 [30] (Fig. 34.40). The dissimilarity of each spectrum with respect to the mean spectrum is plotted in Fig. 34.41a. Two clearly differentiated peaks with maxima around times 46 and 63 indicate the presence of at least two compounds. In this case, the... 

Figure 3 (a) Shows the combinational contribution of AG-j plotted as a function of temperature while, (b) Shows the free volume dissimilarity contribution to AG- plotted as a function of temperature for atmospheric and 180 bars pressure. 

From the family of AG (P, T) curves the projection on the (P, T) plane of the critical lines corresponding to the UCFT for these latexes can be calculated and this is shown plotted in Figure 4. It can be seen that the UCFT curve is linear over the pressure range studied. The slope of the theoretical projection is 0.38 which is smaller than the experimental data line. Agreement between theory and experiment could be improved by relaxing the condition that v = it = 0 in Equation 6 and/or by allowing x to be an adjustable parameter. However, since the main features of the experimental data can be qualitatively predicted by theory, this option is not pursued here. It is apparent from the data presented that the free volume dissimilarity between the steric stabilizer and the dispersion medium plays an important role in the colloidal stabilization of sterically stabilized nonaqueous dispersions. 

LCA and CCK, on the other hand, appear to be strikingly dissimilar. All CCK procedures require at least one quasi-continuous indicator, and if there are none, the investigator has to create such an indicator (e.g., SSMAXCOV procedure). In contrast, LCA does not require continuous indicators and only deals with categorical data. In the case of categorical data, the patterns of interest are usually apparent, so there is no need to summarize the data with correlations. Therefore, LCA evaluates cross-tabulations and compares the number of cases across cells. This shift in representation of the data necessitates other basic changes. For example, LCA operates with proportions instead of covariances and yields tables rather than plots. These differences aside, the two approaches share a lot in common. LCA, like CCK, starts with a set of correlated indicators. It also makes the assumption of zero nuisance covariance-—in the LCA literature this is called the assumption of local independence, and it means that the indicators are presumed to be independent (i.e., uncorrelated) within latent classes. Moreover, LCA and CCK (MAXCOV in particular) use similar procedures for group assignment and both of them involve Bayes s theorem. 

Two objects with different spectral properties, i.e., variation in the slope of spectral reflectance curve of two bands, can be separable with the help of ratio images (Lillesand et al. 2007). In this study standard reflectance data of USGS Spectral Library and John Hopkins University spectral library (Available in ENVI) have been used. To enhance the dissimilarity between different rock types in the scene, plots with a higher reflectance were kept in the numerator and plots with low reflectance were kept in the denominator, while taking the band ratios. Using this approach, a ratio of 5/3 was taken for basalt, 7/3 for peridotite, and 4/2 for vegetation. 

Each object or data point is represented by a point in a multidimensional space. These plots or projected points are arranged in this space so that the distances between pairs of points have the strongest possible relation to the degree of similarity among the pairs of objects. That is, two similar objects are represented by two points that are close together, and two dissimilar objects are represented by a pair of points that are far apart. The space is usually a two- or three-dimensional Euclidean space, but may be non-Euclidean and may have more dimensions. 

The application of Eq. (9) to differential profiles is illustrated in the left-hand plot of Figure 5. With i = 1, it represents the wedge area between the two curves. If the curves do not cross each other, the nominator directly represents the difference of the two AUCs. If they intersect as shown in the example, the choice of absolute differences computes a general dissimilarity index the area difference would be obtained by using signed differences instead of absolute differences. 

Figure 10. Radar plots of the serotonin 5HT2a receptor and ion channel domains likely to be affected by the 5HT2a antagonist sertindole (D4Navl.2 and D4Cav 1.2) and those of two dissimilar channel domains (D3Cav3.3 and DlNavl.5).

The resulting library consisted of 250 compounds representing different compromises between the two conflicting objectives supplied. Figure 3.5 presents a plot of the Pareto-approximation proposed by the software library (circles connected by line). Each of the remaining circles represents a solution from the initial population set after the hard filtering process. The x-axis represents similarity to ER-a ligands and the -axis dissimilarity (1-similarity)... 

The correlation functions plotted in Figs 5.5 and 5.6 present the statistical information about relative spatial distribution of reacting particles. In particular, the distinctive scale emerges here, thus indicating that all dissimilar... 

Fig. 6.16. A logarithmic plot of the joint correlation functions for similar, Xv r, t), and dissimilar, Y(r, t), particles for d = 2 and symmetric (a), Da = Db, and asymmetric (b), Da = 0, cases, respectively. Full curves are Y(r,t), broken and dotted lines XA(r,t) and Xs(r, t). The initial concentration n(t) — 1.0. The dimensionless time Dtjr is 10 (curve 1) ...

The non-equilibrium particle distribution is clearly observed through the joint correlation functions plotted in Fig. 6.47. Note that under the linear approximation [74] the correlation function for the dissimilar defects Y (r, t) increases monotonically with r from zero to the asymptotic value of unity Y(r —y oo,t) = 1. In contrast, curve 1 in Fig. 6.47 (f = 101) demonstrates a maximum which could be interpreted as an enriched concentration of dissimilar pairs, AB, near the boundary of the recombination sphere, r tq. With increasing time this maximum disappears and Y(r, t) assumes the usual smoothed-step form. The calculations show that such a maximum in Y(r, t) takes place within a wide range of the initial defect concentrations and for a random initial distribution of both similar and dissimilar particles used in our calculations X (r, 0) = Y(r > 1,0) = 1. The mutual Coulomb repulsion of similar particles results in a rapid disappearance of close AA (BB) pairs separated by a distance r < L (seen in Fig. 6.47 as a decay of X (r, t) at short r with time). On the other hand, it stimulates strongly the mutual approach (aggregation) of dissimilar particles leading to the maximum for Y(r, t) at intermediate distances observed in Fig. 6.47. 

In Fig. 7.4 the joint correlation functions are plotted for distribution of geminate partners created randomly within narrow interval ro r Rg. Two important conclusions suggest themselves from this figure (i) due to similar and dissimilar reactant correlation back-coupling the narrow peak of Y at short distances is accompanied by the decay in X, (ii) for great doses, n joint correlation functions are quite similar to those observed for uncorrelated distribution, i.e., an aggregation manifests itself mainly at high defect concentrations. 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

N represents the number of sensors in the array. For p = 2, the distance in (10.10) is Euclidian. The protocol is relatively simple. The distance matrix is created from the datapoints and scanned for the smallest values that are then arranged and displayed in the form of a dendrogram (Fig. 10.9 Suslick, 2004) in which the dissimilarity is plotted on the horizontal axis. In a dendrogram, each horizontal line segment represents the distance—that is, the similarity—between samples. Thus, if we want... 

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