Distances defined from data

6 Principal coordinates analysis 31.6.1 Distances defined from data 

Principal coordinates analysis (PCoA) is applied to distance tables rather than to original data tables, as is the case with principal components analysis (PCA). 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

The squared Euclidean distance (also called Pythagorean distance) has been defined in Section 9.2.3  

It can be regarded as a special case of the squared weighted Euclidean distance (Section 30.2.2.1), A property of (weighted) Euclidean distance functions is that the distances between row-items D are invariant under column-centering of the table X  

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Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

FIGURE 9.6 Euclidean distance between two data points in a two-dimensional measurement space defined by the measurement variables x1 and x2. (Adapted from Massart, D.L. and Kaufman, L., The Interpretation of Analytical Chemical Data by the Use of Cluster Analysis, John Wiley Sons, New York, 1983. With permission.)... 

Here, the number density p )(r)is defined as in Eq. [4], but it is normalized by the number density of all atom pairs at this same distance instead of by the number of pairs ij in the whole reference volume. The variable is the number of atom pairs ij found in the data set of protein-ligand complexes, and a is an empirical factor that defines the weight of each observation. This potential is combined with a van der Waals potential as a reference state to compensate for the lack of sampling at short distances and for certain underrepresented atom pairs. Apart from data on 90 protein-ligand complexes used in the original validation, no further application has been published. 

The genus Lupinus comprises several hundred more or less well-defined species, 12 in the Old World and the others in the New World of North, Central and South America. Sequence data indicate that New World lupins apparently derived from Old World species. Long distance dispersal from Old World origin seems to have led to the colonization of the Atlantic part of South America (clade with L. aureonitens, L. albescens and L. paragmrensis) and of North America (see Fig. 7.12a) (Kass and Wink, 1997b). 

Some difficulties in comparing the experimental kinetic data with the outer-sphere reorganization energy calculated from the Marcus formula (28) result from several assumptions made in this theory. The reactant was assumed to have a spherical shape with a symmetric charge distribution. No field penetration into the metal was considered. Also, the spatial dispersion of the dielectric permittivity of the medium was not taken into account. In fact, the positions and orientations of dipoles around a given ion are correlated with each other therefore the reorientation of one dipole, under the influence of the external field, changes to some extent the reorientation of other dipoles within the distance defined by the correlation length. 

Although the diffraction data available from RUF4 are of low quality, probably as a result of the small crystallite size, consequent on the method of preparation, some firm structural conclusions can be drawn. The structure is of the same type as VF4 derived by Becker and Muller from single-crystal studies. The two Ru atoms in the monoclinic unit cell are at 0,0, 0 and /z, /z, /z, the Ru-Ru distances then being simply determined by the unit Cell dimensions, which are accurately defined. From this, each Ru atom is seen to have four close Ru atom neighbors in a roughly square arrangement in the same plane. This Ru-Ru nearest-... 

In view of the maturity of the SANS technique it is surprising that data are still published in arbitrary units that are functions of the timescale of the experiment and/or the sample dimensions (e.g. thickness). Conversion to an absolute scale may be accomplished by multiplying by a calibration constant and, as explained in Section 7.1.2, the absolute cross section dS/dS2(0) is defined [116] as the ratio of the number of neutrons scattered per second into unit solid angle divided by the incident neutron flux (neutrons cm s ) and thus has the dimensions of area (cm ). On normalizing with respect to unit sample volume, dS/df2(2) has units of cm From the above definition, the relationship between the cross section and the measured count rate I(Q) (counts s ) in a detector element with area Aa and counting efficiency s, situated normal to the scattered beam at a distance r from the sample, is given by... 

In creating a PCA model, a best fit hyperplane is placed in the data. Typically, the process data is coplanar with the hyperplane, so the distance of the data point or part from the model plane defines the residual error or what is called DModX. The DModX could also be explained as a measurement of the correlation structure of the data. Large DModX values indicate ineompatibility of the data to the model. 

A two-dimensional slice may be taken either parallel to one of the principal co-ordinate planes (X-Y, X-Z and Y-Z) selected from a menu, or in any arbitrary orientation defined on screen by the user. Once a slice through the data has been taken, and displayed on the screen, a number of tools are available to assist the operator with making measurements of indications. These tools allow measurement of distance between two points, calculation of 6dB or maximum amplitude length of a flaw, plotting of a 6dB contour, and textual aimotation of the view. Figure 11 shows 6dB sizing and annotation applied to a lack of fusion example. 

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