Correlation analysis, transformation

Figure 2.12 Correlation plots of (A) transporter expression and (B) metabolic enzyme expression in duodenum between rat and human. The expression levels of transporters and metabolic enzymes are normalised by GADPH expression, transformed by natural logarithm and absolute values were used in the correlation analysis [121].

Fig. 10.15 Outline of steps performed in the correlation analysis. In the first step the time-varying signal is Fourier transformed to the frequency domain creating auto- and cross-spectral densities. They are then normalized to form coherence...

It has been possible by means of multivariate cross-correlation analysis to include the interaction between the metals which arise as a result of emission and transformation. The assumption is that the metals are transported at the same rate. The multivariate cross-correlation function Rxy(f) expresses a broad maximum at t = 3, i.e. 1.5 h (Fig. 6-20). This means that the mean transport rate of the metals is 3 km h Hydrological data gathered on the same day renders this result plausible ... 

In order to measure the commonness between the two configurations, we have applied the canonical correlation analysis (CCA) [1], The CCA applies linear transformations to the original two sets of coordinate systems to produce new coordinate systems such that the correlation coefficients between y th coordinates in both sets is the largest. CCA applied to the nMDS-generated 2D coordinate systems produced the first coordinates with the correlation coefficient 0.79 and the second with 0.32. These values may be understood as quantitative measures of the commonness between the two different types of... 

Which equation to use in an analysis is for all practical purposes moot since all methods produce equivalent BSA estimates. Bailey and Briars (1996) asked the question Why do all these formulas produce equivalent results when their constant terms [i.e., c, a1 and a2 in Eq. (8.4)] are so different They first analyzed the 401 subjects Gehan and George used to produce their equation and noted that height and weight were highly correlated. Ln-transformed height was a linear function of Ln-transformed weight... 

Ultrasonic correlation analysis in frequency (Fourier transformation of frequency) domain analysis was utilized to measure a thickness of the sample and to image the structure of the material. This technique comprises four processes (1) calculation of the spectrum, (2) division by the power spectrum of a pulse or other component, (3) Fourier transformation into the frequency domain, and (4) analysis and imaging in the frequency domain. Here we obtain much higher resolution in the imaging and thickness measurements by applying the echo analysis developed in earthquake theory [12] and the thickness measurements methods for a thin layer [13,14]. 

Natural bond orbital (NBO) analysis The NBO analysis transforms the canonical delocalized Hartree-Fock (HF) MOs and non-orthogonal atomic orbitals (AOs) into the sets of localized natural atomic orbitals (NAOs), hybrid orbitals (NHOs), and bond orbital (NBOs). Each of these localized basis sets is complete, orthonormal, and describes the wavefunction with the minimal amount of filled orbitals in the most rapidly convergent fashion. Filled NBOs describe the hypothetical, strictly localized Lewis structure. NPA charge assignments based on NBO analysis correlate well with empirical charge measures. ... 

The introduction of the generalized 2D correlation method [8] has made it possible to further extend the utility of 2D correlation analysis beyond the study of simple sinusoidally varying dynamic spectral signals as originally described in Eqs. (1-2) and (1-3). By utilizing the time-domain Fourier transform of spectral intensity variations, a formal definition of 2D correlation spectra resembling Eq. (1-9) can be... 

Application of 2D correlation analysis effectively sorted out the spectral intensity variation of HDPE induced even well below its T. A sequence of the event induced by the increasing temperature was detected to show an apparent additional step in the transformation between the crystals and the amorphous component, namely, the possible partial fusion of the less ordered small crystals of HDPE induced below its was suggested from the patterns appearing in the 2D correlation maps. 

The cross correlation function given in Equation (F6) may be further modified to a form better suited for the two-dimensional correlation analysis with the help of the Wiener-Khintchine theorem [5]. This theorem conveniently relates the cross correlation function with the corresponding Fourier transforms. In the first step, the expression for the dynamic spectrum y(v2, t -I- t) in Equation (F6) is rewritten in terms of the inverse of Fourier transform of y(v2, s )-... 

Empirical linear correlation analysis has also produced results that may be used in predicting the rates of FeS-mediated pollutant transformation based on pollutant thermodynamic or molecular properties. One study (77) performed linear correlation analysis of log kohs values for 8 halogenated aliphatic molecules with five molecular properties one-electron reduction potentials, lowest unoccupied molecular orbital (LUMO) energies, fi ee energies of formation of aqueous phase radicals formed upon one-electron reduction, gas-phase homo-[ytic bond dissociation enthalpies, and aqueous solubilities. Of these parameters, homolytic bond dissociation enthalpies (7)r.x values) were best correlated with log obs values for FeS reductive dechlorination (R =0.82). The correlation between log obs and 7)r x is illustrated in Figure 4. Another parameter shown... 

I Principal Component Analysis (PCA) transforms a number of correlated variables into a smaller number of uncorrelated variables, the so-called principal components. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

A more complete analysis of interacting molecules would examine all of the involved MOs in a similar wty. A correlation diagram would be constructed to determine which reactant orbital is transformed into wfiich product orbital. Reactions which permit smooth transformation of the reactant orbitals to product orbitals without intervention of high-energy transition states or intermediates can be identified in this way. If no such transformation is possible, a much higher activation energy is likely since the absence of a smooth transformation implies that bonds must be broken before they can be reformed. This treatment is more complete than the frontier orbital treatment because it focuses attention not only on the reactants but also on the products. We will describe this method of analysis in more detail in Chapter 11. The qualitative approach that has been described here is a useful and simple wty to apply MO theory to reactivity problems, and we will employ it in subsequent chapters to problems in reactivity that are best described in MO terms. I... 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

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