Logarithmic transformations

Transformed Axis Linear Presentation Transformation Logarithmized Presentation... 

Many kinetic equations can be suitably linearized to the form of Eq. (20). For example, Eq. (1) can be transformed logarithmically, or Eq. (2) can be transformed reciprocally. Two equations proposed for describing pentane-isomerization data (Cl, Jl) are the single site... 

For reasons to be explained below, all retention factors k were transformed logarithmically to In k. Due to the design used, it is possible to check for nonlinear mixing behavior of the mobile phase, i.e. nonlinear variation of In k with composition. Such behavior is indeed present the mean In k of toluene on Cl at mobile phase wml and wal is 1.38, corresponding to a k value of 3.98. The In k value of toluene on Cl at mobile phase ami (the 0.5 0.5 mixture of wml and wal) is 1.10, corresponding to a k value of 3.01. This is a clear difference, which indicates that nonlinearities are present. Hence, there are at least three sources of variation in the mobile-phase mode the relative amounts of the two organic modifiers and nonlinear mixing behavior. 

K) is the Fourier transform of the logarithmic fluctuation of acoustics impedance. 

Because of the double sound path involved in PE measurements of the back wall echo, we approximate the corresponding attenuation at a certain frequency to be twice as large as the attenuation that would be obtained by an ordinary TT measurement. We propose to use the logarithm of the absolute value of the Fourier transform of the back wall echo as input data, i.e... 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Multivariate data analysis usually starts with generating a set of spectra and the corresponding chemical structures as a result of a spectrum similarity search in a spectrum database. The peak data are transformed into a set of spectral features and the chemical structures are encoded into molecular descriptors [80]. A spectral feature is a property that can be automatically computed from a mass spectrum. Typical spectral features are the peak intensity at a particular mass/charge value, or logarithmic intensity ratios. The goal of transformation of peak data into spectral features is to obtain descriptors of spectral properties that are more suitable than the original peak list data. 

When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be described by the data points. From the scatter plot we can take a basic guess as to which type of curve will best describe the X—Y relationship. To aid in the decision process, it is helpful to obtain scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear relationship, the equation has the form of number 7. To facilitate this we frequently employ special graph paper for which one or both scales are calibrated logarithmically. These are referred to as semilog or log-log graph paper, respectively. 

By taking the logarithm of both sides of Eq. (9.34), the Mark-Houwink equation is transformed into the equation of a straight line ... 

The nonlinear constant in these equations cannot be evaluated dkecdy by the methods previously described. Even forms such as these can be handled, however. For example, subtracting a trial value of a fromjy and taking logarithms transforms equation 97 into the linear form ... 

The assumption that the energy can be written as a sum of terms implies that the partition function can be written as a product of terms. As the enthalpy and entropy contributions involve taking the logarithm of q, the product thus transforms into sums of enthalpy and entropy contributions. 

We multiply each term by 0 4343 to transform natural to common logarithms, and obtain finally ... 

This approach is equivalent to the maximum a posteriori (MAP) approach derived by Wallner (Wallner, 1983). The position of the maximum is unchanged by a monotonic transformation and hence further simplification can be achieved by taking the logarithm of Eq. 8... 

Figure 1.17. The 95% confidence intervals for v and Xmean are depicted. The curves were plotted using the approximations given in Section 5.1.2 the /-axis was logarithmically transformed for a better overview. Note that solid curves are plotted as if the number of degrees of freedom could assume any positive value this was done to show the trend / is always a positive integer. The ordinates are scaled in units of the standard deviation.

Figure 1.26. Confidence limits of the standard deviation for p = 0.05 and/- 1. .. 100. The /-axis is logarithmically transformed for a better overview. For example, at n = 4, the true value Ox is expected between 0.62 and 2.92 times the experimental Sx. The ordinate is scaled...

Linearization is here defined as one or more transformations applied to the X- and/or y-coordinates in order to obtain a linear y vs. x relationship for easier statistical treatment. One of the more common transformations is the logarithmic one it will nicely serve to illustrate some pitfalls. 

Error bars defined by the confidence limits CL(y,) will shrink or expand, most likely in an asymmetric manner. Since we here presuppose near absence of error from the abscissa values, this point applies only to y-transformations. A numerical example is 17 1 ( 5.9%, symmetric CL), upon logarithmic transformation becomes 1.23045 -0.02633. .. 1.23045 + 0.02482. 

Figure 2.20. Logarithmic transformations on x- ory-axes as used to linearize data. Notice how the confidence limits change in an asymmetric fashion. In the top row, the y-axis is transformed in the middle row, the x-axis is transformed in the bottom row, both axes are transformed simultaneously.

Both aspects are combined in Fig. (2.20) and Table 2.16, where the linear coordinates are x resp. y, the logarithmic ones , resp. v. Regression coefficients established for the lin/lin plot are a, b, whereas those for the transformed coordinates are p, q. 

LEGEND NN linear regression without data transformation LL idem, using logarithmically transformed axes not interpretable... 

Procedure Use the algorithm given below because of the symmetry of the function, only the part 0 < CP < 0.5 is defined cumulative probability values CP lower than 0.5 are transformed to their (decadic) logarithm, the others are first subtracted from 1.00. The sign is appropriately set to 1 or +1. 

E. A GAUSSIAN WITH A LOGARITHMICALLY EQUIDISTANT GRID We consider now (C.l) but with the transformation... 

The best known approach to measurements with positive skewness is transformation. In environmental data analysis, the measurements are often transformed to their logarithms. In this paper, we consider power transformations with a shift, a set of transformations that includes the log transformation and no transformation at all ( ). These transformations are given by... 

For the analysis, we developed a new method that makes it possible to observe correlated potentials between two trapped particles. The principle is shown in Figure 7.5. From the recorded position fluctuations of individual particles (indicated by the subscripts 1 and 2), histograms are obtained as a function of the three-dimensional position. Since the particle motion is caused by thermal energy, the three-dimensional potential proflle can be determined from the position histogram by a simple logarithmic transformation of the Boltzmarm distribution. Similarly, the... 

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