Spectral logarithm

In Figure 3 we see how the logarithm of the spectral amplitude effects the estimation results. For each component in input data vector, u, we have defined the feature relevance, Fn d), as... 

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. 

Fig. 13. Characteristics of a 50-)J.m long DFB laser, (a) Light-current properties, (b) spectral intensity plotted on a logarithmic scale to better illustrate...

It should be noted that when we compare the brightness of a LGS to a NGS, the result depends on the spectral bandwidth, because the LGS is a line source, whereas the NGS is a continuum one. The magnitude scale is a logarithmic measure of flux per spectral interval (see Ch. 15). This means that a (flat) continuum source has a fixed magnitude, no matter how wide the filter is. In contrast, the magnitude of a line source is smaller for narrower bandpasses. It is therefore advisable to use the equivalent magnitude only for qualitative arguments. The photon flux should be used in careful system analyses. 

The transformation by log double-centering has received various names among which spectral mapping [13], logarithmic analysis [14], saturated RC association model [15], log-bilinear model [16] and spectral map analysis or SMA for short [17]. 

Each of the three approaches will be applied in this section to the transformed retention times of the 23 chalcones with eight chromatographic elution methods in Table 31.2. The transformation is defined by the successive operations of logarithms, double-centering and global normalization which is typical for the method of spectral map analysis (SMA) ... 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

The spectral map shows three distinct poles of specificity. These are respectively the p-receptor (DHM), the 6-receptor (DADLE) and the K-receptor (EKC). The naloxone receptor (NAL) appears to be strongly correlated with the p-receptor (DHM) and, hence, provides little additional information. In spectral map analysis, correlation between variables, as well as similarity between compounds, is evidenced by the proximity of their corresponding symbols. The lines drawn through the three poles of the map represent bipolar axes of contrast. A contrast is defined in this context as a log ratio or, equivalently, as a difference of logarithms. For example, the horizontal axis through the p- and 8-receptors defines the p/6 contrast. Compounds that project on the right side of this axis bind more specifically to the p-receptor, while those that project on the left side possess more... 

Fig. 12. Snapshot from a two-phase DNS of colliding particles in an originally fully developed turbulent flow of liquid in a periodic 3-D box with spectral forcing of the turbulence. The particles (in blue) have been plotted at their position and are intersected by the plane of view. The arrows denote the instantaneous flow field, the colors relate to the logarithmic value of the nondimensional rate of energy dissipation.

X0 is the value of the property in the gas phase. (In practice, X and X0 are often the logarithm of the property in question.) The parameters a and p are measures of a solvent s ability to donate and accept hydrogen bonds, respectively, and tt is an index of its polarity/polarizability. They were initially assigned on the basis of ultraviolet spectral shifts of certain dyes in a variety of solvents, and hence were labeled solvatochromic parameters.186"188... 

Spectral changes in the skeleton are observed when the base is changed into the conjugate acid, i.e., they are clearly observable at acidities of the medium which are some two logarithmic units of acidity greater than that required for half-protonation of the base (> 99% protonation). In the protonation equilibrium of a base B ... 

Figure 1. Logarithm of the spectral density vs. logarithm of the frequency, o>. The lines are the fit of the spectral density for the various models to relaxation...

Fig. 3.3. Theoretical Hertzsprung-Russell diagram. The right-hand scale shows in absolute bolometric magnitude what the left-hand scale expresses as the logarithm of the intrinsic luminosity in units of the solar intrinsic luminosity (Lq = 4 x 10 erg s ). On the horizontal axis, the logarithm of the effective temperature, i.e. the temperature of the equivalent blackbody, is put into correspondence with the spectral type of the star, as determined by the observer. This temperature-luminosity diagram shows the lifelines of the stars as strands combed out like hair across the graph. With a suitable interpretation, i.e. viewed through the explanatory machinery of nuclear physics, it opens the way to an understanding of stellar evolution and its twin science of nucleosynthesis. (Courtesy of Andre Maeder and co-workers.)...

Figure 10.4 Typical time-dependent spectral changes observed for the reaction of cluster [W3S4H3(dmpe)3]PFg with HCl in a CH2CI2 solution at 25.0° C. The data were recorded for 1000 s with a logarithmic time base. (Reproduced with permission from ref. 9.)...

A kind of logarithmic transform, such as In (1 -I- x), is used in spectral maps within row and column centring and global standardization (division by the standard deviation around the mean of all the values of the data matrix). 

Mie calculations with the optical constants of water given in Fig. 10.3 are shown in Fig. 11.3 extinction and absorption are plotted logarithmically, photon energy linearly. The bulk absorption coefficient of water is shown in Fig. 11.3 c. Because many of the extinction features of water and MgO, both of which are insulators, are similar, we present calculations for a single water droplet (in air) with radius 1.0 jam. Size-dependent spectral feature.4 are therefore not obscured as they are for a distribution of radii. 

The calculated extinction spectrum of a polydispersion of small aluminum spheres (mean radius 0.01 jam, fractional standard deviation 0.15) is shown in Fig. 11.4 both scales are logarithmic. In some ways spectral extinction by metallic particles is less interesting than that by insulating particles, such as those discussed in the preceding two sections. The free-electron contribution to absorption in metals, which dominates other absorption bands, extends from radio to far-ultraviolet frequencies. Hence, extinction features in the transparent region of insulating particles, such as ripple and interference structure, are suppressed in metallic particles because of their inherent opacity. But extinction by metallic particles is not without its interesting aspects. 

Absorbance An index of the light absorbed by a medium compared to the light transmitted through it. Numerically it is the logarithm of the ratio of incident spectral irradiance to the transmitted spectral irradiance. 

The terms cepstrum and cepstral come from inverting the first half of the words spectrum and spectral they were coined because often in cepstral analysis one treats data in the frequency domain as though it were in the time domain, and vice versa. The value of cepstral analysis comes from the observation that the logarithm of the power spectrum of a signal consisting of two echoes has an additive periodic component due to the presence of the two echoes, and therefore the Fourier transform of the logarithm of the power spectrum exhibits a peak at the time interval between them. The... 

A wide range of transformations can be applied to spectral data before they are analyzed. The main purpose of transformations is to make the latent variables better available for powerful analysis. One of the most widely used is logarithmic transformation, which is especially useful to make skewed variables more symmetrically... 

Fig. 3.2. The (smoothed) spectral functions derived from the measurements, Fig. 3.1, of the normalized binary absorption coefficients of helium-argon, neon-argon and argon-krypton mixtures at room temperature in a semi-logarithmic grid, Eq. 3.2.

Fig. 3.51. Logarithmic plot of the normalized induced dipole moment correlation function, C(t), for hydrogen-argon mixtures at 165 K. Measurements at 90 amagat ( ) 450 amagat ( ) and 650 amagat (o). The broken lines at small times represents the portion of C(t) affected by the smoothing of the wings of the spectral profiles. Reproduced with permission by the National Research Council of Canada from [109].

Spectral distribution of blackbody radiation. The family of curves is called the Planck distribution after Max Planck, who derived the law governing blackbody radiation. Note that both axes are logarithmic. 

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