Dimensionality of Analytical Signals and Information

The dimensionality of a functional relationship will be defined here (ax-iomatically) by the number of dependent and independent variables in such a function. Therefore, functions of the type a = f(b) are two-dimensional, of the type a = f(bi,b2) three-dimensional, and of the type a = /(fri, b2y... bn) are (n + 1 )-dimensional. The representation of various realizations of only one variable (either y or z) is one-dimensional (Danzer et al. [2002]). 

Dimensionality in analytical chemistry comprises different types of dimensions displayed by different sorts of variables, which can be discrete or continuous  

As a result of analytical measurements, signals are obtained and, in the case of instrumental measurements, signals functions, y = f(z). The record of the signal intensity as a function of the signal position, Fig. 3.8, represents a two-dimensional signal function which can be back-transformed into two-dimensional analytical information, x = /(Q). 

The abscissa in Fig. 3.8 may represent an energy-related scale, e.g., wavelength-, frequency-, or mass/charge coordinates of spectrometers or re- 

Quasi-multidimensional information as shown in Fig. 3.9 (right side) can be obtained in different ways, mostly by sequential measurements, for example  

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Alternative characterization of the dimensionality. The SIMS example demonstrates that the dimensionality of analytical information and of signal functions occasionally follow other principles than those given above, where the dimensionality of a functional relationship is determined by the number of dependent and independent variables in such a function. 

Multispecies analyses require two-dimensional analytical information y = f(x)y see Sect. 3.4, mostly in the form of spectra and chromatograms. By evaluation of various signals or the entire signal function, simultaneous information on several sample components can be obtained (in the extreme case on all the constituents contained in the sample). The relevant quantity that characterizes multicomponent analyses is the information amount,... 

In classical chromatographic methods (excluding hyphenated techniques), analytical signals are two dimensional. Each chromatographic peak may be characterized by two parameters area, which is proportional to the quantity of substance being eluted from column, and its position on the chromatogram (retention time, t, which reflects the interaction between the sorbate (analyte) and sorbent (stationary phase). This interaction is the principal source of information about the chemical nature and structure of the analytes. 

Fig. 3.9. One-dimensional information (in y-direction) of a single signal at a fixed z-value (left-hand side) and quasi-multidimensional information on several analytes AyBy...,N as a sequence of one-dimensional information (right-hand side)... 

Figure 1.3.A shows the scheme for another analytical information hierarchy that is complementary to the previous ones. Thus, gravimetries, titrimetries, classical qualitative analyses and sensors provide onedimensional information of the form F = where x is the signal concerned. On the other hand, instrumental techniques provide two-dimensional information that can be of two types depending on whether the signal (x) is combined with an instrumental parameter (y), time (f) or space (s). Some modem analytical techniques (several of which use hybrid instruments) furnish three-dimensional information by combining signals with one or two instrumental parameters (y, z), time and space. The great...

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