Fuzziness

Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

All the three techniques mentioned above may make use of fuzzy sets and fuzzy logic (for fuzzy classification, fuzzy rules or fuzzy matching) but this does not effect the discussion of the applicability to NDT problems in the next section. 

Neuronal networks are nowadays predominantly applied in classification tasks. Here, three kind of networks are tested First the backpropagation network is used, due to the fact that it is the most robust and common network. The other two networks which are considered within this study have special adapted architectures for classification tasks. The Learning Vector Quantization (LVQ) Network consists of a neuronal structure that represents the LVQ learning strategy. The Fuzzy Adaptive Resonance Theory (Fuzzy-ART) network is a sophisticated network with a very complex structure but a high performance on classification tasks. Overviews on this extensive subject are given in [2] and [6]. 

The Fuzzy-ARTMAP network reaehes the best learning rate of the training data set. This is recognised perfectly with 100% correctness. The exact results are presented in the next table ... 

Table 3 Classifieation results of the Fuzzy-ARTMAP-Network in percent...

These tests generate several Gigabytes of data that are fed into a historical database. Although most of the analysis is performed automatically, human interaction is still needed to compare current and past data. Data are stored on optical CD S s from which the historical data bank are retrieved during field inspections from a mobile unit. Each of these is equipped with a CD-jukebox linked to an analysis station. The jukebox can handle 100 CD s, enough to store all previously recorded data. A dedicated software pre-fetches the historical data and compares it on-line with the newly acquired NDT-data. It is based on fuzzy algorithms applied to signal features. 

Two other atomic properties have been used in the definition of atom type, thereby increasing its fuzziness relative to that in the ap and tt descriptors - atomic log P contribution (yielding hydrophobic pairs, hps, and torsions, hts) and partial atomic charges (charge pairs, cps, and charge torsions, cts). 

Increasing the fuzziness of object description reduces the number of descriptors used and broadens the scope of a similarity search. At the same time, increasing fuzziness may reduce the discriminatory power of desaiptors to unacceptable levels. Therefore it is desirable to be able to control the degree of fuzziness of desaiptors. 

Asymmetric simhaiity measures allow fuzzy super- and substructure searching. A substructure search is defined as looking for structures containing the given query and a superstructure search is defined as looking for structures embedded in the given query. In both cases asymmetric local similarity is estimated. 

The atom pair (ap) and topological torsion (U) descriptors and their fuzzy binding property analogs bp and bt are again selected for illustrative purposes [24, 25]. 

To know about fuzzy sets and fuzzy logic... 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

Figure 9-25. Membership function for the fuzzy set of numbers close to 3.

On the other hand, if we want to characteri2e objects which are described by the rather fuzzy statement "numbers dose to three", we then need a membership function which describes the doseness to three. An adequate membership function could be the one plotted in Figure 9-25 m x) has its maximum value of m x) = 1 for value x = 3. The greater the distance from x to 3 gets, the smaller is the value of m x). until it reaches its minimum m x) = 0 if the distance from x to 3 is greater than say 2, thus for x > 5 or x < 1. 

An important property of a fuzzy set is its cardinality. While for crisp sets the cardinality is simply the number of elements in a set, the cardinality of a fuzzy set A, CardA, gives the sum of the values of the membership function of A, as in Eq. (9). 

Fuzzy logic and fuzzy set theory are applied to various problems in chemistry. The applications range from component identification and spectral Hbrary search to fuzzy pattern recognition or calibrations of analytical methods. 

An overview over different applications of fuzzy set theory and fuzzy logic is given in [15] (see also Chapter IX, Section 1.5 in the Handbook). 

Here, the application of fuzzy logic for multicomponent spectral analysis is described. 

If a spectrum lacks certain Lines or contains extra lines from additional unknown components, or if the true line positions are blurred, fuzzy set theory can improve the matching. 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

Fuzzy logic extends the Boolean logic so as to handle information about truth values which are between absolutely true and "absolutely false . 

F. Ehrentreich, Fuzzy Methods in Chemistry, in Encyclopedia of Computational Chemistry, P.v.R. Schleyer,... 

A series of monographs and correlation tables exist for the interpretation of vibrational spectra [52-55]. However, the relationship of frequency characteristics and structural features is rather complicated and the number of known correlations between IR spectra and structures is very large. In many cases, it is almost impossible to analyze a molecular structure without the aid of computational techniques. Existing approaches are mainly based on the interpretation of vibrational spectra by mathematical models, rule sets, and decision trees or fuzzy logic approaches. 

One variation of rule-based systems are fuzzy logic systems. These programs use statistical decision-making processes in which they can account for the fact that a specific piece of data has a certain chance of indicating a particular result. All these probabilities are combined in order predict a final answer. 

D. H. Rouvray, Fuzzy Loqic in Chemistry Academic Press, San Diego (1997). 

The way the chemist knows that she has methylamine and not ammonium chloride is that she compares the look of the two types of crystals. Ammonium chloride crystals that come from this reaction are white, tiny and fuzzy. The methylamine hydrochloride crystals are longer, more crystalline in nature and are a lot more sparkly. The chemist leaves the methylamine crystals in the Buchner funnel of the vacuum filtration apparatus and returns the filtrate to the distillation set up so it can be reduced one last time to afford a second crop. The combined methylamine hydrochloride filter cake is washed with a little chloroform, scraped into a beaker of hot ethanol and chilled. The methylamine hydrochloride that recrystallizes in the cold ethanol is vacuum filtered to afford clean, happy product (yield=50%). 

Video-Enhanced Contrast. This technique is more expensive but much more effective than any other contrast-enhancing technique (15). Since the 1970s, the development of video processing of microscopical images has resulted in electronic control of contrast. As Shinya InouH, author of a classic text in the field, states "We can now see objects that are far too thin to be resolved, and extract clear images from scenes that appeared too fuzzy, too pale, or too dim, or that appeared to be nothing but noise" (16). The depth of the in-focus field can now be expanded or confined, very thin but very sharp optical sections can be produced, and a vertical succession of these images can be accumulated to reconstmct thicker stmctures in three dimensions (16). 

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