Accountability, fuzzy

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. 

One of the sources of the fuzziness surrounding these concepts may well be the implicit assumption in structure-activity relationship (SAR) studies that molecular structure contains (i.e. encodes) the information on the biological activity of a given compound. Such an assumption cannot be incorrect, since this would imply the fallacy of SAR studies. However, the assumption becomes misleading if not properly qualified to the effect that the molecular structure of a given compound contains only part of the information on its bioactivity. Indeed, what the structure of a compound encodes is information about the molecular features accounting... 

In order to reason using fuzzy data, a way must be found to express rules so that the degree of certainty in knowledge can be taken into account and a level of certainty can be ascribed to conclusions. This is done through fuzzy rules. A fuzzy rule has the form ... 

Values of B calculated from the ordinate intercepts are shown in Fig. 23 as a plot of B/(2q)3 against the number of the Kuhn segments N. For N<4, the data points for the indicated systems almost fall on the solid curve which is calculated by Eq. (78) along with Eqs. (43), (51), (52), and Cr = 0. A few points around N 1 slightly deviate downward from the curve. Marked deviations of data points from the dotted lines for the thin rod limit, obtained from Eq. (78) with Le = L and de = 0, are due to chain flexibility the effect is appreciable even at N as small as 0.5. The good lit of the solid curve to the data points (at N 4) proves that the effect of chain flexibility on r 0 has been properly taken into account by the fuzzy cylinder model. 

Further, ions are not hard, billiard ball like spheres. Since the wave functions that describe the electronic distribution in an atom or ion do not suddenly drop to zero amplitude at some particular radius, we must consider the surfaces of our supposedly spherical ions to be somewhat fuzzy. A more subtle complication is that the apparent radius of an ion increases (typically by some 6 pm for each increment) whenever the coordination number increases. Shannon10 has compiled a comprehensive set of ionic radii that take this into account. Selected Shannon-type ionic radii are given in Appendix F these are based on a radius for O2- of 140 pm for six coordination, which is close to the traditionally accepted value, whereas Shannon takes the reference value as 126 pm on the grounds that it gives more realistic ionic sizes. For most purposes, this distinction does not mat-... 

The fuzzy classification method groups objects into categories without defined boundaries so as to take into account the degree of similarity of the considered object with respect to the others (Du and Sun, 2004). 

Thus, multilinear models were introduced, and then a wide series of tools, such as nonlinear models, including artificial neural networks, fuzzy logic, Bayesian models, and expert systems. A number of reviews deal with the different techniques [4-6]. Mathematical techniques have also been used to keep into account the high number (up to several thousands) of chemical descriptors and fragments that can be used for modeling purposes, with the problem of increase in noise and lack of statistical robustness. Also in this case, linear and nonlinear methods have been used, such as principal component analysis (PCA) and genetic algorithms (GA) [6]. 

If the simultaneous presence of all functional groups Fj, F2,. .. F ,. .. Fm within molecule X is taken into account, then a new fuzzy membership function M-Fi,x(r) for points r of the space belonging to functional group Fj of molecule X can be defined as... 

The ranking of chiral systems such as the foregoing by their structures and associated properties is intuitively obvious, and is arrived at independently of any theory that accounts for these properties. For this reason we are led to the conclusion that chirality in a real system is a primitive fuzzy concept. ... 

Note that the versions of fuzzy sets taken into account in the set /(A, B,.) of definition (70) can be restricted to translated versions only. In this case the proof follows the same steps as before, and the translation-restricted fgp (A,B) scaled fuzzy Hausdorff-type metric is obtained. Alternatively, the allowed rotations can be confined to some angle interval A a, leading to another scaled fuzzy Hausdorff-type metric /op,tr,Aa(" Furthermore, if in addition to translated and rotated versions, reflected versions are also included among the versions in the set /(B .), then one obtains a new version of scaled fuzzy Hausdorff-type metric, f p (A, B). For these metrics, the following general relations hold ... 

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. 

In this work a fuzzy matching procedure is suggested which takes the foregoing uncertainties into account at least in principle. The approach is based on a soft definition of a surface that is defined in terms of membership functions (see Fig. 8) These functions are and, .(r) and they measure to what extent a given space point belongs to the surface and the bulk of a molecule, respectively. The matching of two molecules A and B can then be calculated in many different ways. In a first attempt we used the intersection of two fuzzy sets... 

Figure 17 shows the present-day model of the atom, which takes into account both the particle and wave properties of electrons. According to this model, electrons are located in orbitals, regions around a nucleus that correspond to specific energy levels. Orbitals are regions where electrons are likely to be found. Orbitals are sometimes called electron clouds because they do not have sharp boundaries. When an orbital is drawn, it shows where electrons are most likely to be. Because electrons can be in other places, the orbital has a fuzzy boundary like a cloud. 

The simplest method of accounting for variability consists in identifying the parameters, experimental data, and model output with an average or reference individual [45], One shortcoming of this approach is that although general model behavior is representative, much of the actual population may not be well described. Methods attempting to explicitly account for variability and uncertainty include Monte Carlo simulations [18,46,48], Bayesian population methods [49-51,53,56], the use of fuzzy sets [44,52], and other probability based methods [43], Monte Carlo simulations, which model the parameter variability in terms of probability distributions, are the most common methods. Each individual is characterized by a set of parameters whose values are drawn from a... 

Three cases with one or more objectives from maximization of net present value (NPV) and optimizing two other criteria (1) production delay/advance and (2) flexibility criteria. Multi-Objective GA (MOGA) A fuzzy approach was proposed to account for uncertain demand in the optimization of batch plant design for multiple objectives. Dietz et al. (2007)... 

The second aspect is the appearance, derived by the face, the clothes, and the behavior of the person. Everything that makes a person friendly to you is included in the decision process. You will automatically rank the facts for instance, the eyes might be more important to you than the hair. If all patterns are evaluated and ranked, you will summarize the outcomes. The next step is to include a certain probability (e.g., maybe the person did not smile at you but at another person) and to account for certain fuzziness (e.g., you still cannot be sure whether there was an intent behind the smile). Finally, you will conclude with a decision as to whether to turn around or not. The inference mechanism is again something that has been trained and is affected by experience Maybe you will not turn around because you had a bad experience with a similar situation before. External facts are also accounted for (e.g., you do not have the time to talk to the person). All of these influences have to be covered if we want to create a computer system that mimics human reasoning and decision making. 

Accuracy checks are generally performed by comparing measured data with data from certified reference materials. When measured data are not accurate because of relative or systematic errors, or a lack of precision (noise), the comparison between measured data and reference values cannot lead to any useful conclusion in an expert system. To process larger sets of potential source data for knowledge bases, a method must be used that takes inaccuracies as well as natural fuzziness of experimental data into account — ideally automatically and without the help of an expert. 

The term fuzzy set was first introduced by Zadeh (1965). This is the most adequate model for a description of nondefinitive situations which do not possess sharp boundaries. The mathematical model of the conflict situation, characterized in that the parameters of the empirical interaction potential model XA obtained on the basis of one set of properties do not correspond to the parameters XB obtained from another set of properties, may be represented as a set X of the alternative subsets of parameters with their fuzzy subsets, which map the unsharply formulated criteria i.e., as the system (X, f, fa,. . . , fR, L). In the framework of this problem, one needs to construct the following function in order to account for all possible criteria ... 

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