Fuzzy functional form

NMR, x-ray, Fuzzy Functional Form Technology, bioassay development Integrated Object-oriented PharmacoEngineering (IOPE)... 

In addition to conventional sequence motifs (Prosite, BLOCKS, PRINTS, etc.), the compilation of structural motifs indicative of specific functions from known structures has been proposed [268]. This should improve even the results obtained with multiple (one-dimensional sequence) patterns exploited in the BLOCKS and PRINTS databases. Recently, the use of models to define approximate structural motifs (sometimes called fuzzy functional forms, FFFs [269]) has been put forward to construct a library of such motifs enhancing the range of applicability of motif searches at the price of reduced sensitivity and specificity. Such approaches are supported by the fact that, often, active sites of proteins necessary for specific functions are much more conserved than the overall protein structure (e.g. bacterial and eukaryotic serine proteases), such that an inexact model could have a partly accurately conserved part responsible for function. As the structural genomics projects produce a more and more comprehensive picture of the structure space with representatives for all major protein folds and with the improved homology search methods linking the related sequences and structures to such representatives, comprehensive libraries of highly discriminative structural motifs are envisionable. 

Geneformatics 1999 San Diego, CA Fuzzy functional form modeling for identifying active sites www.genefor matics.com... 

Unfortunately, such methods require the exact placement of atoms within protein side chains and are inapplicable to the inexact, low-resolution predicted structures generated by the state-of-the-art ab initio folding and threading algorithms (see Sections IV-VI). These methods are required when the sequence identity of the sequence of interest to solved structures is too low to use comparative modeling. To address this need, Skolnick and Fetrow have recently developed fuzzy, inexact descriptors of protein functional sites [8]. They are applicable to both high-resolution, experimental structures and low-resolution (backbone RMSD 4—6 A from native) structures. These descriptors are a-carbon-based, fuzzy functional forms (FFFs). Initially, they created FFFs for the disulfide oxidoreductase [8,10] and a/p-hydrolase catalytic active sites [11] (an additional 198 have now been bmlt, with comparable results [234]). 

Recently, a new approach called artificial neural networks (ANNs) is assisting engineers and scientists in their assessment of fuzzy information, Polymer scientists often face a situation where the rules governing the particular system are unknown or difficult to use. It also frequently becomes an arduous task to develop functional forms/empirical equations to describe a phenomena. Most of these complexities can be overcome with an ANN approach because of its ability to build an internal model based solely on the exposure in a training environment. Fault tolerance of ANNs has been found to be very advantageous in physical property predictions of polymers. This chapter presents a few such cases where the authors have successfully implemented an ANN-based approach for purpose of empirical modeling. These are not exhaustive by any means. 

The one-to-one correspondence between the NMR signals and the carbon atoms and the assignment of possibility values through the membership function to each signal transforms the crisp set of vertices V from definition (3) of the graph into a fuzzy set forming a fuzzy molecular graph. 

Fuzzy logic control calculations are executed by using both membership functions of the inputs and outputs and a set of rules called a rule base, as shown in Fig. 16.21. Typical membership functions for the inputs, e and deldt, are shown in Fig. 16.23, where it is assumed that these inputs have identical membership functions with the following characteristics three linguistic variables which are negative (N), positive P), and zero (Z) with trapezoidal, triangular and trapezoidal membership function forms respectively. Input variables e and deldt have been scaled so that the membership functions overlap for the range from -1 to +1. Furthermore, Fig. 16.24 shows the membership functions of the output Aw(r), which are... 

Like all early expert systems, DENDRAL and SHRDLU required exact knowledge to function. The way that expert systems work depends on whether the knowledge that they manipulate is exact ("The temperature is 86°C") or vague ("The temperature is high"). We shall first consider how an ES can use exact knowledge to provide advice. Methods for dealing with ill-defined information form the topic for the next chapter, which covers fuzzy logic. 

The relationships described form the concept (Fig. 5.68) of the FUZZY RULE SET. The membership-functions of the linguistic variables are derived from the experiments that have already been mentioned however, the plausibilities of the RULE SET have not been touched on so far. 

This fuzzy membership function Pxi,L(r) can be written in another form ... 

This fuzzy membership function ppi,x(r) can also be written in a form of a simple density ratio ... 

Two distinct notations are most commonly employed in the literature to denote membership functions. In one of them, the membership function of a fuzzy set A is denoted by p. and its form is... 

Membership functions of any fuzzy binary relation on X Y have the form... 

The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability that several uncertainties must be combined using the rules of probability and that the calculus of probabilities is adequate to handle all situations involving uncertainty. Probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate... that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability, can better be done with probability. 

The mapping of T from the subdomain of temperatures onto the interval [0, 1] is the membership or compatibility function fx T). The form of this function is subject to some relatively nonarbitrary constraints to be consistent with one s primitive notions of hotness, fx T) should increase monotonically and smoothly with T, and it should be roughly sigmoidal in shape. However, since a whole family of curves will fit this description, the choice of numerical values for the parameters of fi(T) will to a considerable extent be an arbitrary one. Therefore, the threshold criteria, which are given by fi(T), are themselves fuzzy. 

Take three fuzzy sets A, B, and C and their a-cuts G (a), Gg(a), and G ia), respectively, for each membership function value a. Assume that the a-cuts GJ.a), Gg(a and G(-(a) depend at least piecewise continuously on the a parameter from the unit interval [0,1], where the intervals of continuity have nonzero lengths and where continuity is understood within the metric topology of the underlying space X. For the three pairs formed from these three fuzzy sets, the ordinary Hausdorff distances h(G (a),Gg(a)), h(Gg(a), Gcia)), and h GJ,a ... 

The definition of a graph given in Eq. (3) shows that it is formed of two sets a set of vertices V and a set of edges E. This definition can be formulated in the context of fuzzy set theory with both sets V and E represented by fuzzy sets (respectively and E to constitute the fuzzy topological graph G = (V E ). This means that each vertex e E and each edge e E may be associated with the membership functions and which map these two sets on the range of real values... 

Each new bond (i,/) created at the current level is thus associated with a fuzzy number m ij). Equation (11) defines a membership function associated with every new bond formed during the generation process. Precautions have been taken in the program so that the term l/R will not introduce an error in those cases where the. / ,yS approach zero (which hardly ever occurs). 

The elassification entropy may be interpreted as measuring the ambiguity assoeiated with a fuzzy partition. A good cluster identification would be indicated by an entropy value close to 0. Various transformations of the validity functionals C and E have been proposed to mitigate their monotonic tendencies. Here we consider a normalization that reduces the validity functional s range of variation to [0,1]. Normalized expressions of C and E assume the forms... 

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