Membership fuzzy observations

For applications we have first to specify universes and fuzzy sets in them. The universe X that is to be specified could be the wavelength, energy or a product yield. The uncertainty of the appearance of the variables could be modeled by a fuzzy set A. As a special case of a fuzzy set we will regard a fuzzy measurement or fuzzy observation. Here ix(x) indicates the degree by which x is to be examined as a result of our actual measurement. The membership function that should be used depends on the application problem. The following practical hints can be given, cf. Otto ... 

Partitioning methods make a crisp or hard assignment of each object to exactly one cluster. In contrast, fuzzy clustering allows for a fuzzy assignment meaning that an observation is not assigned to exclusively one cluster but at some part to all clusters. This fuzzy assignment is expressed by membership coefficients m(/ for each... 

In order to combine the conclusions of several such Fuzzy Inference Systems, they will be translated into belief structures, according to the method proposed in [22]. Each Fuzzy Inference System presented Figure 12 is the association of fuzzification functions (providing a numerical evaluation of the membership of a variable to fuzzy sets) and of rules linking these observations to different classes which can be states or disjunctions of states of the process. 

Most real-world classes do not have sharp boundaries. Between the complete membership or nonmembership of an object in a class we may observe an infinity of intermediate situations. These intermediate situations are said to be fuzzy. Fuzzy set theory as developed by Zadeh permits an object to belong to a cluster with a grade of membership that lies within the interval [0,1]. A class (or a cluster) of objects may be represented as a fuzzy set. Because real-world classes are more fuzzy than... 

The architecture of an ANFIS model is shown in Figure 14.4. As can be seen, the proposed neuro-fuzzy model in ANFIS is a multilayer neural network-based fuzzy system, which has a total of five layers. The input (layer 1) and output (layer 5) nodes represent the descriptors and the response, respectively. Layer 2 is the fuzzification layer in which each node represents a membership. In the hidden layers, there are nodes functioning as membership functions (MFs) and rules. This eliminates the disadvantage of a normal NN, which is difficult for an observer to understand or to modify. The detailed description of ANFIS architecture is given elsewhere (31). 

One of the early applications of fuzzy theory in analytical chemistry was shown in the modeling of straight-line relationship of responses (y) versus concentration (x) and nonlinear cubic spline calibration. Relying on the fuzzy approach, the error in observations was modeled by assigning membership degrees to the set of possible observations. A/,. Subsequently, the predicted concentration values were valued by a membership function without introducing probability based assumptions. The error in the observations was taken into account... 

Zadeh has observed that P( ) can be viewed as the expected value of the characteristic function that defines the set E (Zadeh (1987)). By analogy, he defines the probability of the fuzzy set A as the expected value of the membership function for A ... 

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