Fuzzy modeling examples

For fuzzy modeling or for comparison of fuzzy functions, the fundamentals of fuzzy arithmetics are needed (cf. Figure 8.25). These fundamentals are given, for example, in references [19] or [20]. Applications are known for calibration of analytical methods and for qualitative and quantitative comparison of chromatograms, spectra, or depth profiles. 

Now, we present a small example to show the simplicity of the described fuzzy modeling method. 

Measured non-fuzzy data is one of the primary inputs for the fuzzy logic models. Examples are temperature measurements by thermometers, rainfall by rain-gages, groundwater levels by sounders, etc. Additionally, humans with their fuzzy perceptions could also provide inputs with linguistic statements. 

The Figure 3 shows the example of Training fuzzy model using the three hsted above fuzzy variables as inputs. The Table 2 presents the trapezoidal membership functions parameters (a,b,c,d) of input variables while the trapezoidal membership functions parameters of output variable are shown in Table 3. 

With these models the behavior of a variable in time can be investigated. Also other types of model are possible. Examples are so-called experimental models, or black-box models, such as fuzzy models or neural network models. TTie design of these models proceeds using slightly different sub-steps, which will be discussed later. 

Sometimes, linear techniques can be used to describe non-linear process behavior. An example is a fuzzy model, discussed in chapter 28, which is a combination of local linear models in distinct operating areas. Developing a non-linear model requires much insight and understanding of the developer as to what mechanism underlies the observed data. Application of empirical techniques for modeling non-linear process behavior has therefore become very popular, such as the application of neural networks, described in chapter 27. 

If the process conditions vary over a wide range, there may be a need for a non-linear empirical model. In case of a dynamic non-linear model there are a few possibilities for developing such a model, for example a dynamic neural network or a dynamic fuzzy model. One could also develop a Wiener model, in which the process dynamics are represented by a hnear model, such as a state space model. The static characteristics of the process are then modeled by a polynomial, able to represent the non-linearity. 

In this section three examples fuzzy models will be discussed. 

Neuro fuzzy modeling is a useful technique that combines the advantages of neural networks and fuzzy inference systems. In this approach, the fuzzy model is architecturally the same as a neural network. In this case one could use, for example error back-propagation to train the network to find the parameters of the fuzzy model. The most well-known method is the so-called ANFIS method the Adaptive-Network based Fuzzy Inference System. The method will be explained in this chapter and several examples will be developed as an illustration. 

The number of parameters in the fuzzy sub-models is quite large. One rule of the fuzzy model for the net growth rate, for example, contains about 10 parameters, depending on the type of membership function that is used. Owing to the curse of dimensionahty the number of parameters increases exponentially for systems with higher dimensions. Therefore, only the consequence parameters have been optimized. 

Chapter 5 is structured as follows. Since there exists no direct cause-and-effect relationship between SCM and a company s revenues the following section gives a concise overview of the relevant literature (Chapter 5.1). In the next step a logistics customer service-revenue curve is derived (Chapter 5.2) which determinants are calculated in the next step by a fuzzy-based q)proach (Chapters 5.3). The relevant results from the developed fuzzy model are presented in a numerical example from the consumer goods industry (Chapter 5.4). Finally, in Chapter 5.5 a discussion based on previous experience is offered. The presentation of the determination of the revenue contribution of SCM ends with a short summary in Chapter 5.6. 

The introduced model was improved in a stepwise fashion by conducting exploratory in-depth interviews. Subsequently, the results of one numerical example that was created with an international producer of consumer goods are presented. The in-depth interviews were held in November 2008 in Basel (Switzerland). The two interviewees came from the company s sales and logistics department. In practice, the interviews consisted of three meetings. First, the concept of the fuzzy model was presented and a potential SCI was identified. Second, the adapted questionnaire was completed (cf. questionnaire in Appendix A). Third, the results were evaluated and suggestions for improvement of the model were collected. 

So far, we have considered the QSAR modeling of continuous biological data, that is, where the toxicity value is a number such as an LD50. However, some data are not continuous but are binary (e.g., toxic/ nontoxic) a common example is carcinogenicity, for which test results are almost invariably reported in this way. Clearly, one cannot perform, say, MLR on such classification data (although a method called fuzzy adaptive least squares [70] can be used). A number of methods are available for the modeling of classification data. 

FIGURE 6.18 Cluster validity V(k), see Equation 6.13, for the algorithms fc-means, fuzzy c-means, and model-based clustering with varying number of clusters. The left picture is the result for the example used in Figure 6.8 (three spherical clusters), the right picture results from the analysis of the data from Figure 6.9 (two elliptical clusters and one spherical cluster). 

Examples of nonhierarchical clustering [22] methods include Gaussian mixture models, means, and fuzzy C means. They can be subdivided into hard and soft clustering methods. Hard classification methods such as means assign pixels to membership of only one cluster whereas soft classifications such as fuzzy C means assign degrees of fractional membership in each cluster. 

This section provides an overview of common methods for quantitative uncertainty analysis of inputs to models and the associated impact on model outputs. Furthermore, consideration is given to methods for analysis of both variability and uncertainty. In practice, commonly used methods for quantification of variability, uncertainty or both are typically based on numerical simulation methods, such as Monte Carlo simulation or Latin hypercube sampling. However, there are other techniques that can be applied to the analysis of uncertainty, some of which are non-probabilistic. Examples of these are interval analysis and fuzzy methods. The latter are briefly reviewed. Since probabilistic methods are commonly used in practice, these methods receive more detailed treatment here. The use of quantitative methods for variability and uncertainty is consistent with, or informed by, the key hallmarks of data... 

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