Fuzzy modeling algorithm

In the last decades not only thousands of chemical descriptors but also many advanced, powerful modeling algorithms have been made available, The older QSAR models were linear equations with one or a few parameters. Then, other tools have been introduced, such as artificial neural network, fuzzy logic, and data mining algorithms, making possible non linear models and automatic generation of mathematical solutions. 

An unambiguous algorithm. The intent of this principle is to ensure the transparency of the modeling algorithm. Sometimes, it is a difficult task to satisfy this principle, particularly when complex methods like neural networks or fuzzy logic techniques are used for modeling. 

Parameter determination is accomplished by laboratory measurements or by a least-mean-squares method if reference data are available. With such methods, it is possible to use a current measurement with a low accuracy. This can reduce costs for a SoC measurement capability. Singh et al. [18] and Heinemann [19] describe SoC determination methods based on fuzzy logic. This approach relies on the use of expert knowledge in lieu of complex mathematical models. It is also possible to combine a fuzzy SoC algorithm with other methods to increase the accuracy. 

Ji-Chang L, Chien-Hsing Y (1999) A heuristic error-feedback learning algorithm for fuzzy modeling. IEEE Trans Syst Man Cybernet Part A Syst Humans 29(6) 686-691... 

This paper presents a Fuzzy Clustering of Fuzzy Rules Algorithm (FCFRA) that allows the automatic organisation of the sets of fuzzy IF. .. THEN rules of one fuzzy system in a Hierarchical Prioritised Structure. The proposed FCFRA algorithm has been successfully applied to the modelling of a nonlinear small scale Pilot Plant Reactor. 

The mathematical fundaments for possibilistic fuzzy clustering of fuzzy rules were presented. The P-FCAFR algorithm was used to organize the rules of the fuzzy model of the liquid level inside the Pilot Plant Reactor in the HPS structure. The partition matrix can be interpreted as containing the values of the relevance of the sets of rules in each cluster. This approach is currently showing its potential for modelling and identification tasks, particularly in the fault detection and compensation field. 

The fuzzy means algorithm, which is an efficient method for developing RBF network models for dynamic systems, has already been proposed in a recent publication (Sarimveis et al., 2002). Though this algorithm presents some remarkable advantages... 

In view of the linear form of the consequence part use in fuzzy models, an obvious choice for fuzzy clustering is the fuzzy C-varieties or Gustafson-Kessel algorithm, in which linear or planar clusters are allowed as prototypes to be sought. 

Sub-model identification. Fuzzy clustering provides a good way to identify the fuzzy sub-models. It is an unsupervised learning algorithm and requires little a priori model structure information. In addition, because of the stmcture optimization algorithm, it is insensitive to initialization. It also derives a fuzzy model with independent rules directly from the data, which results in models that are not likely to show anomalous extrapolation behavior. 

Although it is possible to optimize the fuzzy models for // and r sequentially, it is interesting to see how the large-scale algorithm deals with optimization of a large set of parameters in a hybrid model. Therefore, all the parameters of the two fuzzy models will be optimized simultaneously. The result is that a set of 66 parameters will be optimized 48 premise part parameters and 18 consequence part parameters. The premise part parameters are constrained the bounds are set at the initial values 10%. No constraints were placed on the consequence part parameters. The results of the optimization are shown in Fig. 30.11. (for clarity, only some of the measurements are shown). The anomalous behavior has been removed and model offset has been reduced to acceptable levels. 

The next step is to analyze the hybrid model to determine whether the quality requirements are met. The hybrid model will be analyzed with respect to static performance, dynamic performance, complexity, interpretability and process independence. The complexity of the fuzzy models is mainly determined by the number of rales and the rale base. The clustering algorithm in combination with structure optimization provides a way to minimize fuzzy model complexity, which yielded acceptable results. 

Cheng CT, Ou CP, Chau KW (2002) Combining a fuzzy optimal model with a genetic algorithm to solve multi-objective rainfall-runoff model calibration. J Hydrol 268 72-86... 

The solution for model-based clustering is based on the Expectation Maximi-zation (EM) algorithm. It uses the likelihood function and iterates between the expectation step (where the group memberships are estimated) and the maximization step (where the parameters are estimated). As a result, each object receives a membership to each cluster like in fuzzy clustering. The overall cluster result can be evaluated by the value of the negative likelihood function which should be as small as possible. This allows judging which model for the clusters is best suited (spherical clusters, elliptical clusters) and which number of clusters, k, is most appropriate. 

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). 

Domains may be regarded as the basic units for the structure, function and evolution of proteins, but the definition of a domain remains fuzzy. They are most often treated as compact or connected areas that are apparent from a visual inspection of protein models. To avoid subjectivity and ambiguities of visual inspection, computer algorithms have been developed to localize domains. Rashin offered an alternative interpretation domains are stable globular fragments, generated in biochemical experiments that refold autonomously and retain specific functions. He proposed a method for localiz-... 

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