Fuzzy modeling data clustering

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). 

FIGURE 6.20 Cluster validities for the Hyptis data for two to nine clusters analyzed with the methods fc-means clustering, fuzzy clustering, and model-based clustering. For the left plot the original data were used, for the right plot the data were autoscaled. 

What should be done then is to determine regions or operating ranges in which linear relationships can be developed. This is called data clustering, sometimes called fuzzy clustering, it will be discussed in a following section. The region for which the model will be valid corresponds to the premise part of the model. 

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. 

For the identification three data sets and two validation sets were available, which all have different initial bottom holdup and compositions. The fuzzy model will again be identified using fuzzy clustering, for which input-output data are required. 

Hert J, Willett P, Wilton DJ, Acklin P, Azzaoui K, Jacoby E, Schuffenhauer A(2006)New methods for ligand-based virtual screening use of data-fusion and machine-learning techniques to enhance the effectiveness of similarity searching. J Chem Inf Model 46 462-470 Miyamoto S (1990) Fuzzy sets in information retrieval and cluster analysis. Kluwer Academic, Dordrecht... 

Fuzzy clustering will be used to build a model of the net growth rate as a fimction of the substrate concentration Cg and biomass concentration Cx for the hybrid model of the bioreactor. Assume that an input-output data set has been created using all identification experiments and that the Pl-estimator data are available in the form of measurements. This means that input-output data are available of Cs and Cx and estimates of ju and n . The data set can be presented directly to the clustering algorithm. The initial number of clusters was ten. This number was reduced to three by the merging algorithm. The next step is to project the fuzzy partition matrix onto the cs and cx axis, so that parametric membership fimctions can be determined. This is shown in Fig. 30.9. 

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