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7.5.4. Sensor fusion via fuzzy neuronsA broad category of problems of sensor fusion can be formulated as follows. Consider a collection of n senors that provides signals about a possible fault in a system. Each sensor has its own characteristics. For purpose of this study we can assume that the higher the signal of the sensor, the more profound the level of anticipated fault. As the importance of each sensor could be different, we associate with the indication of the sensor a certain level of confidence. Taking this into account, a simple model of sensor fusion can be realized in the form of a single AND neuron in which the connections of the neuron are utilized to model the credibility levels associated with the corresponding sensors. More interesting and far more realistic scenarios emerge when we are interested in incorporating eventual relationships between the indications of the sensors. For instance, consider three sensors whose indications x1, x2, and x3 are fused to infer information about the level of fault (f) of the system. Furthermore we know that there is some relationship between the first and second sensor in the sense that usually the indications of the first one are lower than those reported by the second sensor. This constraint of inclusion relationship has to be accommodated as a part of the model. Note that when x1 is higher than x2 even though both of them are high and should suggest a substantial level of failure, the violation of the satisfaction of the dependency between the evidence furnished by the sensors should lead to a much lower level of the failure - we simply cannot accept their signals as fully credible. These logic specifications can be easily added the model of sensor fusion by adding an extra dominance neuron whose output serves as an additional input to the AND neuron, see Fig. 7.30.
7.6. ConclusionsThe study has been devoted to a broad class of fuzzy neural networks. The underlying processing units embrace high computational flexibility characteristics for artificial neurons while retaining an explicit format of represented domain knowledge (as being driven by triangular norms of highly pronounced logical nature). We have discussed learning in fuzzy neural networks and emphasized the role of domain knowledge in shaping the topology of the architectures, therefore substantially reducing training time. The examples originating from several areas of application illustrate the design procedure. 7.7. Problems7.1. Derive a formula describing the fuzzy neural network with feedback constructed with the use of referential neurons as shown below (assume that the referential neurons complete a disjunctive aggregation). Specify the triangular norms as the product operation and probabilistic sum. Determine a steady state of the network for k → ∞. Simulate the network in software; experiment with some other triangular norms. Try to draw some general conclusions regarding the choice of the triangular norms. How does the temporal response of the network depend upon the initial state x(0)?
7.2. The eigen fuzzy set A of a fuzzy relation R : X × X → [0,1], card (X) = n, satisfies the relationship In the case of the max-min composition (°), (there exists an algorithm that determines A in a finite number of steps. Its generalizations for any s-t composition of A and R is not available. Formulate the problem of finding eigen fuzzy set(s) as a certain optimization task, propose an appropriate fuzzy neural network and develop a complete learning algorithm. Note that an empty set A is a trivial solution to the problem - exercise caution when initiating your learning. Experiment with the network using different learning rates and initial conditions. 7.3. Possibility and necessity measures are two commonly used vehicles used to compare fuzzy quantities (sets) A and B, Poss(A,B), Nec(A,B). Assume that both A and B are defined in a finite universe of discourse (space) X = {x1, x2, , xn}.
7.4. The fuzzy neural network is supposed to model decision regions in the given two-class classification problem, Fig. 7.32. Propose the pertinent architecture of this neural network.
7.5. Determine (plot) characteristics of logic neurons for some other triangular norms (such as, e.g., Lukasiewicz connectives). 7.6. In decision-making we are instantaneously faced with problems of an inherently logical character. For instance, assume that a decision to be made depends on three factors (criteria): energy efficiency, reliability, and accuracy. Assume that all of them are quantified in [0,1] where 1 connotes the highest level of the corresponding criterion. What would be the neuron appropriate to handle this problem? Identify its inputs and output. Justify your selection. If the criterion of efficiency is far more important than that of reliability, what are the values of the corresponding c connections of the neuron?
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