![]() |
|
|||
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|
![]() |
[an error occurred while processing this directive]
7.2. Logic neurons and fuzzy neural networks with feedbackThe logic neurons studied so far realize a static memoryless nonlinear mapping in which the output solely depends upon the inputs of the neuron. In this form, the neurons are not capable of handling dynamical (memory-based) relationships between the inputs and outputs. This aspect might be, however, essential in a proper problem description. As an example, consider the following diagnostic problem. A decision about a systems failure should be issued while one of the systems sensors provides information about an abnormal (elevated) temperature. Observe that the duration of the phenomenon has a primordial impact on expressing the confidence about the failure. If the elevation of the temperature prolongs, the confidence about the failure rises up. On the other hand, some short temporary temperature elevations (spikes) reported by the sensor might be almost ignored (filtered out) and need not to have any impact on the decision about failure. To properly capture this dynamical effect, one has to equip the basic logic neuron with a certain feedback link. A straightforward extension of this nature is schematically illustrated in Fig. 7.9.
The type of neuron with feedback is described accordingly The dynamics of the neuron obtained in this fashion is uniquely defined by the feedback connection (a) as it determines a speed of evidence accumulation x(k). The initial condition x(0) expresses a priori confidence associated with x. After a sufficiently long period of time x(k+1) could take on higher vales in comparison to the level of the original evidence being present at the input. High-order dynamical dependencies to be accommodated by the network, if necessary, have to be taken care of via a feedback loop consolidating several pieces of a temporal information, e.g., One can also refer to the two above relationships as examples of fuzzy difference equations. The limit analysis of the neuron (or network) with feedback allows us to reveal some general dynamical properties of the system. As an example let us study the neuron described by the relationship where a characterizes the feedback loop, a ∈ [0, 1]. Additionally, let the s-norm be given as the probabilistic sum. This yields, Iterating from x(0), that is computing x(1), x(2) , we can unveil the steady-state behavior of the system (assuming that it does exist). The above example is simple enough to analyze it in detail. Let us denote the steady state by x(∞). Then we get Rearranging the terms one obtains and finally As the simulations reveal, Fig. 7.10, for the connection in (0, 1) the neuron does not converge to any specific value but oscillates continuously assuming the states between 0 and 1. In fact, the value x(∞) becomes an average of the states in which the neuron can reside. The same observation holds for the remaining values of the feedback parameter except the fact that lower values of a yield a more profound amplitude of the oscillations.
Let us now study the neuron with the feedback governed by the expression Confining ourselves to the probabilistic sum we derive The steady-state type of analysis leads to the following expression In this case the neuron converges to x(∞) as confirmed by the simulation experiments, Fig. 7.11. Here the speed of convergence depends on the feedback connection.
7.3. Referential logic-based neuronsIn comparison to the AND, OR and OR/AND neurons realizing operations of the aggregative character, the class of neurons discussed now is useful in realizing reference computations. The main idea behind this structure is that the input signals are not directly aggregated as has been done in the aggregative neuron but rather they are analyzed first (e.g., compared) with respect to the given reference point. The results of this analysis (involving such operations as matching, inclusion, difference, dominance) are summarized afterwards in the aggregative part of the neuron as has been described before. In general one can describe the reference neuron as (a disjunctive form of aggregation) or (that is a conjunctive form of aggregation) where the term REF(.) stands for the reference operation carried out with respect to the provided point of reference. Depending on the reference operation, the functional behavior of the neuron is described accordingly (all the formulas below pertain to the disjunctive form of aggregation),
Copyright © CRC Press LLC
![]() |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |