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7.2. Logic neurons and fuzzy neural networks with feedback

The 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 system’s failure should be issued while one of the system’s 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.


Figure 7.9  A logical neuron with feedback

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.


Figure 7.10  States of a fuzzy neuron for several values of the feedback connection; initial condition x(0) = 0.3

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.


Figure 7.11  States of a fuzzy neuron x(k+1) = a OR x(k) for several values of the feedback connection; initial condition x(0) = 0.3

7.3. Referential logic-based neurons

In 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),

(i)  MATCH neuron:

or equivalently

where r ∈ [0,1]n stands for a reference point defined in the unit hypercube. To emphasize the referential character of this processing carried out by the neuron one can rewrite the basic formula in the equivalent form

see also Fig. 7.12.


Figure 7.12  Referential neuron as a superposition of referential and aggregative computation

The use of the OR neuron indicates an “optimistic” (disjunctive) character of the final aggregation. The pessimistic form of this aggregation is produced by using the AND operation.

(ii)  difference neuron: The neuron combines degrees to which x is different from the given reference point g = [g1, g2, …, gn].The output is interpreted as a global level of difference observed between the input x and this reference point,

i.e.,

where the difference operator is defined as a complement of the equality index,

As before, the referential character of processing is emphasized by noting that

(iii)  the inclusion neuron summarizes the degrees of inclusion to which x is included in the reference point f,

(iv)  the dominance neuron expresses a relationship dual to that carried out by the inclusion neuron

where h is a reference point. In other words, the dominance relationship generates the degree to which x dominates h (or, equivalently, h is dominated by x). The coordinatewise notation of the neuron reads as


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