WO1997006493A1

WO1997006493A1 - Process for simulating especially triangular or trapezoidal relevance functions in the transformation of a fuzzy system into a neuronal network

Info

Publication number: WO1997006493A1
Application number: PCT/DE1996/001400
Authority: WO
Inventors: Claudia Müller; Thomas Wolf
Original assignee: Siemens Aktiengesellschaft
Priority date: 1995-08-07
Filing date: 1996-07-26
Publication date: 1997-02-20
Also published as: DE19528984C1

Abstract

In the process of the invention, sigmoid functions (1, 2) are multiplicatively (3, π) linked in the fuzzification component to simulate especially triangular or trapezoidal relevance functions in the transformation of a fuzzy system into a neuronal network. Product neurons (π) are advantageously used for this purpose. The sigmoid functions (1, 2) are preferably each given a set weighting before the multiplicative linking is undertaken, with the weightings advantageously having a value of 1. The invention has the advantage that no undefined negative fuzzy quantities occur and the transformation precision of the neuron structure of the invention is comparable to that of prior art neurofuzzy structures. A further advantage is that learning processes generally take place more rapidly with the multiplicative linking of the invention than in a neuro-fuzzy structure using additively linked sigmoid neurons.

Description

description

Method for emulating, in particular, triangular or trapezoidal membership functions when transforming a fuzzy system into a neural network

Neuro-fuzzy systems represent a combination of fuzzy systems and neural networks. The disadvantages of fuzzy systems and of neural networks can be compensated for in this way. One possibility for such a combination is to transform a fuzzy system into a neural network in order to make the system "learnable", ie self-optimizing. The complex optimization of the fuzzy system can then be automated by optimization algorithms, which can be executed with the neuro-fuzzy system with the aid of a computer.

Various methods are known for converting the components of a fuzzy system into the structures of a neural network. As is well known, a fuzzy system consists of the three components "fuzzification", "set of rules" and "defuzzification". All three can be mapped with certain types of neurons. The basic structure of a neuro-fuzzy system, ie the individual components of a fuzzy system. Systems within a neuro-fuzzy network are shown in FIG. 1.

Either "z" -shaped, "s" -shaped, trapezoidal or triangular membership functions are used for the "fuzzification". They are mapped into a neural network using sigmoid or Gaussian functions. In the example in FIG. 1, symbols are shown symbolically as circles Neurons with sigmoid functions S are used. At the edge of the value range, "z" - or "s" -shaped membership functions can be approximate function. Triangular or trapezoidal membership functions can be simulated by superimposing two sigmoid functions. The superposition is realized by adding a positive and a negative counting sigmoid function. In the example in FIG. 1, the sigmoid function 1 rated with (+1) is combined with the sigmoid function 2 rated with (-1) with a sum neuron 3 to form a trapezoidal membership function y. This is explained in more detail below using the example of FIGS. 2 to 4. Easier, but with the limitation of

Degrees of freedom, trapezoidal membership functions can also be represented with a Gaussian function.

In the fuzzy "set of rules", the values "small", "medium" and "large" provided by the "fuzzification" are linked to one another. This is usually done with AND links, which in practice are provided by a Minimum operator whose direct transmission into a neuron of a neuro-fuzzy network, however, creates difficulties in so-called "learning", ie when fuzzy weight parameters μ and σ are found, in which a stable final state of the system is achieved with an overall error that is as minimal as possible. The reason for this is that the mini mum operator cannot be differentiated in a closed manner. Instead of the minimum operator, when a fuzzy system is mapped into a neural network, the product is often used in the "rule set" component. This has the advantage that it has a smooth transfer function and is continuously differentiable. The product function can be performed directly can be carried out in a neuron and does not prepare one during "learning"

Trouble. In the example in FIG. 1, product neurons π are also used in the "set of rules". Finally, the output variable y is determined in the component "defuzzification" of the fuzzy system by means of a focus method. If the so-called singleton method is used as the focus method, the calculation in the component "defuzzification" is reduced to a normalized one

Sum of the signals from the product neurons π of the "set of rules". This normalized sum can also be transferred to a neuron, which is denoted by ∑ ⁿ in the example in FIG. 1. The abscissa of the singletons are expressed as weights W1. ..W9 the connections between the product neurons π and the output neuron ∑ are saved.

A large number of neural structures are used to represent fuzzy systems.

If, according to a first variant, only sigmoid functions are used for the fuzzification, the basic structure shown in FIG. 1 results for a neuro-fuzzy network. Such structures are known for example from DE 42 09 746 AI. With this structure, the membership functions can be mapped relatively precisely into a neural network. Both “z”, “s”, and trapezoidal as well as triangular functions can be transferred into a neural structure with a minimal error. However, there is a disadvantage in the additive linking of two sigmoid functions, which has already been described above using the example of the sigmoid functions 1, 2 and the sum neuron 3 in FIG. 1. In the so-called “learning process”, the weight parameters μ and σ for the sigmoid functions are varied in the “fuzzification” component. The weight parameter μ determines the position and the weight parameter σ the shape, ie the slope of the increase or decrease of the respective sigmoid function. With this variation of the weight parameters, the two sigmoid functions 1 and 2 of FIG. 1 can be such shift against each other so that interpretation difficulties arise with the relevant fuzzy set y. If the negatively incoming sigmoid function 2 is pushed in front of the positively counting sigmoid function 1, a so-called “negative” fuzzy set y arises. On the one hand, this can result in the learning process “failing”, namely because the system not "converges". No set of weighting factors can then be found which guarantees a stable final state of the system and in which the total error of the system is minimal or even converges towards zero. Fuzzy set created a gap in interpretation, which makes it impossible to transform the neuro-fuzzy system back into a fuzzy system.

With this variant, a fuzzy system can be mapped most precisely into a neuro-fuzzy system. The disadvantage of this structure is, however, that during a learning process it must be prevented that the two sigmoid functions, as shown in FIG. 4, shift too much against each other.

In the diagram in FIG. 2, it is assumed, for example, that the curve shown in a solid line and the left flank of the hatched field corresponds to the sigmoid function 1 rated with +1 and the curve shown in broken line corresponds to the sigmoid function 2 rated in FIG. 1 with -1. The area under the membership function y generated by the sum neuron 3 and similar to a trapezoidal shape is hatched in FIG. 2. On the basis of the variation of the weighting factors described, the sigmoid functions can be pushed towards one another, which is represented by arrows in FIG. 2. This reduces the membership function y. Such a case is shown in FIG. 3, in which the function y, due to the close approximation of the two sigmoid functions, is a greatly reduced, approximately triangular has taken formal form. If the sigmoid functions overlap due to a further shift, an undefined negative fuzzy set occurs. Such a case is shown in Figure 4.

According to a second variant, this problem can be avoided by using Gaussian transfer functions to simulate trapezoidal fuzzy sets. However, this has the other disadvantage that it limits the type of functions that can be represented to symmetrical trapezoids. Different steepnesses of the flanks of membership functions cannot be simulated with Gaussian functions. If crooked trapezoids occurring in a learning process are simulated by Gaussian functions, this causes an increased transformation error.

According to a third variant, edge fuzzy sets with Gaussian neurons are also shown. In this way, a network with uniform membership functions is created again if the functions lying outside the definition area are not taken into account. In the example in FIG. 5, part of the left of the two Gaussian functions lies outside the definition range. If these functions are shifted in the direction of the arrows shown in FIG. 5 by varying the weighting factors during the learning process, definition gaps can now arise at the area edges, since Gauss neurons only act in a limited area. This is shown in Figure 6.

In the third variant, two Gaussian neurons are linked to one product neuron to form the rule set. If this linkage is summarized in an equation and transformed accordingly, one is also formed into a Gaussian neuron two inputs identical neuron structure, if

is. According to a fourth variant, "fuzzification" and "set of rules" can therefore be combined in a Gaussian neuron. This type of neuro-fuzzy system is also referred to as RBF networks (Radial Basis Function - networks). This significantly simplifies network calculations and learning. However, restrictions also occur again, since only radially symmetrical areas can be detected with RBF networks. As a result, the transformation error increases and the reverse transformation is made more difficult.

The accuracy of the transformation from the fuzzy system to the neural network decreases from the first to the fourth variant. In the fourth version, the neural structure is the easiest to use, but it is subject to the greatest restrictions.

In contrast, the invention is based on the object of specifying a method for emulating, in particular, triangular or trapezoidal membership functions, in which, in particular, the definition gaps which occur in the variants shown above are avoided when transforming a fuzzy system into a neural network.

The object is achieved with the method of claim 1.

Advantageous further embodiments are specified in the subclaims.

The invention has the first advantage that all degrees of freedom are retained when a fuzzy system is transformed into a neural network. This has the consequence that the mapping of a membership function having the shape of an oblique trapezoid is also possible without restriction. Another advantage is seen in the fact that when using Weight parameters μ and σ from the entire permissible range of values, fuzzy sets y that can always be interpreted occur during a learning process, that is, undefined negative fuzzy sets are avoided. Another advantage of linking sigmoid functions according to the invention is that a stable, converging system always occurs during a learning process. The overall error of the transformation is small or converges to zero. Finally, there is the advantage that the duration of a learning process is at least comparable to the variants described above and, in some cases, significantly shorter, in order to achieve a sufficiently error-minimal state, ie considerably fewer learning steps are required.

The invention is explained on the basis of the figures briefly listed below and partly already described above. It shows

Fig.l. the individual components of a fuzzy system within a neuro-fuzzy network,

Fig. 2. the superposition of a positive and a negative sigmoid function to an approximately trapezoidal membership function,

Fig. 3. a shift in the sigmoid functions of FIG. 2 caused by variation of the weight factors, the functions being moved towards one another while reducing the result function,

Fig. 4. a shift of the sigmoid functions of FIG. 2 caused by further variation of the weight factors, so that a so-called “negative” fuzzy set occurs, Fig. 5. a possibility of simulating edge fuzzy sets with Gaussian neurons,

Fig. 6. the shift in the Gaussian functions caused by variation of the weight factors

FIG. 5 shows definition gaps at the edges of the area,

Fig. 7. for example the result of the mapping of a theoretically ideal trapezoidal membership function by means of a multiplicative combination of two sigmoid functions according to the invention,

Fig. 8. 2 shows a circuit diagram of the neurons involved in the multiplicative method according to the invention,

Fig. 9. comparable to FIG. 2, the superposition of two sigmoid functions to an approximately trapezoidal membership function using the multiplicative method according to the invention,

Fig. 10. comparable to FIG. 3, a shift in the sigmoid functions of FIG. 9 caused by variation of the weight factors,

Fig. 11. comparable to FIG. 4, such a shift of the sigmoid functions of FIG. 9 caused by further variation of the weight factors, such that a fuzzy set with the permissible size “zero” occurs,

Fig. 12. the course of the error of the overall system, ie its decrease with an increase in the number of learning steps, using the example of approximation of a polynomial using the multiplicative method according to the invention. According to the invention, the left and right flanks of a trapezoidal or triangular membership function are approximated by a sigmoid function and linked to a so-called “product neuron”. FIG. 7 shows, by way of example, the result of the mapping of a theoretically ideal trapezoidal membership function by multiplicative The associated circuit diagram of the neurons involved is shown in Figure 8. The signals from the two sigmoid neurons 1, 2, which are used according to the invention at this point, are preferably supplied with values of fixed weight values of +1, shifting the two neurons now leads to the fact that the membership function becomes zero. Negative fuzzy sets can no longer occur.

In particular, in comparison to the first variant described above, the neuron structure according to the invention has the advantage that the problem of negative fuzzy sets no longer exists and the neuron structure nevertheless has the same transformation accuracy.

Thus, in FIGS. 9 to 11, comparable to the previously explained FIGS. 2 to 4, the mutual displacement of two multiplicatively linked sigmoid functions is shown. According to FIG. 9, these emulate an approximately trapezoidal membership function. A shift according to FIG. 10 does not lead to a negative fuzzy set even if the two sigmoid functions overlap extremely according to FIG. The result function y is then zero.

The multiplicative method according to the invention converged with suitable learning parameters for all cases occurring in practice. The learning is characterized above all by a uniform convergence. When approximating a linear surface and a parabola surface, the error in the play particularly fast. FIG. 12 shows the course of the error of the overall system, ie the decrease in the error as the number of learning steps increases, using the example of the approximation of a polynomial. The approximation of a trapezoid results in a particularly uniform learning process.

It is particularly advantageous that learning processes with the multiplicative linkage according to the invention run in most cases faster than with a neuro-fuzzy structure with additively linked sigmoid neurons. The multiplicative method has the advantage that no problems occur during learning, but the imaging accuracy is as good as that achieved with the standard neuro-fuzzy structure. Sigmoid neurons, which together represent a trapezoidal membership function, can be shifted against one another as desired without any problems of interpretation.

In contrast, e.g. in the first variant described above, the learning time is generally longer than in the multiplicative method according to the invention. In addition, learning with unfavorable parameters or difficult approximation problems actually results in a shift in the membership functions, so that negative membership sets arise. The learning algorithm thus continues to calculate, mostly diverges or at least the error of the overall system fluctuates, so that in itself no meaningful interpretation of the learning results is possible.

Claims

1. A method for emulating particular triangular or trapezoidal membership functions in the transformation of a fuzzy system into a neural network, with sigmoid functions (1,2) being multiplicatively linked (3, π) in the fuzzification component.

2. Method according to Claim 1, product neurons (π) being used for the multiplicative linking.

3. The method according to claim 1 or 2, wherein the sigmoid functions (1, 2) are each evaluated with a fixed weight value before the multiplicative linkage is carried out.

4th The method of claim 3, wherein the fixed weight values are each 1 in size.