WO1997006493A1 - Procede de simulation de fonctions d'appartenance notamment triangulaires ou trapezoidales lors de la transformation d'un systeme a logique floue en un reseau neuronal - Google Patents

Procede de simulation de fonctions d'appartenance notamment triangulaires ou trapezoidales lors de la transformation d'un systeme a logique floue en un reseau neuronal Download PDF

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Publication number
WO1997006493A1
WO1997006493A1 PCT/DE1996/001400 DE9601400W WO9706493A1 WO 1997006493 A1 WO1997006493 A1 WO 1997006493A1 DE 9601400 W DE9601400 W DE 9601400W WO 9706493 A1 WO9706493 A1 WO 9706493A1
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Prior art keywords
functions
fuzzy
sigmoid
transformation
trapezoidal
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PCT/DE1996/001400
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German (de)
English (en)
Inventor
Claudia Müller
Thomas Wolf
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Siemens Aktiengesellschaft
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Publication of WO1997006493A1 publication Critical patent/WO1997006493A1/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/043—Architecture, e.g. interconnection topology based on fuzzy logic, fuzzy membership or fuzzy inference, e.g. adaptive neuro-fuzzy inference systems [ANFIS]

Definitions

  • 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.
  • 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.
  • 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
  • trapezoidal membership functions can also be represented with a Gaussian function.
  • the abscissa of the singletons are expressed as weights W1. ..W9 the connections between the product neurons ⁇ and the output neuron ⁇ are saved.
  • a large number of neural structures are used to represent fuzzy systems.
  • 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.
  • 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.
  • the weight parameters ⁇ and ⁇ for the sigmoid functions are varied in the “fuzzification” component.
  • the weight parameter ⁇ determines the position and the weight parameter ⁇ the shape, ie the slope of the increase or decrease of the respective sigmoid function.
  • 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.
  • this problem can be avoided by using Gaussian transfer functions to simulate trapezoidal fuzzy sets.
  • 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.
  • edge fuzzy sets with Gaussian neurons are also shown.
  • a network with uniform membership functions is created again if the functions lying outside the definition area are not taken into account.
  • 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.
  • the accuracy of the transformation from the fuzzy system to the neural network decreases from the first to the fourth variant.
  • the neural structure is the easiest to use, but it is subject to the greatest restrictions.
  • 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 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 ⁇ and ⁇ 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.
  • 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. 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. 12 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • the learning time is generally longer than in the multiplicative method according to the invention.
  • 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.

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Abstract

L'invention concerne un procédé selon lequel, lors de la transformation d'un système à logique floue en un réseau neuronal, des fonctions sigmoïdes (1, 2) sont combinées de manière multiplicative (3, π) dans la composante de mise en logique floue, pour simuler des fonctions d'appartenance notamment triangulaires ou trapézoïdales. On utilise à cet effet avantageusement des neurones-produits (π). Avant d'être combinées de manière multiplicative, les fonctions sigmoïdes (1, 2) sont toutes pondérées par une valeur de pondération fixe, les valeurs de pondération présentant de préférence une valeur de 1. L'invention présente l'avantage qu'aucune quantité de logique floue négative non définie n'intervient et que la structure neuronale offre une précision de transformation élevée, comparable à celle des structures de logique floue connues. Un autre avantage de l'invention réside dans le fait qu'avec le mode de combinaison multiplicative de l'invention, les processus d'apprentissage se déroulent de manière générale plus rapidement que dans une structure de logique floue neuronale où des neurones sigmoïdes sont combinés de manière additive.
PCT/DE1996/001400 1995-08-07 1996-07-26 Procede de simulation de fonctions d'appartenance notamment triangulaires ou trapezoidales lors de la transformation d'un systeme a logique floue en un reseau neuronal WO1997006493A1 (fr)

Applications Claiming Priority (2)

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DE19528984.6 1995-08-07
DE19528984A DE19528984C1 (de) 1995-08-07 1995-08-07 Verfahren zur Nachbildung von insbesondere dreieck- oder trapezförmigen Zugehörigkeitsfunktionen bei der Transformation eines Fuzzy-Systems in ein neuronales Netz

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106447031A (zh) * 2016-09-27 2017-02-22 西华大学 一种基于区间值模糊脉冲神经膜系统的故障诊断方法及装置

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Publication number Priority date Publication date Assignee Title
DE19703965C1 (de) * 1997-02-03 1999-05-12 Siemens Ag Verfahren zur Transformation einer zur Nachbildung eines technischen Prozesses dienenden Fuzzy-Logik in ein neuronales Netz
DE19703964C1 (de) * 1997-02-03 1998-10-15 Siemens Ag Verfahren zur Transformation einer zur Nachbildung eines technischen Prozesses dienenden Fuzzy-Logik in ein neuronales Netz

Citations (2)

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EP0460642A2 (fr) * 1990-06-06 1991-12-11 Hitachi, Ltd. Méthode et système pour l'ajustement des paramètres d'inférence floue
DE4209746A1 (de) * 1992-03-25 1993-09-30 Siemens Ag Verfahren zur Optimierung eines technischen Neuro-Fuzzy-Systems

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JP2747104B2 (ja) * 1990-10-22 1998-05-06 株式会社東芝 ニューラルネットワーク

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EP0460642A2 (fr) * 1990-06-06 1991-12-11 Hitachi, Ltd. Méthode et système pour l'ajustement des paramètres d'inférence floue
DE4209746A1 (de) * 1992-03-25 1993-09-30 Siemens Ag Verfahren zur Optimierung eines technischen Neuro-Fuzzy-Systems

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HASHIYAMA T ET AL: "A FUZZY NEURAL NETWORK FOR IDENTIFYING CHANGES OF DEGREES OF ATTENTION IN A MULTI-ATTRIBUTE DECISION MAKING PROCESS", PROCEEDINGS OF THE INTERNATIONAL JOINT CONFERENCE ON NEURAL NETWORK (IJCNN), NAGOYA, OCT. 25 - 29, 1993, vol. 1 OF 3, 25 October 1993 (1993-10-25), INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, pages 705 - 708, XP000499235 *
SHIN-ICHI HORIKAWA ET AL: "COMPOSITION METHODS OF FUZZY NEURAL NETWORKS", IECON90, 16TH ANNUAL CONFERENCE OF IEEE INDUSTRIAL ELECTRONICS SOCIETY, POWER ELECTRONICS, EMERGING TECHNOLOGIES, PACIFIC GROVE, NOV. 27 - 30, 1990, vol. 2, 27 November 1990 (1990-11-27), INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, pages 1253 - 1258, XP000216871 *

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN106447031A (zh) * 2016-09-27 2017-02-22 西华大学 一种基于区间值模糊脉冲神经膜系统的故障诊断方法及装置

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