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Local Linear Neuro-Fuzzy Models: Advanced Aspects

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Nonlinear System Identification

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  • The original version of this chapter was revised: This chapter was inadvertently published with an error in equation 14.40 in page 502 which has been corrected now. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-47439-3_30

Abstract

This chapter continues the discussion of local linear neuro-fuzzy models and extends them to local model networks where the local models can be arbitrary and the fuzzy logic interpretation vanishes in the background. A couple of attractive key features of these model architectures are analyzed in this chapter. Two of them shall be highlighted here: The separation of the inputs for rule premises and for the rule consequents (local models) allows to partly overcome the curse of dimensionality because it is often possible to choose the premise input space of low dimension even if the local models are high-dimensional. This feature also will play a crucial role when dealing with dynamic models in Part C. Another nice characteristic is due to the local nature of this model architecture. It allows for extremely robust online learning without unlearning in regions where no new data arrives. Furthermore, the extension to axis-oblique tree construction is discussed and analyzed in great detail. A new algorithm called hierarchical local model tree (HILOMOT) is introduced and compared to LOLIMOT. This opens the door to solving even higher-dimensional problems with this kind of model architectures. On the basis of HILOMOT, a new design of experiments/active learning scheme is proposed that exploits the advantages of the model structure to the fullest. It has been successfully applied to multiple real-world problems that are covered in Part D.

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Notes

  1. 1.

    Strictly speaking the interpolation region of the normalized Gaussians are infinitely large. However, for the degree of accuracy required in any practical consideration, it is virtually equivalent to the interpolation interval [1∕3, 2∕3] of the triangular validity functions.

  2. 2.

    The term “learning” is used if the model possesses a memory in the sense that it does not forget previously learned relationships when the operating conditions change. Thus, here “learning” implies “adaptive nonlinear” plus a mechanism against arbitrary forgetting, e.g., locality.

  3. 3.

    Note that this is not possible for the very first split since the root possesses no parent and it significantly increases the risk of converging to a local optimum.

  4. 4.

    Will be normalized according to (14.39) anyway.

  5. 5.

    Not exactly, see the paragraph about optimism in Sect. 7.2.3.

  6. 6.

    Otherwise a superior white box model could be used.

  7. 7.

    https://en.wikipedia.org/wiki/Optimal_design

  8. 8.

    The widths can be chosen individually for each input dimension but are identical for all kernels.

  9. 9.

    https://en.wikipedia.org/wiki/Active_learning_(machine_learning)

  10. 10.

    This is quite unlikely to happen but for a 1D example due to the suboptimal nature of incrementally building, the tree can occur sometimes.

  11. 11.

    https://en.wikipedia.org/wiki/Inverse-variance_weighting

References

  1. Abou El Ela, A.: Sensorarme Methoden zur Bearbeitung komplexer Werkstücke mit Industrierobotern. Ph.D. Thesis, TU Darmstadt, Reihe 8: Mess-, Steuerungs- und Regelungstechnik, Nr. 824. VDI-Verlag, Düsseldorf (2000)

    Google Scholar 

  2. Aggarwal, C.C., Hinneburg, A., Keim, D.A.: On the surprising behavior of distance metrics in high dimensional space. In: International Conference on Database Theory, pp. 420–434. Springer (2001)

    Google Scholar 

  3. Aleksovski, D., Kocijan, J., Dzeroski, S.: Model tree ensembles for the identification of multiple-output systems. In: Control Conference (ECC), 2014 European, pp. 750–755. IEEE (2014)

    Google Scholar 

  4. Alessio, A., Bemporad, A.: A survey on explicit model predictive control. In: Nonlinear Model Predictive Control, pp. 345–369. Springer (2009)

    Google Scholar 

  5. Atkeson, C.G., Moore, A.W., Schaal, S.: Locally weighted learning. Artif. Intell. Rev. 11(1–5), 11–73 (1997)

    Article  Google Scholar 

  6. Auer, P.: Using confidence bounds for exploitation-exploration trade-offs. J. Mach. Learn. Res. 3, 397–422 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Babuška, R.: Fuzzy Modeling and Identification. Ph.D. thesis, Dept. of Control Engineering, Delft University of Technology, Delft, The Netherlands (1996)

    Google Scholar 

  8. Babuška, R., Fantuzzi, C., Kaymak, U., Verbruggen, H.B.: Improved inference for Takagi-Sugeno models. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 701–706, New Orleans, USA (1996)

    Google Scholar 

  9. Babuška, R., Verbruggen, H.B.: An overview of fuzzy modeling for control. Control Eng. Pract. 4(11), 1593–1606 (1996)

    Article  Google Scholar 

  10. Babuška, R., Verbruggen, H.B.: Fuzzy set methods for local modelling and identification. In: Murray-Smith, R., Johansen, T.A. (eds.) Multiple Model Approaches to Modelling and Control, chapter 2, pp. 75–100. Taylor & Francis, London (1997)

    Google Scholar 

  11. Belz, J., Nelles, O.: Honda project: input selection with local linear models. Technical report, University of Siegen (2015)

    Google Scholar 

  12. Bemporad, A., Borrelli, F., Morari, M., et al.: Model predictive control based on linear programming˜ the explicit solution. IEEE Trans. Autom. Control 47(12), 1974–1985 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benner, M.J., Tushman, M.L.: Exploitation, exploration, and process management: the productivity dilemma revisited. Acad. Manag Rev. 28(2), 238–256 (2003)

    Article  Google Scholar 

  14. Biswas, P., Grieder, P., Löfberg, J., Morari, M.: A survey on stability analysis of discrete-time piecewise affine systems. IFAC Proc. 38(1), 283–294 (2005)

    Google Scholar 

  15. Brandimarte, P.: Low-discrepancy sequences. In: Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics, pp. 379–401 (2014)

    Google Scholar 

  16. Breiman, L.: Hinging hyperplanes for regression, classification, and function approximation. IEEE Trans. Inf. Theory 39(3), 999–1013 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Breiman, L.: Bagging predictors. Mach. Learn. 24(2), 123–140 (1996)

    Article  MATH  Google Scholar 

  18. Breiman, L.: Out-of-bag estimation. Technical report, Citeseer (1996)

    Google Scholar 

  19. Carpenter, G., Grossberg, S.: The ART of adaptive pattern recognition by a self-organizing neural network. IEEE Comput. 21(3), 77–88 (1988)

    Article  Google Scholar 

  20. Cohn, D.: Neural network exploration using optimal experiment design. In: Cowan, J.D., Tesauro, G., Alspector, J. (eds.) Advances in Neural Information Processing Systems, vol. 6, pp. 1071–1083. Morgan Kaufmann, San Francisco (1994)

    Google Scholar 

  21. Cohn, D., Atlas, L., Ladner, R.: Training connectionist networks with queries and selective sampling. In: Touretzky, D.S. (ed.) Advances in Neural Information Processing Systems, vol. 2. Morgan Kaufmann, San Mateo (1990)

    Google Scholar 

  22. Cohn, D., Ghahramani, Z., Jordan, M.I.: Active learning with mixture models. In: Murray-Smith, R., Johansen, T.A. (eds.) Multiple Model Approaches to Modelling and Control, chapter 6, pp. 167–184. Taylor & Francis, London (1997)

    Google Scholar 

  23. Didcock, N., Jakubek, S., Kögeler, H.-M.: Regularisation methods for neural network model averaging. Eng. Appl. Artif. Intell. 41, 128–138 (2015)

    Article  Google Scholar 

  24. Dietterich, T.G.: Ensemble methods in machine learning. In: International Workshop on Multiple Classifier Systems, pp. 1–15. Springer (2000)

    Google Scholar 

  25. Ebert, T., Nelles, O.: A Note on Analytical Gradient Calculation for Hilomot. Technical report, University of Siegen (2013)

    Google Scholar 

  26. Ebert, T., Fischer, T., Belz, J., Heinz, T.O., Kampmann, G., Nelles, O.: Extended deterministic local search algorithm for maximin Latin hypercube designs. In: 2015 IEEE Symposium Series on Computational Intelligence: IEEE Symposium on Computational Intelligence in Control and Automation (2015 IEEE CICA), Cape Town, South Africa (2015)

    Google Scholar 

  27. Ernst, S.: Hinging hyperplane trees for approximation and identification. In: IEEE Conference on Decision and Control (CDC), pp. 1261–1277, Tampa, USA (1998)

    Google Scholar 

  28. Ernst, S.: Nonlinear system identification with hinging hyperplane trees. In: European Congress on Intelligent Techniques and Soft Computing (EUFIT), pp. 659–663, Aachen, Germany (1998)

    Google Scholar 

  29. Fink, A., Fischer, M., Nelles, O.: Supervison of nonlinear adaptive controllers based on fuzzy models. In: IFAC World Congress, vol. Q, pp. 335–340, Beijing, China (1999)

    Google Scholar 

  30. Fischer, M., Nelles, O., Fink, A.: Adaptive fuzzy model-based control. Journal A 39(3), 22–28 (1998)

    MATH  Google Scholar 

  31. Fischer, M., Nelles, O., Isermann, R.: Exploiting prior knowledge in fuzzy model identification of a heat exchanger. In: IFAC Symposium on Artificial Intelligence in Real-Time Control (AIRTC), pp. 445–450, Kuala Lumpur, Malaysia (1997)

    Google Scholar 

  32. Fischer, T., Hartmann, B., Nelles, O.: Increasing the performance of a training algorithm for local model networks. In: World Congress of Engineering and Computer Science (WCECS), pp. 1104–1109, San Francisco, USA (2012)

    Google Scholar 

  33. Fortescue, T.R., Kershenbaum, L.S., Ydstie, B.E.: Implementation of self-tuning regulators with variable forgetting factor. Automatica 17, 831–835 (1981)

    Article  Google Scholar 

  34. Fritzke, B.: Fast learning with incremental radial basis function networks. Neural Process. Lett. 1(1), 2–5 (1994)

    Article  Google Scholar 

  35. Fritzke, B.: Growing cell structures: a self-organizing network for unsupervised and supervised learning. Neural Netw. 7(9), 1441–1460 (1994)

    Article  Google Scholar 

  36. Garulli, A., Paoletti, S., Vicino, A.: A survey on switched and piecewise affine system identification. IFAC Proc. 45(16), 344–355 (2012)

    Google Scholar 

  37. Golub, G., Pereyra, V.: Separable nonlinear least squares: the variable projection method and its applications. Inverse Prob. 19(2), R1 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Golub, G.H., Pereyra, V.: The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10(2), 413–432 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Haber, R.: Nonlinearity tests for dynamic processes. In: IFAC Symposium on Identification and System Parameter Estimation, pp. 409–414, York, UK (1985)

    Google Scholar 

  40. Hafner, M., Schüler, M., Isermann, R.: Fast neural networks for diesel engine control design. In: IFAC World Congress, Beijing, China (to appear in 1999)

    Google Scholar 

  41. Hafner, M., Schüler, M., Nelles, O.: Dynamical identification and control of combustion engine exhaust. In: American Control Conference (ACC), pp. 222–226, San Diego, USA (1999)

    Google Scholar 

  42. Hafner, M., Schüler, M., Nelles, O.: Neural net models for Diesel engines – simulation and exhaust optimization. In: European Congress on Intelligent Techniques and Soft Computing (EUFIT), vol. 1, pp. 215–219, Aachen, Germany (1998)

    Google Scholar 

  43. Halfmann, C., Nelles, O., Holzmann, H.: Semi-physical modeling of the vertical vehicle dynamics. In: American Control Conference (ACC), pp. 1707–1711, San Diego, USA (1999)

    Google Scholar 

  44. Hametner, C., Jakubek, S.: Neuro-fuzzy modelling using a logistic discriminant tree. In: 2007 American Control Conference, pp. 864–869. IEEE (2007)

    Google Scholar 

  45. Hametner, C., Jakubek, S.: Nonlinear system identification through local model approaches: partitioning strategies and parameter estimation. INTECH Open Access Publisher (2010)

    Google Scholar 

  46. Hametner, C., Jakubek, S.: Local model network identification for online engine modelling. Inf. Sci. 220, 210–225 (2013)

    Article  Google Scholar 

  47. Hametner, C., Stadlbauer, M., Deregnaucourt, M., Jakubek, S., Winsel, T.: Optimal experiment design based on local model networks and multilayer perceptron networks. Eng. Appl. Artif. Intell. 26(1), 251–261 (2013)

    Article  Google Scholar 

  48. Hartmann, B., Ebert, T., Nelles, O.: Model-based design of experiments based on local model networks for nonlinear processes with low noise levels. In: American Control Conference (ACC), pp. 5306–5311, 29 2011–July 1 2011 (2011)

    Google Scholar 

  49. Hartmann, B., Moll, J., Nelles, O., Fritzen, C.-P.: Hierarchical local model trees for design of experiments in the framework of ultrasonic structural health monitoring. In: IEEE International Conference on Control Applications (CCA), pp. 1163–1170. IEEE (2011)

    Google Scholar 

  50. Hartmann, B., Nelles, O.: Automatic adjustment of the transition between local models in a hierarchical structure identification algorithm. In: European Control Conference (ECC), Budapest, Hungary (2009)

    Google Scholar 

  51. Hartmann, B., Nelles, O.: Adaptive test planning for the calibration of combustion engines – methodology. In: Design of Experiments (DoE) in Engine Development, pp. 1–16, Berlin, Germany. Expert Verlag (2013)

    Google Scholar 

  52. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer Series in Statistics, 2nd edn. Springer, Berlin (2009)

    Google Scholar 

  53. Hathaway, R.J., Bezdek, J.C.: Switching regression models and fuzzy clustering. IEEE Trans. Fuzzy Syst. 1(3), 195–204 (1993)

    Article  Google Scholar 

  54. Heinz, T.O., Belz, J., Nelles, O.: Design of experiments – combining linear and nonlinear inputs. In: Hoffman, F., Hüllermeier, E., Mikut, R. (eds.) Proceedings 27. Workshop Computational Intelligence, pp. 211–226. KIT Scientific Publishing (2017)

    Google Scholar 

  55. Hu, C., Wan, F.: Input selection in learning systems: a brief review of some important issues and recent developments. In: IEEE International Conference on Fuzzy Systems. FUZZ-IEEE 2009, pp. 530–535. IEEE (2009)

    Google Scholar 

  56. Isermann, R.: Identifikation dynamischer Syteme – Band 1, 2. ed. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  57. Jakubek, S., Keuth, N.: A new training algorithm for neuro-fuzzy networks. In: ANNIIP, pp. 23–34 (2005)

    Google Scholar 

  58. Jang, J.-S.R.: Neuro-Fuzzy Modeling: Architectures, Analyses, and Applications. Ph.D. thesis, EECS Department, Univ. of California at Berkeley, Berkeley, USA (1992)

    Google Scholar 

  59. Jang, J.-S.R.: ANFIS: adaptive-network-based fuzzy inference systems. IEEE Trans. Syst. Man Cybern. 23(3), 665–685 (1993)

    Article  Google Scholar 

  60. Jang, J.-S.R., Sun, C.T., Mizutani, E.: Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence. Prentice Hall, Englewood Cliffs (1997)

    Google Scholar 

  61. Johansen, T.A., Foss, B.A.: Constructing NARMAX models using ARMAX models. Int. J. Control 58(5), 1125–1153 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  62. Junge, T.F., Unbehauen, H.: Real time learning control of an emergency turbo-generator plant using structurally adaptive neural networks. In: IEEE Conference on Industrial Electronics (IECON), pp. 2403–2408, Aachen, Germany (1998)

    Google Scholar 

  63. Kadirkamanathan, V., Fabri, S.: Stable nonlinear adaptive control with growing radial basis function networks. In: IFAC Symposium on Adaptive Systems for Control and Signal Processing (ASCAP), pp. 231–236, Budapest, Hungary (1995)

    Google Scholar 

  64. Klein, P., Kirschbaum, F., Hartmann, B., Bogachik, J., Nelles, O.: Adaptive test planning for the calibration of combustion engines – application. In: Design of Experiments (DoE) in Engine Development, pp. 17–30, Berlin, Germany. Expert Verlag (2013)

    Google Scholar 

  65. Kofahl, R.: Robuste parameteradaptive Regelungen. Fachbericht Nr. 19. Messen, Steuern, Regeln. Springer, Berlin (1988)

    Google Scholar 

  66. Kramer, M.A.: Diagnosing dynamic faults using modular neural nets. IEEE Expert (1993)

    Google Scholar 

  67. Kroll, A.: Fuzzy-Systeme zur Modellierung und Regelung komplexer technischer Systeme. Reihe 8: Mess-, Steuerungs- und Regelungstechnik, Nr. 612. VDI-Verlag, Düsseldorf (1997)

    Google Scholar 

  68. Leonhard, J.A., Kramer, M.A., Ungar, L.H.: A neural network architecture that computes its own reliability. Comput. Chem. Eng. 16(9), 818–835 (1992)

    Google Scholar 

  69. Ljung, L.: System Identification: Theory for the User, 2nd edn. Prentice Hall, Englewood Cliffs (1999)

    Google Scholar 

  70. Lughofer, E.: Evolving Fuzzy Systems-Methodologies, Advanced Concepts and Applications, vol. 53. Springer (2011)

    Google Scholar 

  71. Moll, J., Schulte, R.T., Hartmann, B., Fritzen, C.-P., Nelles, O.: Multi-site damage localization in anisotropic plate-like structures using an active guided wave structural health monitoring system. Smart Mater. Struct. 19(4), 045022 (2010)

    Article  ADS  Google Scholar 

  72. Müller, N., Nelles, O.: Closed-loop ignition control using on-line learning of locally-tuned radial basis function networks. In: American Control Conference (ACC), pp. 1356–1360, San Diego, USA (1999)

    Google Scholar 

  73. Murray-Smith, R.: A Local Model Network Approach to Nonlinear Modeling. Ph.D. thesis, University of Strathclyde, Strathclyde, UK (1994)

    Google Scholar 

  74. Nakamori, Y., Ryoke, M.: Identification of fuzzy prediction models through hyperellipsoidal clustering. IEEE Trans. Syst. Man Cybern 24(8), 1153–1173 (1994)

    Article  Google Scholar 

  75. Nelles, O.: Local linear model tree for on-line identification of time-variant nonlinear dynamic systems. In: International Conference on Artificial Neural Networks (ICANN), pp. 115–120, Bochum, Germany (1996)

    Google Scholar 

  76. Nelles, O.: LOLIMOT – Lokale, lineare Modelle zur Identifikation nichtlinearer, dynamischer Systeme. Automatisierungstechnik (at) 45(4), 163–174 (1997)

    Article  Google Scholar 

  77. Nelles, O.: Structure optimization of Takagi-Sugeno fuzzy models. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 5(2), 161–170 (1998). Special Issue on Applications of New Functional Principles of Fuzzy Systems and Neural Networks within Computational Intelligence

    Google Scholar 

  78. Nelles, O.: Axes-oblique partitioning strategies for local model networks. In: IEEE International Symposium on Intelligent Control, pp. 2378–2383, Munich, Germany (2006)

    Google Scholar 

  79. Nelles, O., Fischer, M.: Lokale Linearisierung von Fuzzy-Modellen. Automatisierungstechnik (at) 47(5), 217–223 (1999)

    Article  Google Scholar 

  80. Nelles, O., Hecker, O., Isermann, R.: Automatic model selection in local linear model trees for nonlinear system identification of a transport delay process. In: IFAC Symposium on System Identification (SYSID), pp. 727–732, Kitakyushu, Fukuoka, Japan (1997)

    Google Scholar 

  81. Nelles, O., Hecker, O., Isermann, R.: Identifikation nichtlinearer, dynamischer Prozesse mit Takagi-Sugeno Fuzzy-Modellen variabler Struktur. In: 4. GI Fuzzy-Neuro-Systeme Workshop, pp. 388–395, Soest, Germany (1997)

    Google Scholar 

  82. Nelles, O., Hecker, O., Isermann, R.: Automatische Strukturselektion für Fuzzy-Modelle zur Identifikation nichtlinearer, dynamischer Prozesse. Automatisierungstechnik (at) 46(6), 302–312 (1998)

    Article  Google Scholar 

  83. Nelles, O., Isermann, R.: Basis function networks for interpolation of local linear models. In: IEEE Conference on Decision and Control (CDC), pp. 470–475, Kobe, Japan (1996)

    Google Scholar 

  84. Paoletti, S., Juloski, A.L., Ferrari-Trecate, G., Vidal, R.: Identification of hybrid systems a tutorial. Eur. J. Control 13(2), 242–260 (2007)

    Article  MATH  Google Scholar 

  85. Park, M.-K., Ji, S.-H., Kim, E.-T., Park, M.: Identification of Takagi-Sugeno fuzzy models via clustering and hough transform. In: Hellendoorn, H., Driankov, D. (eds.) Fuzzy Model Identification: Selected Approaches, chapter 3, pp. 91–161. Springer, Berlin (1997)

    Google Scholar 

  86. Plutowski, M.: Selecting Training Examplars for Neural Network Learning. Ph.D. thesis, University of California, San Diego, USA (1994)

    Google Scholar 

  87. Pucar, P., Millnert, M.: Smooth hinging hyperplanes: a alternative to neural nets. In: European Control Conference (ECC), pp. 1173–1178, Rome, Italy (1995)

    Google Scholar 

  88. Ronco, E., Gawthrop, P.J.: Incremental model reference adaptive polynomial controller network. In: IEEE Conference on Decision and Control, pp. 4171–4172, New York, USA (1997)

    Google Scholar 

  89. Runkler, T.A., Bezdek, J.C.: Polynomial membership functions for smooth first order Takagi-Sugeno systems. In: GI-Workshop Fuzzy-Neuro-Systeme: Computaional Intelligence, pp. 382–387, Soest, Germany (1997)

    Google Scholar 

  90. Sanner, R.M., Slotine, J.-J.E.: Gaussian networks for direct adaptive control. IEEE Trans. Neural Netw. 3, 837–863 (1992)

    Article  Google Scholar 

  91. Schmidt, M., Nelles, O.: Filtering and deriving signals using neural networks. In: American Control Conference (ACC), pp. 2730–2731, Philadelphia, USA (1998)

    Google Scholar 

  92. Schwarz, R.: Rekonstruktion der Bremskraft bei Fahrzeugen mit elektromechanisch betätigten Radbremsen. Ph.D. Thesis, TU Darmstadt, Reihe 12: Verkehrstechnik/Fahrzeugtechnik, Nr. 393. VDI-Verlag, Düsseldorf (1999)

    Google Scholar 

  93. Schwarz, R., Nelles, O., Isermann, R.: Verbesserung der Signalgenauigkeit von Raddrehzahlsensoren mittels Online-Kompensation der Impulsradfehler. Automatisierungstechnische Praxis (atp) 41(3), 35–42 (1999)

    Google Scholar 

  94. Schwarz, R., Nelles, O., Isermann, R., Scheerer, P.: Verfahren zum Ausgleich von Abweichungen eines Raddrehzahlsensors. Patent DE 197 21 488 A1 (1999)

    Google Scholar 

  95. Schwarz, R., Nelles, O., Scheerer, P., Isermann, R.: Increasing signal accuracy of automatic wheel-speed sensors by on-line learning. In: American Control Conference (ACC), pp. 1131–1135, Albuquerque, USA (1997)

    Google Scholar 

  96. Settles, B.: Active learning literature survey. Univ. Wisconsin Madison 52(55–66), 11 (2010)

    Google Scholar 

  97. Thrun, S.B.: The role of exploration in learning control. In: Handbook of Intelligent Control: Neural Fuzzy and Adaptive Approaches. Van Nostrand Reinhold (1992)

    Google Scholar 

  98. Tibshirani, R.J.: Degrees of freedom and model search. arXiv preprint arXiv:1402.1920 (2014)

    Google Scholar 

  99. Tulleken, H.J.A.F.: Grey-box modelling and identification using prior knowledge and Bayesian techniques. Automatica 29(2), 285–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  100. Yoshinari, Y., Pedrycz, W., Hirota, K.: Construction of fuzzy models through clustering techniques. Fuzzy Sets Syst. 54, 157–165 (1993)

    Article  MathSciNet  Google Scholar 

  101. Zhou, Z.-H., Wu, J., Tang, W.: Ensembling neural networks: many could be better than all. Artif. Intell. 137(1), 239–263 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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    Oliver Nelles

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Nelles, O. (2020). Local Linear Neuro-Fuzzy Models: Advanced Aspects. In: Nonlinear System Identification. Springer, Cham. https://doi.org/10.1007/978-3-030-47439-3_14

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