研究生: |
謝鴻琳 Horng-Lin Shieh |
---|---|
論文名稱: |
以模糊群聚為基礎的強健式函數趨近法 A Fuzzy-Clustering-Based Robust Approach of Function Approximation |
指導教授: |
楊英魁
Ying-Kuei Yang |
口試委員: |
黃漢邦
none 連耀南 none 鄧洪聲 none 蘇仲鵬 none 孫宗瀛 none 吳傳嘉 Chwan-Chia Wu 黎碧煌 Bih-Hwang Lee |
學位類別: |
博士 Doctor |
系所名稱: |
電資學院 - 電機工程系 Department of Electrical Engineering |
論文出版年: | 2006 |
畢業學年度: | 94 |
語文別: | 中文 |
論文頁數: | 177 |
中文關鍵詞: | 模糊模式 、函數趨近 、強健式 |
外文關鍵詞: | Fuzzy modeling, Function Approximation, Robust |
相關次數: | 點閱:136 下載:1 |
分享至: |
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報 |
近年來,在科學應用上,對一個未知的系統行為函數建模(modeling)一直是非常重要的研究課題。但是當取得的系統知識不完整、口語化的知識表達或是以專家知識描述系統的操作行為等等,便無法以一個完整的數學模式對系統行為函數來建模。這個問題一直到模糊理論(fuzzy theory)被提出來,才有突破性發展,並逐漸發展成為系統建模的有效工具。
但是,在實際應用系統的取樣資料中,由於各種因素影響,常常使得取樣資料中存有一般雜訊(noise)或大誤差(outlier)的資料。若將這些雜訊或大誤差資料直接使用在系統行為函數的建模上,往往使得所建立的系統模式與實際的系統行為產生很大的誤差,這種現象稱為「過度適應」(overfitting)。
為了解決這個問題,本文提出一個非監督模糊建模方式,可直接由具有雜訊及大誤差的輸入-輸出取樣資料中擷取模糊規則,以建立系統行為的近似函數。所提方法主要有兩個特色:(1) 提出一個強健式 fuzzy c-means(RFCM)演算法,以降低雜訊及大誤差的影響。(2)提出一個模糊資料篩選器(FDS),可尋找資料轉折特徵點,藉以將一個非線性系統分割成片段的子系統組合。因此,可以建立起一個較少規則及較少誤差的Takagi and Sugeno模糊模式。本文中利用多個實驗證明所提方法可以用在不同的資料領域上,且在效能上,比其他方法好。
In recent scientific applications, modeling an unknown system is a very important research subject. However, it is difficult to well model a system by mathematics model when the acquired system knowledge is incomplete, linguistic interpretations or expertise descriptions by experts. It had been no breakthrough on solving the problem until the proposal of fuzzy theory that has become one of efficient methodologies on system modeling.
Due to influence of various factors, the sampling data used for system modeling often include noises and outliers. If such sampling data is directly being used to model a system, there will be a big difference, named overfitting, on the system behavior between the resultant modeled system and the actual system.
To overcome this problem, this thesis presents an unsupervised fuzzy model construction approach to extracting fuzzy rules directly from numerical input–output sampling data for nonlinear systems bound with noises and outliers. There are two core ideas in the proposed method: (1) The robust fuzzy c-means (RFCM) algorithm is proposed to greatly mitigate the influence of data noises and outliers; and (2) A fuzzy-based data sifter (FDS) is proposed to locate good turning-points to partition a given nonlinear data domain into piecewise clusters so that a Takagi and Sugeno fuzzy model can be constructed with fewer rules and less errors. Several experiments are illustrated and their results have shown the proposed approach has good performance in various kinds of data domains with data noises and outliers
參考書目
1.R. Babuska and H.B. Verbruggen, ”Constructing Fuzzy Models by Product Space Clustering,” Fuzzy Model Identification. Springer-Verlag, 1997.
2.Chen-Chia Chuang, Shun-Feng Su, Jin-Tsong Jeng, and Chih-Ching Hsiao, “Robust Support Vector Regression Networks for Function Approximation With Outliers,” IEEE trans. On neural networks, 13(6), pp1322~1330, 2002.
3.D. S. Chen and R. C. Jain, “A robust back-propagation learning algorithm for function approximation,” IEEE Trans. on Neural Networks,5, pp467–479, 1994.
4.C.-C. Chuang, S.-F. Su, and C.-C. Hsiao, “The annealing robust backpropagation (ARBP) learning algorithm,” IEEE Trans. on Neural Networks, 11, pp1067–1077, 2000.
5.F. E. H. Tay and L. Cao, “Application of support vector machines in financial time series forecasting,” Int. J. Manage., 2001, pp309–317.
6. J. A. K. Suykens, J. De Brabanter, L. Lukas, and J. Vandewalle, “Weighted least squares support vector machines: Robustness and sparse approximation,” Neurocomputing, 48, pp85-105,2002
7.V. David Sanchez, “Robustization of a learning method for RBF network,” Neurocomputing 9, pp85~94,1995
8.莊鎮嘉, Robust Approaches of modeling under outlier, 台科大博士論文,2000
9. L.X.Wang and J.M. Mendel,"Generating fuzzy rules by learning from examples," IEEE Trans. on Syst. Man, Cybern., 22, pp1414-1427,1992
10.T. P. Hong and C.Y. Lee, ”Induction of fuzzy rules and membership functions from training examples,” Fuzzy Sets and System, 84, pp33-47, 1996.
11.K. Nozaki, H. Ishibuchi,and H.Tanaka ”A simple but powerful heuristic method for generating fuzzy rules from numerical data,” Fuzzy Sets and System, 86, pp251-270, 1997.
12.R. Thawonmas and S. Abe,“Function approximation based on fuzzy rules extracted from partioned numerical data,” IEEE Trans. on Syst. Man, Cybern. B, 29, pp525-534, 1999.
13.J. Hollatz, ”Fuzzy Identification Using Methods of Intelligent Data Analysis,“ Fuzzy Model Identification. Springer-Verlag. 1997.
14.J.–S. Roger Jang, ”ANFIS: Adaptive-Network-based Fuzzy Inference Systems,” IEEE Trans. on Syst. Man, Cybern. 23(03), pp 665-685,1993.
15.R. J.–S. Jang, C. –T. Sun,E. Mizutani ,Neuro-fuzzy and Soft Computing, Prentice Hall,1997
16.Shiqian Wu, Meng Joo Er, and Yang Gao” A Fast Approach for Automatic Generation of Fuzzy Rules by Generalized Dynamic Fuzzy Neural Network,"IEEE Trans. on Fuzzy Systems, 9 (4), pp578-594,2001
17.Y.H. Lin and G. A. Cunningham ,”A new approach to fuzzy-neural system modeling," IEEE Trans. on Fuzzy Systems, 3, pp190-197,1995
18.孫宗瀛, “模糊系統自動建模:使用強化學習機制的放射撞奇函數類神經網路為基礎的模糊推理系統”, 台科大博士論文, 2001.
19.Frank Hoppner, Frank Klwawonn, Rudolf Kruse and Thomas Runkler, Fuzzy Cluster Analysis, John Wiley & Sons, Inc., New York, 1999
20.M.Sugeno and T. Yasukawa,”A fuzzy-logic-based approach to qualitative modeling,” IEEE Trans. On Fuzzy Systems,1, pp7~31,1993
21.M. Sugeno and K. T. Kang, “Structure identification of fuzzy model,” Fuzzy Sets and systems, 28, 1988.
22.R.S. Sun, “Rule-base structure identification in an adaptive-network-based fuzzy inference system,” IEEE trans. On Fuzzy systems, 2, pp64~73,1994.
23.P. Lindskog, “Fuzzy identification from a grey box modeling point of view,” in fuzzy model identification: selected approaches, H. Hellendoorn, D. Driankov (Eds.) Springer-Verlag Berlin Heidelberg, 1997.
24.R. Babuska and H. B. Verbruggen, “An overview of fuzzy modeling for control,” Control Engineering Pratice, 4(11), pp1953~1606, 1996.
25.L. A. Zadeh, “Fuzzy sets,“ Information Control, 8,pp338~353, 1965.
26.L. A. Zadeh, “The concept of a linguistic variable and its application for approximate reasoning I,II,III,” Information Sciences, 8,pp199~p251, pp301~p357, 9, pp43~80, 1975
27.L. A. Zadeh, “Fuzzy logic, Neural network, and soft computing,” Communications of the ACM, 37(3), pp77-84, 1994
28.L. A. Zadeh, “The role of fuzzy logic in modeling, identification and control,” Modeling Identification and Control, 15(3), pp191-203, 1994
29.J.L. Castro, “Fuzzy systems with defuzzification are universal approximators,” IEEE Trans. on Syst. Man, Cybern, 26, pp149-152, 1996
30.J.L. Castro, “Fuzzy logic controllers are universal approximators,” IEEE Trans. on Syst. Man, Cybern. 26, pp629-635,1995
31.B. Kosko, "Fuzzy Systems are Universal Approximators," IEEE Trans. on Computers, 43, pp1329-1333,1994.
32.L.-X. Wang, "Fuzzy systems are universal approximators," Proc. IEEE International Conf. On Fuzzy Systems, San Diego, pp1163-1170, 1992.
33.L.-X. Wang, "Universal approximation by hierarchical fuzzy system," Fuzzy sets and Systems, 93, pp223-230, 1998.
34.H. Ying, Y.-S. Ding, S.-K. Li and S.-H. Shao, "Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy systems as universal approximators," IEEE Trans. on Syst. Man, Cybern., 29,pp508-514,1999.
35.H. Ying, "'General SISO Takagi-Sugeno Fuzzy Systems with Linear Rule Consequent are Universal Approximators," IEEE Trans. On Fuzzy Systems, 6(4), pp582-587, 1998.
36.X.J. Zeng and M. G Singh, "Approximation theory of fuzzy systems - MIMO case, " IEEE Trans. on Fuzzy Systems, 3, pp219-235,1995.
37.X.J. Zeng and M. G. Singh, "Approximation properties of fuzzy systems generated by the min inference, " IEEE Trans. on Syst. Man, Cybern., 26, pp187-193,1996.
38.Yuji Yoshida, “Fuzzy stopping in continuous-time dynamic fuzzy systems,” Fuzzy Sets and Systems,132, pp291 -301, 2002
39.Tapan Kr. Dind, Kumar S. Ray and Mihir Kr. Chakraborty, “Fuzzy relational calculus approach to multidimensional pattern classification,” Pattern Recognition, 32, pp973-995, 1999
40.R.E.Mercer, J.L.Barron, A.A.Bruen, D.Cheng,” Fuzzy points: algebra and application,” Pattern Recognition, 35, pp1153 –1166, 2002
41.C.C. Lee, "Fuzzy logic in control systems: fuzzy logic controller, part I," IEEE Trans. on Syst. Man, Cybern., 20(2), pp404-418, 1990.
42.C.C. Lee, "Fuzzy logic in control systems: fuzzy logic controller, part II,” IEEE Trans. on Syst. Man, Cybern., 20(2), pp419-435, 1990.
43.Chen-Chia Chuang, Shun-Feng Su, and Song-Shyong Chen, “Robust TSK Fuzzy Modeling for Function Approximation With Outliers,” IEEE Trans. on Fuzzy Systems, 9(6), pp810-821,2001
44.J.Zhao , V.Wertz , and R.Gorez, “A Fuzzy Clustering Method for the Identification of Fuzzy Models for Dynamic Systems,” IEEE International Symposium on Intelligent Control,16-18, pp172-177, 1994
45.E.H. Mamdani and S. Assilian “An Experiment in Linguistic Synthesis with A Fuzzy Logic Controller.” International J. of Man-Machine Studies, 7(1), pp1-13, 1975
46.M. Sugeno and K. T. Kang, "Structure identification of fuzzy model," Fuzzy Sets and Systems, 28, 1988.
47.T. Takagi and M. Sugeno, "Fuzzy Identification of Systems and its Application to Modeling and Control," IEEE Trans. on Syst. Man, Cybern., 15(l), pp116-132, 1985.
48.R. Babuska and H.B. Verbruggen, “An Overview of Fuzzy Modeling for Control,” Control Eng. Practice, 4(11), pp1593-1606, 1996
49. E. Kim and M. Park and S. Ji,“A New Approach to Fuzzy Modeling”,IEEE Trans. On Fuzzy Systems,5(3), pp328-337,1997
50.Hao Ying and Guanrong Chen, “Necessary Conditions for Some Typical Fuzzy Systems as Universal Approximators,” Automatica, 33(7), pp1333~1338, 1997
51.R. Babuska and H.B. Verbruggen, “An Overview of Fuzzy Modeling for Control,” Control Eng. Pratice, 4(11), pp 1593~1606, 1996
52.L. A. Zadeh, “Outline of a new approach to the analysis of complex system and decision process,” IEEE Trans. on Syst. Man, Cybern.,3, pp28~44,1973
53.P.J.Costa Branco and J.A.Dente, “Fuzzy systems modelingin practice,” Fuzzy Sets and Systems, 121, pp73-93, 2001
54.Klir and Folger, Fuzzy sets, Uncertainty, and Information, Prentice-Hall, New Jersey, 1988
55.Postlethwaite, B.E., "Empirical comparison of methods of fuzzy relational identification", IEE Proceedings, Part D, 138(3), pp199-206. 1991
56.C.C. Lee, “Fuzzy logic in control systems: fuzzy logic controller, part II,” IEEE Trans. on Syst. Man, Cybern., 20(2), pp419-435, 1990.
57.E. H. Mamdani, “Application of fuzzy logic to approximate reasoning using linguistic systems,” Fuzzy Sets and Systems, 26, pp1182-1191, 1977.
58.J. Yen and R. Langari, Fuzzy Logic: Intelligence, Control, and Information, Prentice Hall, Englewood Cliffs, NJ, 1999.
59.D.S. Chen and Ramesh C.Jain, “A Robust Back Propagation Learning Algorithm for Function Approximation,” IEEE Trans. on Neural Networks, 5, 1994.
60.S.-T. Li and E. Leiss, “On Noise-immune RBF Networks,” in Radial Basis Function Neural Networks: Recent Developments in Theory and Applications, Springer Verlag, 2001,
61.J.-N. Hwang, “Nonparametric Multivariate Density Estimation: A Comparative Study,” IEEE Trans. Signal Processing, 42, pp2795-2810, 1994.
62.H.L. Shashidhara, S.Lohani, and V.M. Gadre, “Function Learning using Wavelet Neural Networks,” Proceedings of IEEE International Conference on Industrial Technology, 2, pp335-340, 2000.
63.Trevor Hastie and Werner Stuetzle, “Principle Curves,” Journal of the American Statistical Association, pp502-516, 1989
64.Sathyakama Sandilya and Sanjeev R. Kulkarni, ”Principal Curves With Bounded Turn,” IEEE Trans. On Information Theory,48(10), pp2789-2793, 2002
65. K. Honda, N. Sugiura, and H. Ichihashi, "Robust Local Principal Component Analyzer with Fuzzy Clustering," Proceedings of the International Joint Conference on Neural Networks, 1, pp732 - 737, 2003 .
66.V. Vapnik, “The Nature of Statistical Learning Theory”, Springer-Verlage , London, UK, 1995.
67.J. A. K. Suykens , J.De Brabanter, L. Lukas, J. Vandewalle, “Weighted least squares support vector machines: robustness and sparse approximation,” Neurocomputing, 48, pp85-105,2002
68.T-T. Frie ,R.F.Harrison, “A kernel-based Adaline for function approximation, ” Intelligent Data Analysis, 3, pp307-313,1999
69. Chen-Chia Chuang, Jin-Tsong Jengb, Pao-Tsun Lin, Annealing robust radial basis function networks,” Neurocomputing, 56, pp123-139, 2004
70.V. Vapnik, S. Golowich, and A. J. Smola,”Support vector method for function approximation, regression estimation, and signal processing”, in M. Mozer, M. Jordan, and T. Petsche (eds.), Neural Information Processing Systems, 9. MIT Press, Cambridge, MA, 1997,
71.E. Ruspini, “A new approach to clustering,” Information Control.15, pp22–32, 1969.
72.J. C. Bezdek, Pattern Recognition With Fuzzy Objective Function Algorithms. New York: Plenum Press, 1981.
73.K. Honda, N. Sugiura, and H. Ichihashi, "Robust Local Principal Component Analyzer with Fuzzy Clustering," Proceedings of the International Joint Conference on Neural Networks, 1, pp732 - 737, 2003.
74.H. Yan, "Fuzzy Curve-Tracing Algorithm," IEEE Trans. on Syst. Man, Cybern. Part B: Cybernetics, 31(5), pp768-780, 2001
75.R.N. Davé, “Use of the Adaptive Fuzzy Clustering Algorithm to Detect Lines in Digital Images,” Intelligent Robots and Computer Vision VIII, 1(192), pp 600-611, 1989
76.F. L. Lewis, S.Q. Zhu, and K. Liu, “Function Approximation by Fuzzy Systems, ” Proceedings of the American Control Conf.pp3760-3764, 1995
77.Jesús González, Ignacio Rojas, Héctor Pomares, Julio Ortega, and Alberto Prieto, “A New Clustering Technique for Function Approximation” IEEE Trans. on Nerual Networks, 13(1), pp131-142, 2002
78.R.N. Davé and K. Bhaswan, “Adaptive Fuzzy C-Shells Clustering and Detection of Ellipses,” IEEE Trans. Neural Networks, 3(5), pp643-662, 1992
79.R. Krishnapuram, H. Frigui, and O. Nasraoui, “Fuzzy and Possibilistic Shell Clustering Algorithms and Their Application to Boundary Detection and Surface Approximation,” IEEE Trans. on Fuzzy Systems, 3(1), pp29-60, 1995
80.Marcello Salmeri, Arianna Mecattini, and Riccardo Rovantti, " Function Approximation using non-normalized SISO Fuzzy Systems," International Journal of Approximate Reasoning, 26, pp211-231, 2001
81.R. Krishnapuram, O. Nasraoui, and H. Frigui, “The Fuzzy C-Spherical Shells Algorithms: A New Approach,” IEEE Trans. on Neural Networks, 3, pp663–671, 1992
82.K. Chang and J.Ghosh, “Principal Component for nonlinear feature extraction and classification,” Proc. SPIE:Application of artificial Neural Network in Image Processing III, pp120-129,1998
83. B. Kegl, A. Krzyzak, T. Linder and K. Zeger, "Learning and Design of Principal Curve," IEEE Trans. on Pattern Analysis and Machine Intelligence, 22(3),pp.281-297, 2000
84.B. Kegl and A. Krzyzak, "Piecewise linear skeletonization using principal curves," 15th International Conference on Pattern Recognition, 3 pp131-134, 3-7, 2000
85.J.B. Chen and Zurbenko, “Nonparametric boundary detection,” Commun. Stattist.-Theory and Method, 26(12), pp2999-3014,1997
86.Y.Linde, A.Buzo, and R.M.Gray, “An algorithm for vector quantizer design,” IEEE trans. On Communications, 28, pp84–95,1980.
87.W.S.Cleveland, “Robust locally weighted regression and smoothing scatter plots,” Journal of the American Statistical Association, 74, pp829–835,1979.
88.J.D.Ban eldandA.E.Raftery,“Ice floe identification in satellite images using mathematical morphology and clustering about principal curves,” Journal of the American Statistical Association, 87, pp7–16,1992.
89.R.Singh, M.C.Wade ,and N.P. Papanikolopoulos, “Letter-level shape description by skeletonization in faded documents,” in Proceedings of the Fourth IEEE Workshop on Applications of Computer Vision, pp121–126,1998.
90.K.Reinhard and M.Niranjan, “Subspace models for speech transitions using principal curves,” Proceedings of Institute of Acoustics, 20(6), pp53–60,1998.
91.K.Chang and J.Ghosh, “Principal curve classifier a nonlinear approach to pattern classification,” IEEE International Joint Conference on Neural Networks, pp 695–670, 1998.
92.Dug Hun Honga, Changha Hwang,” Support vector fuzzy regression machines,” Fuzzy Sets and Systems, 138, pp271–281, 2003
93. C. J. C. Burges, “A tutorial on support vector machines for pattern recognition, “ Data Mining Knowledge Discovery 2(2), pp121–167, 1998.
94.Francis E.H. Tay and L.J.Cao, “Modified support vector machines in financial time series forecasting,” Neurocomputing, 48, pp847–861, 2002
95.Weida Zhou, LiZhang and Licheng Jiao, “Linear programming support vector machines,” Pattern Recognition 35, pp2927–2936, 2002
96.Colin Campbell,” Kernel methods: a survey of current techniques,” Neurocomputing, 48, pp63–84, 2002
97.John Yen, “Fuzzy Logic—A Modern Perspective,” IEEE Trans. on Knowledge and Data Engineer.11(1), pp153-165, 1999
98.Frank Höppner, and Frank Klawonn,"A Contribution to Convergence Theory of Fuzzy c-Means and Derivatives", IEEE Trans. on fuzzy set 11(5), pp 682~694 2003
99.X. L. Xie and G. Beni, "A Validity Measure for Fuzzy Clustering," IEEE Trans. Pattern Anal. Machine Intell., 13. pp841–847,1991.
100.R. J. Hathaway and J. C. Bezdek," Fuzzy c -Means Clustering of Incomplete Data," IEEE Trans. on Syst. Man, Cybern. Part B: Cybernetics, 31(5), pp735~744, 2001
101.R. R. Yager and D. P. Filev, "Approximate Clustering Via the Mountain Method," IEEE Trans. on Syst. Man, Cybern, 24(8), pp1279~1284. 1994
102.Thomas A. Runkler and James C. Bezkek, ”Alternating Cluster Estimation: A new Tool for Clustering and Function Approximation,” IEEE Trans. On Fuzzy Systems, 7(4), pp377-393, 1999
103.J. Moody and C. J. Darken, “Fast learning in networks of locally-tuned processing units,” Neural Computing, 1(2), pp281–294,1989.
104.E. L. Sutanto, J. D. Masson, and K.Warwick, “Mean-tracking clustering algorithm for radial basis function center selection,” Int. J. Control. 67(6), pp961–977, 1997.
105.R. N. Dave and R. Krishnapuram, "Robust Clustering Methods: A Unified View," IEEE Trans. on Fuzzy system, 5(2), pp270-293, 1997
106.R.N.Davé, “Characterization and detection of noise in clustering,” Pattern Recognition letter, 12(11), pp657-664, 1991
107.R. Krishnapuram, O. Nasraoui, and J. M. Keller, “A possibilistic approach to clustering,” IEEE trans. on Fuzzy System, 1, pp98-110, 1993
108.M. Brani, V. Cappellini, and A. Mecocci, “Commoms on “A possibilistic approach to clustering”,” IEEE trans. on Fuzzy system, 4(3), pp393-396, 1996
109.C.C. Lee, P.C. Chung, J.R. Tsai and C.I Chang, “Robust radial basis function neural networks,” IEEE Trans. on Syst. Man, Cybern. Part B: Cybernetics, 29(6), pp674-685, 1999
110.W. Y. Wang, Tsu-Tian Lee, Ching-Lang Liu and Chi-Hsu Wang, “Function approximation using fuzzy neural networks with robust learning algorithm,” IEEE Trans. on Syst. Man, Cybern. Part B: Cybernetics, 27(4), pp740-747, 1997
111.R. Tibshirani, “Principal Curves Revisited,” Stat. Comput., 2, pp183–190, 1992
112.R. Krishnapuram, H. Frigui, and O. Nasraoui, “Fuzzy and Possibilistic Shell Clustering Algorithms and Their Application to Boundary Detection and Surface Approximation,” IEEE Trans. on Fuzzy Systems, 3(1), pp29-60, 1995
113.C. R. Wylie and A. Barrett, Advanced Engineering Mathematics, pp840~pp846, McGraw-Hill, New York, 1982
114.R. Singh, V. Cherkassky, and N. Papanikolopoulos, “Self-Organizing Maps for the Skeletonization of Sparse Shapes,” IEEE Trans. on Neural Networks, 11(1), pp 241~248, 2000
115.G.. A. Rovithakis and M. A. Christodoulou, “Adaptive Control of Unknown Plants Using Dynamical Neural Networks,” IEEE Trans. on Syst. Man, Cybern., 24, pp400–412, 1994.
116.R. Yager and D. P. Filev, Essentials of Fuzzy Modeling and Control. Wiley, New York, 1994.
117.M. Zhihong, X. H. Yu, and Q. P. Ha, “Adaptive Control Using Fuzzy Basis Function Expansion for SISO Linearizable Nonlinear Systems,” in Proc. 2nd ASCC, pp695–698, Seoul, Korea, 1997.
118.J. T. Spooner and K. M. Passino, “Stable Adaptive Control Using Fuzzy Systems and Neural Networks, ” IEEE Trans. on Fuzzy System, 4, pp339–359, 1996.
119.顏月珠,現代統計學, 三民書局,民國82年2月
120.Walpole and Myers, Probability and Statistics,Prentice Hall,1998