本論文提出一個新的方法，使用給定的輸入-輸出訓練資料集來建構一個第二型類神經模糊系統。資料經過此系統的模糊推論後，輸出一個第二型模糊集合，透過型態降階(type reduction)演算法及解模糊方法可得到一個明確的輸出值(crisp output)。Karnik 與 Mendel 提出一個區間第二型型態降階演算法，稱為KM演算法，KM演算法只能對區間第二型模糊集合進行降階。Liu 基於KM演算法提出一個第二型型態降階演算法，其概念是使用α-截集將第二型模糊集合分成數個區間第二型模糊集合，每一個區間第二型模糊集合又稱為α-平面，然後再使用KM演算法對每個α-平面進行型態降階。由於KM演算法的切換點初始值並不夠好，導致Liu所提方法的計算量比較大。本論文提出一個強化式型態降階演算法來改善此問題。我們利用上一個α-平面的型態降階結果，當成這次α-平面型態降階的初始值，大幅地窄化KM演算法的解空間，進而加速型態降階的速度。實驗模擬的結果顯示，本論文所提出的方法可以在不影響準確度的情況下減少型態降階所需要的時間。

建構一個第二型類神經模糊系統包含兩個階段：架構鑑別(structure identification)與參數鑑別(parameter identification)。在架構鑑別階段，一個自建構式模糊分群演算法被用來將輸入-輸出訓練資料分成數個模糊群聚，其概念是利用在輸入及輸出空間上的相似度測試來產生模糊群聚，而每個模糊群聚的歸屬函數則是根據其所包含的資料之平均值及標準差來定義。之後，結合奇異值分解最小平方估計法，每一個模糊群聚可以被萃取出一條第二型TSK型式的模糊“若-則”法則，以組成一個初步的第二型模糊法則庫，此法則庫可以直接使用於第二型模糊推論，也可以根據這些模糊法則來建立一個第二型TSK型式的模糊類神經網路。在參數鑑別階段，我們發展一個混合式學習演算法來對模糊類神經網路進行學習，以調整這些模糊法則。此演算法結合了粒子群優法(particle swarm optimization, PSO)及奇異值分解最小平方估計法，分別用來調整前鑑部參數及後鑑部參數。實驗結果顯示，經由混合式學習演算法修正過的第二型模糊法則可以增加預測精確度，而且本論文所提第二型類神經模糊系統優於區間第二型類神經模糊系統及第一型類神經模糊系統。除此之外，我們應用第二型類神經模糊系統進行台灣股市預測，實驗結果也顯示我們的預測系統優於其他方法。

We propose a novel approach for building a type-2 neuro-fuzzy system from a given set of input-output training data. For an input pattern, a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type-2 fuzzy set which is then defuzzified by applying a type reduction algorithm. Karnik and Mendel proposed an algorithm, called KM algorithm, to compute the centroid of an interval type-2 fuzzy set efficiently. Based on this algorithm, Liu developed a centroid type-reduction strategy to do type reduction for type-2 fuzzy sets. A type-2 fuzzy set is decomposed into a collection of interval type-2 fuzzy sets by α-cuts. Then the KM algorithm is called for each interval type-2 fuzzy set iteratively. However, the initialization of the switch point in each application of the KM algorithm is not a good one. In this thesis, we present an improvement to Liu's algorithm. We employ the result previously obtained to construct the starting values in the current application of the KM algorithm. Convergence in each iteration except the first one can then speed up and type reduction for type-2 fuzzy sets can be done faster. The efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results.

Constructing a type-2 neuro-fuzzy system involves two major phases, structure identification and parameter identification. We propose a method which incorporates self-constructing fuzzy clustering algorithm and a SVD-based least squares estimator for structure identification of type-2 neuro-fuzzy modeling. The self-constructing fuzzy clustering method is used to partition the training data set into clusters through input-similarity and output-similarity tests. The membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster. Then applying SVD-based least squares estimator, a type-2 fuzzy TSK IF-THEN rule is derived from each cluster to form a fuzzy rule base. After that a fuzzy neural network is constructed. In the parameter identification phase, the parameters associated with the rules are then refined through learning. We propose a hybrid learning algorithm which incorporates particle swarm optimization and a SVD-based least squares estimator to refine the antecedent parameters and the consequent parameters, respectively. We demonstrate the effectiveness of our proposed approach in constructing type-2 neuro-fuzzy systems by showing the results for two nonlinear functions and two real-world benchmark datasets. Besides, we use the proposed approach to construct a type-2 neuro-fuzzy system to forecast the daily Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX). Experimental results show that our forecasting system performs better than other methods.

論文審定書. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
致謝. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and purposes . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Contributions of this thesis . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Type-1 fuzzy logic systems . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Interval type-2 fuzzy logic systems . . . . . . . . . . . . . . . . . . . 7
1.5 Basic type-2 fuzzy concepts . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 SVD-based least squares estimator . . . . . . . . . . . . . . . . . . . 18
1.7 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Type Reduction for Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . 27
2.1 Karnik-Mendel algorithm . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Liu’s type reduction algorithm . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Improved type reduction algorithm . . . . . . . . . . . . . . . . . . . 31
2.3.1 Improved algorithm . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Some observations . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Complexity comparison . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 An example for improved type reduction . . . . . . . . . . . . 36
3 Type-2 Neuro-Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 The architecture of type-2 TSK-based neuro-fuzzy system . . . . . . 46
3.1.1 An example for type-2 TSK-based neuro-fuzzy system . . . . 50
3.2 Structure identification . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Self-constructing fuzzy clustering algorithm . . . . . . . . . . 53
3.2.2 Type-2 TSK-based fuzzy rules extraction . . . . . . . . . . . . 56
3.2.3 An example for structure identification . . . . . . . . . . . . . 58
3.3 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Refining antecedent parameters . . . . . . . . . . . . . . . . . 62
3.3.2 Refining consequent parameters . . . . . . . . . . . . . . . . . 64
3.3.3 An example for parameter identification . . . . . . . . . . . . 66
4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Experimental results for type reduction . . . . . . . . . . . . . . . . . 71
4.1.1 Experiment I . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.2 Experiment II . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Experimental results for neuro-fuzzy systems . . . . . . . . . . . . . . 87
4.2.1 Experiment III . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 Experiment IV . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.3 Experiment V . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3 Application for TAIEX forecasting . . . . . . . . . . . . . . . . . . . 104
4.3.1 Experiment VI . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.2 Experiment VII . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.3 Experiment VIII . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.4 Experiment IX . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendix A Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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