符號區間資料分析已成功應用於廣泛的領域，包括金融、工程和環境科學，使其成為許多研究人員在處理現實世界情境中常見的不確定性和不精確性時的一種有價值的工具。本論文從快速收斂的角度提出了符號區間改良模糊劃分模糊C均值(IIFPFCM)分群演算法與符號區間一般化改良模糊劃分模糊C均值(IGIFPFCM)分群演算法，並分別結合了歐氏距離(Euclidean distance)、城市街區距離(City block distance)和豪斯多夫距離(Hausdorff distance)。所提出的六種方法在收斂速度方面均比傳統的區間模糊C均值(IFCM)分群演算法在符號區間資料中分群更快。針對符號區間資料的大群和小群劃分問題，提出的方法也具有更好的表現。因在傳統的區間模糊C均值方法會受到離群值的影響，本論文探討了此問題。通過實驗模擬對比分析驗證，所提出的結合城市街區距離和豪斯多夫距離測量的符號區間改良模糊劃分模糊C均值分群演算法與符號區間一般化改良模糊劃分模糊C均值分群演算法為比較適合處理具有離群值的符號區間資料。
一般而言，豪斯多夫距離是考慮兩個集合之間的最大距離，使其對離群值的敏感度較低。此外模糊分群在處理資料時常常遇到雜訊和模糊性等挑戰。因此豪斯多夫距離通過考慮兩個集合之間的最大距離，而不僅僅是平均距離或中心之間的距離，提供了一定程度的抵抗力，以應對這些挑戰。這種穩健性使其在處理模糊和不確定的資料方面非常有效。所以在論文中，我們提出了基於豪斯多夫距離的符號區間一般化改良模糊劃分模糊C均值分群演算法，用於符號區間資料分析。最後符號區間資料分析將傳統統計方法擴展到分析複雜資料類型，如區間，對於不確定或聚合資料非常有用。在這些資料集中，雜訊問題是不可避免的。本論文亦針對符號區間資料的分群進行了討論，重點是處理雜訊。本論文提出了基於豪斯多夫距離的符號區間一般化改良模糊劃分模糊C均值分群演算法，該演算法利用競爭學習處理符號區間資料，具有更好的強健性和收斂性能。此外由於豪斯多夫距離考慮的是最壞情況（最遠的點），而不是平均距離，因此該演算法對資料集中的離群值的敏感度較低。根據實驗模擬統計結果顯示，基於豪斯多夫距離的符號區間一般化改良模糊劃分模糊C均值分群演算法對於具有大量離群值和雜訊的符號區間資料在學生氏t分佈下具有更好的收斂性和效率表現。同時對符號區間資料進行了實驗結果評估，在性能的收斂性和時間效率的統計結果顯示，所提出的演算法具有更好的結果。

Symbolic interval data analysis has been successfully applied in a wide range of fields, including finance, engineering, and environmental science, making it a valuable tool for researchers dealing with the uncertainty and imprecision common in real-world situations. This thesis proposes two types of interval improved fuzzy partitioning FCM clustering algorithms from the perspective of rapid convergence: the symbolic interval improved fuzzy partitioning FCM (IIFPFCM) clustering algorithm and the symbolic interval generalized improved fuzzy partitioning FCM (IGIFPFCM) clustering algorithm, independently incorporating Euclidean distance, City block (Manhattan) distance, and Hausdorff distance, respectively. The six proposed methods demonstrate faster convergence than traditional IGIFPFCM interval FCM (IFCM) clustering algorithm in symbolic interval data clustering. Additionally, these methods perform better in the clustering of both large and small groups of symbolic interval data. Since traditional interval FCM methods are affected by outliers, this thesis addresses this issue. Through experimental comparative analysis with simulation, it is shown that the independently proposed symbolic IIFPFCM and IGIFPFCM clustering algorithms combined with City block distance and Hausdorff distance are more suitable for handling symbolic interval data with outliers.
In general, Hausdorff distance considers the maximum distance between two sets, making it less sensitive to outliers. Besides, fuzzy clustering often encounters challenges such as noise and fuzziness in data. Hausdorff distance provides a degree of resistance to such challenges by considering the maximum distance between two sets rather than just the average distance or distance between centroids. This robustness makes it effective in handling fuzzy and uncertain data. Hence, in this paper Hausdorff distance is proposed on IGIFPFCM clustering algorithm for symbolic interval data analysis (SIDA). In general, the SIDA extends traditional statistics to analyze complex data types like intervals, useful for imprecise or aggregated data. In these datasets, noise issues are inevitable. This paper addresses clustering for SIDA, focusing on handling noise. This paper proposes the IGIFPFCM clustering algorithm under Hausdorff distance clustering algorithm, which uses competitive learning to handle symbolic interval data with improved robustness and convergence performance. Besides, this algorithm is less sensitive to small perturbations or outliers in the datasets due to the Hausdorff distance considering the worst-case scenario (the farthest point) rather than averaging distances, which can be skewed by outliers. From the experimental simulation results, the statistical results of convergence and efficiency on performance show that the proposed IGIFPFCM under Hausdorff distance clustering algorithm has better results for SIDA with large outliers and noise under Student's t-distribution. At the same time, an evaluation of the experimental results on symbolic interval data shows that the proposed algorithms achieve better convergence performance and time efficiency compared to traditional methods.

摘要......i
Abstract......iii
誌謝......v
目錄......vi
表目錄......viii
圖目錄......ix
符號說明......xiv
第1章 簡介......1
1.1 符號區間資料概述......1
1.2 分群演算法概述......2
1.3 研究動機與貢獻......2
1.4 論文的組織......3
第2章 基礎背景......5
2.1 符號區間資料的定義和距離測量......5
2.2 學生氏 t 分佈......5
2.3 改良模糊劃分模糊C均值分群演算法......6
2.4 一般化改良模糊劃分模糊C均值分群演算法......7
第3章 符號區間改良模糊劃分模糊C均值分群演算法......9
3.1 基於歐氏距離的符號區間改良模糊劃分模糊C均值分群演算法......9
3.2 基於城市街區距離的符號區間改良模糊劃分模糊C均值分群演算法......12
3.3 模擬結果......16
3.4 本章結論......47
第4章 符號區間一般化改良模糊劃分模糊C均值分群演算法......48
4.1 基於歐氏距離的符號區間一般化改良模糊劃分模糊C均值分群演算法......48
4.2 基於城市街區距離的符號區間一般化改良模糊劃分模糊C均值分群演算法......51
4.3 模擬結果......56
4.4 本章結論......91
第5章 基於豪斯多夫距離的符號區間一般化改良模糊劃分模糊C均值分群演算法......92
5.1 基於豪斯多夫距離的符號區間改良模糊劃分模糊C均值分群演算法......92
5.2 基於豪斯多夫距離的符號區間一般化改良模糊劃分模糊C均值分群演算法......97
5.3 模擬結果......101
5.4 本章結論......120
第6章 結論......121
參考文獻......122
Extended Abstract......127
Abstract......127

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