由於金融市場的多變性，投資組合最佳化至今仍是相當吸引人的研究主題。過去幾十年間，許多不同的演化式演算法針對不同的投資組合亦不斷的被提出，其中一種就是多樣群組投資組合。然而，本研究發現現存的多樣群組投資組合最佳化技術仍有三個問題待解決，分別是：如何設定合適的群組數、演化過程耗時、投資組合風險差異過高。故為解決這些問題，本論文使用群組遺傳演算法提出兩個多樣群組股票投資組合最佳化方法。
第一個方法主要用於解決前兩個問題。針對設定合適的群組數問題，所提的方法首先透過股東權益報酬率(ROE)與本益比(P/E)兩屬性將股票分群後，之後設計分群效度因子(cluster validation factor)並將之當成適合度函數的一部份，使演算法可以自動搜尋較佳之分群結果。為解決演化過程耗時問題，在方法一則設計暫存染色體(Temporary chromosome)透過降低需要評估的組合數使演化過程得以加速。
第二個方法則著手解決投資組合風險差異過高問題。首先，方法中設計風險比例因子(Risk ratio factor)計算多樣群組股票投資組合可產生之最大組合風險。接著，所提的演算法結合自我調適之交配(Adaptive crossover)、突變(Adaptive mutation)運算與排序為基礎之輪盤選擇法(Rank-based roulette wheel selection)以達更好的搜尋能力。
最後，實驗透過31與50家公司之真實股市資料驗證所提方法的效率與效能確實優於現存的最佳化技術。與現存方法比較顯示，方法一不但在獲利上可達到相似結果，在執行時間上亦可減少約原來的百分之八十五。而方法二所找出之多樣群組投資組合其風險上也有明顯降低。

Due to the variety of financial markets, stock portfolio optimization is an attractive research topic. In the past decades, many evolutionary-based algorithms have been proposed to optimize different types of stock portfolios, and one of them is named diverse group stock portfolio (DGSP). However, this study found that three problems remain to be solving in the existing DGSP approaches. They are how to set an appropriate group size, evolution process is time-consuming, and difference between risks of portfolios is too high. To solve these problems, two approaches by grouping genetic algorithms (GGA) are proposed for optimizing DGSPs in this thesis.
The first approach is used to deal with the first two problems. For setting an appropriate group size, the two attributes, Return on Equity (RoE) and Price Earnings Ratio (P/E), are utilized to group stocks. Then, cluster validation factor, which is used as a part of fitness function, is designed to derive better stock groups. To solve time-consuming problem, a temporary chromosome is designed to reduce number of stock portfolios should be evaluated to speed up the evolution process.  
The second approach is then proposed to handle the third problem. It first designs risk ratio factor to calculate the maximum risk of a given DGSP. Then, by combining adaptive crossover, adaptive mutation, and rank-based roulette wheel selection, the second approach has higher searching ability to find better solution. 
At last, experimental results on the two real datasets that contain 31 and 50 stocks were made to verify the two proposed approaches are effective and efficient. Comparing with the existing approach, the results show that the first approach can not only reach similar return but also reduce execution time up to 85%. The risk of optimized DGSP by second approach is significantly lower than that by the existing approaches.

Contents
CHAPTER 1	INTRODUCTION	1
1.1 Problem Definition and Motivation	1
1.2 Contributions	4
1.3 Reader’s Guide	6
CHAPTER 2	RELATED WORK	7
2.1 The M-V model	7
2.2 Metaheuristics for portfolio optimization	8
2.3 Metaheuristics for Clustering	10
2.4 Cluster Validation Indices	11
CHAPTER 3	DIVERSE GROUP STOCK PORTFOLIO OPTIMIZATION APPROACH WITHOUT SETTING A GROUP NUMBER	14
3.1 Motivation	14
3.2 Components of Proposed Approach	16
3.2.1 Clustering Attributes	16
3.2.2 Encoding Scheme	17
3.2.3 Temporary Chromosome	19
3.2.4 Initial Population	22
3.2.5 Fitness Evaluation	23
3.2.6 Genetic Operations	25
3.3 Algorithm for First Approach	27
3.4 An Example	30
CHAPTER 4	A ENHANCED ALGORITHM FOR OPTIMIZING DIVERSE GROUP STOCK PORTFOLIO	42
4.1 Motivation	42
4.2 Elements of the proposed algorithm	44
4.2.1 Encoding Scheme	45
4.2.2 Initial Population	47
4.2.3 Fitness Evaluation	47
4.2.4 Genetic Operations	53
4.3 Algorithm for Second Approach	59
4.4 An Example	62
CHAPTER 5	EXPERIMENTAL RESULTS	76
5.1 Experimental Results for Approach (I)	76
5.1.1 Experimental Datasets	76
5.1.2 Effectiveness of The Proposed Approach	79
5.1.2.1 Experimental Results for Different Desired Stock	80
5.1.2.2 Analysis of the Derived GSP	85
5.1.3 Efficiency of Proposed Approach	88
5.2 Experimental Results for Approach (II)	90
5.2.1 Experimental Datasets	91
5.2.2 Effectiveness Of The Proposed Approach	91
5.2.3 Efficiency of Proposed Approach	100
CHAPTER 6	CONCLUSION AND FUTURE WORKS	103
REFERENCES	106

List of Figures
Figure 1. Encoding scheme of chromosome Cq	17
Figure 2. An example of the encoding scheme with three parts.	18
Figure 3. An example of a chromosome with four parts	19
Figure 4. An example of a chromosome from the previous approach	20
Figure 5. An example of a chromosome with relevant information	20
Figure 6. An example of a temporary chromosome	22
Figure 7. Encoding scheme of chromosome Cq.	45
Figure 8. An example of the encoding scheme.	46
Figure 9. The pie chart of the five chromosome in Table 10	58
Figure 10. The stock price series for 31 companies from the beginning of 2010 to the end of 2014	77
Figure 11. The stock price series for 50 companies from the beginning of 2012 to the end of 2014	78
Figure 12. The average fitness over 100 generations using elitist selection	80
Figure 13. The average fitness of the second approach with rank-based roulette wheel selection over 10 runs	92
Figure 14. The average risk of both approaches for 200 generations	96

List of Tables
Table 1. The relationship between group size and number of combinations.	21
Table 2. The 10 stocks used in the example	31
Table 3. The portfolio satisfaction for each chromosome	35
Table 4. The group balance for each chromosome	35
Table 5. The dissimilarity matrix for the proposed approach	35
Table 6. The diversity values for each chromosome	36
Table 7. The Davies-Bouldin Index for every chromosome	37
Table 8. The fitness for each chromosome	38
Table 9. Two generated DGSPs with similar ROI but different risk values	43
Table 10. Survival probabilities of five chromosomes	58
Table 11. The 10 stocks used in the example	62
Table 12. The portfolio satisfaction for each chromosome	66
Table 13. The group balance for each chromosome	66
Table 14. The dissimilarity matrix for the proposed approach	67
Table 15. The diversity values for each chromosome	67
Table 16. The values of Davies-Bouldin index for every chromosome	69
Table 17. The fitness for each chromosome	70
Table 18. The total fitness values for all the chromosomes	70
Table 19. The fitness f'(Cq) for all chromosomes after conversion for rank-based selection	71
Table 20. The probabilities of selection based on ranking	71
Table 21. The parameter settings for 31 and 50 company data sets	79
Table 22. The experimental results for the Davis-Bouldin Index approach on 31 stocks	80
Table 23. The experimental results for the C Index approach on 31 stocks	81
Table 24. The experimental results for the Dunn Index approach on 31 stocks	81
Table 25. The experimental results for the Silhouette approach on 31 stocks	82
Table 26. Approach using Davis-Bouldin Index with 50 stocks	84
Table 27. Approach using Silhouette index with 50 stocks	84
Table 28. Approach using Dunn index with 50 stocks	84
Table 29. Approach using C index with 50 stocks	85
Table 30. The initial and derived GSP for 31 stocks	86
Table 31. The initial and derived DGSP for 50 stocks	87
Table 32. The efficiency of previous and proposed approach 1 for 31 stocks	89
Table 33. The efficiency of Approach 1 using 50 stocks	90
Table 34. The experimental results for approach 2 using 31 stocks	93
Table 35. The experimental results for approach 2 using 50 companies	93
Table 36. The risk for 10 runs using the original approach	94
Table 37. The risk for approach 2 using 31 stocks	95
Table 38. The risk for the original approach using 50 stocks dataset	96
Table 39. The risk for proposed approach 2 using 50 stocks	97
Table 40. The initial and derived DGSP for 31 companies	98
Table 41. The initial and derived DGSP for approach 2 using 50 stocks	99
Table 42. The efficiency of approach 2 using 31 stocks	101
Table 43. The efficiency of approach 2 using 50 stocks	102

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