泰勒近似法於灰色預測及其在教育資訊與測驗統計之應用

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(1)國立臺中教育大學教育資訊與測驗統計研究所博士論文. 指導教授：許天維永井正武. 博士博士. Taylor Approximation Method in Grey Prediction and Its Applications in Educational Information and Measurement 泰勒近似法於灰色預測及其在教育資訊與測驗統計之應用. 研究生：阮福海. 中. 華. 民. 國. 一. ○. 四. 撰. 年. 五. 月.

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(3) National Taichung University of Education Graduate Institute of Educational Information and Measurement Ph.D. Dissertation. Dissertation Advisors: Prof. Tian-Wei Sheu Prof. Masatake Nagai. Taylor Approximation Method in Grey Prediction and Its Applications in Educational Information and Measurement 泰勒近似法於灰色預測及其在教育資訊與測驗統計之應用. By Nguyen Phuoc Hai. Taiwan, May 2015.

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(5) Acknowledgements The academic work of a Ph.D. candidate is about to end along with the completion of the dissertation and writing these acknowledgements, and starting a new page for my research. Firstly, I would like to express my special appreciation to my advisors: Prof. Tian-Wei Sheu and Prof. Masatake Nagai, for their mentoring, encouragement, and inspiration. My research advisors have not only helped me with this research but also provided many suggestions for the writing of the dissertation. Moreover, I am also grateful to all other committee members: Prof. Chin-Tsai Lin, Prof. Jiang-Long Lin, Prof. Kun-Li Wen, Prof. Jung-Chin Liang, and Prof. Chaang-Yung Kung, for their valuable suggestions on my research as well as the help that they have provided during my years of studying abroad. I would like especially to thank Prof. Hui-Chung Ho, who helped me in times of difficulty, for the attention and giving words of encouragement. Thanks to all the teachers in Graduate Institute of Educational Information and Measurement, National Taichung University of Education, helped me and guided enthusiastically in the learning process. In addition, I also want to thank those of my classmates: Phung-Tuyen Nguyen, Duc-Hieu Pham, Ching-Pin Tsai, Hsiu-Jye Chiang, and Wei-Ling Liu, who have supported and helped me a lot in the research process. The joys and sorrows of the past years are forever in my heart. Finally, I would like to thank my family for their unconditional support during my years of studying abroad. Especially, I thank my wife who has accompanied me during these years for taking good care of our children. This helped me to finish the dissertation without having to worry too much.. Phuoc-Hai Nguyen Taiwan, May 2015.

(6) 誌謝隨著論文的完成及致謝的起筆，研究生的生涯就此告一段落，畢業的時刻，心中除了雀躍也充滿了深深的感謝。這篇論文能夠順利完成，首先最要感謝的，是我的兩位指導教授許天維老師和永井正武老師。感謝兩位老師的耐心指導，從論文選題、引導論文的研究方向到之後的文獻探討及結果的討論分析，都讓兩位老師費心許多。每當在我遇到問題或瓶頸時，均能適時地給予我建議與幫助。同時也要謝謝口試委員林進財教授、林江龍教授、溫坤禮教授、梁榮進教授、與龔昶元教授，謝謝您們對論文的用心建議與指正，才使得論文能更佳完善周延。在我研究所的生涯中，特別感謝何慧群老師在我遇上挫折、氣餒時的鼓勵與照顧，謝謝所上所有老師們的指導與關懷。此外，我還要感謝一群情誼深厚、真心相挺的好友阮逢選、范德孝、蔡清斌、姜秀傑、劉維玲，他們是幫助我最多的好同學；也感謝我們班上的所有同學，謝謝你們對我的照顧與鼓勵。過往種種真是點滴在心頭……。最後，要感謝我的家人，感謝我的老婆，謝謝你不時的陪伴與支持，謝謝你替我照顧好我們的孩子，讓我能無後顧之憂專心在論文寫作上。. 阮福海謹誌中華民國 104 年 5 月.

(7) Abstract The purpose of this study proposes to apply Taylor approximation method in grey prediction (TAMGP) to educational information and measurement. TAMGP is developed based on the combination of Taylor approximation method and grey prediction of grey system theory. In 1982, grey system theory was first proposed by Deng. In recent years, the grey system theory has also been successfully employed in various prediction applications. It has become a very effective method of solving uncertainty problems under discrete data and incomplete information. In educational information and measurement, when the number of data in the system is not enough for traditional statistical methods, the application of grey system theory can get good results in which grey prediction models play a very important role for prediction problems. However, there is a problem that the predicted accuracy of grey models is unsatisfied. The coefficients of the prediction models are not the optimal coefficients. Therefore, using TAMGP can obtain the most optimal prediction values by multi-times approximate calculation. The research approach adopted in this dissertation includes theoretical study and experimental study related to Taylor approximation method in grey prediction to improve the accuracy of the previous prediction models. The findings from this study are shown as follows: (1) Applying TAMGP in educational information and measurement for prediction problems, especially using Taylor approximation method in RaschGSP IRT to improve the accuracy and to optimize coefficients α, β and γ of RaschGSP IRT. (2) Using the combination of TAMGP and GRA to predict and evaluate the academic achievement of students, and using the combination of TAMGP, GRA, and ROC to build setting the standard for tests. (3) Developing a MATLAB toolbox based on TAMGP for the purpose of the study and learning of grey system theory. The experimental results showed that TAMGP, GRA, and ROC are actually useful for prediction problems, evaluation, and setting the standard for tests of uncertainty systems and incomplete information in educational information and measurement. Keywords: Taylor approximation method, Grey prediction, Grey system theory, Educational information and measurement, GRA, ROC, MATLAB toolbox I.

(8) 摘要本研究提出泰勒近似法於灰色預測及其在教育資訊與測驗統計之應用，泰勒近似法於灰色預測是結合泰勒近似方法與灰色預測在灰色系統理論發展而成的研究方法。灰色系統理論 (Grey System Theory) 為鄧聚龍於 1982 年所提出。近年來，灰色系統理論已成功地用於各種預測的應用。它已成為解決離散數據，不完整信息與不確定性問題的一種非常有效的方法。在教育資訊與測驗統計中當系統的數據以傳統統計方法不能夠有效的計算時，應用灰色系統理論可以得到良好的結果，其中灰色預測模型對預測的問題具有非常重要的功能。然而，應用灰色預測模型仍存在一個問題就是預測的準確性還不夠令人滿意。因灰色系統理論之灰色預測精度係數仍不是最佳係數，故使用泰勒近似法於灰色預測可以通過多次近似計算，得到預測的最優價值。本研究所採用的研究方法包涵理論分析與實證研究跟泰勒近似法於灰色預測有關，以改進預測模型的準確性。本研究結果具有以下的貢獻：一、應用泰勒近似法於灰色預測及其在教育資訊與測驗統計所預測的問題，特別是使用泰勒近似法於 RaschGSP IRT 可以改進 RaschGSP IRT 的準確度和最優化係數 α，β 和 γ。二、使用泰勒近似法於灰色預測結合灰關聯分析(Grey Relational Analysis, GRA)，來預測和評估學生的成就時，同時使用泰勒近似法於灰色預測結合灰關聯分析與接收者操作特徵(Receiver Operating Characteristic, ROC)，來建立測試標準設定。三、達成了研究和學習灰色系統理論的目的，基於泰勒近似法於灰色預測建立了一個 MATLAB 工具箱。實驗結果顯示，泰勒近似法於灰色預測、灰關聯分析與接收者操作特徵對系統不明確性及資訊不完整性之預測問題、評估、測試標準應用在教育資訊與測驗統計具有實際的效用價值。. 關鍵字：泰勒近似方法、灰色預測、灰色系統理論、教育資訊與測驗統計、灰關聯分析、接收者操作特徵、MATLAB 工具箱. II.

(9) Summary Purpose The purpose of this study proposes to apply Taylor approximation method in grey prediction (TAMGP) to educational information and measurement. Method Taylor approximation method in grey prediction is developed based on the combination of Taylor approximation method from approximation optimization theory and grey prediction models from grey system theory. The combined models can obtain the most optimal predicted values by multi-times approximate calculation. Results (1) Applying TAMGP in educational information and measurement for prediction problems, especially using Taylor approximation method in RaschGSP IRT to improve the accuracy and to optimize coefficients α, β and γ of RaschGSP IRT. (2) Using the combination of TAMGP and GRA to predict and evaluate the academic achievement of students, and using the combination of TAMGP, GRA, and ROC to build setting the standard for tests. (3) Developing a MATLAB toolbox based on TAMGP for the purpose of the study and learning of grey system theory. Conclusions Taylor approximation method in grey prediction is actually useful for prediction problems of uncertainty systems and incomplete information in educational information and measurement. It not only can apply to educational information and measurement but also can contribute to educational engineering and kansei engineering.. III.

(10) 總結研究目的本研究目的提出泰勒近似法於灰色預測及其在教育資訊與測驗統計之應用。. 研究方法泰勒近似法於灰色預測是基於泰勒近似法從近似優化理論，結合灰色預測模型從灰色系統理論之發展。將合併的模型透過多次近似計算，而獲得最優化的預測值。. 研究結果一、應用泰勒近似法於灰色預測與教育資訊測驗統計對於所預測的問題，特別是使用泰勒近似法於 RaschGSP IRT 可以改進 RaschGSP IRT 的準確度和最優化係數 α，β 和 γ。二、使用泰勒近似法於灰色預測結合灰關聯分析(Grey Relational Analysis)來預測和評估學生的成就時，同時使用泰勒近似法於灰色預測結合灰關聯分析與接收者操作特徵(Receiver Operating Characteristic)，來建立測試標準設定。三、達成了研究和學習灰色系統理論的目的，基於泰勒近似法於灰色預測建立一個 MATLAB 工具箱。. 研究結論泰勒近似法於灰色預測對系統不明確性及資訊不完整性之預測問題在教育資訊與測驗統計其實是有用的。它不但可以應用在教育資訊與測驗統計，而且在今後的教育工學與感性工學研究上，也有其貢獻效果。. IV.

(11) Contents English Abstract................................................................................................. Page I. Chinese Abstract................................................................................................. II. English Summary............................................................................................... III. Chinese Summary............................................................................................... IV. Contents............................................................................................................... V. List of Tables....................................................................................................... IX. List of Figures..................................................................................................... XI. Notations............................................................................................................. XIII. Chapter 1 Introduction................................................................................... 1. 1.1 Research Background and Motivation................................................... 1. 1.2 Research Purpose and Objectives........................................................... 3. 1.3 Research Method and Research Flowchart............................................ 4. 1.4 Research Questions................................................................................ 5. 1.5 Definition of Terms................................................................................ 5. 1.6 Summary of Research Contribution....................................................... 8. 1.7 Limitations of Research......................................................................... 9. 1.8 Overview of Research............................................................................ 9. Chapter 2 Literature Review......................................................................... 11. 2.1 Research Papers Related to Improving Grey Models............................ 11. 2.2 Grey Model............................................................................................ 19. 2.2.1 GM(1,1)....................................................................................... 19. 2.2.2 Grey Verhulst Model................................................................... 21. 2.2.3 GM(2,1)...................................................................................... 23. 2.2.4 GM(1,n)...................................................................................... 27. 2.2.5 GM(2,n)...................................................................................... 30. 2.2.6 GM(0,n)...................................................................................... 34. 2.3 Taylor Approximation Method in Grey Prediction…........................... 36. V.

(12) 2.4 Error Analysis Method........................................................................... 40. 2.5 Z Scores and T Scores............................................................................ 41. 2.5.1 Z Scores....................................................................................... 41. 2.5.2 T Scores....................................................................................... 42. 2.6 Grey Relational Analysis........................................................................ 42. 2.7 Receiver Operating Characteristic......................................................... 43. 2.7.1 Sensitivity and specificity with their calculation......................... 44. 2.7.2 ROC Curve.................................................................................. 45. 2.8 Setting the Standard for Tests................................................................. 46. 2.9 Conclusion of Chapter 2......................................................................... 46. Chapter 3 TAM in RaschGSP IRT................................................................. 49. 3.1 Item Response Theory............................................................................ 49. 3.2 RaschGSP IRT........................................................................................ 50. 3.3 Taylor Approximation Method in RaschGSP IRT.................................. 56. 3.4 Conclusion of Chapter 3......................................................................... 59. Chapter 4 Experimental Methods.................................................................. 61. 4.1 Research Design..................................................................................... 61. 4.2 Significance of the Research.................................................................. 62. 4.3 Research Methods.................................................................................. 63. 4.3.1 Using TAMGP to Predict the Number of Students and Teachers for Admission, and the Number of Foreign Students Studying in Taiwan………. 63. 4.3.2 Using Taylor Approximation Method in RaschGSP IRT…........ 64. 4.3.3 Using the Combination of TAMGP and GRA to Predict and Evaluate the Academic Achievement of Students................................................ 4.3.4 Using the Combination of TAMGP, GRA, and ROC to Set the Standard for Tests................................................................................................ 4.3.5 Building a MATLAB Toolbox Based on Taylor Approximation Method in Grey Prediction…............................................................................... 4.4 Conclusion of Chapter 4......................................................................... VI. 66 67 69 80.

(13) Chapter 5 Applications of TAMGP............................................................... 81. 5.1 Applying TAMGP to Educational Information and Measurement.......... 81. 5.1.1 Using TAM in GM(1,1), GVM, and GM(2,1) ........................... 81. 5.1.2 Using TAM in GM(1,1) and GM(1,n) ....................................... 85. 5.1.3 Using TAM in GM(1,n) and GM(2,n) ....................................... 89. 5.1.4 Using TAM in GM(0,n).............................................................. 91. 5.2 Using Taylor Approximation Method in RaschGSP IRT……................ 93. 5.3 Using the Combination of TAMGP and GRA...................................... 98. 5.4 Using the Combination of TAMGP, GRA, and ROC........................... 102. 5.5 Conclusion of Chapter 5.......................................................................... 106. Chapter 6 A MATLAB Toolbox for TAMGP................................................ 107. 6.1 Graphical User Interface of MATLAB for TAMGP............................... 108. 6.1.1 Using GUI of MATLAB for TAMGP1....................................... 108. 6.1.2 MATLAB code for TAMGP1..................................................... 112. 6.1.3 Using GUI of MATLAB for TAMGP2....................................... 115. 6.1.4 MATLAB code for TAMGP2...................................................... 117. 6.2 Graphical User Interface of MATLAB for Taylor Approximation Method in RaschGSP IRT…................................................................................. 120. 6.2.1 Using GUI of MATLAB for TAM in RaschGSP IRT................... 120. 6.2.2 MATLAB Code for TAM in RaschGSP IRT................................ 122. 6.3 Graphical User Interface of MATLAB for the Combination of TAMGP and GRA............................................................................................................... 124. 6.3.1 Using GUI of MATLAB for TAMGP and GRA.......................... 124. 6.3.2 MATLAB Code for TAMGP and GRA........................................ 126. 6.4 Graphical User Interface of MATLAB for the Combination of TAMGP, GRA, and ROC..................................................................................................... 127. 6.4.1 Using GUI of MATLAB for TAMGP, GRA, and ROC............... 127. 6.4.2 MATLAB Code for TAMGP, GRA, and ROC............................. 129. 6.5 Conclusion of Chapter 6.......................................................................... 130. VII.

(14) Chapter 7 Conclusions and Recommendations............................................ 133. 7.1 Conclusions............................................................................................. 133. 7.2 Recommendations................................................................................... 134. References........................................................................................................... 135. Appendix 1 - Personal Resume.......................................................................... 149. Appendix 2 - Academic Papers of Author........................................................ 151. Appendix 3 - Honorary Awards........................................................................ 157. VIII.

(15) List of Tables Table 2-1. Page Research papers related to improving GM(1,1)............................. 12. Table 2-2. Research papers related to improving grey Verhulst model............ 15. Table 2-3. Research papers related to improving GM(2,1).............................. 16. Table 2-4. Research papers related to improving GM(1,n).............................. 17. Table 2-5. Confusion matrix............................................................................ 44. Table 3-1. GSP chart........................................................................................ 51. Table 4-1. The prediction results and the accuracy of three prediction models............................................................................................. 72. Table 4-2. The prediction results and the accuracy of two prediction models.. 74. Table 4-3. Parameters α, β, γ, and error analysis............................................... 75. Table 4-4 Table 4-5 Table 5-1. The academic achievement in Mathematics of 30 students in six semesters in junior high school....................................................... The academic achievement of senior high school students in the previous five semesters and test score (a part of measured data)... The prediction results and the accuracy of three prediction models……………………………………………………………. 77 79 84. Table 5-2. The number of teachers and students for admission……………... 85. Table 5-3. Results of testing data……………………………………………. 86. Table 5-4. The predicted results and the accuracy of the T-GM(1,1) model.... 86. Table 5-5. The predicted results and the accuracy of the T-GM(1,n) model…. 88. Table 5-6. The number of teachers and students for admission……………... 89. Table 5-7. The predicted results and the error analysis of two prediction models……………………………………………………………. 90. Table 5-8. The academic achievement of 24 students in three subjects……... 92. Table 5-9. Table 5-9 The results of GM(0,n) and T-GM(0,n). 93. Table 5-10. GSP chart for class A…………………………………………….. 95. Table 5-11. Parameters and error analysis (class A)…………………………... 96. IX.

(16) Table 5-12. GSP chart for class B…………………………………………….. 97. Table 5-13. Parameters and error analysis (class B)…………………………. 98. Table 5-14. T Scores for the academic achievement of 30 students………….. 99. Table 5-15 Table 5-16 Table 5-17 Table 5-18. The predicted results, error analysis, Gamma values for 30 students…………………………………………………………... Ranking the academic achievement of students based on Gamma values…………………………………………………………….. The predicted results, error analysis, Gamma values, test scores and state for 286 students (a part of the results)…………………. Hypothetical data for the sensitivity and specificity at various cutoff scores (a part of the results)…………………………………... X. 100 101 103 104.

(17) List of Figures Page Fig. 1-1. Research flowchart of dissertation................................................... 4. Fig. 2-1. Flowchart of Taylor approximation method in grey prediction….... 40. Fig. 2-2. Z scores and T scores in a normal distribution................................. 41. Fig. 3-1. A three-parameter logistic model item characteristic curve............. 50. Fig. 3-2. Test results of the students plotted against the pass-fail categories... 52. Fig. 3-3. Test results plotted against probability of allocation to pass-fail categories………………………………………………………….. 53. Fig. 3-4. Logistic regression curve of test results………………………….... 54. Fig. 3-5. Flowchart of Taylor approximation method in RaschGSP IRT....... 58. Fig. 4-1. Framework of the dissertation.......................................................... 61. Fig. 4-2. Research flowchart for TAMGP....................................................... 64. Fig. 4-3 Fig. 4-4 Fig. 4-5. Research flowchart for Taylor approximation method in RaschGSP IRT…............................................................................ Research flowchart for the combination of TAMGP and GRA....... Research flowchart for the combination of TAMGP, GRA and ROC.................................................................................................. 65 66 68. Fig. 4-6. Progress of designing the MATLAB toolbox.................................. 69. Fig. 4-7. GUI of MATLAB for T-GM(1,1), T-GVM, and T-GM(2,1)........... 71. Fig. 4-8. GUI of MATLAB for T-GM(1,n) and T-GM(2,n)........................... 73. Fig. 4-9 Fig. 4-10. GUI of MATLAB for Taylor approximation method in RaschGSP IRT….............................................................................................. GUI of MATLAB for predicting and evaluating the academic achievement of students.................................................................. 74 76. Fig. 4-11. GUI of MATLAB to set the standard for tests................................. 78. Fig. 5-1. Graphs for results and error analysis based on the T-GVM model…. 82. Fig. 5-2. Graphs for results and error analysis based on the T-GM(2,1) model……………………………………………………………... XI. 82.

(18) Fig. 5-3. Graphs for results and error analysis based on the T-GM(1,1) model……………………………………………………………... 83. Fig. 5-4. T-RGSP graph for 30 students (class A)………………………….. 95. Fig. 5-5. T-RGSP graph for 25 problems (class A)…………………………. 96. Fig. 5-6. T-RGSP graph for 30 students (class B)………………………….. 97. Fig. 5-7. T-RGSP graph for 25 problems (class B)…………………………. 98. Fig. 5-8. The ROC curve and the area under the ROC curve (AUC)…….... 105. Fig. 6-1. Graphical user interface of the MATALAB toolbox……………... 108. Fig. 6-2 Fig. 6-3. Using GUI of MATLAB for T-GM(1,1), T-GVM, and T-GM(2,1) to input data………………………………………………………. Using GUI of MATLAB for T-GM(1,1), T-GVM, and T-GM(2,1) to test data……………………………………………………….... 109 110. Fig. 6-4. The results and the graphs for the T-GM(1,1) model…………….. 110. Fig. 6-5. The results and the graphs for the T-GVM model………………... 111. Fig. 6-6. The results and the graphs for the T-GM(2,1) model…………….. 112. Fig. 6-7. Using GUI of MATLAB for T-GM(1,n) and T-GM(2,n) to input data………………………………………………………………... 115. Fig. 6-8. Results, parameters, error analysis, and graphs for T-GM(1,n)…... 116. Fig. 6-9. Results, parameters, error analysis, and graphs for T-GM(2,n)…... 116. Fig. 6-10. GUI of MATLAB for Taylor approximation method in RaschGSP. 121. Fig. 6-11. The results of TAM in RaschGSP IRT for students………….…… 121. Fig. 6-12. The results of TAM in RaschGSP IRT for problems………….….. 122. Fig. 6-13. Using GUI of MATLAB for TAMGP – GRA to input data………. 124. Fig. 6-14. The results of TAMGP and GRA for 30 students…………………. 125. Fig. 6-15. Using GUI of TAMGP – GRA to save the results………………… 125. Fig. 6-16. Using GUI of MATLAB for TAMGP – GRA – ROC to input data 127. Fig. 6-17 Fig. 6-18. The results of setting the standard for tests based on GUI of MATLAB for the combination of TAMGP, GRA, and ROC…..….. 128. Using GUI of TAMGP – GRA to save the graph……..………….. 129. XII.

(19) Notations h. Step length. x0. In GRA system, reference vector. xi. In GRA system, inspected vector. xij. In S-P chart, Item response result of student i for item j, i  1,2,  , m ; j  1, 2 ,  , n. AGO. In GM(m,n) calculation, Accumulated Generating Operation. AUC. In ROC analysis, Area under the ROC curve. CPj. In S-P chart, Caution index for problem j. CSi. In S-P chart, Caution index for student i. GM(m,n). Grey model, where m is the order of the difference equation and n is the number of variables. GRA. Grey Relational Analysis. GSP. In GSP chart, Grey Student Problem. GUI. Graphical User Interface. GVM. Grey Verhulst Model. H. Adjustment coefficient. IAGO In GM(m,n) calculation, Inverse Accumulated Generation Operation J. In ROC analysis, Youden index. K. Updated times. MAPE Mean Absolute Percentage Error. Pj. In S-P chart, Problem-number,. Q. The predicted error XIII. j  1, 2 ,  , n.

(20) RMSPE Root Mean Square Percentage Error ROC. Receiver Operating Characteristic. Se. In ROC analysis, Sensitivity. Si. In S-P chart, Student-number, i  1,2,  , m. Sp. In ROC analysis, Specificity. TAM. Taylor Approximation Method. TAMGP Taylor Approximation Method in Grey Prediction aˆ. Coefficients of grey model. xi(0). In GM(m,n) calculation, the raw data sequence, where i  1,2,  , m. xi(1). ( 0) The 1-AGO sequence of xi , where i  1,2,  , m. xˆ i(0). In GM(m,n) calculation, the predicted data sequence, where i  1,2,  , m. z i(1). The background values, where i  2,3, , m. ɛ. Tolerance error.  (0) (k) Class ratio, where k  2,3, , m.  0i. In GRA system, Absolute value of the difference between x0 and xi. 0i. In GRA system, Localized grey relational grade (LGRG). XIV.

(21) Chapter 1 Introduction This chapter is an overall introduction of the study. It first highlights the research motivation of the paper through using Taylor approximation method in grey prediction and its applications in educational information and measurement. Also, the methods of improving the predicted accuracy related to the grey models are introduced. The research purpose and objectives are then discussed, followed by the research questions. Next, the explanations of terms are introduced, and the significance of this paper is subsequently pointed out. Finally, an overview of the study has been presented.. 1.1 Research Background and Motivation In educational information and measurement, prediction plays a very important role, it provides to the educational administrators with important information which is the basis for accelerating the construction, upgrading and rehabilitation of new and existing infrastructure in education, and enhancing the capacity to deliver infrastructure in education to continue to attract and efficient use of resources, bring international cooperation for training and scientific research. This is critical for improving the overall quality of teaching and learning, and improving the quality of education across the country. In addition, it provides the important information for educational administrators to proactively propose the appropriate policy, and building the educational development strategy in accordance with the new conditions. The predicted data are required reliable and high accuracy to contribute to the success in the educational development of the country. In educational information and measurement, when the number of data in the system is not enough for traditional statistical methods, the application of grey system theory can get good results in which grey prediction models have a very important role for prediction problems. Grey prediction models have the advantages of establishing a model with few data and uncertain data and has become the core of grey system theory. Prediction is to analyze the developing tendency in the future according to the past facts. 1.

(22) Most of the prediction methods need a large amount of history data, and use the statistic method to analyze the characteristics of the system. Grey prediction models are the essential part of the grey system theory, GM(m,n) denotes a grey model, where m is the order of the difference equation and n is the number of variables. In recent years, grey model has been successfully used in finance, physical control, engineering and economics. The advantages of grey model include: (1) It can be used in circumstances with relatively little data; as low as four observations were reported to estimate the outcome of an unknown system; (2) It can use a first-order differential equation to characterize a system. Therefore, only a few discrete data are sufficient to characterize an unknown system. However, many researchers have pointed that there were some problems occurred that the predicted accuracy of grey prediction models were unsatisfied, the coefficients of grey prediction models were not the optimal coefficients, and the prediction precision of the grey models was not stable, they have performed a lot of researches for this to improve the predicted accuracy. In 2008, Li and co-workers proposed the T-GM(1,2) model, it was established based on Taylor approximation method to enhance the accuracy of prediction for GM(1,2) (Li et al., 2008b). The experimental results showed that the TGM(1,2) model has high predictable power for asphalt pavement permanent deformation. In 2011, Li and co-workers used the T-GM(1,2) model to develop a prediction model of steel's tensile strength with the Brinell hardness acting as a leading indicator for a high temperature (Li et al., 2011). In 2014, Sheu and co-workers used Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1) (Sheu et al., 2014b). They also used the combination of GM(1,1) and Taylor approximation method to predict the academic achievement of student (Sheu et al., 2014c). In 2014, Nguyen and co-workers used Taylor approximation method in grey system theory to predict the number of teachers and students for admission in Vietnam (Nguyen et al., 2014b). They also used Taylor approximation method in grey system theory to predict the number of foreign students studying in Taiwan (Nguyen et al., 2014a). From the researches presented above show that using Taylor approximation method 2.

(23) in grey prediction can obtain the most optimal prediction values by multi-times approximate calculation. Taylor approximation method which combines the Taylor development with the least squares method is an approximate calculation method of multi-times to obtain the optimal parameters and it can make the convergent error reduce to the minimum. Taylor approximation method in grey prediction can be adjusted repeatedly until reaching the optimal values and can make the predicted error reduce to the minimum. In this study, Taylor approximation method in grey prediction is proposed to apply in educational information and measurement for prediction problems of uncertainty systems and incomplete information, it is also suggested to apply in many different fields such as engineering, economics, and medicine.. 1.2 Research Purpose and Objectives The purpose and objectives of this study are as follows: (1) This study introduces Taylor approximation method in grey prediction to improve the accuracy of grey prediction models from grey system theory. Next, the study proposes to apply Taylor approximation method in grey prediction to educational information and measurement for prediction problems of uncertainty systems, especially using Taylor approximation method in RaschGSP IRT to improve the accuracy and to optimize coefficients α, β and γ of RaschGSP IRT. (2) This study proposes using the combination of Taylor approximation method in grey prediction and grey relational analysis to predict and evaluate the academic achievement of students. The study also proposes using the combination of Taylor approximation method in grey prediction, grey relational analysis and receiver operating characteristic to set the standard for tests. (3) This study proposes to build a MATLAB toolbox based on Taylor approximation method in grey prediction for the purpose of the study and learning of grey system theory. The MATLAB toolbox will not only help user to process data quickly and accurately but also can display results and graphs on a graphical user interface visually. 3.

(24) 1.3 Research Method and Research Flowchart The dissertation is mainly studied based on the quantitative method, experimental method, comparative method, and analytical method. In general, it was conducted according to Fig. 1-1. Start Research motivation and purpose. Related literature review and discussion Grey system theory No Taylor approximation method in grey system theory (TAMGP) Applications of TAMGP in educational information and measurement Building a MATLAB toolbox. Results discussion Yes End. Fig. 1-1 Research flowchart of the dissertation Research flowchart of the dissertation is summarized as follows: Firstly, research motivation and purpose are determined for the prediction problems of uncertainty systems in educational information and measurement. Secondly, related 4.

(25) literature review and discussion are studied to select grey prediction models of grey system theory. However, the predicted accuracy of grey models is unsatisfied, the coefficients of the prediction models are not the optimal coefficients. Thirdly, Taylor approximation method is used in grey prediction to improve the accuracy of the prediction models. Then this method is used to apply for the prediction problems of uncertainty systems in educational information and measurement. Fourthly, a MATLAB toolbox is developed to help processing data quickly and accurately. Finally, the results are discussed to draw conclusions.. 1.4 Research Questions Below are the major research questions to be answered in this study. How to get good results for data of educational information and measurement when the number of data in the system is not enough for large sample size of statistical methods? There is a problem that the predicted accuracy of grey prediction models is unsatisfied. The coefficients of grey prediction models based on grey system theory are not the optimal coefficients. How to improve the predicted accuracy of grey prediction models? Can researchers use the combination of Taylor approximation method in grey prediction, grey relational analysis, and receiver operating characteristic in prediction, evaluation, and setting the standard for tests? How to build a MATLAB toolbox based on Taylor approximation method in grey prediction for the purpose of the study and learning of grey system theory, and this toolbox not only can help to process data quickly and accurately but also can display results and graphs on a graphical user interface visually?. 1.5 Definition of Terms To guide the preparation of this study, the following terms of grey system theory, GSP chart and RaschGSP IRT, Taylor approximation method, grey relational analysis, and receiver operating characteristic are clearly defined prior to the body of the 5.

(26) dissertation. In this study, Taylor approximation method in grey prediction is used to improve the predicted accuracy of grey models, and Taylor approximation method in RaschGSP IRT is also used to enhance the accuracy of RaschGSP IRT curves. Grey relational analysis is used to calculate the Gamma values for students and problems in RaschGSP IRT. Grey relational analysis is also used to calculate the Gamma values of the academic achievement of students. The results obtained from Taylor approximation method in grey prediction combine with GRA to evaluate the academic achievement of students, and they are the basis for dividing students into master and non-master groups based on the Gamma values to set the standard for tests. After the test score distributions of these two groups are determined, the possible cut scores are established. The Youden indexes corresponding to possible cut scores are calculated to pick the optimal cut score. ROC graph is plotted and AUC is also determined to set the standard for tests. Grey System Theory Deng proposed grey system theory to study the uncertainty of a system. As far as information is concerned, the systems which lack information, such as structure message, operation mechanism and behavior document, are referred to as grey systems. The grey system theory includes five major parts, which include grey prediction, grey relation, grey decision, grey programming and grey control. Grey model is the core of grey system theory, which collects available data to obtain the internal regularity without using any assumptions. The intention is to make forecasting useful for decision and policy makers who need future predictions. GSP Chart and RaschGSP IRT GSP chart, designed by Nagai in 2010, is combined with GRA and S-P chart, becoming a method to identify the uncertain causes. GSP chart can help to analyze problem more specifically based on mathematic and scientific data (Sheu et al., 2013). RaschGSP IRT proposed in this research adapts GRA and the theoretical structure of S-P chart can show examinees’ reaction when they are answering items and can 6.

(27) measure students’ discrimination α, the average of item difficulty β, and the worst grade γ. Through GSP chart and distribution of γ, the classification of students and items is identified (Sheu et al., 2014; Tzeng et al., 2012). Taylor Approximation Method (t ) If a function f(x) has derivatives of order t, that is f ( x) . dt f ( x) exists, then dxt. for any constant a, the Taylor polynomial of order t about a is (Novaprateep, Neamprem, & Kaneko, 2012; Ren, Zhang, & Qiao, 1999). f (k ) (a) ( x  a) k k! k 0 t. (1-1). gt ( x)  . While the Taylor polynomial was introduced as far back as beginning calculus, the major theorem from Taylor is that the remainder from the approximation, namely. f (x)  gt (x) , tends to 0 faster than the highest-order term in gt (x) . (t ) If f (a) . lim x a. dt f ( x) | xa exists, then dxt. f ( x)  g t ( x) ( x  a) t. 0. (1-2). Grey Relational Analysis (GRA) GRA can treat uncertain, multiple, discrete and incomplete information effectively. GRA can not only count and quantize the discrete data, but also make them ordinal to be analyzed. It uses a specific concept of information which defines situations with no information as black, and those with perfect information as white. However, neither of these idealized situations ever occurs in real world problems. In fact, situations between these extremes are described as being grey, hazy or fuzzy. Therefore, a grey system means that a system in which part of information is known and part of information is unknown.. 7.

(28) Receiver Operating Characteristic (ROC) ROC analysis is a useful tool for evaluating the performance of diagnostic tests and more generally for evaluating the accuracy of a statistical model such as logistic regression and discriminant analysis, it classifies the objects into one of two categories, diseased or non-diseased, corresponding to the tests in education context, those two states can be achieved qualification and non-achieved qualification. Therefore, for applying the ROC method in educational assessment, the concepts of sensitivity and specificity have to be defined.. 1.6 Summary of Research Contribution The contribution of the study can be summarized as follows: In this study, Taylor approximation method in grey prediction is proposed to apply for prediction problems of uncertainty systems in educational information and measurement. The results will provide the important information for educational administrators to proactively propose the appropriate policy and to build the educational development strategy in accordance with the new conditions in the future. This study uses Taylor approximation method in RaschGSP IRT to enhance the accuracy of RaschGSP IRT curves for students and problems. In addition, this study uses the combination of Taylor approximation method in grey prediction and grey relational analysis to predict and evaluate the academic achievement of students. This is critical for improving the overall quality of teaching and learning and improving the quality of education. This study also uses the combination of Taylor approximation method in grey prediction, grey relational analysis, and receiver operating characteristic to set the standard for tests. This method is not only objective, but also helps teachers to make independent decisions, the result obtained objective and reliable, it helps educators more convenient to establish the standard for tests in the teaching process. Moreover, this study also develops a MATLAB toolbox for the prediction models based on Taylor approximation method in grey prediction. This toolbox has many advantages such 8.

(29) as: easy to use, time-saving, accurate and clearly visual output. Especially, the user can save the results as an EXCEL file and the graphs as an image file (JPG). To sum up, Taylor approximation method in grey prediction is actually useful for the predicted problems of uncertainty systems when the number of data is not enough for statistical methods. In addition, the MATLAB toolbox not only helps to process data quickly and accurately, but also displays the results on a graphical user interface visually. Moreover, the results and proposed methods in this dissertation could be expanded as the basis for future research.. 1.7 Limitations of Research The limitations of reasearch are summarized as follows: This study only uses common prediction models to predict and evaluate in educational information and measurement. In the future, it will be suggested to develop new prediction models based on Taylor approximation method in grey prediction. In this study, the graphical user interface of the MATLAB toolbox is not good to display graphs with high resolution. Thus, the research is suggested to develop in order to achieve better efficiency in the future. For the prediction models, this study only compares the results obtained from grey models and Taylor approximation method in grey prediction. In the future, they will be compared with other prediction models.. 1.8 Overview of Research This dissertation is comprised of 7 chapters. Chapter 1 introduces research background and focus of the study: research purpose and objectives, research method and research flowchart, research questions, definition of terms, summary of research contribution, limitations of this study, and overview of this study. Chapter 2 is a description of the literature relevant to research papers related to improving grey models will be introduced, then the predicted accuracy of grey models are discussed, and Taylor approximation method 9.

(30) in grey prediction is proposed to apply in educational information and measurement. Next, grey models, Taylor approximation method in grey prediction, error analysis method, Z scores and T scores, grey relational analysis, receiver operating characteristic, and setting the standard for tests are presented. In chapter 3, item response theory, RaschGSP IRT, and Taylor approximation method in RaschGSP IRT are introduced. Chapter 4 is Taylor approximation method in grey prediction is proposed to apply for prediction problems of uncertainty systems in educational information and measurement. Especially, using Taylor approximation method in RaschGSP IRT to enhance the accuracy of RaschGSP IRT curves for students and problems. The combination of TAMGP and grey relational analysis is proposed to predict and evaluate the academic achievement of students. The combination of TAMGP, grey relational analysis, and receiver operating characteristic also is proposed to use setting the standard for tests. Chapter 5 is the discussion of research results, details the results and findings of the present study, including the applications of Taylor approximation method in grey prediction in educational information and measurement; using Taylor approximation method in RaschGSP IRT to improve the accuracy of RaschGSP IRT curves; the combination of Taylor approximation method in grey prediction, grey relational analysis, and receiver operating characteristic in prediction, evaluation, and setting the standard for tests. Chapter 6 describes a MATLAB toolbox based on Taylor approximation method in grey prediction. It includes Graphical User Interface (GUI) of MATLAB for Taylor approximation method in grey prediction, GUI of MATLAB for Taylor approximation method in RaschGSP IRT, GUI of MATLAB for the combination of Taylor approximation method in grey prediction and GRA, and GUI of MATLAB for the combination of Taylor approximation method in grey prediction, GRA, and ROC. In chapter 7, conclusions and recommendations are drawn. Main themes based on the research questions are discussed, followed by suggestions for future research.. 10.

(31) Chapter 2 Literature Review This chapter is a description of the literature relevant to Taylor approximation method in grey prediction. Firstly, research papers related to improving grey models are presented; Next, grey models from grey system theory are introduced, which include GM(1,1), grey Verhulst model (abbreviated as GVM), GM(2,1), GM(1,n), GM(2,n), and GM(0,n); Then, Taylor approximation method in grey prediction is explained by the algorithm of T-GM(m,n); Secondly, error analysis methods are also introduced to calculate the accuracy of the prediction models; Z scores, T scores, grey relational analysis, receiver operating characteristic, and setting the standard for tests are also presented. Finally, conclusions of this chapter are drawn.. 2.1 Research Papers Related to Improving Grey Models Grey system theory was initiated in 1982 by Deng. It can effectively deal with incomplete and uncertain information (Deng, 1989; Liu & Forrest, 2007). In grey system theory, if the system information is fully known, the system is called a white system, while the system information is unknown, it is called a black system. A system with partial information known and partial information unknown is grey system (Liu & Lin, 2006, 2010). It included five major parts that are grey prediction, grey relation, grey decision, grey programming, and grey control (Li et al., 2010; Li et al., 2008b). Grey models are the essential part of the grey system theory, GM(m,n) denotes a grey model, where m is the order of the difference equation and n is the number of variables. In recent years, grey model has been successfully used in finance, physical control, engineering and economics. The advantages of grey model include: (1) It can be used in circumstances with relatively little data; as low as four observations were reported to estimate the outcome of an unknown system; (2) It can use a first-order differential equation to characterize a system. Therefore, only a few discrete data are sufficient to characterize an unknown system. However, there 11.

(32) is a problem that the predicted accuracy of grey model is unsatisfied. The coefficients of the prediction model are not the optimal coefficients. Many researchers have performed a lot of researches for this to improve the predicted accuracy. In this study, research papers related to improving GM(1,1), grey Verhulst model, GM(2,1), and GM(1,n) are introduced as follows: Table 2-1 Research papers related to improving GM(1,1) Researchers (Year) Tien and Chen (1997) Hsu and Wen (1998) Wen, Huang, and Wen (2000) Hsu and Chen (2003) Tsaur (2005) Li, Yamaguchi, and Nagai (2007) Lin and Lee (2007) Chen (2008) Li, Yamaguchi, Nagai, and Masuda (2008a) Wang, Dang, and Liu (2008) Lin, Lee, and Chang (2009) Tien (2009b) Chen (2010) Li et al. (2010) Pi, Liu, and Qin (2010) Zeng, Liu, and Xie (2010). The main content of paper The indirect measurement of fatigue limits of structural steel by the deterministic grey dynamic model DGDM(1,1,1). They combined residual modification and residual Markov-chain sign estimation to improve the accuracy of the original models. They analyzed a predicted error in using GM(1,1) based on the parameter α. Using the criterion of α to decide the optimal value of α has both theoretical and practical possibilities. Using a technique that combines residual modification with artificial neural network sign estimation is proposed. The fuzzy system derived from collected data is considered by the fuzzy grey controlled variable to derive a fuzzy GM(1,1). They proposed a new dynamic analysis model which combines the first-order one-variable grey differential equation model from grey system theory and Markov chain model from stochastic process theory. They called the improved model as T-MCGM(1,1). They proposed a novel prediction model termed MFGMn(1,1). Chen proposed Nonlinear Grey Bernoulli Model, which is a nonlinear differential equation with power n. They proposed a new prediction analysis model which combines GM(1,1) model from grey system theory and time series ARIMA model from statistics theory. The generated model is called as M3PARGM(1,1) model. They proposed a new method for optimizing background value in GM(1,1). They proposed a new model named EFGMm(1,1) by eliminating the error term resulted from the traditional calculation of background value with an integration equation to substitute for such error term. Tien proposed a grey prediction model called first-entry GM(1,1), abbreviated as FGM(1,1) Chen proposed a new grey model DPGM(1,1) based on the analysis of the relationship between GM(1,1) and DGM(1,1). To improve the prediction accuracy of GM(1,1), They proposed a prediction model P-3spGM(1,1). The original GM(1,1) model is improved by using three methodologies of the 3-points average technology and the residual modification. Zeng proposed a prediction model of interval grey number based on DGM(1,1). (table continues) 12.

(33) Table 2-1 Research papers related to improving GM(1,1) (continued) Researchers (Year) Chen, Guo, and Lo (2011) Kong, Liu, and Wei (2011) Chen, Sun, and Liu (2012) Huang (2012) Huo and Zhan (2012) Li, Masuda, and Nagai (2012) Li and Wei (2012) Pao, Fu, and Tseng (2012) Truong and Ahn (2012) Wang et al. (2012) Cui et al. (2013) Li, Masuda, and Nagai (2013) Wang (2013) Guo et al. (2014) Li, Masuda, and Nagai (2014c) Li, Masuda, and Nagai (2014b) Li, Masuda, and Nagai (2014a) Liu et al. (2014) Qu, He, and Jia (2014). The main content of paper They proposed a hybrid grey model termed as EGM(1,1), which adopting exponential series to identify the residual error series resulted from grey model. They proposed a kind of data processing method to make the class ratio of the transformed sequence approximate the minimum or maximum class ratio of the original sequence for being better suited to construct R-ODGM(1,1) model and M-ODGM(1,1) model. The parameter is optimized based on genetic algorithm method in the unbiased GM(1,1) power model to minimize the average relative proportional error of accuracy. They applied GM(1,1) with adaptive levels of α (hereafter GM(1,1)-α model) to provide a concise prediction model. They proposed an improved GM(1,1), which used Fourier series to correct the residual of original value and predictive value, and reconstructed GM(1,1) white background value based on genetic algorithm. They proposed a prediction model MC-T-ESGM(1,1). They established the optimization of GM(1,1) direct model. They proposed a numerical iterative method to optimize the parameter of NGBM. They proposed a prediction method based on the modified grey model with first order - one variable - MGM(1,1). To improve the prediction accuracy, an adaptive parameter learning mechanism is applied to SFGM(1,1) model to develop a new model named APL-SFGM(1,1). They proposed a novel grey prediction model termed NGM model and its optimized model. They proposed an improved hybrid optimization model based on GM(1,1) to develop the prediction model in power systems. The improved model is defined as T-MC-RGM(1,1). Wang proposed a new method for optimizing Nash nonlinear grey Bernoulli model (Nash NGBM(1,1)). A novel grey NGM(1,1,k) self-memory coupling prediction model is put forward in order to promote the predictive performance. They proposed an improved grey model to acquire high-control system performance. They proposed an improved grey model to predict Japan's domestic and overseas automobile production. They proposed a novel grey prediction model to enhance the performance of prediction for the amount of fixed-line and cellular phone subscribers in Japan. They developed a optimization model for the GM(1,1) model problem which includes optimization of initial and background values. They proposed an adaptive multi-variable optimized GM(1,1) based on cuckoo search algorithm. (table continues) 13.

(34) Table 2-1 Research papers related to improving GM(1,1) (continued) Researchers (Year) Xiao, Guo, and Mao (2014) Zhang et al. (2014) Sheu et al. (2014c) Sheu et al. (2014b) Nguyen et al. (2014b) Nguyen et al. (2014a). The main content of paper The extension form GGM(1,1) based on the fractional order accumulated generating is put forward and its theoretical significance is analyzed. They proposed an improved Nash nonlinear grey Bernoulli model termed PSO–NNGBM(1,1) by using a particle swarm optimization algorithm. Using the combination of GM(1,1) and Taylor approximation method to Predict the academic achievement of student. Using Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1). Using Taylor approximation method in grey system theory to predict the number of teachers and students for admission. Using Taylor approximation method in grey system theory to predict the number of foreign students studying in Taiwan.. According to Table 2-1, grey system theory was firstly proposed by Deng in 1982. Since then, it has become a very popular technique with its applications on the partially unknown parameters, variables etc. The GM(1,1) model is one of the most frequently used grey prediction model. This model is a time series prediction model, encompassing a group of differential equations adapted for parameter variance, rather than a first order differential equation. Its difference equations have structures that vary with time rather than being general difference equations. However, many researchers have pointed that using the GM(1,1) model may face the predicted accuracy of the GM(1,1) model was unsatisfied. First of all, a real system will grow at different speeds during the whole period, but it is difficult for the original GM(1,1) model to reflect real growth trends among the different periods, since it is just suitable for one exponential growth rule. Secondly, it has been proven that this model is not suitable for long-term prediction, since the absolute value of model coefficient is too large it may lead to a larger prediction error. In order to solve these problems, many researchers have performed a lot of researches for this to improve the predicted accuracy of the GM(1,1) prediction model. (Hsu and Chen, 2003). The GM(1,1) model has been applied to many real life systems such as social, economic and technical systems (Jian, Wakamatsu, & Feng, 1991; Liang, Liu, & Li, 2014; Wu, Hsiao, & Tsai, 2008; Yang, 1993; Zhu, 2014). 14.

(35) Table 2-2 Research papers related to improving Grey Verhulst Model Researchers (Year) Guo, Song, and Ye (2005) Hsu (2008) Wang and Song (2008) Wang, Dang, and Liu (2009) Liu and Bi (2010) Zhang (2012) Wang et al. (2013) Zeng et al. (2013) Zhou (2013) Kordnoori, Mostafaei, and Kordnoori (2014) Sheu et al. (2014b). The main content of paper They used the grey Verhulst model on time series error corrected for the port throughput forecasting. This paper tended to set up a saturated analysis model of the population of Taiwan by utilizing grey Verhulst model and GM(1,1) model. They imported grey system Verhulst model theory, established the manpower management forecast model for transforming maintenance system. They proposed unbiased grey Verhulst model. Recursive solutions are given under two initial conditions of the unbiased model. The Verhulst model with remedy and its application in forecasting quantity of student taking entrance examination to college. Zhang analyzed the cause of grey Verhulst model’s inaccuracy, the new formula of grey derivative is strutted and the unbiased grey Verhulst model is given in this paper. They proposed the time-delayed Verhulst model and then establish a grey time-delayed Verhulst model using the method of grey differential equations. They proposed the Verhulst model of interval grey number based on Information decomposing and model combination. Zhou proposed a new time series prediction model for the time series growth in S-type or growth being saturated. They suggested a new Grey Verhulst model and Fourier residual Grey Verhulst model to improve the predicted accuracy. Using Taylor approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1).. Verhulst model was first proposed by Germany biologist Verhulst to describe some increasing process like “S” curve which has saturation. It has been extensively used in numerous applications to explain the phenomenon of population increasing, living creature breeding and its individual growth. Grey Verhulst model, which is a first-order one-variable grey differential equation and also a time series model. It is a special grey prediction model which is developed to deal with the simulation for small sample data sequence with the characteristic of approximate single peak. This model is capable of simulating the time sequence data with the characteristic of saturated S curved. In recent years, it has been widely applied in some research fields as throughput forecasting, the population prediction of Taiwan, and predicting the road traffic accident (Guo et al., 2005; Hsu, 2008; Mohammadi et al., 2011). 15.

(36) Table 2-3 Research papers related to improving GM(2,1) Researchers (Year) Li, Yamaguchi, and Nagai (2005). Li et al. (2007). Kong and Wei (2009). Huang and Wei (2010). Li et al. (2010) Li et al. (2011). Xu et al. (2011). Yong and Wei (2011). Deng et al. (2012) Shao and Su (2012) Nguyen et al. (2014a) Sheu et al. (2014a). The main content of paper They proposed a new method of GM according to Laplace transform in frequency domain. This proposal method is solved to nth order differential equation in respect of the solution of the equation. They proposed a new calculation methods of derivative and background value z to enhance the predicted power according to cubic spline function. They called the improved prediction as T3spGM. Based on the grey differential equation of DGM(2,1), the solution was inconsistent with the connotation expression, and the differential restored value was also inconsistent with the inverseaccumulating restored value. So the optimized DGM(2,1) was presented, which had the white exponential superposition. For S-shaped sequence, they proposed GM(2,1) and time series combined model while non-monotonic oscillations and saturated the residual series model were set, sequence meeting the conditions of residual modeling. They proposed a new calculation of initial value and derivative to enhance the predicted power according to cubic spline function. They called the improved prediction as T-3spGM(2,1). They proposed the improved grey dynamic model GM(2,1), a second order single variable grey model, to enhance the forecasted accuracy. They called the proposed model as 3spGM(2,1) model. They presented an approach to the least squares solution to grey Verhulst model, and verified its feasibility by numerical examples. They also presented the least squares solutions of grey models GM(1,1) and GM(2,1). They optimized the grey derivative and background value of nonequigap DGM(2,1) model by calculating the definite integral of the whitened differential equation, and then, established a new nonequigap DGM(2,1) model. They improved the classical non-equidistant GM(1,1). To be more specific, n-AGO transformation took the place of the original data, which was called as mGM(n,1). Based on the solution structure of white differential equation of DGM(2,1) model, they deduced the new 2-order grey derivative expression. Using Taylor approximation method in grey system theory to predict the number of foreign students studying in Taiwan. Using GM(2, 1) and T-GM(2, 1) to predict the number of students for admission.. According to Table 2-3, the GM(2,1) model is a kind of grey model which is constructed by grey derivative and second-order grey derivative. GM(2,1) model can make up some defect of GM(1,1) model, but it still has its own defects. In order to solve 16.

(37) these problems, many researchers have performed a lot of researches for this to improve the predicted accuracy of the GM(2,1) prediction model. Table 2-4 Research papers related to improving GM(1,n) Researchers (Year) Wu and Chen (2005) Han and Xu (2007) Li et al. (2008b). Tien (2008) Shao and Wei (2009) Tien (2009a) Li et al. (2011) Tien (2011) Tien (2012) Che, Luo, and Liu (2013) Guo, Xiao, and Forrest (2013) Hui, Li, and Shi (2013) Luo (2013) Luo and Liu (2013b) Luo and Liu (2013a) Luo, Liu, and Yi (2013) Shen and Sun (2013). The main content of paper They proposed an integrated prediction method using GMC(1,n) combined with the improved GRA. The parameter q of MGM(1,n) has been optimized, and a multivariable grey model (MGM(1,n,q)) based on genetic algorithm. This improved prediction model combined the first-order two variables grey differential equation model (abbreviated as GM(1,2) model) from grey system theory and Taylor approximation method from approximation optimization theory. Tien proposed a new prediction model, the interval grey dynamic model by convolution integral with the first-order derivative of the 1-AGO data and n series related, abbreviated as IGDMC(1,n). They reconstructed a new GM(1,n) model equation by using a kind of optimized background value. Tien proposed a new prediction model called the deterministic grey dynamic model with convolution integral (DGDMC(1,n)). They developed a prediction model of steel's tensile strength with the Brinell hardness acting as a leading indicator for a high temperature. They abbreviated the combined model as T-GM(1,2). Tien proposed a new model, which is called the first-pair-of-data GMC(1,n), abbreviated as FGMC(1,n). The algorithm of GMC(1,n) is applied to explain why the existing GM(1,n) model is incorrect. They proposed a new non-equidistant multivariable new information MGRM(1,n) model. They proposed a novel comprehensive adaptive grey model CAGM(1,n) in order to overcome the disadvantages existing in the traditional GM(1,n). They proposed a new method to choose optimal forecast variable number and data sample size for multi-variable grey model. Luo established a new information optimizing model can be used to build model in non-equal interval & equal interval time sequences. They established MGRM(1,n) model can be used in non-equal interval & equal interval time series. They established a non-equidistant multivariable MGRM(1,n). Based on accumulated generating operation of reciprocal number, a non-equidistant multivariable new information optimization MGRM(1,n) model was put forward which was taken the mth component as the initialization. They proposed an optimized grey system multivariate GM(1,n) model based on the optimization of background values of GM(1,n). (table continues). 17.

(38) Table 2-4 Research papers related to improving GM(1,n) (continued) Researchers (Year) Wang (2013) He, Che, and Liu (2014) Li et al. (2014) Luo, Che, and Liu (2014) Luo and Liu (2014b) Luo and Liu (2014a) Wang (2014) Wang and Pei (2014) Yang, Deng, and Jiang (2014b) Yang, Deng, and Jiang (2014a) Nguyen et al. (2014b). The main content of paper In order to create the model of differential equations of system using limited and poor datum in the simulation of system dynamic the inverse GM model is proposed by Wang. They established a new non-equidistant multivariable MGM(1,n) model can be used in equidistance & non-equidistance model. They established MGRM(1,n) model can be used in non-equal interval and equal interval time series. A non-equidistant multivariable grey model MGRM(1,n) was built through applying reciprocal accumulated generating operation. Non-equidistant multivariable MGRM(1,n) model based on accumulated generating operation of reciprocal number was put forward which was taken the first component as the initialization. They analyzed the building method of background value in grey model MGRM(1,n). Wang introduced nonlinear parameters into GMC(1,n) model and additionally apply a convolution integral to produce an improved forecasting model here designated as NGMC(1,n). To improve the modelling accuracy of GDMC(1,n), n interpolation coefficients (taken as unknown parameters) are introduced into the background values of the n variables. They constructed the new-information background values of the multivariable non-equidistant new-information grey model MGRM(1,n) using the modeling method of progressive optimization. They proposed a constructing method of the background values of non-equidistant MGRM(1,n) model, and the background value constructing formula was given. Using Taylor approximation method in grey system theory to predict the number of teachers and students for admission.. According to Table 2-4, Li and co-workers proposed the T-GM(1,2) model in 2008 (Li et al., 2008b). In 2011, Li and co-workers used the T-GM(1,2) model to develop a prediction model of steel's tensile strength with the Brinell hardness acting as a leading indicator for a high temperature (Li et al., 2011).. In 2014, Sheu and co-workers used Taylor. approximation method to improve the predicted accuracy of GM(1,1), GVM, and GM(2,1) (Sheu et al., 2014b). They also used the combination of GM(1,1) and Taylor approximation method to predict the academic achievement of student (Sheu et al., 2014c). In 2014, Nguyen and co-workers used Taylor approximation method in grey system theory to predict the number of teachers and students for admission (Nguyen et al., 2014b), and predict the number of foreign students studying in Taiwan (Nguyen et al., 2014a). 18.

(39) 2.2 Grey Model In grey systems theory, GM(m,n) denotes a grey model, where m is the order of the difference equation and n is the number of variables. Grey models can predict the future outputs of systems with relatively high accuracy without a mathematical model of the actual system. Although various types of grey models can be mentioned, most of the previous researchers have focused their attention on the GM(1,1) model in their predictions because of its computational efficiency. In this section, the common grey models are introduced including GM(1,1); grey Verhulst model; GM(2,1); GM(1,n); GM(2,n); and GM(0,n).. 2.2.1 GM(1,1) Before using grey model, the initial data have to be tested based on (2-1) whether the initial data consistent with the prediction model (Nagai & Yamaguchi, 2004). If the (0)  initial data have n  4, x  R , and.  (0) (k ) . 2 x(0) (k 1)   2  and  ( 0 ) ( k )   e n 1 , e n 1  ( 0)   x (k )  . (2-1). (0) where k  2,3,, m;  (k) is called class ratio.. In grey system theory, the GM(1,1) model is one of the most widely used model. In general, GM(1,1) requires four observations to build a prediction model. Assume that is the original sequence as follows.. x(0)  (x(0) (1), x(0) (2),, x(0) (m)). (2-2). where m is the total number of modeling data. The accumulated generating operation (AGO) formation of x (1 ) is defined as:. x(1)  (x(1) (1), x(1) (2),, x(1) (m)) (1) (0) where x (1)  x (1) , and. 19.

(40) t. x (1) (t )   x ( 0 ) (i ) , t  1,2,  , m i 1. The GM(1,1) model can be constructed by establishing a first order differential (1) equation for x (t ) as. dx(1) (t)  ax(1) (t)  b dt. (2-3). Then the discrete form of the GM(1,1) differential equation model is expressed as. x(0) (t)  az(1) (t)  b. (2-4). . . (1) (1) (1) (1) where z  z (2), z (3),, z (m) are called background values of. dx(1) (t) dt. and calculated by. z(1) (t)  0.5(x(1) (t 1)  x(1) (t)),. t  2,3,  , m. Coefficients a and b are called the developing coefficient and grey input, respectively. In practice, coefficients a and b are not calculated directly from Eq. (2-3). Therefore, the solution of Eq. (2-5) can be obtained by using the least square method. That is T aˆ  a, b  (BT B) 1 BT Y.   z (1) (2)  (1)  z (3) B    (1)  z (m). (2-5). 1  1   1. . Y  x ( 0) (2), x ( 0) (3),, x ( 0) (m). . T. Then, the modeling values of Eq. (2-3) are obtained as b b xˆ (1) ( k  1)  ( x ( 0 ) (1)  ) e  ak  a a. Applying the inverse accumulated generation operation (IAGO). xˆ (0) (k 1)  xˆ (1) (k 1)  xˆ (1) (k) ,. k  1, 2 ,  , m. The prediction sequence can be obtained as (Deng, 1989) 20.

(41) xˆ ( 0 ) ( k  1)  ( x ( 0 ) (1) . b )(1  e a ) e  ak , a. k  1, 2 ,  , m. . .. (2-6). . (0) (0) (0) (0) (0) where xˆ (1)  x (1) , xˆ (1), xˆ (2),, xˆ (m) are called the GM(1,1) fitted. values, while. xˆ. (0). . (m 1), xˆ(0) (m  2),, xˆ(0) (m  j). are called the GM(1,1). predicted values. MATLAB Algorithm for GM(1,1) procedure GM(1,1) (0) (0) (0) (0) input x ( 0 ) ; % x ( 0 ) is the original data x  (x (1), x (2),, x (m)).. Testing data; % Testing data to show the initial data consistent with the prediction model. for i = 1 to m do Calculating x (1 ) based on the method of accumulated generating operation. end do for i = 1 to m-1 do Calculating background values z (1 ) . end do T Using the least square method to calculate coefficients aˆ  a, b .. for i = 1 to m+j do Calculating the fitted values and the predicted values xˆ ( 0 ) . end do Error analysis with MAPE and RMSPE. output x ( 0 ) ; xˆ ( 0 ) ; MAPE and RMSPE. end. 2.2.2 Grey Verhulst Model Grey Verhulst Model was first proposed by Germany biologist Verhulst to describe some increasing process like “S” curve which has saturation. It has been extensively used 21.