Taylor Approximation Method in Grey Prediction and Its Applications in Educational Information ...

177 Pages·2015·4.99 MB·English

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 博士
永井正武 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 撰
中 華 民 國 一 ○ 四 年 五 月

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

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
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
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
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
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
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