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研究生： 黃偉誠
Huang, Wei-Cheng
論文名稱： 使用交替投影法估計出特定結構下的最近矩陣
Alternating projection methods for calculating the nearest structured matrix
指導教授： 林敏雄
Lin, Matthew M.
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 25
中文關鍵詞： 交替投影法 、相關矩陣 、半正定矩陣
外文關鍵詞： Alternating projection method, correlation matrix, positive semi-definite matrix
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    • 在財務金融的領域當中，尋找最接近的相關矩陣是一種觀察股票形式以及此相關矩陣跟初始環境下形成的對稱矩陣之間的關聯性的學問。而在我們這篇論文當中，則是依此問題加上特定條件來建構出兩種最接近的矩陣，第一種是和初始矩陣相同對角線元素的半正定矩陣，第二種是找出幾個含有與原本對稱矩陣相同的非對角線的固定元素的相關矩陣。

    In some finance industry problems, we usually compute the nearest correlation matrix to a given symmetric matrix due to the relationship between the correlation matrix and the behaviors of the stock. Corresponding to the given symmetric matrix, we focus, in this paper, on finding the nearest correlation matrix with two constraints. The first is controlling the matrix with some elements being the same as the given matrix. The second constraint is restricting the matrix in a form with all diagonal elements equaling to the given matrix. Our work is based on alternating projection methods trying to find out the global optimal solution.

    1 Introduction................................................1 2 Alternating projection......................................2 2.1 The correlation matrix with fixed elements...............4 2.2 The nearest positive semi-definite matrix with fixed diagonal elements............................................6 2.3 Projections..............................................6 3 Numerical examples.........................................8 3.1 Example 1: Find the nearest correlation matrix with some fixed elements in W−norm, where W is diagonal................9 3.2 Example 2: Find the nearest correlation matrix with some fixed elements in W−norm, where W is symmetric positive definite....................................................11 3.3 Example 3: Find the nearest positive semi-definite matrix with fixed diagonal elements in W−norm, where W is diagonal..13 3.4 Example 4: Find the nearest positive semi-definite matrix with fixed diagonal elements in W−norm, where W is symmetric positive definite............................................14 4 Conclusion.................................................16 5 Reference...................................................18 6 Appendix...................................................19 6.1 Example : Finding ̃S in Theorem 2.6.......................19 6.2 The nearest correlation matrix with fixed elements.......20 6.3 The nearest positive semi-definite matrix with fixed diagonal elements............................................23

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