以最小平方法處理有限離散型條件分配相容性問題

Abstract

給定兩個有限離散型條件分配，我們可以去探討有關相容性及唯一性的問題。Tian et al.(2009)提出一個統合的方法，將相容性的問題轉換成具限制條件的線性方程系統(以邊際機率為未知數)，並藉由 l_2-距離測量解之誤差，進而求出最佳解來。他們也提出了電腦數值計算法在檢驗相容性及唯一性時的準則。\n 由於 Tian et al.(2009)的方法是把邊際機率和為 1 的條件放置在線性方程系統中，從理論的觀點來看，我們認為該條件在此種做法下未必會滿足。因此，本文中將邊際機率和為 1 的條件從線性方程系統中抽離出來，放入限制條件中，再對修正後的問題求最佳解。\n 我們提出了兩個解決問題的方法：(一) LRG 法；(二) 干擾參數法。LRG 法是先不管機率值在 0 與 1 之間的限制，在邊際機率和為 1 的條件下，利用 Lagrange 乘數法導出解的公式，之後再利用 Rao-Ghangurde 法進行修正，使解滿足機率值在 0 與 1 之間的要求。干擾參數法是在 Lagrange 乘數法公式解中有關廣義逆矩陣的計算部份引進了微量干擾值，使近似的逆矩陣及解可快速求得。理論證明，引進干擾參數所增加的誤差不超過所選定的干擾值，易言之，由干擾參數法所求出的解幾近最佳解。故干擾參數法在處理相容性問題上，是非常實用、有效的方法。從進一步分析Lagrange 乘數法公式解的過程中，我們也發現了檢驗條件分配\"理論\"相容的充分條件。\n 最後，為了驗證 LRG 法與干擾參數法的可行性，我們利用 MATLAB 設計了程式來處理求解過程中的運算，並以 Tian et al.(2009)文中四個可涵蓋各種情況的範例來解釋說明處理的流程，同時將所獲得的結果和 Tian et al. 的結果做比較。

Given two finite discrete conditional distributions, we could study the compatibility and uniqueness issues. Tian et al.(2009) proposed a unified method by converting the compatibility problem into a system of linear equations with constraints, in which marginal probability values are assumed unknown. It locates the optimum solution by means of the error of l_2 - discrepancy. They also provided criteria for determining the compatibility and uniqueness. Because the condition of sum of the marginal probability values being equal to one is in Tian et al.s’linear system, it might not be fulfilled by the optimum solution. By separating this condition from the linear system and adding into constraints, we would look for the optimum solution after modification.\n We propose two new methods: (1) LRG method and (2) Perturbation method. LRG method ignores the requirement of the probability values being between zero and one initially, it then uses the Lagrange multipliers method to derive the solution for a quadratic optimization problem subject to the sum of the marginal probability values being equal to 1. Afterward we use the Rao-Ghangurde method to modify the computed value to meet the requirement.\n The perturbation method introduces tiny perturbation parameter in finding the generalized inverse for the optimum solution obtained by the Lagrange multipliers method. It can be shown that the increased error is less than the perturbation value introduced. Thus it is a practical and effective method in dealing with compatibility issues. We also find some sufficient conditions for checking the compatibility of conditional distributions from further analysis on the solution given by Lagrange multipliers method. \n To show the feasibilities of LRG method and Perturbation method, we use MATLAB to device a program to conduct them. Several numerical examples raised by Tian et al.(2009) in their article are applied to illustrate our methods. Some comparisons with their method are also presented.