Generating Singular Multi-Normal Random Vector by Using Square Root Matrix Approach

奇特多元常態分佈隨機向量產生法－運用平方根矩陣法

10.6338/JDA.200812_3(6).0002

李泰明(Tai-Ming Lee)

奇特多元常態分佈 ； Cholesky分解法 ； 平方根矩陣法 ； 蒙地卡羅模擬 ； Singular multi-normal distribution ； Cholesky decomposition ； Square root matrix ； Monte Carlo simulation

Journal of Data Analysis

3卷6期（2008 / 12 / 01）

17 - 29

英文

當共變矩陣或相關矩陣為半正定時，多元常態分佈隨機向量的聯合分配稱為奇特多元常態。當隨機向量有此特性時，傳統的Cholesky分解法將遇到數值上的障礙，原因是此分解法的共變異矩陣必須為正定才能使用。在此文章中作者提出了平方根矩陣的方法來克服這個問題，並詳細說明如何運算一個半正定共變異矩陣的平方根矩陣。利用平方根矩陣的方法可以分解這個矩陣，藉以完成奇特多元常態分佈的蒙地卡羅模擬。運用數值的例子和圖解，可以說明及展示此方法的正確性，及奇特多元常態分佈的機率性質。一些創新的繪圖技術也同時呈現奇特性的意義。

The joint distribution of a random vector is called singular multi-normal distribution, when the corresponding variance covariance matrix or the correlation matrix is only positive semi-definite. To generate the random vector with this property, traditional Cholesky decomposition method is failed because it needs the variance covariance matrix to be positive definite. In this paper, I propose a square matrix approach to overcome this problem. Specifying a particular positive semi-definite variance covariance matrix, the square root matrix approach can still decompose this matrix and perform the Monte Carlo simulation of singular multi-normal distribution. Several numerical example and graphical representation are illustrated as well as some probability evaluation. The techniques to present the singularity are developed and applied to the numerical examples.

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