數值分析法應用在NORTA多變量模擬產生法的前置作業上

NORTA Initialization for Random Vector Generation by Numerical Methods

顏瑞池(Jui-Chih Yen)

指導教授 ： 陳慧芬

中原大學/工學院/工業工程研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200494

摘要

摘 要
數值分析法應用在NORTA多變量模擬產生法的前置作業上
顏瑞池

本論文探討模擬產生多變量觀察值的方法。在模擬模式的輸入模式中可能包含n個具相關性的隨機變數(或稱n維多變量)，例如在股市中，大盤指數與期貨指數會互相影響，如何產生這組具有相關性的隨機變數即是多變量模擬產生法的問題。NORTA（NORmal To Anything）為多變量模擬產生法的一種，此法可產生具指定邊際分配及相關矩陣的多變量觀察值，邊際分配不限定於連續型或標準統計分配，因此可廣泛應用。在使用NORTA之前，前置作業為必須解n(n-1)/2個一元方程式使得產生的多變量具有指定的相關係數值，而方程式的函數值是一個二微積分值。

解決NORTA前置作業問題的方法有模擬法、數值分析法及解析法三種，但解析法須在某些特殊情形下才可使用，例如所有指定邊際分配都是常態分配。在模擬法方面，Chen (2001) 提出解決NORTA前置作業的模擬法，Chen將解方程式的問題視為隨機求根的問題，即利用函數估計值(而非真正值)來解方程式，缺點是計算時間可能較長。在數值分析法方面參考文獻較少，本論文的研究目的在提出一有效率的數值分析法。

我們使用數值分析法來解n(n-1)/2個一元方程式的問題，因為方程式的函數值為二維積分值，數值分析法分成兩部分:積分及解方程式部分。在積分部分，因為當指定的相關係數的值很接近1 (或-1)時，二維積分函數中的標準常態雙變量密度函數會非常陡峭，只有在45度線(或135度線)附近有較高值，其他地區的密度函數值則幾乎接近零。在這種情況下，數值積分法的誤差可能非常高。因此我們把二維函數畫分五個區域，每個區域採效率較高的Gaussian quadrature法來計算積分值。在解方程式部分，我們以二分法及牛頓法合併來求解。

我們以模擬實驗來評估積分及求解部分的精確度，實驗結果顯示當指定邊際分配的偏度越小則我們積分方法的精確度會越高；反之，越差。模擬實驗結果也顯示當指定邊際分配偏度較小時，數值分析法較模擬法精確且計算時間短，但當偏度大時，數值分析法計算時間雖然還是很快，但誤差會較大，此時模擬法為較佳方法。

關鍵詞：多變量模擬產生法，NORTA法，隨機求根，數值分析法，Gaussian quadrature法，牛頓法

關鍵字

多變量模擬產生法 ； 牛頓法 ； 數值分析法 ； NORTA法 ； Gaussian quadrature法 ； 隨機求根

並列摘要

ABSTRACT

NORTA Initialization for Random Vector Generation by Numerical Methods

Jui-Chih Yen

We propose a numerical method for generating observations of a n-dimensional
random vector with arbitrarily specified marginal distributions and correlation matrix.
Our random vector generation (RVG) method uses the NORTA (NORmal To Any-
thing) approach. NORTA generates a random vector by first generating a standard
normal random vector. Then, transform it into a random vector with specified marginal
distributions. During initialization for NORTA, n(n-1)/2 nonlinear equations need
to be solved to assure that the generated random vector has the specified correlation
structure. The root-finding function is a two-dimensional integral.

For NORTA initialization, there are three approaches: analytical, numerical, and
simulation. The analytical approach is exact but applicable only for special cases, such
as normal random vectors. Chen (2001) uses the simulation approach to solve the
n(n-1)/2 equations by treating it as a stochastic root-finding problem, solving equa-
tions using only the estimates of the function values. The disadvantage is that the
computation time is usually longer than the numerical approach.

We use the numerical approach to solves these equations. Since the root-finding
function is a two-dimensional integral, our numerical method includes two parts:
integration and root-finding. For integration, when the specified correlation is close to 1 or-1, the bivariate normal density function in the integrand is steep; the density is high
along the 45 or 135 degree line and almost zero everywhere else. In this situation, the
numerical integration error could be large. Therefore, we divide the integration area
to five parts. The efficient Gaussian-quadrature integration method is used for each
part. For rootfinding, the combination of the bisection and Newton’s methods is used
to guarantee convergence.

Simulation experiments are conducted to evaluate the accuracy of the numerical
integration and root-finding methods. The results show that the numerical integration
method is quite accurate when the skewness of the specified marginal distribution is
small. When the skewness is high, the integration method may have large errors. The
simulation results also show that our numerical RVG method is more accurate and
efficient than Chen’s simulation method when the skewness is small. When the skew-
ness is high, the numerical method is still faster but less accurate. In this case, the
simulation method is a better choice.

Keywords: multivariate random vector generation, NORTA, stochastic root finding,
numerical analysis, Gaussian quadrature, Newton’s method

並列關鍵字

NORTA ； numerical analysis ； Newton’s method ； stochastic root finding ； Gaussian quadrature ； multivariate random vector generation

參考文獻

[4] Chen, H. "Initialization for norta: Generation of random vectors with specified marginals and correlations". INFORMS Journal on Computing

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[8] Clemen, R. T. and Reilly, T. "Correlations and copulas for decision and risk analysis". Management Science, 45, 208-224, 1999.

[11] Kendall, M. G. and Stuart, A. "The Advanced Theory of Statistics" (Vol. 1, 3rd ed.). Gri.n, London, U.K., 1967.