| Summary: | 碩士 === 中原大學 === 工業工程研究所 === 90 === 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
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