Computational Efficiency of Polar and Box Muller Method: Using Monte Carlo Application
CATEGORY: APPLIED MATHEMATICS
Computational Efficiency of Box-Muller and Polar Method Using Monte-Carlo Application
by : Joy V. Lorin-Picar
Mathematics Department
Davao del Norte State College, New Visayas, Panabo City picar_joy@yahoo.com ABSTRACT The efficiency of Mean Square Error (MSE) of the random normal variables generated from both the Marsaglia Polar Method and Box-Muller Method was examined for small and large n with Monte-Carlo application using MATHLAB. The empirical results showed that MSE of the random normal variables using the Marsaglia Polar Method approaches zero as n becomes larger. Moreover, when run in MATHLAB, the Box-Muller method encountered some problems like: a) it runs slow in generating its MSE because of many calls to the math library; b) it has numerical stability problems when x1 is very close to zero; as a consequence of b, as n becomes large, there are serious problems if you are doing stochastic modeling and generating millions of numbers. Apparently, the Polar Method computes the MSE faster even when n is large, since it does the equivalent of the sine and cosine geometrically without a call to the trigonometric function library.
Keywords: Mean Square Error (MSE), Marsaglia Polar Method, Box-Muller Method, Monte-Carlo application
A. INTRODUCTION The topic of generating Gaussian pseudo-random numbers given a source of uniform pseudo-random numbers comes up more frequently. There are many ways of solving this problem but this paper focuses through the Box-Muller and Marsaglia Polar Methods. If we have an equation that describes our desired distribution function, then it is possible to use some mathematical manipulations based upon the fundamental transformation law of probabilities to obtain a transformation function for the distributions. This transformation takes random variables from one distribution as inputs and outputs random variables in a new distribution function. One of the
Computational Efficiency of Box-Muller and Polar Method Using Monte-Carlo Application
by : Joy V. Lorin-Picar
Mathematics Department
Davao del Norte State College, New Visayas, Panabo City picar_joy@yahoo.com ABSTRACT The efficiency of Mean Square Error (MSE) of the random normal variables generated from both the Marsaglia Polar Method and Box-Muller Method was examined for small and large n with Monte-Carlo application using MATHLAB. The empirical results showed that MSE of the random normal variables using the Marsaglia Polar Method approaches zero as n becomes larger. Moreover, when run in MATHLAB, the Box-Muller method encountered some problems like: a) it runs slow in generating its MSE because of many calls to the math library; b) it has numerical stability problems when x1 is very close to zero; as a consequence of b, as n becomes large, there are serious problems if you are doing stochastic modeling and generating millions of numbers. Apparently, the Polar Method computes the MSE faster even when n is large, since it does the equivalent of the sine and cosine geometrically without a call to the trigonometric function library.
Keywords: Mean Square Error (MSE), Marsaglia Polar Method, Box-Muller Method, Monte-Carlo application
A. INTRODUCTION The topic of generating Gaussian pseudo-random numbers given a source of uniform pseudo-random numbers comes up more frequently. There are many ways of solving this problem but this paper focuses through the Box-Muller and Marsaglia Polar Methods. If we have an equation that describes our desired distribution function, then it is possible to use some mathematical manipulations based upon the fundamental transformation law of probabilities to obtain a transformation function for the distributions. This transformation takes random variables from one distribution as inputs and outputs random variables in a new distribution function. One of the
References: 1. L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986. 2. Everett F. Carter, Jr., The Generation and Application of Random Numbers, Forth Dimensions (1994), Vol. 16, No. 1 & 2. 3. G. E. P. Box and Mervin E. Muller, A Note on the Generation of Random Normal Deviates, The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611 4. Kloeden and Platen, Numerical Solutions of Stochastic Differential Equations, pp. 11–12 5. Sheldon Ross, A First Course in Probability, (2002), pp. 279–81.