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The pattern block method for generating random numbers : Reformulation and generalization of the Ziggurat method using conditional random variables

Kensuke Ishitani

Abstract

The Ziggurat method is an efficient rejection sampling method for generating $1$-dimensional normally distributed random numbers. This paper proposes the pattern block method, which is a generalization of the Ziggurat method. The pattern block method can be used to generate random numbers from multimodal density functions or multidimensional distributions. We demonstrate the effectiveness of the pattern block method through several examples.

The pattern block method for generating random numbers : Reformulation and generalization of the Ziggurat method using conditional random variables

Abstract

The Ziggurat method is an efficient rejection sampling method for generating -dimensional normally distributed random numbers. This paper proposes the pattern block method, which is a generalization of the Ziggurat method. The pattern block method can be used to generate random numbers from multimodal density functions or multidimensional distributions. We demonstrate the effectiveness of the pattern block method through several examples.
Paper Structure (7 sections, 2 theorems, 40 equations, 9 figures)

This paper contains 7 sections, 2 theorems, 40 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The pattern block method
  3. Applications to generate $1$-dimensional random numbers
  4. The Ziggurat method
  5. The density function with singularities
  6. Application to generate $2$-dimensional random numbers
  7. Conclusion

Key Result

Lemma 2.1

For $A\in \mathcal{E}$, we have

Figures (9)

  • Figure 1: $f(x)$ and its pattern block
  • Figure 2: Ziggurat method
  • Figure 3: $f(x)$ and its pattern block $B_1, B_2, \cdots, B_8$
  • Figure 4: Histogram and density function
  • Figure 5: $f(x_1, x_2)=b_0$
  • ...and 4 more figures

Theorems & Definitions (3)

  • Lemma 2.1
  • proof
  • Theorem 2.1