2018-04-01 开胃学习数学系列 - 随机数生成

True and pseudo random numbers

True random number generators (T-RNG) observe random physical processes 真随机数发生器（T-RNG）观察随机物理过程：

• often involves quantum effects: radioactive decay, electrical/thermal noise etc. 通常涉及量子效应：放射性衰减，电/热噪声等
• costly and slow 代价高昂，速度慢
• mostly used in cryptography that requires a high level of security 主要用于需要高度安全性的密码学

Pseudo random number generators (P-RNG) are computer software that
伪随机数发生器（P-RNG）是计算机软件：

• produce number sequence that only appears to be random 产生看起来随机的序列
• cheap and fast 便宜又快捷
• can be initialized (seeded) to different states 可以被初始化（播种）到不同的状态
• states can be stored and recovered, convenient for MC 状态可以存储和恢复，方便MC

Pseudo random number generators 伪随机数生成器

• Building good P-RNG requires highly specialized skills:
• must pass rigorous statistical tests 必须通过严格的统计测试
• numerical scientists have devoted their careers to build good P-RNGs
• always use a good P-RNG libarary and never write your own
  使用一个好的P-RNG libarary并且永远不要自己写一个
• random() in Excel is implemented incorrectly, don't use it for anything serious
  在Excel中的random执行是不正确的，不要严重使用它

Mersenne Twister (MT19937)

The most popular P-RNG:

• often the default P-RNG for MC ：the default choice of Python's numpy.random.uniform
• very long cycle of 2^ 19937 − 1 and execellent statistical properties
• suitable for MC applications, but not good for cryptography
• very fast: ~100M random numbers per second on a modern PC

Non-uniform random numbers

Box-muller

Normal random number is the most important of all:

• most quant Finance models are driven by Brownian motions
• fast and accurate inversion of normal CDF weren't known till recently

Box-muller is a classic method to transform independent uniform RNs to independent normal RNs:

• takes 2 independent uniforms and produce 2 independent normals
  把两个独立的
• its polar form is faster and more stable, which is behind Python's numpy.random.normal
• the polar form is a rejection algorithm: the number of output normal RNs is less than the input uniforms
  • about 21% of input uniforms are rejected (wasted)

Normal inversion

Numerical inversion of normal CDF is an effective alternative to genrate normal random numbers:

• fast and accurate numerical inversion of the normal CDF were discovered recently
• based on polynomial approximation, accuracy is ~$10^{-9}$
• every uniform RN produce a normal RN output, more convenient than the polar Box-muller
• works well with low discrepency sequences (next lecture)
  Normal inversion is often preferred over the Box-muller.

How to create correlated normal RNs from independent normals RNs?