- How to simulate an area of a circle
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Variance reduction techniques
- authentic variable
- momentum matching
- control variate
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importance sampling
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$$E_{f(x)}[h(x)] = E_{g(x)}[\frac{h(x)f(x)}{g(x)}]$$ - sample from g if hf/g has smaller variance than h(x)
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- Low-discrepancy Sequence
- x,y and x, z correlation 0.8, whats the max/min correlation for z, y?
- cosine similarity
- positive semi-definite property of correlationmatrix
- generate correlated random variables
- Cholesky decomposition
$$x = \mu + R^T z$$ - SVD decomposition
$$x = \mu + UD^{\frac{1}{2}}z$$
- Cholesky decomposition
- generate random variables with distribution
- normal distribution - use Central Limit Theorem
- inverse transformation method
- rejection method
- How to shuffle n numbers with every sequence equal probability
- by sorting + random number sequence : O(nlogn)
- Knuth Shuffle: sample without putting back (each 1/n! probability)
- read data in sequence, how to ensure each data has equal probability to be sampled
- pick the first, change with 1/2 probability if there is a second ... pick n th and keep with probability n/(n+1), change with 1/(n+1)
- give Z-U(0,1) how to draw N(0,1) variable
- give Z-N(0,1) how to draw multi-variate (N(mu, sigma)) variable
- P.D.E
- Implicit method, explicit method, Crank-Nickolson Method