Motivation
QMCPy provides several low-discrepancy constructions, but does not appear to provide a Randomized Latin Hypercube Sampling (RLHS) distribution.
RLHS is a variance reduction technique that ensures uniform coverage of each dimension while preserving the unbiasedness and convergence properties of standard uniform random sampling.
Proposed feature
Add a Randomized Latin Hypercube Sampling (RLHS) discrete distribution following the existing QMCPy distribution interfaces and conventions.
The construction is based on
$$
X_i^j = \frac{\Pi_j(i) - U_i^j}{n}, \quad i = 1, \dots, n,; j = 1, \dots, d,
$$
where $\Pi_j$ are independent uniform random permutations of ${1, \dots, n}$, and $U_i^j \sim \mathcal{U}([0, 1[)$ are i.i.d. and independent of the permutations.
Each point satisfies $X_i \sim \mathcal{U}([0, 1[^d)$, so all results for uniform random sequences apply (unbiased estimators, convergence rate, etc.). In addition, RLHS provides:
- Better space-filling properties than pure random sampling.
- Exact stratification in each dimension, ensuring that every coordinate stratum is represented exactly once per dimension.
- Reduced variance for integrands that are monotonic or additive.
Proposed implementation
- Add the new distribution to the appropriate QMCPy module.
- Follow the existing distribution API.
- Add unit tests for the mathematical construction and edge cases.
- Add documentation and examples.
- Add references to the relevant literature.
I would be interested in implementing this feature.
Motivation
QMCPy provides several low-discrepancy constructions, but does not appear to provide a Randomized Latin Hypercube Sampling (RLHS) distribution.
RLHS is a variance reduction technique that ensures uniform coverage of each dimension while preserving the unbiasedness and convergence properties of standard uniform random sampling.
Proposed feature
Add a Randomized Latin Hypercube Sampling (RLHS) discrete distribution following the existing QMCPy distribution interfaces and conventions.
The construction is based on
where$\Pi_j$ are independent uniform random permutations of ${1, \dots, n}$ , and $U_i^j \sim \mathcal{U}([0, 1[)$ are i.i.d. and independent of the permutations.
Each point satisfies$X_i \sim \mathcal{U}([0, 1[^d)$ , so all results for uniform random sequences apply (unbiased estimators, convergence rate, etc.). In addition, RLHS provides:
Proposed implementation
I would be interested in implementing this feature.