latin hypercube sampling

**Latin Hypercube Sampling (LHS)** is a **stratified sampling technique that divides each factor's range into $N$ equal intervals and places exactly one sample point in each interval** — ensuring marginal uniformity for every factor while maintaining good space-filling properties in the full-dimensional space. **How LHS Works** - **Stratification**: Divide each factor range into $N$ equal probability intervals. - **Random Placement**: Place one point randomly within each interval for each factor. - **Permutation**: Randomly permute the assignments across factors to create the design. - **Optimization**: Optimized LHS (MaxiMin, correlation-minimizing) improves multi-dimensional uniformity. **Why It Matters** - **Marginal Coverage**: Guarantees that every region of each variable is sampled — no gaps. - **Efficient**: Provides better coverage than random sampling with the same number of points. - **Standard Practice**: The default sampling method for computer experiments, sensitivity analysis, and Monte Carlo studies. **LHS** is **fair sampling across all dimensions** — ensuring that every factor's range is evenly covered regardless of sample size.