Latin Hypercube Sampling
Latin hypercube sampling stratifies every input dimension so a modest sample covers the space evenly, improving on plain random sampling.
Stratified in every dimension
Latin hypercube sampling (LHS) divides each input's range into N equal-probability intervals and draws exactly one sample from each interval, then pairs the per-dimension draws at random. The result is that every input is evenly represented across its whole range - no interval is over- or under-sampled - which random sampling cannot guarantee for small N.
The Latin square analogy
The name comes from the Latin square: in a grid, exactly one point per row and per column. LHS generalizes this to many dimensions, so each one-dimensional projection of the sample is perfectly stratified. This gives excellent marginal coverage from few points, a major gain for expensive models.
How it is built
- Split each of the d inputs into N equal-probability bins
- Draw one value from each bin per input, giving N values per dimension
- Randomly permute the values in each dimension
- Combine column by column to form N sample points
Improving the pairing
Random pairing can still leave spurious correlations between inputs or leave gaps in the joint space. Optimized LHS variants fix this: maximin LHS maximizes the minimum distance between points, and orthogonal-array-based LHS controls low-dimensional projections. These refinements matter most when fitting surrogates that depend on joint structure.
Strengths and limits
LHS reduces estimator variance for functions dominated by main effects and is a standard default for both UQ sampling and surrogate training designs. Its guarantee is per-dimension, not per joint region, so in high dimensions it does not ensure uniform coverage of the full space the way low-discrepancy sequences aim to. It is also naturally a one-shot design, though extensible variants exist.
In practice
LHS is the common starting design for Kronos surrogate builds and Monte Carlo studies: a Latin hypercube over the uncertain physics inputs seeds the training set for models of the machines, after which adaptive sampling adds points where the surrogate is weakest.