Latin hypercube
Space-filling: one sample per row and column.
Latin hypercube sampling spreads experimental points to fill the factor space evenly, one sample per row and per column of every factor's range. It is the right way to sample a high-dimensional space for surrogate models and computer experiments, where the classic corner-based designs do not fit.
What it measures
Latin hypercube sampling (LHS) is a space-filling design, built for a different goal than the classical factorial and response-surface designs:
- The construction: divide each factor's range into n equal intervals, and place n samples so that each interval of each factor contains exactly one sample. In two dimensions that is the Latin-square property, one point per row and per column; in higher dimensions it generalizes to the hypercube.
- What it optimizes for: even coverage of the whole space with a chosen number of runs, rather than precise estimation of a low-order polynomial at specific design points. Good LHS variants also maximize the minimum distance between points or minimize correlation between factors.
- Where it fits: surrogate modeling (training a Gaussian process or other flexible model), computer experiments and simulations, and any problem with enough factors that a full factorial's run count is hopeless.
How to read the output
Choose the sampling strategy by what the data is for. If the goal is to fit a known quadratic and read off main effects and interactions, a structured response-surface design (factorial, CCD, Box-Behnken) is more efficient at that specific job. If the goal is to learn an unknown, possibly wiggly response across many factors, to train a surrogate you will then optimize or explore, you want even coverage, and that is what LHS provides where a corner-based design would leave the interior unsampled. Read an LHS design for its space-filling quality: points spread out, no large empty gaps, low correlation between factors. The number of runs is yours to set, which is part of why it suits expensive simulations and large factor counts.
A real use case
A team is building a surrogate model that maps seven electrode and process parameters to predicted cycle life, so they can later optimize across all seven without running thousands of cells. A full factorial in seven factors is out of the question, and a response-surface design assumes a quadratic shape the true relationship may not have. A Latin hypercube of, say, 60 runs spreads cells evenly across the seven-dimensional space, giving the Gaussian process surrogate even coverage to learn from rather than dense clusters and empty regions. The trained surrogate then drives the optimization, and the space-filling sample is what made it trustworthy across the whole window instead of only near a few corners.
Common mistakes
- Using LHS to estimate specific polynomial coefficients efficiently. For fitting a known low-order model, a structured design does that job with fewer runs.
- Accepting a raw random LHS without checking its space-filling quality. A plain Latin hypercube can still leave clusters and gaps; optimized variants (maximin, low correlation) are better.
- Choosing too few samples for the dimensionality, so the space is not actually filled and the surrogate has blind regions.
- Ignoring factor correlation in the generated design, which can confound the surrogate if two factors end up moving together by chance.
- Treating the surrogate as truth where the samples are sparse. Even good coverage is thin in high dimensions; the model's uncertainty still matters.
Even coverage for surrogates and many-factor problems
Niobia generates space-filling Latin hypercube designs for the factors and run budget you set, spreading samples evenly across the space so a surrogate model has the coverage it needs to learn an unknown response. It pairs naturally with the Gaussian process surrogate it fits, which turns the space-filling sample into a mean-plus-uncertainty model of the whole window, and with Bayesian optimization, which then chooses each subsequent run adaptively from that model. For many-factor problems where a full factorial cannot fit, this is the sampling approach Niobia reaches for, and it steers you to a structured factorial or response-surface design instead when the goal is estimating a specific low-order model.
Frequently asked
How is Latin hypercube sampling different from a factorial design?
A factorial places runs at fixed corners to estimate a low-order polynomial precisely; LHS spreads runs to fill the whole space evenly with a chosen number of samples. Factorials are for estimating known effects; LHS is for learning an unknown response across many factors, typically to train a surrogate model.
When should I use LHS?
When you have many factors (a full factorial is infeasible), when you are training a flexible surrogate like a Gaussian process, or when the response may not be a simple quadratic so a fixed response-surface design would impose the wrong shape. Space-filling coverage is the goal in all three.
How many runs does a Latin hypercube need?
You set it, which is part of the appeal for expensive experiments and simulations. The guidance is to scale with dimensionality so the space is genuinely filled; a common heuristic is roughly ten samples per factor as a starting point, then check the design's space-filling quality.