Latin Hypercube Sampling

Application Context

When using Latin Hypercube Sampling (LHS) to sample from multiple variables, maintaining independence between variables is critical. To ensure independence, each variable must randomly select sampled values from different sampling intervals. For example, in one iteration, variable #1 samples from stratum #4, while variable #2 samples from stratum #22, and so forth. This approach guarantees both randomness and independence, thereby preventing unintended correlations between variables.

Latin Hypercube Sampling represents a more efficient sampling method that offers substantial advantages in terms of sampling efficiency and runtime (due to reduced iteration counts). These benefits are particularly pronounced in PC-based simulation environments like @RISK. LHS proves especially valuable for analyzing input probability distributions that contain low-probability outcomes. By enforcing the inclusion of extreme events in simulations, LHS ensures that rare events are accurately represented in the simulation output.

When low-probability outcomes are particularly important, running analyses that specifically simulate the impact of these rare events on output distributions is highly beneficial. In such cases, the model exclusively simulates the occurrence of low-probability events, setting their probability to 100%. This approach isolates low-probability outcomes, enabling direct study of their resulting effects.

Key Technology

The core technology of Latin Hypercube Sampling lies in stratifying the input probability distribution. Stratification involves dividing the cumulative probability scale (from 0 to 1.0) into equal intervals along the cumulative curve. Samples are then randomly drawn from each interval or "stratum" of the input distribution. This sampling method forces representation of values from each interval, thereby reconstructing the input probability distribution. In implementation terms, when sampling m points from an n-dimensional vector space, LHS algorithms typically:

• Divide each dimension into m equally probable intervals
• Randomly permute the interval assignments for each dimension
• Sample once from each interval using random or center-point selection
• Combine dimensions through random pairing of sampled values

Simply put, when drawing m samples from an n-dimensional vector space, this method ensures independence while reducing computational time for sampling operations.