FFRD Sampling Demos

Overview

This repository demonstrates two methodologies for improving computational efficiency in FFRD stochastic storm transposition (SST) simulations:

1. k-Nearest Neighbors (k-NN) Subsampling
   Reduces computational burden by selecting representative storm events while preserving multivariate dependence. We stratify in transformed Z-space (Gaussian copula) and subsample within each stratum, then reconstruct frequency curves using the law of total probability.

2. Importance Sampling (IS)
   Focuses computational effort on storm realizations that meaningfully intersect the watershed. Synthetic toy problems illustrate how adaptive and mixture-based IS can dramatically improve efficiency over uniform sampling by targeting regions of interest.

Part 1: k-NN Subsampling

Concept

The k-NN subsampling approach transforms high-dimensional marginal flows into standardized normal variates (Z-space) via plotting positions. A Gaussian copula captures inter-site dependence in Z-space. We then stratify the joint Z-domain into probability bins, and for each bin, select the K nearest neighbors from a large Monte Carlo ensemble. Finally, we apply the law of total probability to reconstruct marginal and joint frequency curves.

Implementation

• Toy Example (Test_kNN_SubSampling_ToyProblem)

  1. Generate multivariate Normal samples at 5 synthetic sites.
  2. Convert each marginal to Weibull plotting positions and Z-space.
  3. Stratify joint Z into 10 bins and subsample K=100 events per bin via k-NN.
  4. Reconstruct AEP curves using weighted exceedance counts.

• Case Study (Test_kNN_SubSampling_Kanawha_14_sites)

  1. Load 25,040 raw SST events across 14 sites.
  2. Estimate marginal plotting positions and fit a D-dimensional Gaussian copula.
  3. Stratify joint Z into 10 bins and subsample K=100 events per bin.
  4. Build site-specific flow frequency curves at pre-defined AEP levels.

Key Strengths

• Multivariate Preservation: Maintains dependence structure via Gaussian copula in Z-space.
• Variance Reduction: Stratified subsampling reduces estimator variance compared to crude MC at equal sample size.
• Computational Savings: Only K·bins runs needed, instead of N, to approximate frequency curves.

Limitations

• Curse of Dimensionality: In high D, nearest-neighbor neighborhoods become less meaningful without very large N (e.g., ≥100,000).
• Bin and K Selection: Empirical choice of bin count and K trades off bias vs. variance: too few neighbors → high variance; too many → biased smoothing.
• Copula Assumption: Gaussian copula may under-represent tail dependence present in true hydrologic extremes.
• Tail Strata Coverage: Extreme bins may lack sufficient samples, leading to duplicated neighbors or poor representation.

Potential Improvements

• Dimensionality Reduction: Apply PCA or t-SNE to Z-space before k-NN to focus on dominant variance directions [1][2].
• Adaptive Binning: Define strata so each contains roughly equal Monte Carlo counts for balanced variance.
• Alternative Sampling: Use clustering (e.g., k-means) to select representative centroids, or direct multivariate importance sampling to draw from tail-focused copula distributions.
• Tail-Copulas: Replace Gaussian copula with t-copula or extreme-value copulas for better tail dependence modeling [3].

References

1. Loftsgaarden, D. O., & Quesenberry, C. P. (1965). A nonparametric estimate of a multivariate density function. Annals of Mathematical Statistics.
2. Devroye, L., & Wagner, T. J. (1977). The L₁ convergence of nearest neighbor density estimates. Annals of Statistics.
3. Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in hydrology: a review. Water Resources Research.

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Part 2: Importance Sampling

Concept

Importance Sampling (IS) reroutes sampling effort to high-impact regions (e.g., storms overlapping a watershed). By choosing an appropriate proposal distribution (often a truncated normal or adaptive mixture) IS can dramatically reduce the number of simulations needed for reliable estimates.

Project Structure

The test class Test_SST_ImportanceSampling includes the following regions:

1. SST Toy Integration Problem

Simple geometric integration of a rectangular watershed using:

• Test_BasicMonteCarlo_Integration: baseline MC estimate using uniform sampling.
• Test_ImportanceSampling_Integration: basic IS with truncated normal proposals.
• Test_ImportanceSampling_Integration_ReducedSamples: IS with fewer samples.
• Test_Adaptive_ImportanceSampling_Integration_ReducedSamples: adaptive proposal update via weighted sample moments.

2. SST Toy Parametric Simulation Problem

Storms are sampled with varying footprints and rainfall depth:

• Test_BasicMonteCarlo_Parametric_Simulation: classic MC with variable storm sizes.
• Test_ImportanceSampling_Parametric_Simulation_Reduced_Samples: IS with truncated normal proposals.
• Test_Adaptive_ImportanceSampling_Parametric_Simulation_ReducedSamples: adaptive proposal using depth-weighted moment matching and EM-updated mixture weights.

3. SST Toy Nonparametric Simulation Problem

Storm depth derived from overlap area and storm intensity:

• Test_BasicMonteCarlo_Nonparametric_Simulation: classic MC simulation.
• Test_ImportanceSampling_Nonparametric_Simulation_Reduced_Samples: IS using fixed truncated normal proposals.
• Test_Adaptive_ImportanceSampling_Nonparametric_Simulation_ReducedSamples: adaptive IS with depth-based weighting and mixture modeling.

Key Features

• Simple Toy Problems: Simple examples are provided to demonstrate the sampling techniques.
• Adaptive Moment‐Matching: Proposal updates via weighted sample moments [4].
• Mixture Proposals: EM‐based mixture weights for flexible tail coverage [5].
• Truncated Normal Sampling: Efficiently targets the watershed‐intersection region.
• Frequency Curve Generation: Computes depth‐AEP curves via weighted exceedance.

Limitations and Future Extensions

• All spatial domains are idealized (square basin and storms), as shown in the image above.
• These sampling approaches need to be extended to support real-world watershed geometry and rainfall data.
• Explore options for irregularly shaped watersheds and transposition domains.

References

4. Cappé, O., Douc, R., Guillin, A., Marin, J.-M., & Robert, C. P. (2008). Adaptive importance sampling in general mixture classes. The Annals of Statistics, 36(4), 1947–1976.
5. Bugallo, M. F., Elvira, V., Martino, L., Luengo, D., Míguez, J., & Djuric, P. M. (2017). Adaptive Importance Sampling: The Past, the Present, and the Future. IEEE Signal Processing Magazine, 34(4), 60–79.

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Dependencies

• Numerics library suite:
  • Numerics.Data.Statistics
  • Numerics.Distributions
  • Numerics.MachineLearning
  • Numerics.Sampling
  • Numerics.Data

Provides utilities for multivariate distributions, stratified sampling, k-NN algorithms, and probability transforms.
GitHub: USACE-RMC/Numerics

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Running the Tests

Use MSTest (or compatible framework) to execute all [TestMethod] routines. Results are printed via Debug.WriteLine in CSV format (AEP,Value) for easy plotting.

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Usage and Output

• Metrics Reported: Estimated mean, standard error, and depth-frequency (or flow-frequency) curves.
• Visualization: Plot results on log-probability axes to assess tail performance.

Authors

Haden Smith
USACE Risk Management Center
[email protected]

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