EAD Model
Expected Annual Damage Mode
Overview
The EAD (Expected Annual Damage) mode computes the expected damage from flooding by analytically integrating over uncertainty. It provides deterministic, smooth objective values that are well-suited for optimization.
In many risk management contexts, a decision-maker needs to evaluate thousands of candidate strategies quickly. EAD mode serves this need by replacing stochastic sampling with analytical integration, producing a single deterministic cost for each policy evaluation.
How It Works
EAD uses two-level integration:
\[ \text{EAD} = \int p_h(h) \cdot \mathbb{E}[\text{damage} \mid h] \, dh \]
Inner Expectation (Analytical)
For a given surge height \(h\), the expected damage integrates over dike failure analytically:
\[ \mathbb{E}[\text{damage} \mid h] = p_{fail}(h) \cdot d_{failed}(h) + (1 - p_{fail}(h)) \cdot d_{intact}(h) \]
This eliminates the stochastic dike failure uncertainty exactly – no sampling is needed.
Outer Expectation (Numerical)
The surge distribution is a stationary GEV, parameterized by location, scale, and shape in the EADScenario. The expected damage is integrated over this distribution using the integration method on EADConfig:
- Adaptive quadrature (
QuadratureIntegrator): Uses QuadGK for precise numerical integration - Monte Carlo sampling (
MonteCarloIntegrator): Samples surge heights from the distribution
When to Use EAD
EAD mode is the right choice when:
- You want fast, deterministic evaluations for optimization
- You need smooth objective landscapes (no sampling noise)
- You are doing risk-neutral policy comparison (expected cost minimization)
- You want the simulation to be RNG-independent (with quadrature integration)
Comparison with Stochastic Mode
| Property | EAD | Stochastic |
|---|---|---|
| Dike failure | Integrated analytically | Sampled (Bernoulli) per event |
| Output | Expected damage (deterministic with quadrature) | Realized damage (varies with RNG) |
| Speed | Faster per evaluation | Needs many replicates |
| Variance info | No | Yes |
| Tail risk | No | Yes |
| Convergence | Immediate (quadrature) | By Law of Large Numbers |
For static policies, EAD mode converges to the mean of Stochastic mode outcomes by the Law of Large Numbers.
Both modes give the same expected cost in the limit. The key difference is that EAD provides a single deterministic number, while Stochastic mode reveals the full distribution of outcomes.
Integration Methods
Quadrature
QuadratureIntegrator(rtol=1e-6) uses adaptive Gauss-Kronrod quadrature from QuadGK.jl. It integrates over the surge distribution’s quantile range (0.01% to 99.99%).
- Deterministic: Same result regardless of RNG seed
- Precise: Adaptive error control via relative tolerance
- Best for: Optimization, parameter sweeps, sensitivity analysis
Monte Carlo
MonteCarloIntegrator(n_samples=1000) samples surge heights from the distribution.
- Stochastic: Varies with RNG seed (but converges with more samples)
- Flexible: Works with any distribution, including complex mixtures
- Best for: Validation, distributions where quadrature struggles
Use quadrature for most purposes. Monte Carlo is mainly useful for validation or exotic distributions.
See Details for worked examples with visualizations.