7.5. Ordinal Fragility Functions

postprocessor.calculate_ordinal_fragility(imls, edps, damage_thresholds, intensities=np.round(np.geomspace(0.05, 10.0, 50), 3))[source]

Fits an ordinal (cumulative) probit model to estimate fragility curves for different damage states.

This function estimates the probability of exceeding various damage states using an ordinal regression model based on observed Engineering Demand Parameters (EDPs) and corresponding Intensity Measure Levels (IMLs).

Parameters:

  • imls (array-like) – Intensity measure levels corresponding to the observed EDPs.

  • edps (array-like) – Engineering Demand Parameters (EDPs) representing structural responses.

  • damage_thresholds (array-like) – Damage state thresholds for classifying exceedance levels.

  • intensities (array-like, optional) – Intensity measure levels for which fragility curves are evaluated (default: np.geomspace(0.05, 10.0, 50)).

Returns:

poes – A 2D array of exceedance probabilities (CDF values) for each intensity level. Shape: (len(intensities), len(damage_thresholds) + 1), where the last column represents the probability of exceeding the highest damage state.

Return type:

numpy.ndarray

References

1) Lallemant, D., Kiremidjian, A., and Burton, H. (2015), Statistical procedures for developing earthquake damage fragility curves. Earthquake Engng Struct. Dyn., 44, 1373-1389. doi: 10.1002/eqe.2522.

2) Nguyen, M. and Lallemant, D. (2022), Order Matters: The Benefits of Ordinal Fragility Curves for Damage and Loss Estimation. Risk Analysis, 42: 1136-1148. https://doi.org/10.1111/risa.13815

Theoretical Background

The ordinal regression approach treats damage state assignment as an ordered categorical outcome, jointly fitting all damage states in a single model while respecting their natural ordering (Nguyen & Lallemant, 2022).

Ordered response model

Let \(D \in \{0, 1, \ldots, k\}\) be the observed damage state (0 = no damage, \(k\) = highest damage state). The cumulative exceedance probability for damage state \(i\) is:

\[P(D \geq i \mid \text{IM}) = g^{-1}(\alpha_i - \beta \ln \text{IM}), \quad i = 1, \ldots, k\]

where \(\alpha_i\) are ordered thresholds satisfying \(\alpha_1 \leq \alpha_2 \leq \cdots \leq \alpha_k\), \(\beta\) is a shared slope common to all damage states, and \(g\) is a link function (logit or probit).

Shared slope constraint

Unlike fitting each damage state independently, ordinal regression enforces a single slope \(\beta\) across all damage states. This prevents fragility curve crossings and improves statistical efficiency by using all damage observations simultaneously.

Fragility curves

The probability of being in exactly damage state \(i\) is:

\[P(D = i \mid \text{IM}) = P(D \geq i \mid \text{IM}) - P(D \geq i+1 \mid \text{IM})\]

The exceedance fragility curves \(P(D \geq i \mid \text{IM})\) are obtained directly from the fitted cumulative model.