4.2. Efficiency — Modified Cloud Analysis

imselection.compute_efficiency_mca(cloud_dict)

Classic efficiency βD|IM from a Modified Cloud Analysis result.

Efficiency is defined as the standard deviation of the residuals of the log-linear cloud regression, σ (sigma). A lower value indicates a more efficient intensity measure.

Parameters:

cloud_dict (dict) – Output of postprocessor.process_mca_results.

Returns:

• dict with keys

• * 'beta_D_given_IM' — regression residual sigma

• * 'method' — 'MCA'

Theoretical Background

Efficiency measures the dispersion of structural demand conditioned on the intensity measure (Luco & Cornell, 2007). A more efficient IM produces tighter demand predictions, reducing the required number of analyses.

Definition

For MCA, efficiency is quantified by the residual standard deviation of the log-log cloud regression:

\[\beta_{D|IM} = \sqrt{ \frac{1}{N-2} \sum_{j=1}^{N} \bigl[\ln(\text{EDP}_j) - \ln(a) - b\,\ln(\text{IM}_j)\bigr]^2 }\]

where \(a\) and \(b\) are the OLS regression coefficients of \(\ln(\text{EDP})\) on \(\ln(\text{IM})\), and \(N\) is the number of non-collapse records.

A smaller \(\beta_{D|IM}\) indicates that IM explains more of the record-to-record variability in demand — i.e. the IM is more efficient.

Example

from openquake.vmtk.imselection import imselection

ims = imselection()
# cloud_dict is the output of postprocessor.process_mca_results()
result = ims.compute_efficiency_mca(cloud_dict)
print(f"Efficiency (beta_D|IM) = {result['efficiency']:.4f}")