4.2. Efficiency — Modified Cloud Analysis

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.