4.6. Relative Score Method — Modified Cloud Analysis

Theoretical Background

The Relative Score Method (RSM) quantifies the information gain from using one intensity measure over another using a relative sufficiency metric expressed in bits (Ebrahimian & Jalayer, 2021).

Kullback–Leibler divergence

The sufficiency of IM₂ relative to IM₁ is measured by the Kullback–Leibler (KL) divergence between the demand distributions conditioned on each IM. For two lognormal distributions \(D|IM_1 \sim \ln\mathcal{N}(\mu_1, \sigma_1^2)\) and \(D|IM_2 \sim \ln\mathcal{N}(\mu_2, \sigma_2^2)\), the KL divergence is:

\[D_{KL}(f_1 \| f_2) = \ln\frac{\sigma_2}{\sigma_1} + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2\sigma_2^2} - \frac{1}{2}\]

Relative Sufficiency Measure

The RSM of IM₂ relative to IM₁, averaged over the record set, is:

\[\text{RSM}(\text{IM}_2 \mid \text{IM}_1) = \frac{1}{N} \sum_{j=1}^{N} \ln\frac{f_{D|\text{IM}_1}(\text{EDP}_j)} {f_{D|\text{IM}_2}(\text{EDP}_j)}\]

expressed in nats (divided by \(\ln 2\) to convert to bits). A positive RSM means IM₂ is the more sufficient IM.

For MCA, the conditional demand distributions are derived from the cloud regression residuals evaluated at each record’s demand and IM level.