Frequency Models

CRML frequency models describe how often loss events occur for assets or risk components.

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1. Poisson model

Mathematically:

$$N \sim \text{Poisson}(\lambda)$$

where:

• $N$ = number of events in a fixed period (e.g., 1 year)
• $\lambda > 0$ = expected number of events per period

In CRML:

model:
  frequency:
    model: poisson
    parameters:
      lambda: 0.3

Runtime mapping (reference implementation):

from crml.frequency import sample_poisson_frequency

freq = sample_poisson_frequency({"lambda": 0.3}, size=n_assets)

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2. Gamma–Poisson (Negative Binomial)

To model over-dispersion (variance > mean), CRML uses a Gamma–Poisson model:

$$\lambda \sim \text{Gamma}(\alpha, \beta), \quad N \mid \lambda \sim \text{Poisson}(\lambda)$$

Marginally, $N$ follows a Negative Binomial distribution.

In CRML:

model:
  frequency:
    model: gamma_poisson
    parameters:
      alpha_base: 1.2
      beta_base: 1.5

Reference runtime:

from crml.frequency import sample_gamma_poisson_frequency

params = {"alpha": 1.2, "beta": 1.5}
freq = sample_gamma_poisson_frequency(params, size=n_assets)

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3. Hierarchical Gamma–Poisson (QBER-style)

In QBER-like models, $\alpha$ and $\beta$ may themselves depend on asset-level features (e.g., criticality index CI):

$$\alpha_i = f_\alpha(\text{CI}_i), \quad \beta_i = f_\beta(\text{CI}_i)$$

The full hierarchical model (conceptually):

$$\begin{aligned} \lambda_i &\sim \text{Gamma}(\alpha_i, \beta_i) \\ N_i \mid \lambda_i &\sim \text{Poisson}(\lambda_i) \end{aligned}$$

CRML expresses only the base form:

model:
  frequency:
    model: hierarchical_gamma_poisson
    parameters:
      alpha_base: 0.8
      beta_base: 1.3

The reference runtime currently interprets this similarly to gamma_poisson but can be extended to use CI-dependent parameters.

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4. Practical guidance

• Use Poisson for simple, independent, low-variance event counts.
• Use Gamma–Poisson when you observe fat-tailed or over-dispersed count data across assets.
• Use hierarchical forms when you have entropy-based criticality indices or other asset-level features.