Classical Power Transform Model (ClassicalPowerTransform)

The Classical Power Transform model (Shoun, in prep [1]) is a parametric tail model akin to the GeneralizedBondy and Sherman models that is designed to arbitrate between an exponential decay model, Clark’s square root model [2], and Sherman’s inverse power model [3] in a data-driven way. Our ClassicalPowerTransform model is mathematically expressed as:

\[\begin{split}\begin{align} \log ATA_{j} &\sim \mathrm{Normal(\mu_{j}, \sigma^2)}\\ \mu_{j} &= \beta_{\text{int}} + \beta_{j} \text{L}_j\\ \beta_{j} &= \lambda - 1 - \beta_{\text{slope}}\\ \text{L}_j &= j^{\lambda-1} / \lambda\\ \beta_{\text{int}} &\sim \mathrm{Normal}(\beta_{\text{int}, \text{loc}}, \beta_{\text{int}, \text{scale}})\\ \beta_{\text{slope}} &\sim \mathrm{Normal}(\beta_{\text{slope}, \text{loc}}, \beta_{\text{slope}, \text{scale}})\\ \log \sigma^2 &\sim \mathrm{Normal}(\sigma^{2}_{\text{loc}}, \sigma^{2}_{\text{scale}})\\ \beta_{\text{int}, \text{loc}} &= 0.0\\ \beta_{\text{int}, \text{scale}} &= 100.0\\ \beta_{\text{slope}, \text{loc}} &= 0.0\\ \beta_{\text{slope}, \text{scale}} &= 10.0\\ \sigma^{2}_{\text{loc}} &= -4.0\\ \sigma^{2}_{\text{scale}} &= 5.0\\ \lambda &= \text{user input} \in [0,1] \end{align}\end{split}\]

where \(\bf{ATA}\) is a vector of age-to-age factors that capture how losses change across development. Unlike other loss development and tail models, the ClassicalPowerTransform model is fitted directly to age-to-age factors in a two-stage fashion. When fitting the model, we first estimate \(\bf{ATA}\) by fitting the TraditionalChainLadder model first (with use_volume_weighting=True). The estimated \(\bf{ATA}\) are then extracted and then fitted given the model specification above.

In the ClassicalPowerTransform model, the parameter \(\lambda\) is a user-specified parameter that determines the shape of the tail curve. When \(\lambda = 1\), the model is equivalent to an exponential decay model on the age-to-age factors. When \(\lambda = 0.5\), the model is equavalent to Clark’s square root decay model on age-to-age factors. Finally, when \(\lambda = 0\), the model is equivalent to Sherman’s inverse power decay model on the age-to-age factors. Therefore, by change the value of \(\lambda\), the user can change how heavy the implied tails are in the model.

Typically, tail models like above model are fitted to only the window of development lags \(j \in [\rho_1, \rho_2]\), where \((\rho_1, \rho_2) \in {2,...,M}, \rho_1 < \rho_2\), are chosen by an analyst based on where the tail process is assumed to begin and end. In practice, this can be accomplished my mutating/clipping the triangle as a preprocessing step before fitting.

Model Fit Configuration

The ClassicalPowerTransform model above is fit using the following API call:

model = client.tail_model.create(
    triangle=...,
    name="example_name",
    model_type="ClassicalPowerTransform",
    config={ # default model_config
        "loss_definition": "paid",
        "lambda_": 1.0, # defaults to exponential decay shape
        "priors": None, # see defaults below
        "recency_decay": 1.0,
        "seed": None
    }
)

The ClassicalPowerTransform model accepts the following configuration parameters in config:

  • loss_definition: Name of loss field to model in the underlying triangle (e.g., "reported", "paid", or "incurred"). Defaults to "paid".

  • priors: Dictionary of prior distributions to use for model fitting. Default priors are:

{
    "dev_slope_offset__loc": 0.0,
    "dev_slope_offset__scale": 10.0,
    "sigma__loc": -4.0,
    "sigma__scale": 5.0,
    "dev_intercept__loc": 0.0,
    "dev_intercept__scale": 100.0,
}

  • recency_decay: Likelihood weight decay to down-weight data from older evaluation dates. Defaults to 1.0, which means no decay. If set to a value between 0.0 and 1.0, the likelihood of older evaludation dates will be downweighted by a geometric decay function with factor recency_decay. See Geometric decay weighting for more information.

  • seed: Random seed for model fitting.

Model Predict Configuration

The ClassicalPowerTransform model is used to predict future losses using the following API call:

predictions = model.tail_model.predict(
    triangle=...,
    config={ # default config
        "max_dev_lag": None,
        "include_process_noise": True,
    }
    target_triangle=None,
)

Note that although the ClassicalPowerTransform model is specified with age-to-age factors as the target variable, predictions are generated and returned to the user as losses.

Above, triangle is the triangle to use to start making predictions from and target_triangle is the triangle to make predictions on. For most use-cases, triangle will be the same triangle that was used in model fitting, and setting target_triangle=None will create a squared version of the modeled triangle. However, decoupling triangle and target_triangle means users could train the model on one triangle, and then make predictions starting from and/or on a different triangle. By default, predictions will be made out to the maximum development lag in triangle, but users can also set max_dev_lag in the configuration directly.

The ClassicalPowerTransform prediction behavior can be further changed with configuration parameters in config:

  • max_dev_lag: Maximum development lag to predict out to. If not specified, the model will predict out to the maximum development lag in triangle. Note that GeneralizedBondy can be used to make predictions for development lags beyond the last development lag available in the training triangle, as there is a mechanism in the model to extrapolate out age-to-age beyond the training data.

  • eval_resolution: the resolution of the evaluation dates in the tail. Defaults to the evaluation date resolution in triangle. If triangle is from a single evaluation date, falls back to the resolution of the training data.

  • include_process_noise: Whether to include process noise in the predictions. Defaults to True, which generates posterior predictions from the mathematical model as specified above. If set to False, the model will generate predictions without adding process noise to the predicted losses. Referring to the mathematical expression above, this equates to obtaining the expectation given \(\mu_{ij}\) while not including the observation error \(\sigma^2\).