Telematics Combined Actuarial Neural Networks for Cross-Sectional and Longitudinal Claim Count Data

TL;DR

Novel cross-sectional and longitudinal claim count models for vehicle insurance built upon the combinedd actuarial neural network (CANN) framework proposed by Wüthrich and Merz are presented and exhibit superior performance compared to log-linear models that rely on manually engineered telematics features.

Abstract

We present novel cross-sectional and longitudinal claim count models for vehicle insurance built upon the Combined Actuarial Neural Network (CANN) framework proposed by Mario Wüthrich and Michael Merz. The CANN approach combines a classical actuarial model, such as a generalized linear model, with a neural network. This blending of models results in a two-component model comprising a classical regression model and a neural network part. The CANN model leverages the strengths of both components, providing a solid foundation and interpretability from the classical model while harnessing the flexibility and capacity to capture intricate relationships and interactions offered by the neural network. In our proposed models, we use well-known log-linear claim count regression models for the classical regression part and a multilayer perceptron (MLP) for the neural network part. The MLP part is used to process telematics car driving data given as a vector characterizing the driving behavior of each insured driver. In addition to the Poisson and negative binomial distributions for cross-sectional data, we propose a procedure for training our CANN model with a multivariate negative binomial (MVNB) specification. By doing so, we introduce a longitudinal model that accounts for the dependence between contracts from the same insured. Our results reveal that the CANN models exhibit superior performance compared to log-linear models that rely on manually engineered telematics features.

Figures (6)

• Figure 1: Number of contracts per vehicle.
• Figure 2: CANN architecture for the Poisson specification. The MLP's preactivation output value $a_1^{(4)}$ is added to the log-linear model's preactivation output value $\langle \boldsymbol{x}, \boldsymbol{\beta} \rangle$ before being transformed with the softplus function $\zeta(\cdot)$. The resulting $\mu$ value is compared to the ground truth $y$ using Poisson cross-entropy loss. The architecture shown employs a 3-hidden-layer MLP, but can be customized with any number of layers.
• Figure 3: CANN architecture for the negative binomial specification. The MLP's preactivation output value $a_1^{(4)}$ is added to the log-linear model's preactivation output value $\langle \boldsymbol{x}, \boldsymbol{\beta} \rangle$ before being transformed with the softplus function $\zeta(\cdot)$ to obtain the $\mu$ value of the negative binomial distribution. The $\phi$ value is obtained by transforming a real-valued parameter $w_\phi$ through the softplus function. The resulting parameters $\mu$ and $\phi$ are then compared to the ground truth $y$ using negative binomial cross-entropy loss. The architecture shown employs a 3-hidden-layer MLP, but can be customized with any number of layers.
• Figure 4: CANN architecture for the MVNB specification. The MLP's preactivation output value $a_1^{(4)}$ is added to the log-linear model's preactivation output value $\langle \boldsymbol{x}, \boldsymbol{\beta} \rangle$ before being transformed with the softplus function $\zeta(\cdot)$ to obtain the $\mu$ value of the negative binomial distribution of Equation (\ref{['eq:gen_negbin']}). The $\phi$ value is obtained by transforming a real-valued parameter $w_\phi$ through the softplus function. To obtain $\alpha$, the sum of past claims $\Sigma^{(y)}$ is added to the $\phi$ parameter, while for $\gamma$, the sum of past $\mu$ values $\Sigma^{(\mu)}$ is added to the same $\phi$ parameter. The resulting distribution parameters $\mu$, $\alpha$ and $\gamma$ are then compared to the ground truth $y$ using negative binomial cross-entropy loss. The architecture shown employs a 3-hidden-layer MLP, but can be customized with any number of layers.
• Figure 5: Importance scores of the 20 most important variables obtained for the MVNB CANN model.