A Portfolio-Anchored Frequency-Severity Risk Index for Trip and Driver Assessment Using Telematics Signals

Abstract

In this paper, we propose a novel frequency-severity joint trip-level risk index that combines the frequency of abnormal driving patterns with a severity component reflecting how extreme such behavior is relative to a portfolio-level baseline. Severity is quantified through an inverse-probability penalty that increases with the rarity of observed tail extremes, rather than being interpreted as a claim size. Based on high-frequency telematics data, we construct a multi-scale representation of longitudinal acceleration using the maximal overlap discrete wavelet transform (MODWT), which preserves localized driving patterns across multiple time scales. To capture severity as tail rarity, we model the portfolio distribution using a Gaussian-Uniform mixture with a layered tail structure, where Gaussian components describe typical driving behavior and the tail is partitioned into ordered severity layers that reflect increasing extremeness. We develop a likelihood-based estimation procedure that makes inference feasible for this mixture model. The resulting severity layers are then used to construct multi-layer tail counts (MLTC) at the trip level, which are modeled within a Poisson-Gamma framework to yield a closed-form posterior risk index that jointly reflects frequency and severity. This conjugate structure naturally supports sequential updating, enabling the construction of dynamically evolving driver-level risk profiles. Using the UAH-DriveSet controlled dataset, we demonstrate that the proposed index enables reliable discrimination across behavioral driving states, identification of high-risk trips, and coherent ranking of drivers, yielding a purely behavior-driven risk measure suitable for actuarial ratemaking and potentially mitigating fairness concerns associated with traditional covariates.

Figures (4)

• Figure 1: Driver 1 motorway example: longitudinal acceleration densities (left) and trip trajectories with pooled left-tail thresholds marked at the 0.5%, 0.1%, and 0.05% quantiles (right), illustrating how tail observations are turned into MLTC events.
• Figure 2: Selecting the decomposition depth $J$: The raw longitudinal acceleration for an aggressive trip of Driver 4 on Motorway (top), level 6/7/8 MODWT coefficient trajectories (middle), and the corresponding variance contributions across levels (bottom), illustrating that risk-relevant variations become sparse at deeper levels and motivating the choice of $J{=}6$.
• Figure 3: Fitted Gaussian--Uniform mixture under BIC for the portfolio distribution of aggregated wavelet coefficients: the Gaussian bulk captures typical driving dynamics, while multi-layer Uniform tails define ordered severity layers; rug marks along the axis show the assignment of individual coefficients.
• Figure 4: Binary classification (normal vs. risky) using MLTC-based inputs: balanced accuracy under repeated stratified $K$-fold CV and LODO (top), and out-of-fold distributions of $P(\texttt{risky})$ with the nested-CV threshold (bottom), comparing Total frequency, Layer-wise frequency, and Severity-weighted frequency representations.