Feature extraction from telematics car driving heatmaps

Abstract

Insurance companies have started to collect high-frequency GPS car driving data to analyze the driving styles of their policyholders. In previous work, we have introduced speed and acceleration heatmaps. These heatmaps were categorized with the K-means algorithm to differentiate varying driving styles. In many situations it is useful to have low-dimensional continuous representations instead of unordered categories. In the present work we use singular value decomposition and bottleneck neural networks (autoencoders) for principal component analysis. We show that a two-dimensional representation is sufficient to re-construct the heatmaps with high accuracy (measured by Kullback–Leibler divergences).

Appendix: KL divergence, revisited

In this appendix we briefly revisit the KL divergence. Denote by $\mathcal{X} \subset {\mathbb {R}}^J$ the $(J-1)$-unit simplex. Consider $k$ independent and identically distributed trials among $J$ classes providing a multinomial distribution $\pi \in \mathcal{X}$ given by the discrete probability weights

$$\begin{aligned} p(k_1,\ldots , k_J) ~ = ~{k \atopwithdelims ()k_1, \ldots ,k_J} ~ \prod _{j=1}^J \pi _j^{k_j} ~ \mathbb {1}_{\{\sum _{j=1}^J k_j =k\}}, \end{aligned}$$

for $k_j \in {\mathbb {N}}_0$, $j=1,\ldots , J$. The deviance statistics of an observation $(k_1,\ldots ,k_J)$ of that multinomial distribution is given by

$$\begin{aligned} D\left( (k_1,\ldots , k_J),\pi \right)&= 2 \left[ \sum _{j=1}^J k_j \log \left( \frac{k_j}{k}\right) - \sum _{j=1}^J k_j \log \pi _j \right] \\&= 2 k \sum _{j=1}^J \frac{k_j}{k} \left( \log \left( \frac{k_j}{k}\right) - \log \pi _j \right) . \end{aligned}$$

In Sect. 3 we have defined the empirical distributions on the $(J-1)$-unit simplex by setting $x_j=k_j/k$ which, of course, provides $\varvec{x}=(x_1,\ldots , x_J)'\in \mathcal{X}$. Doing this transformation we receive

$$\begin{aligned} D\left( (k_1,\ldots , k_J),\pi \right) =- 2 k \sum _{j=1}^J x_j \log \frac{\pi _j}{x_j}= 2k ~d_\mathrm{KL}\left( \varvec{x}\Vert \pi \right) . \end{aligned}$$

Thus, by minimizing the KL divergence in (3.3), we minimize the corresponding deviance statistics, which provides the maximum likelihood estimator of the network parameter $\theta$ under independent multinomial models (having $J$ classes) for drivers $i=1,\ldots , n$. This additionally assumes that all drivers have identical weights $k$. If the latter is not appropriate we may replace the average KL divergence in (3.3) by a weighted counterpart

$$\begin{aligned} \mathcal{L}^w_\mathrm{KL}\left( \theta , (\varvec{x}_i)_{i=1:n}\right) = \sum _{i=1}^n w_i~ d_\mathrm{KL}\left( \varvec{x}_i \Vert \pi (\varvec{x}_i)\right) , \end{aligned}$$

with weights $w_i\ge 0$ satisfying $\sum _{i=1}^n w_i=1$.