Classification of scale-sensitive telematic observables for riskindividual pricing

Not the insured risk, i.e. the vehicle, but the vehicle driver causes insurance losses. A driver recognition is therefore essential for an adequate assessment of the factual, underlying risk.

Therefore, a sophisticated logic for measuring and transmission is the decisive quality factor, clarified further in Weidner and Transchel [23].

At this point fraud prevention, i.e. an extensive detection of data manipulation, should already be started. For example, a comparison of the measured data of a single vehicle and measurements of the whole ensemble is conceivable for this.

Since we are not interested in back-transforming examined frequency data, the choice of normalization is somewhat arbitrary. When looking at the self-duality of \(F(F(f(x))) = f(x)\), then a choice of \(1/\sqrt{(}n_k)\) for both the transform itself and the conjugate transform respectively is more reasonable from a mathematical point of view.

Although the random variables generally represent discrete signals whose information parameters take only a finite number of values, the differentiability of the scoring function s(X) can be guaranteed by means of interpolation, provided that sufficient values within certain limits are available.

In particular, this means that the classification is executed separately for each scale. Classes from different scales can resemble each other (despite the lack of comparability) by reason of similar contents, but the risk assessment of those classes can vary according to the context.

Only half of the evolution coefficients are necessary for representing of DFT-results due to Hermitian symmetry (see Sect. 2.2). Therefore, since the driving maneuver is recorded with 4 Hz, Fig. 4 (right) represents the magnitude of Fourier coefficients for frequencies from 0 to 2 Hz. Both, zero and folding frequency are neglected here because they do not provide any information about the maneuver structure.

The following relationship exists for frequency f [Hz] and period duration T [s]: \(T = \frac{1}{f}\).

VO (EU) Nr. 267/2010.

Since the driving behaviour naturally differs between different road types, we chose this particular selector to highlight the significance of the Fourier analysis of driving profiles. Other conditions and contexts are also possible, for example differentiation with respect to weather/light conditions or time of day.

For the DFT, we reduced the degree of detail to the mean value of 4 seconds, i.e. to a frequency of 0.25 Hz; we thus abstract the information to the required precision at the section level. Therefore, by reason of Hermitian symmetry (see Sect. 2.2), Fig. 5 (below) represents the magnitude of Fourier coefficients for frequencies from 0 to 0.125 Hz. The zero frequency, which generally reflects the homogeneity of the telematic measurements, is also shown.

Instead of processing the signal data directly, one can resort to model the auto-correlation function instead, given by \(R_{m_k^{(j)}}(\tau ) = \frac{1}{n_k} \sum _{t=1}^{n_k}m_k^{(j)}(t)m_k^{(j)}(t+\tau )\), where \(\tau \in \mathbb {N}\) is the probed period length. However, to address the question when to use which quantity, there is no sufficient portfolio data available to define figures of merit for this particular case and this comment should be considered for brevity.

We fitted the DFT spectrum after the following considerations: Due to Parseval’s theorem, the total energy of the process (and thus, in the discrete case, the sum of all Fourier coefficients) in the frequency domain matches the integral over the velocity data in positional space. As a consequence from the fact that frequency is proportional to the inverse period, we fit the velocity transform by \(F(x) \sim \mu _{k,2}^{(j)} \cdot e^{-0.57 \cdot x^{0.5}}\), where the scaling factor \(\mu _{k,2}^{(j)}\) is the coefficient of the greatest period length, due to the expectation that higher frequencies must contain exponentially less energy in order to get a bounded expression. It appears as an exponentially less steeper line in the log plots. Although the longitudinal acceleration transform depends linearly on the velocity transform, its fit is almost linear due to the fact that the range of the coefficients is reduced compared to the velocity transform. It is thus sufficient to fit it by a linear expression as mean value for each Fourier coefficient per a portfolio average to indicate the expected distribution according to the central limit theorem that appears as a slightly exponentially decreasing function in the log plots. The dependent part of the lateral acceleration transform shows, like the longitudinal dimension, a limited range. Therefore, even without direct connection with the energy as in the case of the longitudinal acceleration, we fit it by a linear expression too.

As basic property of Fourier analysis, the longitudinal acceleration behaviour must differ from the velocity coefficients only by a constant as the former is the time derivative of the latter. As such, plotting and analyzing both measurement parameters is redundant if they are derived from the same source. If, however, as is the case with some of the emerging hardware solutions, velocity is obtained from GPS measurement whereas acceleration data is measured by respective acceleration sensors, the comparison at hand provides either redundancy or plausibility checks of either device.

“Smoothness” of the DFT spectrum of the acceleration behaviour is meant here as in the sense of small coefficients.

The main point of investigation in Weidner et al. [24] was to determine representative driving styles differentiating the velocity and longitudinal acceleration behaviour for each trip. In their research, they find two groups of driving styles depending on the velocity process—with the same or similar acceleration and deceleration value distribution—those with lower (i.e. up to 50 km/h in the mean) and those with higher velocity values (i.e. more than 50 km/h in the mean), whereas no differentiation regarding road type was made.

It is worth noting that naturally, the available amount of data for the night time is two orders of magnitudes smaller than at day time, such that all conclusions must be thoroughly checked against both portfolio size and total number of trips, especially if conclusions are drawn with respect to large personal injury and material damage occurring at night time. Consequently, for a risk-adequate scoring it is necessary to compare the distribution of the individual ratio of driving styles (per hour) with the portfolio or segment of choice, weighted with the risk associated with each driving style and time of day.