In this section we introduce a prediction model for the number of collisions *A*(*t*) between bicyclists and motorized vehicles within a time frame *t*. The only predictor variable taken into account by the model is the variation of the total volume *X* of bicyclists in the region under consideration. This can result in an appropriate model as long as the volume of motorized vehicles stays approximately constant within the considered time span.

For the analysis we consider a dependence of the current collisions *A*(*t*) not only on the current bicyclist volumes *X*(*t*) but also on the average volume \(X_{\tau } = \frac {1}{\tau } \sum _{i=0}^{\tau - 1} X(t-i)\) in the previous *τ* time frames. The corresponding model is introduced in Section 2.1.

Furthermore, to apply the proposed models to empirical data on crashes and bicyclist volumes, we develop a model for the daily, weather-dependent variation of bicyclist volumes based on monthly counts at a few locations in Section 2.2. In Section 2.3, we conceive a test for the proposed method for the recovery of temporal correlations.

### 2.1 Models with memory

Several statistical works have presented results, which suggest that the assumption of a linear dependence of the number of collisions *A* on the traffic volume *X* is not appropriate. Often a non-linear relation of the form

$$ A \sim \alpha X^{\gamma} $$

(1)

is considered instead. The reported values of the exponent *γ* lie between *γ*=0.3 and *γ*=0.65 [7]. The case of linear dependence, i.e., *γ*=1 in (1), corresponds to a physical particle-model where entities of types *X* and *Y* are assumed to collide randomly with a probability *P*∼*X*·*Y*.

The SiN hypothesis sees the reason for an increased safety, i.e., a decreased risk *R*=*A*/*X*, in an increased traffic volume *X*. The main reason to postulate a causal relation of this form (and not an NbS effect, see Introduction) is that one may expect an elevated attentiveness or experience of motorists with regard to the VRUs under consideration. We introduce a measure *E*(*t*) for the current magnitude of attention, resp. experience, at time *t*. The value of *E*(*t*) is allowed to depend on previous *τ* volumes, i.e., *X*(*t*),*X*(*t*−1),...,*X*(*t*−*τ*+1), which is indicated by adding a subscript as in *E*_{τ}(*t*), where appropriate. Further, we stipulate that in absence of any experience, i.e., *E*=0, the “particle model” is correct.

This leads us to assume a model of the form

$$ A \sim \alpha X/(1+E_{\tau}). $$

(2)

Below we assume that the experience *E* is proportional to a power *η* of the average volume *X*_{τ} within the last *τ* time frames. That is, we consider

$$ E_{\tau,\eta} = X_{\tau} (t)^{\eta} = \left(\frac{1}{\tau} \sum_{k=0}^{\tau-1} X(t-k) \right)^{\eta}. $$

(3)

Note that, at least for large values of *X*, the resulting model captures (1) as a special case if *τ*=1 and *η*=1−*γ*, because

$$ \frac{\alpha X}{1+E_{1,1-\gamma}} =\frac{\alpha X}{1+X^{1-\gamma}} \approx \frac{\alpha X}{X^{1-\gamma}} = \alpha X^{\gamma}. $$

(4)

Figure 1 illustrates the most important feature of the memory model, which is an inert reaction to changes in the bicycle volumes. It shows a hypothetical time line of bicycle volumes [black curve], which exhibits an increase around *t*=0. The blue curve shows the behaviour of a model without memory (*τ*=1), which follows the bicycle volumes instantaneously, while the red curve illustrates the inertia of the system’s adaptation to the increased volumes in the presence of a memory *τ*=10. In both cases *γ*=0.5.

It seems appropriate to differentiate between different time scales for the SiN effect. Firstly, an immediate appearance of the effect would correspond to an immediately increased attention raised by the elevated presence of VRUs. For instance, an increased visibility of a group of pedestrians would have an immediate impact on their individual safety. Note that for a purely immediate mechanism a model with memory can not be expected to provide a significantly better fit.

On the other hand, if the SiN mechanism involved a longer time interval of conditioning or learning for the motorists, this would be indicated by improved model fits for the appropriate memory timespan. This equips us with a mean to deduce the time scale of the effect if it was present: It corresponds to the memory size *τ* that leads to the best model fit.

It seems adequate to distinguish at least between two non-immediate time scales for a possible SiN effect. The first is connected to a process of adjustment to an increased average number of VRUs due to seasonally changing conditions (on the first sunny days of spring a motorist might initially not expect many bicyclists since a longer period of low bicycle traffic volumes preceded that day). The related general behavioural adjustment, which might be hypothesized, is a change towards a more careful driving mode. This process would probably not take effect simultaneously and immediately on all drivers and therefore an inert, delayed adaptation of the traffic system as a whole would be visible in the risk if the mechanism had a significant influence on the collision frequency and no concurring factors were present neutralizing this influence.

As a process on an even slower time scale, a first familiarization with increasing numbers of VRUs due to shifts in societal mode choices would be conceivable and would express itself in the quality of models involving a larger memory.

Concerning the NbS effect, the modelling would rather aim at predicting the bicyclist volumes by past values of the individual risk, i.e.,

$$ X(t)\sim f\left(\sum_{k=0}^{\tau-1}R(t-k)\right). $$

(5)

For the identification of the model parameters we assume a negative binomial distribution of crashes, which allows controlling for overdispersion often observed in crash frequencies [15, 16].

### 2.2 A bicycle volume model

In order to study the dependence of collision frequencies on the overall bicycle traffic volumes on a level of high temporal resolution, a comprehensive record of daily bicycle volumes (usually available in the form of bicycle counts) is required.

In many municipalities, this data is rather sparse. In Berlin, for instance, counts are available only for one day per month at a few locations [see Section 3.1 for details]. In many cases the availability of weather data, which can reasonably be assumed as the most important short term factor for the variation of bicycle volumes, is much more abundant, though. Here we present an approach to reconstruct daily bicycle volumes for a region of interest from sparse count data with the aid of daily weather information.

To this end we decompose the total daily bicycle volume *X*(*t*) as

$$ X(t)=P(t)(1+W(t)), $$

(6)

where *P*(*t*) is a slowly varying average volume and *W*(*t*) is the current deviation from the average, which is assumed to be determined predominantly by the weather.

The local values of *P*(*t*) at the count locations are extracted from the available time series for the reported counts, *x*_{j}, *j*=1,2,..., by a local regression (LOESS) *P*∼*X*, see Fig. 3a. From this we obtain samples

$$w_{j}=\frac{x_{j}}{P(t_{j})}-1 $$

for the weather induced deviation *W*(*t*), which should have a comparable magnitude across the different count locations. As predictors of *W*(*t*) we considered the daily mean temperature *T*(*t*) (in deg*C*), the daily hours of sunshine *S*(*t*), and the daily height of precipitation *N*(*t*) (in *mm*). In this study the functional dependence was assumed to be of the form

$$ W=\alpha+\beta f(c_{N} N)\cdot f(c_{T} (T-\theta))\cdot (1+a_{S} f(c_{S} S))/(1+a_{S}), $$

(7)

where *f*(*x*)=(1+ exp(*x*))^{−1}.

### 2.3 Recovering delayed correlations in synthetic data

In this section we describe a test for the capability of the proposed memory models to reveal temporal relations in time series. For the purpose of evaluation we generated samples *z*(*t*) synthetically, which were drawn from negative binomial distributions *Γ*_{r,m(t)} with mean values

$$ m(t)=\frac{\alpha X(t)}{1+E_{\tau, \eta} (t)}, $$

(8)

and shape *r*=10 (parameter of the Gamma distribution in the Gamma-Poisson mixture). For the evaluation we chose *X*(*t*) as the normalized (i.e., <*X*>=100) daily predicted bicycle volume in Berlin with *t* ranging over the years 2013 and 2014, and *E*_{τ,η}(*t*) denotes the corresponding measure of experience, see (3). The parameters *α*=0.55 and *η*=0.2 have been fixed as we were interested in the abilities of the model to recover *τ*. To this end, several sets of synthetic crash data were created, with *τ* ranging within 1,...,20 days of memory. Qualitatively, all synthetic data sets coincided well with the observed crash numbers, see Fig. 2a-c.

As a next step, a model of the form (2) was fitted to the synthesized data with different values of *τ*=*τ*_{model} in order to determine the capability of the modelling to recover the parameters from the data, in particular the chosen values for *τ*. The model fit was accomplished using a maximum likelihood estimation (MLE). Hence, we sought for the maximum of

$$ \left(\tau_{\text{model}},\eta,r,\alpha\right)\mapsto L\left(\tau_{\text{model}},\eta,r,\alpha\right)\,=\,\log\left(\prod_{t} \Gamma_{r,m(t)}z(t) \right)\!. $$

(9)

Here, the predictors *m*(*t*) for the mean were given in a non-standard form as in (8) with *τ*=*τ*_{model} and *Γ*_{r,m(t)}(*z*(*t*)) is the probability mass of the negative binomial distribution in the sampled value *z*(*t*).

The models usefulness for revealing temporal correlation may be assessed by its capability to recover the real value of *τ*, which was estimated as the one leading to an optimal MLE. The results are reported in Section 3.2.