# Elucidating Super-Compressible States in Inner Structure of Vehicular Flow
## 8 Statistical Compressibility in Two Lanes Separated  
The data that we analyze in this section (2.35 mill events in total) were gauged during reconstruction works of Czech Expressway D1 (direction from P'reov to Brno) near city Vyskov during June, July, and August 2016, i.e., before COVID-19. The originally four-lane highway was narrowed and a barrier was installed between the fast and main lanes (in the direction toward Brno) to prevent vehicles from changing lanes. In the area of reconstruction, the maximum permitted speed was reduced to 80 km per hour. Here, three sets of radars (Wavetronics SmartSensor HD), having an accuracy: 0.1 s for time measurements and 1 km/h for velocity measurements, have been located--as visualized in Figure 10. For purposes of measurements, a 16-kilometers-long segment near the reconstruction area has been chosen (see the Radar 2 in Figure 10). Detectors 3 and 4 represent an optimal choice for our intentions. Figure 9: Gray crosses and pink plus signs show the quality of the estimates that have been performed using three-parametric estimations that allow for the existence of super-compressible states. To highlight the quality of such estimates, we re-plot the original estimates (blue and green curves) based on the sub-compressibility assumption—see Figure 3.  
Data required for the following statistical analyses have been processed with the help of the standard 3 s unification procedure. This means that only data from the same segment of the fundamental diagram (i.e., with similar values of traffic density and intensity) were evaluated together. Subsequently, data samples of inter-vehicular clearances have been re-scaled to a mean value equal to one. The data pre-processed by the above procedure have been then subjected to statistical estimations based on the minimum distance method (MDE). Specifically, we have used the metric  
\[\varkappa(\alpha,\beta,\lambda)=\left(\int_{\mathbf{R}}|H(x)-G(x|\alpha,\beta,\lambda)|^{2}\,\mathrm{d}x\right)^{1/2} \tag{29}\]  
measuring the \(L_{2}\) distance between empirical cumulative distribution functions \(H(x)\) and \(G(x|\alpha,\beta,\lambda)\) which are associated with the empirical histogram \(h(x)\) of clearances and density (28), respectively. The result of such statistical testing is the finding that the proposed three-parameter GIG model replicates the real courses of the histogram functions \(h(x)\) very convincingly and, moreover, relatively independently of the traffic density and the selected traffic lane. Thus, the distribution (28) represents a compact and robust tool for describing the traffic micro-structure of the traffic stream. But this is a relatively well-known fact (see [6; 11; 12; 20; 23]). However, our main objective in this part of the text is to analyze the compressibility of vehicular traffic in reconstruction areas. To be specific, we intend to use empirical data to decide  
* Whether the safety precautions enforced by reconstruction constraints suppress the presence of super-compressible traffic states;
* Whether super-compressible states occur only in the fast lane as they do when there are no traffic restrictions;
* Whether, when the maximum speed of vehicles in the reconstruction area is sharply limited, the quantitative differences between the fast and slow lanes disappear. For completeness, we add that samples containing trucks were excluded from the analysis. The presence of trucks would fundamentally change the behavior of the entire system, as understandable. Subsequently, the interval frequency \(\mathcal{N}_{L}\) has been analyzed in the same segmentation way as the random variable \(\mathcal{R}\) in the previous part of the article. Significant emphasis has been put on the asymptotic behavior of statistical rigidity \(\Delta(L)\). Individual values of rigidity enumerated for values \(L\) lying between 1 and 10--i.e., beyond the area \(L\in(0,1)\) of the curvilinear trend of rigidity--have been subjected to standard regression analysis.