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Classification of mobility by survival models and entropy of information

Classification of mobility by survival models and entropy of information
 

The development of ICT technologies has had a great impact in the study of the dynamic and statistical properties of mobility in modern cities thanks to the possibility of collecting large georeferenced databases that contain information on individual mobility.
The main consequence of this development was the change of point of view in the analysis of phenomena related to mobility. Before the scientific community used a Eulerian point of view in which the study of mobility was carried out through observations located in single roads and averaged over time to measure the average flow and density of vehicles and deduce a fundamental diagram. Now ICT-based databases provide information on the trajectories carried out by a sample of individuals using different means of transport and it is necessary to develop analysis techniques that adapt to a Lagrangian viewpoint for the study of mobility. In the case of urban mobility, it is also necessary to develop statistical observables to understand when we are in the presence of phenomena that reflect a macroscopic change in the state of mobility, or a phase transition in the physical sense of the term.
This approach opens new interesting perspectives for understanding traffic problems and for developing new governance policies that allow an improvement in the quality of life in cities.
Let's take as an example the problem of measuring the degree of congestion of a city: the observation that in a given street, or crossroads, a queue has been generated could be used as a measure of the degree of congestion of the urban network only if we assume we know the mobility demand of different classes of citizens and the existence of a Wardrop balance for the state of mobility (ie each individual behaves in a rational way, achieving optimal mobility with respect to his knowledge of the behavior of other users). Empirical evidence has shown that it is extremely difficult to justify such assumptions, which can also be wrong in many cases. Taking the Lagrangian point of view the question becomes: what is the effect of the degree of congestion of a city on the dynamic trajectories achieved by individuals and on the behavior of individuals themselves. Currently we are not able to give a satisfactory answer to this question if not that the traveling speed in the urban road network decreases as the traffic load in the network itself increases, in a non-linear way.
The fundamental point is the development of new models that allow an interpretation of statistical laws from a Lagrangian point of view to measure quantities in relation to the behavior of individuals. This problem has had recent contributions from the Physics of Complex Systems based on survival models and on the concept of information entropy (Lempel-Ziv entropy).
The survival models have been applied to understand the distribution of mobility times related to different means of transport. In a mechanical statistical approach to the analysis of individual mobility, time can be considered an amount corresponding to energy [1]. Let P (t) be the probability that an observed path has a duration greater than a given time t, we propose to introduce the following model to classify the empirical distribution of travel durations [2]:

 

where the function t (t) defines the so-called hazard function.
The logistic form of this function is typical of decision models as it represents mathematically a threshold effect. This model depends on three parameters that take on a precise meaning and allow to classify the observed mobility: tc is a characteristic time that measures a duration where the function t (t) defines the so-called hazard function.
The logistic form of this function is typical of decision models as it mathematically represents a threshold effect. This model depends on three parameters that take on a precise meaning and allow to classify the observed mobility: tc is a characteristic time that measures a duration considered convenient for the chosen means of transport, measures the typical temporal distance from a goal that alpha to -1 measures the typical temporal distance from a goal that an individual reaches and ß -1 is a time scale that represents the distance characteristic of the destinations in the considered urban context. We note that for t »tc the hazard function tends to the constant ß and we have P (t) ~ exp (-ßt) which coincides with the Maxwell-Boltzmann distribution with ß-1 which plays the role of temperature. In Figure A we show how this model interpolates both the empirical distribution of travel times for journeys by car and the distribution of times for bicycle journeys recorded in the city of Bologna.

 

Figure A - (left) distribution of journey times of car routes in the city of Bologna (histogram) and interpolation with the model (continuous line) (journeys reconstructed using GPS data in May 2011 - Octotelematics database); (right) distribution of journey times of bicycle routes in the city of Bologna (histogram) and interpolation with the model (continuous line) (journeys reconstructed using GPS data - database Bellamossa 2018 Municipality of Bologna).

The model parameters are different in the two cases considered: in particular for cars tc it is equal to 3 minutes while for bicycles it is 8 minutes while alpha -1 is estimated 1.5 minutes for cars and 2.5 minutes for bicycles.
Taking into account the speed ratio between the two types of transport, these results suggest that in Bologna cars and bicycles respond to the same demand for small-scale mobility. Finally, if we consider that the parameter ß-1 has a value of 30 minutes for cars and 13 minutes for bicycles, this reflects the fact that the car is still used for routes of longer duration than the bicycle even if the presence of a queue larger than expected in the distribution of cycling routes could suggest the presence of a small fraction of individuals who use the bicycle as the main transport tool.

 

Figura B - (left) distribution of the Lempel-Ziv entropy calculated on all trajectories of cyclists longer than 15 minutes on a 200-meter spatial scale with a 10-second time step; (right) the same restricting the analysis to the trajectories that start from within the historic center

We can therefore classify the typology of the paths using the concept of entropy of information of Lempel-Ziv that measures the compressibility of the symbolic coding of a trajectory [3]. In other words, if we divide the city into different sectors (for example a partition in squares of 200 meters on each side) and we associate a symbol to each sector, it is possible to encode a trajectory by associating the symbol corresponding to the square in which the trajectory is located in each given time interval, the Lempel-Ziv entropy corresponds to the ratio between the length of the compressed signal with the Lempel-Ziv algorithm and the original signal length and is measured in bits per character. Figure B shows the analysis carried out on the trajectories of cyclists in the city of Bologna using the trajectories longer than 15 minutes with a sampling of 10 seconds per trajectory. The results show the existence of two types of mobility:
a low entropy mobility most likely of origin-destination type and a high entropy mobility that can be associated to a random component in the trajectories. The analysis restricted to the trajectories carried out in the historic center of Bologna shows that the origin-destination component is actually more present in the historic center than in the whole city.
This fact can be interpreted with the fact that the bicycle is a transport tool that satisfies the demand for mobility in the center, while it is used for local movements in the periphery preferring the car or a public transport for longer journeys. The information entropy is therefore a good indicator to distinguish the characteristics of the demand for mobility at the base of the trajectories observed through ICT technologies.

 

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