United States Department of Transportation
i
The Modified Brandeis Dice for the Traffic States Dependency: The I-Dice Problem
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2020-06-01
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Abstract:Brandeis dice example is a toy example discussed by Jaynes to explain the maximum entropy principle. A die is tossed many times and an average number of spots is revealed. If other than 3.5, die is obviously not an honest one. Probabilities are then assigned for the next toss. This example is contested on the finite sampling grounds. The authors use Maximum Relative Entropy (MRE) in Jaynes’ Brandeis dice example to model traffic regimes (states) in a traffic flow. A die-like hypercube (I-dice) is generated, and instead of the spots, unit kinetic energies are given on each and every face of the cube. The number of faces for this cube is the same as the number of states for the traffic flow of a given highway segment. The faces of the I-dice represent the congested, intermediate and free-flow regimes. Probability distributions are generated via maximum relative entropy principle, a modified version of the Jaynes’ MaxEnt principle. The prominent feature of this hypercube, which is called I-dice by the authors, is that it generates probabilities of the traffic states à la Boltzmann-Gibbs statistical mechanics. The probabilities for the states at each lane are computed via MRE. It is found that the probabilities do not match the observed frequencies. MRE imposes a more uniform distribution of probabilities for the speeds than the observed ones. As a result, for each state, the new speed classification is suggested by I-dice. The authors propose that I-dice may very well be used as a test-bed to check randomness in traffic flow.
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