PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum-Inspired Modeling of Driving Behavior

Mohammad Elayan, Omid Armantalab, Wissam Kontar

arXiv:2608.25907v1cs.LGeess.SYstat.ML

TL;DR

Driver behavior is heterogeneous, time-varying, contextual, probabilistic, and difficult to represent without pre-specifying variable interactions. The paper models each driver with an evolving density matrix whose interactions are learned from data, recovering interpretable regimes and reproducing known traffic phenomena while supporting classical models and autonomous-vehicle prediction.

  • Problem

    Existing representations struggle to capture driver heterogeneity, temporal change, context, probability, and learned variable interactions simultaneously without imposing a fixed form.

  • Method

    Each driver is represented as an evolving density matrix in a nonlinear feature space, with behavioral structure discovered through unsupervised training.

  • Results

    The framework recovered three interpretable profiles corresponding to free flow, transitional following, and congested stop-and-go, and reproduced the fundamental diagram and hysteresis.

  • Takeaways & Limitations

    The representation supplies context-dependent parameters to classical car-following models and provides autonomous vehicles with live behavioral reads and short-horizon motion forecasts.

  • Takeaways & Limitations

    The study was evaluated on a single freeway corridor and modeled only passenger vehicles.

Abstract

from arXiv · show

Driver behavior is heterogeneous, context-dependent, and changes over time, and these properties shape the traffic phenomena we observe. Most models, however, fix in advance which behavioral variables interact and how. Behavior outside that form is absorbed as noise, while models flexible enough to capture it tend to lose interpretability. We introduce a quantum-inspired representation of driver behavior that combines properties usually treated separately or in part: it is continuous, probabilistic, context-dependent, history-dependent, and represents interactions among behavioral variables as learned from data. Each driver is encoded as an evolving density matrix, providing a unified representation of behavioral uncertainty, temporal evolution, and context-dependent behavioral variation. Trained without supervision on the I-24 MOTION dataset, the framework recovers three interpretable driving profiles representing three regimes: free flow, transition, and congestion. The profiles capture the behavioral range of the data and the smooth transitions drivers make between regimes as conditions change. The same representation also reproduces known macroscopic phenomena, aligning with the fundamental diagram and reproducing hysteresis loops. We also show how the representation supports practical use: it supplies context-dependent parameters to classical car-following models, and gives an autonomous vehicle a live behavioral read of the surrounding drivers with a short-horizon forecast of their motion. The framework points toward models of traffic that are interpretable and trustworthy by construction. We release an open-source toolkit on GitHub (https://github.com/mselayan/quantum-driver-representation) spanning data processing, training, inference, and analysis.

1. Introduction

The paper develops a density-matrix representation that learns behavioral interactions from data while combining driver heterogeneity, temporal change, context, probability, and history. Applied to I-24 MOTION, it yields interpretable behavioral structure connected to macroscopic traffic phenomena and practical modeling.

  • Existing traffic models pre-specify interacting variables and functional forms, treating behavior outside those forms as noise or residual variation.
  • The framework represents each driver as an evolving density matrix in a nonlinear feature space, enabling learned interactions among behavioral and contextual variables.Random Fourier features provide the nonlinear lift, while the density matrix encodes the evolving driver state.
  • The learned microscopic representation connects individual driver behavior to aggregate traffic phenomena, including the fundamental diagram, traffic waves, hysteresis, and stop-and-go behavior.This macroscopic structure emerges from the data rather than being imposed by the model.
  • It combines continuous, probabilistic, context-dependent, and history-dependent behavior in a unified representation rather than imposing fixed behavioral labels.
  • Unsupervised training discovers interpretable behavioral profiles, within-profile modes, and continuous transitions as observed behavior and local context change.
  • The trained representation supplies behavioral quantities for car-following and lane-changing models and supports context-aware autonomous-vehicle planning around surrounding drivers.

2. Motivation of the Study

The study motivates a density-matrix representation to capture structured differences between drivers, context-dependent changes, and interactions among behavioral variables while remaining interpretable.

  • Driver heterogeneity: Drivers exhibit distinct following styles, including differences that share nearly identical mean speed, headway, and acceleration.The simulation includes four followers responding to the same leader disturbance, with d2 and d3 differing despite similar summary statistics.
  • Context dependence: As density rises, drivers change along different paths, with one transitioning early and gradually and another later and more sharply.Both drivers settle into a low-speed regime, but their responses to the same increasing-density context differ.
  • Representation comparison: Static and fixed-mixture representations cannot capture within-driver changes because they use fixed behavioral classes or time-invariant mixture weights.The density-matrix representation instead carries profile weights at each instant, with profiles and transitions learned from data rather than imposed by an IDM or predefined functional form.
  • Variable interactions: Acceleration depends jointly on headway and speed, with the intermediate-speed range exhibiting a nonlinear relationship that separate profiles can preserve.The density-matrix representation captures the intermediate range as its own profile rather than averaging it into the others.
  • Study motivation: The proposed framework combines shared behavioral profiles, evolving states, and eigendecomposition to balance flexibility with interpretability.The motivating examples target differences between drivers, adaptation to changing conditions, and interactions among driving variables.

3. Methodology

The framework represents each driver with an evolving density matrix over nonlinear behavioral features, combining probabilistic profiles, context dependence, temporal change, and learned behavioral interactions.

  • Framework: Each driver is represented by an evolving density matrix rather than a fixed parameter vector or discrete label.The representation is designed to capture both differences across drivers and changes within a driver as context evolves.
  • Framework: Quantum-inspired structure encodes observations as normalized states, uses Born-rule measurements, and represents uncertainty with valid density matrices.The matrices are symmetric, positive semidefinite, and trace-normalized.
  • Behavioral representation: Density matrices represent distributions over possible behaviors while preserving relationships among behavioral modes, including interactions between speed and spacing.This allows one object to encode both behavioral range and the structure linking behaviors.
  • Behavioral representation: Different drivers can combine profiles differently, profile mixtures can change over time, and each profile can contain multiple behavioral modes.The framework therefore captures heterogeneity across drivers, temporal evolution within drivers, and multimodal profiles.
  • Variable selection: The selected variables are speed, spacing, jerk, local traffic density, speed entropy, and acceleration entropy.The three behavioral variables describe motion, while the three contextual variables describe surrounding demand and variability; averaged surrounding speeds were excluded as redundant.
  • Feature representation: Random Fourier features expand speed, spacing, and jerk into nonlinear combinations without pre-specifying which variables interact.Each observation is then converted into a normalized feature vector and a rank-one matrix that can be accumulated into profiles and evolving states.
  • Estimation: The model estimates profiles and state contributions by minimizing negative log-likelihood with an entropy penalty on profile eigenvalues.The penalty encourages profiles to retain multiple behavioral modes; its strength is set to γ = 4.

4. Data and Training

The framework is trained on high-resolution, spatially dense I-24 MOTION trajectories spanning a complete free-flow-to-congestion-to-recovery cycle, using sequential stateful optimization and three density-matrix profiles.

  • Dataset: I-24 MOTION records every vehicle across multiple freeway lanes at high frequency, providing trajectories and local context over varied traffic conditions.The dataset covers a 4.33-mile Interstate 24 segment near Nashville using a 294-camera network at 25 Hz.
  • Dataset: The westbound subset contains 519,665 trajectories, or 67.3% of 771,946 total trajectories, and spans a complete free-flow-to-congestion-to-recovery cycle.The observed speed field includes dense stop-and-go waves during the congested interval.
  • Dataset: The modeled population excludes semis and trucks and retains sedan, midsize, van, and pickup trajectories with comparable passenger-vehicle dynamics.The eligible classes comprise 35.3% sedans, 34.9% midsize vehicles, 2.9% vans, and 16.1% pickups.
  • Dataset: Eligible trajectories show broad behavioral variation, with speeds ranging from near stationary to 102 ft/s at the 90th percentile and a median of 49.9 ft/s.All four lanes exhibit a coherent slowdown with modest cross-lane differences, while spacing is concentrated at short following distances but spans the observation window.
  • Training: Training uses D = 100 random Fourier features, K = 3 profiles, and rank-10 profile factors while minimizing the penalized negative log-likelihood.Behavioral and contextual variables are standardized before training.
  • Training: Observations are processed in ordered chunks of 5,000 frames so each driver’s evolving state carries across chunks within an epoch.Adam uses learning rate 0.005 for 15 epochs, with α = 0.2 and γ = 4 fixed while η is learned.
  • Evaluation: The full framework reaches NLL = 0.880, compared with NLL = 2.230 for a static one-matrix model and approximately 4.605 for an uninformative model.Relative to the static model, the full framework makes observed behavior approximately 3.9 times more likely on average.

5. Behavioral Analysis

The framework separates three behaviorally distinct profiles and explains them through variable interactions, internal modes, and context-dependent evolution. Profile 2 uniquely decomposes into a dominant congested-following state plus brief acceleration and braking events.

  • Internal structure: Profiles 1 and 3 are effectively rank one, while Profile 2 is multimodal with purity P(ρ2) = 0.734.Profiles 1 and 3 have purity values P(ρ1) = 0.999 and P(ρ3) = 0.997; Profile 2’s leading eigenvalue is 0.852.
  • Profile relationships: Profiles 2 and 3 are closer in profile space because they represent neighboring stages of a congestion cycle, whereas Profile 1 is structurally distant from both.Profile 1 reaches 93% and 97% of maximum distance from Profiles 2 and 3, while Profiles 2 and 3 are separated by 35% of the maximum distance.
  • Profile separation: Profile 1 is fastest with the largest spacing, Profile 2 is slowest with the shortest spacing, and Profile 3 lies between them.Mean speed and spacing are 62.2 ± 28.5 ft/s and 193.7 ± 127.9 ft for Profile 1; 27.4 ± 20.7 ft/s and 117.6 ± 103.6 ft for Profile 2; and 53.5 ± 26.5 ft/s and 154.5 ± 113.1 ft for Profile 3.
  • Behavioral interactions: The learned interactions distinguish profiles even when marginal jerk distributions do not: Profile 1 couples spacing with jerk, whereas Profile 3 couples speed with jerk.The strongest mutual-information couplings are 0.228 for spacing–jerk in Profile 1 and 0.133 for speed–jerk in Profile 3.
  • Inside Profile 2: Profile 2 contains a dominant steady-following mode and brief acceleration and braking modes, with jerk marking transitions between these events.Mode 1 carries 85.2% of Profile 2’s mass and accounts for 93% of its active frames; Mode 3 is rare at λ3 = 0.034 and 0.3% of frames.
  • State evolution: Context activates different profiles: smooth, low-entropy traffic favors Profile 1, unsettled moving traffic favors Profile 3, and rising density favors Profile 2.The trajectory shows smooth transitions from free flow through transition to congested stop-and-go without supplied regime labels.

6. Connection to Macroscopic Traffic Phenomena

The learned profiles align with established macroscopic traffic structure despite being inferred without aggregate traffic measures or regime labels. They also reproduce a negative hysteresis loop whose profile composition reflects traffic history during entry into and recovery from congestion.

  • Fundamental diagram: Profile 1 occupies low-density, high-speed flow, Profile 3 approaches capacity, and Profile 2 occupies the high-density congested branch.The profiles appear in the same order as the traffic regimes: free flow, transition, and congestion.
  • Fundamental diagram: The alignment emerges without training on the density-flow relationship or traffic-regime labels, with weakest separation near overlapping branches around capacity.Some off-regime observations remain, and sharper separation could be obtained with more than K = 3 profiles, but K = 3 is retained for interpretability and distinctness.
  • Hysteresis: The stop-and-go wave produces a negative hysteresis loop, while profile composition progresses from Profile 1 before the wave to Profile 3 during entry and Profile 2 inside congestion.During recovery, composition shifts back toward Profiles 1 and 3.
  • Hysteresis: Equal speeds can correspond to different profile compositions: entering congestion favors Profile 3, while leaving it retains more Profile 2 because recent stop-and-go history persists.At similar speeds, Point 4 and Point 8 have sharply different compositions; Point 9 at 53 mph retains substantial Profile 2 compared with Point 1.

7. Applications to Driver and Autonomous Vehicle Models

The representation supplies evolving, context-dependent behavioral information to car-following models and supports live interpretation of surrounding drivers for autonomous vehicles. In an IDM example, state-derived targets adapt as traffic moves through free flow, transition, and congestion.

  • Integration with Driver Models: The driver state defines a distribution over behavior from which average behavioral quantities, including speed and spacing, can be read.The probability distribution is defined through the Born rule.
  • Integration with Driver Models: IDM receives evolving target parameters v0(t), T(t), and s0(t) from the driver state while retaining its original equation.The state supplies target parameters, whereas response parameters remain separately specified.
  • Integration with Driver Models: In the example, the representation-supported follower tracks the leader through free flow, stop-and-go, recovery, and changing behavioral profiles.Profile composition shifts from Profile 1 through Profile 3 to Profile 2 as context changes.
  • Integration with Driver Models: A variant represents acceleration directly, removing the need for target or response parameters and any fixed car-following functional form.The state supplies the likely acceleration from current driver state and context.
  • Autonomous and Mixed Traffic: For autonomous and mixed traffic, the representation provides each surrounding vehicle with a compact, continuous, history-aware behavioral read and uncertainty over profiles.Applying it across the neighborhood reveals heterogeneous profile mixtures among vehicles exposed to similar spatial context.
  • Autonomous and Mixed Traffic: The framework supports short-horizon motion forecasting, although the demonstrated forecast reused a representation trained without a forecasting objective and used linear input extrapolation.A predictive version would learn future context and driver-state evolution jointly.

8. Conclusion and Future Work

The paper concludes that evolving density-matrix states can learn interpretable, context- and history-dependent driver behavior while connecting microscopic profiles to macroscopic traffic phenomena. It also identifies scope, timing, tuning, simulation, prediction, and benchmarking boundaries for future work.

  • Conclusion: Each driver is represented by an evolving density matrix whose continuous, probabilistic, context-dependent, and history-dependent structure is learned without pre-specifying variable interactions.Behavioral profiles, internal modes, and profile mixtures emerge through unsupervised training.
  • Conclusion: On I-24 MOTION, the framework recovered three profiles corresponding to free flow, transitional following, and congested stop-and-go.The profiles also contain modes distinguishing steady following from release and braking events in congestion.
  • Conclusion: The learned profiles align with the fundamental diagram, while state history dependence reproduces hysteresis observed through a kinematic wave.The paper presents this microscopic-to-macroscopic aggregation as a validation principle for behavioral representations.
  • Applications: The state supplies target parameters to classical car-following models and provides autonomous vehicles with live surrounding-driver reads and near-future motion forecasts.These applications are demonstrated through an IDM example and neighborhood-level profile compositions.
  • Limitations: The evaluation uses one freeway corridor and passenger vehicles, so the recovered profiles should not be assumed to transfer to intersections, weaving, or vulnerable-road-user interactions.Broader environmental coverage is needed to test generalization.
  • Limitations: The representation models longitudinal behavior only, so lane-changing requires lateral variables and a mixed discrete-continuous formulation.Lane-changing effects appear only indirectly through contextual variables.
  • Limitations: Behavior and context are modeled at the same time step, although drivers respond after variable delays; the assumption is reasonable here but remains an approximation.The contextual variables are spatially local and change smoothly relative to the state update.
  • Limitations: Several settings were selected empirically for I-24 MOTION, requiring the selection procedure to be repeated in other environments.The appendix ablation supports the chosen settings for this dataset but not their transfer elsewhere.

Author Contribution Statement

The author contribution statement assigns responsibility across study design, data processing, analysis, interpretation, drafting, and editing. The accompanying appendix material describes a transparent procedure for selecting behavioral and context variables.

  • Author Contributions: Study conception and design were conducted by Elayan, Armantalab, and Kontar.
  • Author Contributions: Data processing, analysis, interpretation, drafting, and editing were completed by the listed authors, with no data collection contribution reported.All authors reviewed and approved the final manuscript.
  • Variable Selection: The variable-selection procedure evaluated twenty literature-based candidates using four criteria across four equal-sized speed regimes.The criteria measure within-context spread, between-context shift, temporal evolution, and cross-driver consistency.
  • Variable Selection: Redundant candidates were removed when pairwise correlation exceeded 0.85, and arithmetic and geometric composite means ranked the remaining variables.The geometric mean favors variables that perform consistently across all criteria.

A.3. Results

The results identify which candidate variables best capture heterogeneous, context-sensitive driving behavior in congested I-24 MOTION traffic. The selected variables balance distinct behavioral dimensions and avoid redundant measurements.

  • Results: 10,000 trajectories were scored in the congested westbound I-24 MOTION subset against a neighborhood constructed from the full traffic stream.
  • Results: Speed-related variables B1, C3, and C4 ranked highest but were mutually correlated at approximately |r| ≈ 0.85–0.96.This correlation reflects the close correspondence between ego speed and surrounding speed in congestion.
  • Results: Jerk B3 and entropy measures C6 and C8 were the main candidates showing temporal evolution, while most variables were temporally weak.The result suggests largely stationary behavior over time under congestion.
  • Results: Omnidirectional density C1 ranked among the strongest context variables when measured over a 150 m perception radius.
  • Results: The retained variables cover motion, spacing, jerk, density, and surrounding traffic disorder, with the strongest retained correlation equal to 0.610.Forward-speed variables C3 and C4 were excluded because they duplicated ego speed; entropy measures had |r(C6, C8)| = 0.25.

B.1. Selecting the Number of Profiles (K)

The analysis selects K = 3 because it captures distinct free-flow, transition, and congestion regimes without redundant profiles. Fit improves most from K = 2 to K = 3, with no further improvement at larger K.

  • K = 3 reduces per-observation NLL from 0.90 at K = 2 to 0.88, with no further improvement for K = 4 or K = 5.
  • At K = 2, transitional and congested traffic are conflated, whereas K = 3 introduces a meaningful regime between free flow and congestion.
  • Increasing K beyond three does not uncover additional regimes; K = 4 largely duplicates a transitional profile, while K = 5 adds a diffuse profile.
  • Distinct context-activation patterns at K = 3, versus largely mirrored patterns at K = 4 and K = 5, support using three profiles.
  • Figure B1 compares K ∈ 2, 3, 4, 5 using eigenvalue composition, dominant modes in speed–spacing space, context activation, and per-observation NLL.

B.2. Selecting the Regularization Weight (γ)

The regularization choice is determined by profile eigenvalue structure rather than likelihood, which remains essentially flat across the tested γ values. γ = 4 preserves supported multimodality while maintaining coherent, interpretable profiles.

  • The per-observation NLL is essentially flat across the tested γ range, so likelihood does not select a preferred regularization value.
  • At γ ∈ {0, 0.5, 1}, all three profiles collapse near rank one, with dominant eigenvalues above 0.995.
  • At γ = 4, the congested profile has λ1 = 0.852, while its next four eigenvalues carry a further 14% of the weight.
  • At γ = 8, the leading eigenvalue falls to 0.663 and the multimodal profile becomes too diffuse to describe a coherent regime.
  • Context coefficients remain stable across γ, so regularization changes within-profile mode structure rather than which contexts activate profiles.
  • The study therefore uses γ = 4 because it alone recovers supported multimodal structure while retaining profile interpretability.

B.3. Selecting the Behavioral Variables

The full behavioral set (v, ∆s, j) is selected because it recovers distinct regime signatures and preserves multimodal structure. Ablations show that dropping variables weakens profile structure or shifts contextual assignments.

  • NLL values are not directly comparable across behavioral-variable ablations because models represent distributions over different numbers of variables.
  • Figure B3 compares (v, ∆s), (v, j), and (v, ∆s, j) using eigenvalue spectra, dominant modes, context coefficients, and NLL.
  • With (v, ∆s), three profiles remain, but the congested profile loses most of its eigenvalue spread.
  • With (v, j), spacing is unavailable for organizing free flow, transition, and congestion in the speed–spacing plane.
  • Dropping a behavioral variable substantially changes one profile’s beta signs and magnitudes, indicating reassignment of contextual signals.
  • Only (v, ∆s, j) recovers three distinct speed–headway signatures together with a spread eigenspectrum for congestion.

B.4. Selecting the Context Variables

The full context set is selected because acceleration entropy provides the largest fit improvement and helps distinguish free flow from transitional following at common density. It also yields more informative, differentiated context-activation patterns.

  • The full context set reduces per-observation NLL to 0.88, compared with 0.91 using density alone or density plus speed entropy.
  • Figure B4 compares density alone, density with speed entropy, and the full density–entropy context set through spectra, modes, activations, and NLL.
  • Adding acceleration entropy produces the largest fit improvement and is consistent with distinguishing free flow from transitional following under common density.
  • All context sets produce one multimodal profile, but density-only alternatives place the spread in free flow rather than congestion.
  • Including more context variables makes activation coefficients more informative, with profiles responding to different combinations instead of density alone.
  • The update cost scales as O(D2), making D = 200 approximately 4× and D = 300 approximately 9× as costly as D = 100.
  • At D = 100, 200, and 300, per-observation NLL is 0.88, 0.87, and 0.91, respectively, while profile structure remains stable.
  • Figure B5 reports the same profile diagnostics across D ∈ {100, 200, 300} alongside NLL and projected wall-clock costs.
Loading 2608.25907v1…