Trajectory-Level Scoring Methods

• Trajectory-level scoring is the quantitative evaluation of entire sequences by aggregating metrics to assess global behavior, safety, and compliance.
• It integrates symbolic methods, machine learning models, and probabilistic techniques to address diverse applications like hardware verification, autonomous driving, and anomaly detection.
• Hybrid scoring approaches combine interpretable rules with data-driven insights, enabling robust, scalable assessment of trajectories in complex, real-world scenarios.

Trajectory-level scoring refers to the quantitative assessment or ranking of entire sequences—trajectories—where each sequence represents an ordered collection of states, locations, actions, or events unfolding in time or space. Unlike pointwise or step-level evaluation, trajectory-level scoring aggregates or derives metrics from the holistic behavior of a trajectory, allowing for insight into the global quality, safety, realism, compliance, anomaly, or desirability of the entire sequence. This paradigm is prevalent across diverse domains, such as formal hardware verification, recommendation and planning, autonomous driving, robotics, sequential decision-making, tracking, crowd simulation, and anomaly detection. The literature reveals a variety of scoring methodologies, from symbolic abstraction and probabilistic metrics to learned and interpretable scoring functions, each tuned for particular requirements and operating constraints.

1. Symbolic and Formal Approaches to Trajectory Scoring

In formal hardware verification, symbolic trajectory evaluation (STE) exemplifies trajectory-level scoring rooted in lattice-based abstraction and temporal reasoning. STE traditionally encodes system properties using symbolic trajectory formulas that marry Boolean conditions with a next-step temporal operator. Conventional bit-level STE operates with values in {0, 1, X}, but this quickly becomes intractable for wide datapaths because the lattice of possible system states grows exponentially with bit width.

To address scalability, the word-level STE engine STEWord (Chakraborty et al., 2015) introduces automatic atomization of words (partitioning into subwords called atoms) and invalid-bit encoding, deriving shallow abstract lattices from RTL without resorting to bit-blasting. Each atom—typically an entire multi-bit slice not accessed partially—has a value or is marked as invalid (X). This abstraction yields a lattice with drastically fewer elements while preserving sufficient precision for most operators and enables efficient computation of least upper bounds (lub) in trajectory simulation. Scoring in this context refers to assessing how well symbolic simulation of the design matches the expected trajectory properties, with abstractions allowing for scalable and systematic evaluation across long and wide signal trajectories.

2. Machine Learning and Data-driven Methods

Trajectory-level scoring frequently leverages learned or data-driven models, especially in domains where patterns are too complex or context-dependent for purely symbolic solutions.

• Recommendation and Route Planning:

For trajectory recommendations (e.g., tourist routes), scoring incorporates both pointwise preferences (such as POI attributes) and sequence-dependent transitions. A machine learning model (rankSVM) (Chen et al., 2016) learns to rank POIs using feature vectors that encode POI characteristics and their relationship to start/end query and candidate POIs, with additional transition probabilities inferred via Markov chains over observed POI-to-POI transitions. The overall trajectory score for a given route is a weighted sum of ranking probabilities and transition likelihoods. Evaluation metrics extend beyond bag-of-POIs F₁ scores to “pairs-F₁,” which captures both POI selection and their visiting order.

• Driver Safety Scoring:

Scoring methods for driver safety based on trajectory data (Wang et al., 2018) use extracted features reflecting driving habits, aggressive maneuvers, and traffic violations. Features are then weighted and combined via a random forest classifier to estimate risk, with driver scores reflecting the likelihood of future violations—demonstrated via long-term simulations.

• Counterfactual and Modular Scoring:

TraCE (Clark et al., 2023) introduces a generalized model-agnostic modular scoring framework using counterfactual explanations for sequential decision processes. The trajectory score at each step synthesizes geometric information—angle (directional alignment with a target trajectory) and landing (distance to a target state)—to provide a continuous, interpretable evolution metric (range [–1, 1]) reflecting progress toward or away from a defined outcome. Such techniques enable both instantaneous and cumulative tracking of improvement, independent of black-box predictive models.

3. Probabilistic and Generative Approaches

Generative probabilistic modeling underpins robust trajectory-level scoring in multiple contexts:

• Multi-object Tracking Metrics:

The Probabilistic Trajectory GOSPA (PTGOSPA) metric (Xia et al., 18 Jun 2025) generalizes the trajectory-level GOSPA metric to explicitly account for both state estimation and existence probability uncertainties. PTGOSPA is constructed as a multidimensional assignment problem between sequences of Bernoulli densities (representing per-step existence and state distributions). Cost decomposition covers expected localization error, existence mismatch error, missed detections, false positives, and track switches, and can be efficiently approximated via linear programming relaxation.

• Generative Models for Tracking and Prediction:

Probabilistic autoregressive models (e.g., ArTIST (Saleh et al., 2020)) provide scores by evaluating the likelihood of an observed trajectory under a learned distribution of “natural” motions, with applications in tracking, inpainting missing detections, and human motion prediction. Diffusion-based approaches for trajectory generation (e.g., DICE (Choi et al., 2023), SceneDM (Guo et al., 2023), GTRS (Li et al., 7 Jun 2025)) sample diverse future trajectories from a noisy latent space, then rank them using dedicated scoring modules—often leveraging attention mechanisms—to select those trajectories most likely under real-world constraints (collision avoidance, smoothness, compliance with road rules).

4. Rule-Based, Logical, and Interpretable Scoring

Trajectory-level evaluation is increasingly focused on generating interpretable scores that encode safety, compliance, or preference criteria:

• Temporal Logic in Autonomous Driving:

FLoRA (Xiong et al., 17 Feb 2025) learns scoring rules as logical formulas (in LTL_f, finite linear temporal logic) parameterized and optimized from real driving demonstrations. The logic structure is differentiable, supporting learning over temporal predicates (safe distance, acceleration, lane keeping, etc.), soft selection of logical/temporal operators, and aggregation of rules for comprehensive, interpretable scoring of candidate trajectories.

• Rule Correction for Robustness:

In operational settings where data distributions shift (e.g., due to disasters), neuro-symbolic frameworks (Xi et al., 2023) can learn rules over movement trajectories to detect and correct classifier errors, improving robust classification under domain shift without retraining base models.

• Multi-criteria and Simulation-Supervised Scoring:

Decoupled frameworks, such as HMAD (Wang et al., 29 May 2025), generate diverse trajectory proposals and then evaluate each using a learned scorer trained on simulation-supervised ground-truth metrics (e.g., no at-fault collision, drivable area compliance, comfort, extended PDM score). Extended PDM combines multiplicative and weighted additive penalties across critical metrics, allowing nuanced, context-dependent selection of the optimal trajectory.

5. Domain-specific Metrics and Validation

Trajectory-level scoring is frequently domain-tailored:

• Crowd Simulation and Perceptual Validation:

Crowd simulation leverages composite feature-based metrics (QF) validated by user studies to score realism of simulated trajectories (Daniel et al., 2021). QF combines weighted cost functions over a broad set of individual, local, and global trajectory features—statistically matched to reference human trajectory data and validated against human perceptual judgments.

• Anomaly Detection in Graph-constrained Domains:

GETAD (Mbuya et al., 22 Sep 2025) integrates road network graphs, road segment semantics, and historical transitions to learn feature-enriched node embeddings via multi-head Graph Attention Networks. A Transformer sequence decoder then models segment transitions probabilistically, with trajectory-level anomaly scores given by Confidence Weighted Negative Log Likelihood (CW-NLL), emphasizing token-level deviations made with high model certainty.

• Human-machine Shared Control:

In cooperative trajectory planning (Schneider et al., 22 Oct 2024), scoring emerges naturally via iterative negotiation and arbitration: human and machine agents propose trajectories, score them according to explicit cost functions and system state, then converge on a “joint” trajectory optimizing collective criteria. This agreement process is crucial for safety and stability in applications such as advanced driver assistance systems.

6. Performance, Interpretability, and Generalization

Performance of trajectory-level scorers is assessed by both classical and novel metrics—e.g., improvement in driving score (HMAD), pairs-F₁ (trajectory order in recommendation), cross-entropy to ground truth preferences (reward learning from language (Yang et al., 8 Oct 2024)), and robust differentiation between similar critical trajectories (GTRS refinement (Li et al., 7 Jun 2025)). Recent developments emphasize interpretability, formalized rule extraction, and model-agnostic designs that permit easy integration as post-processing modules in larger pipelines.

Generalization—across sensors, scenario diversity, and data drift—is achieved by combining coarsely discretized static vocabularies, fine-grained diffusion-generated trajectory proposals, vocabulary dropout, structured sensor augmentation, and refinement via self-distillation frameworks. This enables deployment of robust scorers even under sub-optimal real-world conditions.

7. Future Directions and Open Challenges

Challenges persist in balancing abstraction and precision (STEWord), integrating negative examples for scoring rules (FLoRA), seamlessly combining symbolic rules with learned modules (EDCR frameworks), efficiently calibrating and improving agent trajectories in long-horizon environments (STeCa (Wang et al., 20 Feb 2025)), and scaling to high-dimensional, multi-agent, and graph-constrained domains (GETAD, SceneDM).

A plausible implication is that future directions will focus on hybrid systems: modular, interpretable, and uncertainty-aware scorers that are both data-efficient and suitable for online adaptation; deeper integration of domain-specific semantics with generic learning architectures; and more comprehensive validation frameworks spanning simulated, synthetic, and real-world driver, pedestrian, or user behaviors. Trajectory-level scoring is poised to remain central in applications where robust sequential decision making, safety, and compliance are critical.