Trajectory Metrics: AMD & AMV

• Trajectory distribution metrics are quantitative tools that characterize the statistical, geometric, and probabilistic properties of trajectory ensembles in areas like exoplanet dynamics and multi-agent prediction.
• AMD quantifies orbital excitation and long-term system stability, while AMV extends this by decomposing angular momentum deficits into magnitude and directional components.
• These metrics provide actionable insights into model calibration and predictive uncertainty, impacting trajectory prediction benchmarks, exoplanet system architecture, and multi-object tracking.

Trajectory distribution metrics are quantitative tools for evaluating and analyzing the statistical properties of sets of trajectories, commonly used in multi-agent prediction, exoplanetary system architecture, and multi-object tracking. Core examples include the Angular Momentum Deficit (AMD) and Angular Momentum Vector (AMV) metrics in exoplanetary dynamics, as well as the Average Mahalanobis Distance (AMD) and Average Maximum Eigenvalue (AMV) for probabilistic trajectory prediction models. These metrics rigorously characterize not only the fidelity—but also the geometric and probabilistic structure—of trajectory ensembles.

1. Formal Definitions and Mathematical Construction

Angular Momentum Deficit (AMD):

For an $N$-planet exoplanetary system orbiting a star of mass $M_\star$, with planets indexed by $j=1,...,N$ and possessing respective masses $M_{p,j}$, the AMD quantifies deviation from perfectly circular, coplanar orbits. Define the mass ratio $\mu_j = M_{p,j}/M_\star$ and the circular angular momentum

$\Lambda_j = \mu_j \sqrt{G M_\star a_j}$

where $a_j$ is the semimajor axis. The system-wide AMD is

$$\mathrm{AMD}_{\rm tot} = \sum_{j=1}^N \Lambda_j [1 - \sqrt{1-e_j^2}\cos i_{m,j}]$$

where $e_j$ and $i_{m,j}$ are the eccentricity and mutual inclination of planet $M_\star$0 (He et al., 2020).

Angular Momentum Vector (AMV):

While AMD is a scalar, AMV generalizes it to a vector deficit,

$M_\star$1

where $M_\star$2 and $M_\star$3 is the canonical angular-momentum vector for planet $M_\star$4, and $M_\star$5 is the normal to the orbital plane. $M_\star$6 is the total for circular coplanar configuration. AMV magnitude quantifies excitation; direction encodes the inclination of invariable plane (He et al., 2020).

Average Mahalanobis Distance (AMD):

Given a set of stochastic predicted trajectories, for agent $M_\star$7 and time $M_\star$8, let $M_\star$9 model samples be $j=1,...,N$0. Fit a Gaussian (or GMM) to the samples:

• Mean: $j=1,...,N$1
• Covariance: $j=1,...,N$2

Mahalanobis distance to the ground-truth $j=1,...,N$3:

$j=1,...,N$4

Then

$j=1,...,N$5

(Mohamed et al., 2022).

Average Maximum Eigenvalue (AMV):

Let $j=1,...,N$6 denote the largest eigenvalue of $j=1,...,N$7; the AMV is

$j=1,...,N$8

(Mohamed et al., 2022).

2. Physical and Statistical Interpretation

AMD quantifies the orbital excitation in exoplanetary systems, and acts as a predictor of long-term dynamical stability. A low AMD corresponds to orbits near-circular and coplanar; higher values indicate stronger eccentricity and inclination excitation. The scalar nature reduces full architecture to a single excitation number (He et al., 2020).

AMV extends AMD by retaining both magnitude and direction, allowing dissection of excitation into in-plane (eccentricity-driven) and out-of-plane (inclination-driven) components. The vector structure encodes not just how much angular momentum is "missing," but in which direction relative to the circular-coplanar reference (He et al., 2020).

Average Mahalanobis Distance (AMD, in prediction): By normalizing the deviation between predicted and true trajectories using the covariance structure of predictions, AMD expresses how "surprising" the ground truth is relative to the predictive distribution, in units adapted to local uncertainty ("$j=1,...,N$9-units") (Mohamed et al., 2022).

Average Maximum Eigenvalue (AMV): This scalar summary of the largest variance direction in the predictive covariance encapsulates model uncertainty. A larger AMV indicates a long tail or diffuse spread in at least one direction of the predictive trajectory distribution. It is invariant to translation, focusing exclusively on geometric dispersion (Mohamed et al., 2022).

3. Computational Procedures

Exoplanet System Metrics (AMD, AMV)

• Compute $M_{p,j}$0 for each planet using masses and semimajor axes.
• Obtain eccentricity $M_{p,j}$1 and mutual inclination $M_{p,j}$2.
• Sum according to the AMD formula. For AMV, compute orbit normals, scale vectors, sum, and take the difference against the reference vector.

Distributional Trajectory Metrics (AMD, AMV)

1. For each agent and prediction timestep:
   • Fit a Gaussian or GMM to the sample set $M_{p,j}$3.
   • Compute mean $M_{p,j}$4 and covariance $M_{p,j}$5.
   • For each $M_{p,j}$6, calculate Mahalanobis distance to ground-truth.
   • Compute largest eigenvalue of $M_{p,j}$7 (for 2x2 matrices, via analytic quadratic roots).
2. Aggregate across all $M_{p,j}$8 to obtain AMD and AMV.
3. Regularize $M_{p,j}$9 as $\mu_j = M_{p,j}/M_\star$0 for numerical stability, $\mu_j = M_{p,j}/M_\star$1 (Mohamed et al., 2022).

Pseudocode in (Mohamed et al., 2022) provides implementational details for batch computation, matrix inversion, and eigenvalue extraction for $\mu_j = M_{p,j}/M_\star$2 covariances.

4. Theoretical Properties, Sensitivity, and Limitations

Stability and Anti-correlation (Exoplanetary AMD):

AMD-stable configurations sharply constrain eccentricity and inclination distributions. Median eccentricity and mutual inclination scale with intrinsic multiplicity $\mu_j = M_{p,j}/M_\star$3 as $\mu_j = M_{p,j}/M_\star$4 and $\mu_j = M_{p,j}/M_\star$5. This anti-correlation originates from dynamical stability boundaries imposed by AMD-criticality: more crowded systems require lower $\mu_j = M_{p,j}/M_\star$6 and $\mu_j = M_{p,j}/M_\star$7 to avoid orbit crossing/collision (He et al., 2020).

Calibration and Robustness (Predictive AMD/AMV):

ADE/FDE metrics may be insensitive to broad or multi-modal distributions or to systematic distributional shifts ("lucky" outlier samples may dominate Best-of-N ADE). In contrast, AMD tracks both distance and spread rigorously; shifting all predicted samples by $\mu_j = M_{p,j}/M_\star$8 increases AMD linearly, but ADE may remain nearly unchanged. AMV penalizes overconfident models with tiny variances, precluding trivial solutions where artificially inflated covariances would drive AMD to zero (Mohamed et al., 2022).

For a $\mu_j = M_{p,j}/M_\star$9-dimensional normal, AMD$\Lambda_j = \mu_j \sqrt{G M_\star a_j}$0 is $\Lambda_j = \mu_j \sqrt{G M_\star a_j}$1-distributed when the ground-truth is itself drawn from the predictive distribution, so confidence regions are interpretable in terms of classical statistical tests (Mohamed et al., 2022).

5. Applications and Empirical Findings

Exoplanetary System Architecture:

He et al. show that maximum-AMD forward modeling, combined with detection bias modeling (SysSim), reproduces the full array of observed Kepler system statistics—multiplicity, radius uniformity, period-normalized duration ratios, and size ordering—without invoking heterogeneous populations (no distinct high-inclination class required). AMD-stability delivers a unified architectural framework for interpreted distributions and anti-correlations (He et al., 2020).

Trajectory Prediction Benchmarks:

Empirical studies on ETH/UCY datasets illustrate that models with similar ADE/FDE can have vastly different AMD/AMV, reflecting differences in predictive calibration. For example, an "ExpertTraj" model can achieve minimal ADE but highly inflated AMD (38 m), signifying narrow predictions missing the true distribution, whereas Trajectron++ and Social-Implicit offer calibrated AMD/AMV values with competitive displacement errors (Mohamed et al., 2022).

Model	ADE/FDE (m)	AMD	AMV
S-GAN	0.58/1.18	2.47	0.361
S-STGCNN	0.44/0.75	2.39	0.104
Trajectron++	0.19/0.41	1.84	0.183
ExpertTraj	0.17/0.35	38.4	0.004
Social-Implicit	0.33/0.67	1.63	0.174

Note: ExpertTraj, despite a lower ADE, is miscalibrated as evidenced by its extremely high AMD.

6. Extensions and Generalizations

• Multi-object Tracking: The metric of (García-Fernández et al., 2016) defines a family of p-norm optimal assignment-based set metrics, handling missed/false targets and track switches in a trajectory assignment context. The construction is distinct but related by its ambition to quantify trajectory-set agreement over time and scenarios.
• Random Set Formulation: For random finite sets of trajectories, expected AMD/AMV (or assignment metrics) can be used as metrics on spaces of trajectory-set probability distributions given integrability/finite moment conditions (García-Fernández et al., 2016).
• Vector-valued Deficits: The conceptual extension from AMD (scalar) to AMV (vector) enables finer partition of excitation and misalignment, both in celestial mechanics and in probabilistic trajectory evaluation frameworks (He et al., 2020).

7. Implementation Considerations

• Covariance Regularization: Ensures numerical stability for predictive AMD/AMV.
• Batch Operations: Tensorized computation suitable for deep-learning batch inference (Mohamed et al., 2022).
• Eigenanalysis: Efficient closed formula exists for $\Lambda_j = \mu_j \sqrt{G M_\star a_j}$2 matrices common in planar trajectory tasks.
• Gaussian Mixture Models: Use of BIC to select component count for sample sets, GMM forms of AMD/AMV can capture multimodality while providing analytic tractability (Mohamed et al., 2022).
• Exoplanetary Metrics: AMD/AMV require detailed orbital element catalogs and robust mass-radius scaling for planet assignment (He et al., 2020).

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Trajectory distribution metrics such as AMD and AMV—across both dynamical and probabilistic paradigms—are foundational for evaluating system architectures, trajectory prediction quality, and calibration. Their rigorous mathematical definitions and practical sensitivity to statistical structure make them indispensable in contemporary astrostatistical and machine learning research (He et al., 2020, Mohamed et al., 2022, García-Fernández et al., 2016).