PaperlayerView exact revision on arXiv

Source-linked AI summary

Kairos: Grounded Forecasting of Presence and Directional Flow in 4D Scene Graphs

Iacopo Catalano, Julio A. Placed, Javier Civera, Jorge Peña Queralta

arXiv:2609.27467v1cs.ROcs.AI

TL;DR

Long-term planning needs forecasts of both pedestrian presence and directional motion at future times, but existing approaches tend to provide scalar activity forecasts or time-invariant directional distributions. Kairos stores these quantities on reconstructed voxels and forecasts them over time. Across real environments, its forecasts are competitive with dedicated occupancy models, and its learned state remains consistent under loop-closure corrections.

  • Problem

    Existing motion representations forecast future activity as a scalar or describe full directional flow only as a time-invariant average, limiting forecasts of how a location’s flow changes over time.

    Source: §I · p. 1

  • Method

    Kairos extends a 3D scene graph with per-voxel spectral forecasts of presence and the full directional distribution, plus pairwise dependence between adjacent voxels.

    Source: §I · p. 2, §V · p. 5, E. Flow Dependence · p. 8

  • Results

    Kairos is competitive with a dedicated occupancy model trained on the full detection stream despite observing a small fraction, and its learned state remains consistent under loop-closure corrections.

    Source: §IX · p. 18

  • Takeaways & Limitations

    Kairos provides future-time presence and directional-flow forecasts anchored to traversable locations in the reconstructed geometry.

    Source: §IX · p. 17

    Cumulative-mean predictors correct permanent flow changes at a rate of one over the accumulated sample count, and the method does not distinguish such changes from sampling noise.

    Source: §IX · p. 18

Abstract

from arXiv · show

Long-term autonomy in human-populated environments requires anticipating whether and how people will move at times a robot has not yet observed. Existing representations of pedestrian motion face a tradeoff: they either forecast future activity, reducing each location to a scalar rate, or model the full directional distribution, holding it fixed in time. We present Kairos, a predictive directional-flow memory that extends a hierarchical 3D scene graph (3DSG) to a 4D scene graph (4DSG). Every observed voxel of the reconstructed geometry stores a directional mixture and a presence rate, and spectral predictors forecast, for any future query time, both the probability that people are present and the full directional distribution of their motion. Pairwise flow dependence between adjacent voxels supports conditional queries, and per-voxel predictive variances yield calibrated credible intervals that tighten as observations accumulate. We evaluate Kairos on three real pedestrian environments: a robot-collected campus dataset, a shopping mall, and a station concourse recorded continuously for eleven months. Its learned state remains consistent under loop-closure corrections, and its forecasts are competitive with dedicated occupancy and flow models trained on the full detection stream, although Kairos learns from only the small fraction available to a patrolling robot. Finally, we validate the representation on a downstream encounter-probability planning task, where plans computed over the Kairos forecasts encounter more people than plans computed over any time-invariant map at an equal success rate. We provide the code at https://github.com/IacopomC/kairos.

I. INTRODUCTION

Predictive planning needs flow forecasts that are directional, time-conditioned, and grounded in reconstructed geometry, but prior representations do not provide all three together. Kairos extends a 3D scene graph with full-distribution forecasts over time, keyed to reconstructed voxels and kept consistent under map correction.

A. Maps of Dynamics

  • Earlier occupancy models forecast scalar activity, while directional models commonly represent a fixed long-run average rather than changes over time.Kairos applies a spectral predictor to each mixing weight of a per-voxel directional mixture and to occupancy rate.Source: §I · p. 1, A. Maps of Dynamics · pp. 2–3

I. INTRODUCTION

  • Kairos forecasts the complete directional flow distribution over time on reconstructed voxels, keeping estimates consistent as the map is corrected.It extends a hierarchical 3D scene graph to a 4D scene graph and updates each voxel from a fixed-size summary.Source: §I · p. 2
  • A separate presence channel distinguishes locations with no agents from locations where agents have not yet been observed.Forecasts are aggregated up the hierarchy and exposed on the navigational graph for planner use.Source: §I · p. 2
  • Pairwise dependence between adjacent voxels supports conditional flow queries, while predictive variances and observation scatter provide calibrated credible intervals.Source: §I · p. 2

A. Maps of Dynamics

  • Other learned approaches relax spectral, online-update, or grid limits, but their forecasts are not keyed to the robot’s corrected reconstructed geometry.Kairos keys each channel to active reconstruction voxels and re-keys them after backend corrections.Source: A. Maps of Dynamics · p. 3

B. Dynamics in 3D Scene Graphs

  • Existing scene graphs track state changes, individual agents, or aggregate flow, but the cited flow layers either discard the full distribution, lack temporal forecasting, or attach estimates by node proximity.Kairos forecasts full directional flow and anchors it to reconstruction voxels rather than navigational-node proximity.Source: §I · p. 1, B. Dynamics in 3D Scene Graphs · p. 3

III. PROBLEM FORMULATION

  • The problem setting uses sparse, irregular detections of moving agents and characterizes flow by horizontal velocity; stationary agents and in-place rotations are outside the model’s scope.Source: §III · p. 3

A. Dynamics Field

The dynamics field represents pedestrian activity by both occupancy intensity and a conditional distribution over heading and speed. Kairos seeks to forecast this distribution-valued field at future times from a bounded-size online state, assuming temporal recurrence.

A. Dynamics Field

  • The field separates occupancy intensity, whose integral gives expected agent counts, from the flow distribution of agents present at a location.The flow distribution is defined where occupancy intensity is positive and is jointly normalized over heading and speed.Source: A. Dynamics Field · p. 4

B. Estimation Problem

  • The online estimator keeps a state whose size does not grow with detections, so memory scales with the extent of the environment covered.A readout maps that state to a forecast at a future time and position.Source: B. Estimation Problem · p. 4
  • A useful robot forecast must capture motion direction, temporal change, and where the estimate applies on the robot’s map.The field’s distribution-valued output and future-time query encode the directional and time-conditioned requirements.Source: B. Estimation Problem · p. 4
  • Future forecasts are distribution-valued over heading and speed, evaluated at query times beyond the observation horizon.Forecasting beyond observed times is well-posed only under recurrence; the model decomposes intensity into recurrent and temporally unstructured residual components, and forecasts only the recurrent part.Source: B. Estimation Problem · p. 4

C. Grounding

Kairos grounds forecasts in the evolving scene graph’s reconstructed geometry so estimates remain addressable as the map changes. Its state models occupancy and directional flow over voxels and supports cross-cell flow queries.

C. Grounding

  • The readout maps covered positions to scene-graph elements, making forecasts addressable by the graph rather than external coordinates.The graph is an online hierarchy built over a TSDF geometry layer, with navigational nodes, traversability edges, and room nodes.Source: C. Grounding · p. 4, 1) Voxels · p. 5, 2) Navigational Nodes · p. 5

IV. SCENE GRAPH SUBSTRATE

B. Grounding on the Graph

  • Re-keying voxel forecasts after map corrections relocates where they apply while leaving accumulated statistics otherwise unchanged.When a correction maps multiple voxels to one, their statistics are pooled.Source: C. Grounding · p. 4, B. Grounding on the Graph · p. 5

V. MODELING SPATIO-TEMPORAL FLOW DYNAMICS

  • Kairos makes future-time estimation tractable by using piecewise-constant voxel fields, separate occupancy and flow channels, fixed directional components, and scalar spectral predictors.Evidence sharing shrinks sparsely observed cell weights toward neighborhood weights, while pairwise dependence relates flow between adjacent cells.Source: §V · p. 5

A. Piecewise-Constant Field

Kairos stores a fixed-size state per voxel and represents flow with a semi-wrapped Gaussian mixture over heading and speed. It forecasts changing directional weights with online spectral predictors, while maintaining speed means as running averages.

A. Piecewise-Constant Field

  • Each query retrieves a voxel’s fixed-size state, and piecewise constancy makes occupancy intensity uniform within that voxel.The cell’s expected agent count corresponds to intensity up to the known cell volume.Source: A. Piecewise-Constant Field · p. 5

B. Flow Dynamics Model

  • With K=8 fixed heading slots, each voxel’s mixture varies across voxels in mean speed and varies over time through its mixing weights.The angular component is wrapped while the speed component is an untruncated Gaussian; the stated speed range makes negative-speed mass negligible.Source: B. Flow Dynamics Model · pp. 5–6, Figure 2 · p. 6

C. Temporal Forecasting

  • Only the mixing weights are forecast; per-direction speed means instead use responsibility-weighted running averages, with a pooled global-mean prior until a slot has enough observations.Each slot predictor uses a scalar time series based on traversal shares, so its estimate depends neither on detection rate nor on cell dwell time.Source: C. Temporal Forecasting · pp. 6–7
  • Online spectral predictors represent each scalar trajectory with a mean and periodic harmonics, updating from timestamped samples without fixed-interval aggregation.At query time, the model selects the candidate order with the lowest accumulated one-step prediction error, defaulting to order 0 until a higher order improves it.Source: C. Temporal Forecasting · p. 6, 1) Model Order Selection · p. 7
  • Predicted mixing weights are clipped at zero and renormalized to the probability simplex before reconstructing the future flow distribution.The dominant heading and speed are then the weight-averaged component means at the query time.Source: C. Temporal Forecasting · p. 7

1) Model Order Selection:

The predictor returns uncertainty with each forecast, combining uncertainty in its coefficients with the spread of observed samples. Its state remains constant-memory, while a diagonal posterior approximation omits coefficient correlations.

1) Model Order Selection:

  • Predictive uncertainty combines uncertainty remaining in the predictor’s coefficients with the spread of samples around the tracked value.The coefficient uncertainty contracts as samples accumulate.Source: 2) Predictive Uncertainty · p. 7
  • A diagonal Gaussian posterior approximation tracks one variance per mean or harmonic coefficient, enabling O(F) state and rank-one updates.The approximation omits correlations induced by basis functions that are not orthogonal over the sample times.Source: 2) Predictive Uncertainty · p. 7
  • The forecast variance weights each harmonic’s coefficient variance by its basis function at the query time.The variance state covers all F candidate frequencies, even though readout uses the M retained frequencies.Source: 2) Predictive Uncertainty · p. 7
  • Sample spread is estimated from a running second moment and combined with a prior variance, whose weight decays as κ/(κ + C).A cell reports the prior spread until it has evidence of its own, then reports its measured spread thereafter.Source: 2) Predictive Uncertainty · p. 7

D. Evidence Sharing

Evidence sharing stabilizes sparsely observed voxel flows using nearby traversable cells, while pairwise couplings represent dependencies between neighboring flows. Together, these mechanisms support conditional directional queries across adjacent voxels.

D. Evidence Sharing

  • The neighborhood includes traversable voxels in the same or connected navigational-node support sets, so evidence is not shared across obstacles.Each voxel enters the neighborhood at most once.Source: D. Evidence Sharing · p. 7
  • With few crossings, a voxel’s estimate moves toward an evidence-weighted neighborhood mean; with many, it moves toward its own mean.The two distributions are combined convexly, preserving a distribution over slots.Source: D. Evidence Sharing · p. 7

E. Flow Dependence

  • Independent voxel marginals can permit flow combinations inconsistent with continuous motion, so Kairos models dependence between adjacent voxels with pairwise couplings.The model assumes local dependence: non-adjacent voxels are conditionally independent given the voxels between them.Source: E. Flow Dependence · p. 8

1) Joint and Conditional:

  • A coupling reweights the product of two adjacent voxels’ mixing weights: positive values raise a slot pair’s joint probability, while negative values lower it.At zero coupling, the joint distribution is the independent product; conditioning on one voxel’s slot sharpens the neighbor’s weights toward co-occurring slots.Source: 1) Joint and Conditional · p. 8

2) Estimating the Coupling:

Kairos learns time-varying couplings from paired observations of adjacent voxels and separately estimates presence and flow. Its presence model uses occupancy-derived rates to return a joint distribution over presence and motion.

2) Estimating the Coupling:

  • Adjacent-voxel detections within a 1 s coherence window form pairs; consecutive sampling windows accumulate their slot responsibilities into co-occurrence tensors.Windows are timestamped at their midpoint for temporal prediction, and their nominal length is set from the shortest candidate period with 12 samples per cycle.Source: 2) Estimating the Coupling · p. 8
  • The coupling estimate is a pointwise mutual-information log-ratio of observed co-occurrence to expected co-occurrence under independence; sparse evidence with smoothing defaults toward independence.A window with one pair yields zero coupling, so windows below the minimum pair count remain open past their nominal span.Source: 2) Estimating the Coupling · p. 8

F. Presence Estimation

  • Kairos combines presence probability with the flow distribution conditional on presence to return a joint over presence and velocity.The presence probability is derived from occupancy intensity over the queried cell and horizon.Source: F. Presence Estimation · pp. 8–9
  • The presence model treats each voxel as an M_t/G/∞ queue, with Poisson arrivals at a time-varying rate, general crossing times, and unbounded concurrent service.The authors note that fine cells hold about one agent at a time, while crowds appear as multiple occupied voxels.Source: F. Presence Estimation · pp. 8–9
  • A Gamma–Poisson estimate uses detections and visible frames to estimate occupancy, then forecasts the visible detection rate with the spectral predictor.Once visible time dominates the prior exposure, occupancy is approximately detections divided by visible frames, including empty frames.Source: F. Presence Estimation · p. 9

VI. HIERARCHICAL FLOW REPRESENTATION

Kairos lifts voxel-level flow and presence estimates into the scene graph, aggregating them into navigational-node summaries and annotating edges with directional flow scores. These attributes expose time-conditioned flow through a planner-facing interface.

1) Node-Level Aggregation:

  • Each navigational node aggregates flow voxels in its support set using occupancy-weighted mixtures; heading weights aggregate by slot, while pooled speed means preserve slot means and approximate spread.The heading mixture itself does not depend on the presence-query horizon.Source: 1) Node-Level Aggregation · pp. 9–10
  • Region presence is computed from expected occupancy and a region crossing time, rather than as a union of cell presences that could count one transiting agent repeatedly.The node metadata stores occupancy so a query can supply its own horizon at read time.Source: 1) Node-Level Aggregation · p. 10
  • Node metadata exposes occupancy, dominant heading, directional mixture, and concentration, enabling route scoring by expected encounters and preferred direction while retaining the full mix.Concentration measures how sharply occupied voxels commit to a heading, from 1/K to 1.Source: 1) Node-Level Aggregation · p. 10

2) Edge Flow Annotation:

  • For each navigational edge, forward and reverse alignment scores project slot headings onto the edge direction and scale the result by endpoint occupancy.The scores augment edges with average speed and bidirectionality.Source: 2) Edge Flow Annotation · p. 10

3) Consistency under Optimization:

Kairos re-aligns and fuses its learned state when backend optimization corrects map geometry, and adds a layer of motion-semantic archetypes above navigational nodes. The evaluation spans datasets, observability regimes, and flow-model baselines.

3) Consistency under Optimization:

  • On backend optimization, voxel states move with corrected geometry and headings are resampled onto fixed global slots; fused cells pool accumulated estimates and counts.The described consistency holds for locally rigid yaw corrections; planar slots cannot represent out-of-plane rotation.Source: 3) Consistency under Optimization · p. 10

B. Spatial Motion Semantics

  • The motion-archetype layer groups adjacent navigational nodes into Static, Unimodal, Bimodal, Intersection/Multimodal, or Diffuse classes.The classes are assigned from presence and dominant-slot counts.Source: B. Spatial Motion Semantics · p. 10

VII. EXPERIMENTAL EVALUATION

  • The evaluation compares predicted flow under identical scoring and sparse embodied coverage, tests map-correction consistency and presence forecasting, measures added computation, and includes encounter-probability planning.The flow comparisons include both grid- and graph-based baselines.Source: §VII · p. 11

1) Datasets:

  • The dataset suite combines a robot-collected campus dataset for end-to-end embodied feasibility with mall and station data supplying long-horizon recurrence.The station concourse was recorded continuously for eleven months; the mall trajectories cover 92 days across roughly a year.Source: 1) Datasets · p. 11

2) Evaluation Protocol:

  • Training and testing are temporally separated: the campus split spans weeks, while mall and station tests use held-out days after training dates.Candidate periods are set to each dataset’s recurrence scale.Source: 2) Evaluation Protocol · p. 11

3) Observability Regimes:

  • In the embodied regime, the simulated robot observes about 4% of ATC detections and 7% of HB; full visibility trains on every detection.The main results use embodied observations; full-visibility results are reported only where marked.Source: 3) Observability Regimes · p. 11

1) Baselines:

  • Flow baselines span offline directional density, grid-based temporal prediction, and scene-graph methods using binned temporal flow or online mixtures without forecasting.The four named methods are CLiFF-map, STeF-Map, Aion, and Rheos.Source: 1) Baselines · p. 11

2) Evaluation Metrics:

Kairos is evaluated on flow prediction, sparse embodied coverage, and resilience to loop-closure corrections. Its results show strong flow forecasts under sparse coverage and preservation of learned state under map correction.

2) Evaluation Metrics:

  • Flow evaluation uses MLPD for density, circular CRPS for heading distance, and speed MAE on matched detections only.Uncovered detections receive a shared uniform-density floor for MLPD; heading-only methods are scored only on heading.Source: 2) Evaluation Metrics · pp. 11–12

3) Results:

  • On TBD, Kairos achieves the best MLPD in every channel and lowest speed MAE despite embodied training against baselines with full observability.On ATC and HB, STeF-Map has slightly better heading MLPD, while the two methods are at parity on distance-aware CRPS; Kairos also predicts speed and presence.Source: Table III · p. 12, 3) Results · p. 12
  • On TBD, Rheos trails Kairos by 4.2 nats on joint MLPD, with near-degenerate online covariance estimates concentrating density on a one-dimensional ridge.CLiFF-map and Rheos are reported only on TBD because of corpus-scale fitting or scoring costs.Source: 3) Results · p. 12

C. Robustness Under Sparse Embodied Coverage

D. Loop Closure Robustness

  • Loop closure keeps heading MLPD within 0.03 nats of the reference, while uncorrected drift degrades it by 0.11 nats and raises CRPS from 0.71 to 0.75.With correction, median dominant-direction difference is 9.5° and a quarter of nodes exceed 22.5°; without correction, the median is 22.8° and half exceed that threshold.Source: Table V · p. 13, D. Loop Closure Robustness · pp. 13–14

E. Presence Estimation

Kairos forecasts windowed presence using time-varying estimates and is evaluated against static-rate and spectral baselines. Its presence and planning results show performance close to full-visibility alternatives in the tested settings, with computational costs reported separately.

E. Presence Estimation

  • Kairos forecasts whether a cell contains at least one detection during a future window, evaluated at horizons from 5 to 600 seconds.The window is capped at 600 seconds so it remains shorter than HB’s shortest candidate period.Source: E. Presence Estimation · p. 14

1) Baselines:

  • Presence baselines include a training-frequency constant rate, full-visibility per-cell FreMEn, and Kairos ablations isolating its dwell correction and periodic component.The static-rate ablation holds the rate constant and removes gate-selected harmonics.Source: 1) Baselines · p. 14

2) Evaluation Metrics:

  • Presence forecasts are scored with MLPP and Brier-score reliability and resolution; reliability and resolution are reported from 60 seconds onward because they separate estimators there.Reliability measures the gap between forecast probabilities and observed frequencies, while resolution measures how those frequencies vary around the overall occupancy rate.Source: 2) Evaluation Metrics · p. 14

3) Results:

  • In embodied evaluation, Kairos has the best MLPP at every horizon, from −33 at 5 s to −460 at 600 s, and is more informative and better calibrated than its static-rate variant.At 600 s, resolution is 58.9 versus 44.3 and reliability is 4.6 versus 9.0; the margins grow with horizon as windows integrate more of the daily profile.Source: 3) Results · p. 14

F. Computational Cost

  • The flow layer adds 0.43 ms per frame, while node aggregation and deformation remapping add 4.1% per backend re-optimization.The measured front end runs within the 200 ms key-frame budget, and enabling the layer leaves its timing unchanged.Source: F. Computational Cost · p. 15

G. Downstream Task: Encounter-probability Planning

  • Encounter-probability planning uses a finite-horizon dynamic program to choose whether to wait or move, maximizing predicted activity accumulated before a 600-second deadline.Episodes replay recorded pedestrians, and only presence is evaluated because encounters depend on presence.Source: G. Downstream Task:… · p. 15, 2) Evaluation Protocol · p. 15

3) Results:

  • On HB, time-conditioned planning sources encounter 26.8 people with Kairos and 27.4 with FreMEn, versus 23.5 with the static rate.Success is 92% for Kairos and 94% for FreMEn, with overlapping confidence intervals; Kairos uses about 7% of the HB detection stream compared with FreMEn’s full record.Source: Table VIII · p. 15, 3) Results · p. 15

2) Evaluation Protocol:

  • Planning results use one fixed home node, so the reported day-to-day confidence intervals do not measure sensitivity to the robot’s start location.Success-rate intervals are t-intervals over held-out days, the unit of replication.Source: 2) Evaluation Protocol · p. 15

A. Temporal forecasting of the direction weights

This section evaluates temporal direction forecasts, evidence sharing, adjacent-voxel coupling, and predictive uncertainty. The results support temporal weights, sharing for sparsely observed cells, evidence-dependent coupling, and calibrated intervals that can under-cover when evidence is thin.

A. Temporal forecasting of the direction weights

  • Temporal weights improved joint MLPD by 0.0117 nats embodied and 0.0139 nats at full visibility on HB.Gains were positive at all but three hours, with marginal dips no greater than 0.007; improvements were largest at 05:00, 13:00, and 23:00.Source: Table IX · p. 16, A. Temporal forecasting of the… · p. 16

B. Evidence Sharing

  • Evidence sharing improved MLPD most in sparse strata: 0.761 on TBD and 0.527 in HB’s sparsest stratum, versus no change in HB’s dense stratum.Across all three datasets, no stratum lost more than 0.0001; the neighborhood contribution decreases as a voxel accumulates crossings.Source: B. Evidence Sharing · p. 16, Table X · p. 16

C. Flow Dependence

  • Coupling reduced conditional heading error by 31.0° on ATC and 56.5° on HB, while joint-MLPD gains ranged from +0.02 on TBD to +0.63 on HB.The conditional heading metric showed the larger effect; TBD’s pair density remained within 0.02 of the uncoupled reference.Source: C. Flow Dependence · p. 16, Table XI · p. 17
  • Learned coupling beat the fixed prior only with dense per-edge evidence, and gains concentrated on similar-flow neighbors; divergent pairs could have negative joint gain.On ATC, the learned increment over the fixed prior shifted from −0.02 embodied to +0.10 at full visibility.Source: C. Flow Dependence · p. 17

1) Speed:

  • Speed-interval coverage was 0.76 and 0.95 on ATC and 0.66 and 0.95 on HB against nominal 0.68 and 0.95, with ECE 0.06 and 0.01.The intervals were conservative at 68% and exact at 95%; widths reflected site-specific speed scatter without per-site parameters.Source: 1) Speed · p. 17, Table XII · p. 17

2) Weights:

  • Weight-interval coverage was 0.707 and 0.935 on ATC and 0.699 and 0.964 on HB against nominal 0.68 and 0.95, with ECE 0.022 and 0.011.On ATC, epistemic variance accounted for 0.7% of predictive variance above 200 training crossings, 19.0% below 200, and 52.9% below 50; coverage still fell to 0.832 in the thinnest stratum against nominal 0.95.Source: 2) Weights · p. 17, Table XII · p. 17

IX. CONCLUSION

  • Kairos’s presence forecasts were competitive with a dedicated occupancy model trained on the full detection stream, remained consistent under loop-closure corrections, and improved person-finding outcomes over time-invariant maps.On the robot-collected dataset, it achieved the best scores in every channel and was the only compared method to predict speed, joint, and presence densities together.Source: §IX · p. 18

APPENDIX A TEMPORAL STRUCTURE OF PEDESTRIAN FLOW

The appendix measures how pedestrian activity, heading, and speed vary over candidate temporal cycles across real environments. Activity varies substantially by time and site, while directional and speed variation is more localized and depends on the selected period.

A. Datasets and Candidate Periods

  • The analysis uses six environments with differing temporal coverage and tests candidate daily, weekly, and shorter cycles suited to each dataset.ATC and the station concourses support calendar-based daily and weekly analyses; TBD contains about 213 minutes of short sessions, while other datasets have distinct timestamp constraints.Source: Appendix A · p. 20, A. Datasets and Candidate Periods · p. 20

B. Evaluation Protocol

  • Evaluation partitions sites into 0.4 m cells and temporal phase bins, counts cell passages rather than repeated detections, and compares phase-specific distributions with pooled references and repeated-half divergence.The signal-to-reference ratio is interpreted against a null near 0.3, with ratios above 3 indicating variation beyond repeated-observation effects.Source: B. Evaluation Protocol · p. 20

C. Variation in Activity

  • Activity varied hourly at all sites; median cell-level signal-to-reference ratios were near 150 at HB and PB, while activity divergence was 0.34 bits at ATC and about 0.13 bits at HB and PB.ATC’s larger divergence reflects recording only during opening hours, including hours of zero activity; HB and PB were recorded continuously.Source: C. Variation in Activity · p. 21

D. Variation in Heading

  • Heading rhythms differed by site: ATC varied hourly in 197 of 200 cells, HB’s hourly variation was localized, PB varied by morning versus afternoon, and Eindhoven by a 30-minute cycle.Edinburgh and TBD showed no hourly variation above the threshold.Source: D. Variation in Heading · p. 21

E. Variation in Speed

  • Speed varied detectably by hour at ATC, but its mean changes were only 0.02–0.06 m/s against the model component standard deviation of 0.3 m/s.HB’s site-level signal-to-reference ratio was 1.6 and TBD’s was 0.2, both below the stated threshold.Source: E. Variation in Speed · p. 21

F. Interpretation and Model Design

The interpretation links observed temporal structure to separate presence and flow estimates and design choices for speed and directional components. Additional comparisons show temporal-weight gains on ATC and a trade-off between directional sharpness and coverage relative to STeF-Map.

F. Interpretation and Model Design

  • Activity and heading vary differently, supporting separate estimation of presence and conditional flow; directional changes are localized, while hourly speed changes are small relative to component width.The interpretation also connects crossing-based heading estimates with presence estimates that account for visible time and dwell.Source: F. Interpretation and Model Design · pp. 21–22

APPENDIX B HEADING AND SPEED DEPENDENCE WITHIN A

  • Within-slot heading–speed correlations have no consistent sign and explain 1.9%–5.8% of speed variance, while speed differs across directional slots.Because the component covariance is shared across voxels, the near-zero aggregate within-slot dependence supports diagonal covariance; per-slot means represent across-slot speed differences.Source: Appendix B · p. 22

A. Contribution of Temporal Weights on ATC

B. Heading Difference with STeF-Map

  • Kairos’s broader angular components spread probability onto uncommon directions, whereas STeF-Map concentrates mass more sharply on common directions; their circular CRPS scores were nearly indistinguishable.On HB, Kairos’s heading MLPD was higher for detections below 0.25 m/s but lower by 0.08–0.15 nats between 0.5 and 2 m/s.Source: B. Heading Difference with STeF-Map · pp. 22–23

A. Presence Estimation by Hour of Day at HB

Presence forecasting differs across HB and ATC, while speed-interval adaptation adjusts uncertainty in opposite directions at the two sites. Weight-interval evaluation also notes limits from sparse HB strata and approximate count-based calibration.

A. Presence Estimation by Hour of Day at HB

B. Presence Estimation on ATC

  • On ATC, the static-rate variant beats temporal Kairos at every window: MLPP is −1719 versus −1887 at 600 s in the embodied regime and −459 versus −1042 at full visibility.The constant base rate has the best embodied MLPP from 60 s onward, while per-cell rates perform better at 5 and 10 s; at full visibility, the static-rate variant exceeds the base rate by 75 at 60 s and 242 at 600 s.Source: B. Presence Estimation on ATC · p. 23, Table XVII · p. 24

APPENDIX E PREDICTIVE UNCERTAINTY

  • For weight intervals, HB has only 64 evaluation windows below 50 training crossings, too few to support comparison with ATC; coverage there is 0.969 against nominal 0.95.The predictive count calibration is approximate because the target sums fractional responsibilities; propagation omits cross-slot covariance, and voxel-day resampling addresses correlated crossings.Source: Appendix E · p. 24
  • Adapted variance yields coverage 0.76 and 0.95 on ATC and 0.66 and 0.95 on HB, versus constant-prior coverage of 0.79 and 0.96 on ATC and 0.55 and 0.84 on HB.Mean interval width decreases from 0.60 to 0.54 m/s on ATC and increases from 0.60 to 0.80 m/s on HB; the constant 0.3 m/s prior therefore fits one site or the other, not both.Source: Appendix E · p. 24, Table XIX · p. 24
Loading 2609.27467v1…