A Comparative Study of Spline-Based Trajectory Reconstruction Methods Across Varying Automatic Vehicle Location Data Densities

Jake Robbennolt; Sirajum Munira; Stephen D. Boyles

REVIEW 4 major objections 5 minor 28 references

A Comparative Study of Spline-Based Trajectory Reconstruction Methods Across Varying Automatic Vehicle Location Data Densities

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Across 13 spline-based reconstruction methods and two data densities, the velocity-constrained Hermite spline with monotonicity enforcement (VCHIP-ME) gives the best accuracy-per-compute balance for transit AVL trajectories.

desk verdict A solid hold-out comparison of spline methods for bus AVL data, but the unreported smoothing hyperparameters and a circular baseline in Table 4 undercut the stronger claims. read the letter →

arxiv 2509.00119 v1 pith:IY4EUAVM submitted 2025-08-28 cs.RO

Jake Robbennolt , Sirajum Munira , Stephen D. Boyles This is my paper · ORCID

classification cs.RO

keywords AVLdatatrajectoryreconstructionsplineinterpolationHermitemonotonicityvelocity-awaretransitoperationsdensity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper compares 13 spline-based methods for reconstructing bus trajectories from automatic vehicle location (AVL) data, logged every ~16 seconds (sparse) or ~6 seconds (dense) along congested downtown routes in Austin, Texas. It asks which method best recovers the true path and speed between logged points, and how much data density matters. The central finding is that velocity-aware interpolation consistently beats position-only methods, that smoothing the data often hurts accuracy in stop-and-go urban traffic, and that enforcing a monotonic (never-backward) trajectory is essential. The velocity-constrained Hermite spline with monotonicity enforcement (VCHIP-ME) comes out on top: it needs no tuned parameters, matches recorded positions and speeds, guarantees monotonicity, and runs in milliseconds, making it suitable for real-time use. The paper also shows dense data roughly quadruples position accuracy over sparse data, arguing for investment in higher-frequency AVL collection.

What carries the argument

VCHIP-ME (velocity-constrained Hermite interpolation with monotonicity enforcement). It constructs a cubic polynomial on each interval between consecutive AVL records whose four coefficients are fixed by matching recorded position and velocity at both endpoints. The recorded velocities serve as tangent slopes, giving smooth, differentiable position and velocity curves. Monotonicity is then forced by applying the Fritsch–Carlson criterion (α² + β² ≤ 9) to the velocities: if the ratio of neighboring velocities to the interval secant slope falls outside the admissible circle, the tangents are scaled down so the position curve never decreases. This yields a parameter-free, O(n) method that is th

What would settle it

Run the same 13-method comparison on an independent transit dataset with the smoothing parameters re-tuned for each method; if any smoothing-based method (V-SPLINE-ME, LOCREG-PCHIP-V, PCHIP-VCHIP) then beats VCHIP-ME on both position RMSE and monotonic success rate, the paper's recommendation would be overturned.

Watch

Extended reading notes

Core claim

The paper's central claim is that VCHIP-ME — cubic Hermite interpolation in which the tangents are the recorded velocities, with those tangents then clipped by the Fritsch–Carlson circle constraint to force monotonicity — is the best general-purpose trajectory reconstruction method for transit AVL data among the thirteen tested. Against position-only baselines (linear, PCHIP, local regression) it cuts position RMSE by roughly a third on sparse data and a quarter on dense data while keeping velocity error comparable. Against the more elaborate velocity-aware smoothers (V-SPLINE variants, local-regression hybrids), it matches or slightly trails their accuracy but is about two orders of magnitu

Load-bearing premise

The comparison assumes the tunable parameters of the smoothing methods (neighborhood sizes, penalty weights, velocity weights) were set fairly and not overfit to the Austin data, but the paper does not report those parameter values or the tuning procedure.

Editorial extensions

If this is right

Transit agencies reconstructing bus trajectories from AVL data can adopt VCHIP-ME as a default: it requires no parameter tuning, runs fast enough for on-vehicle or real-time use, and produces physically plausible monotonic trajectories.
Investment in higher-frequency AVL collection (6-second or better intervals) is directly justified: it yields 4–6× lower position error and 2–3× lower velocity error, with the best reconstruction methods showing the largest gains.
Smoothing-based reconstructions should be used with caution in congested urban corridors; the paper finds they can obscure true stop-and-go dynamics, although they may help on data with more measurement noise.
Velocity-aware reconstruction should be adopted whenever speed data is available, since position-only methods cap out at substantially higher errors even with monotonicity enforcement.
For intersection-level performance metrics (travel time, speed variability, deceleration), dense data plus VCHIP-ME reduces mean absolute percentage errors to under 1% for the main quantities, enabling reliable before-after infrastructure evaluations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next experiment the paper does not run is to inject controlled noise or GPS dropouts into the dense Austin data; if VCHIP-ME's edge shrinks when measurements are noisy, the 'smoothing is unhelpful' conclusion would be specific to the paper's relatively clean, pre-filtered dataset rather than general.
Because VCHIP-ME is parameter-free, it should transfer across transit agencies without recalibration; one could test this immediately by running the same code on another city's AVL feed and comparing against the paper's reported error levels.
The monotonicity enforcement mechanism is a general ingredient: the same Fritsch–Carlson clipping could be added to any velocity-aware interpolant, which suggests V-SPLINE-ME and LOCREG-PCHIP-V might approach VCHIP-ME's accuracy if their smoothing parameters were omitted entirely.
The paper's per-intersection metric framework (300 ft upstream of signals) could be repurposed as a stop- and signal-level validation standard for future reconstruction studies, since it exposes method differences that global RMSE hides.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this is a careful empirical comparison of thirteen spline trajectory reconstruction methods on a large Austin bus AVL dataset. The headline result—velocity-aware methods beat position-only methods, and the simple parameter-free VCHIP-ME gives the best accuracy per compute—is largely supported by the hold-out evaluation. But the paper's secondary claim that smoothing is 'generally unhelpful or even detrimental' is not supported, because the tuning parameters for every smoothing method are never reported.

The paper does several things well. It uses two data densities (16.5s vs 6s gaps), a large sample of 7,620 complete trajectories, and a hold-out scheme that leaves out 5% of points per trajectory for position and velocity error. The methods are described in enough detail that they could be reimplemented, and the computation-time comparison is practically useful. The intersection-level metrics (travel time, speed volatility, deceleration) connect reconstruction to real agency decisions, which is a nice contribution.

The load-bearing problem is the unreported hyperparameters. LOCREG uses k; PCHIP-VCHIP uses alpha; LOCREG-V and LOCREG-PCHIP-V use kx, kv; V-SPLINE variants use gamma, eta, and mu. None are disclosed, and no tuning procedure is described. Without those, the comparison between tuned smoothing methods and parameter-free VCHIP-ME is not a fair test. A different, equally defensible parameter choice could reorder the ranking—the top gaps are small relative to trajectory-level standard deviations of 34–50 ft, and V-SPLINE actually has lower position RMSE in both densities. This problem is real, and the stress-test note is right.

There are two more soft spots. First, Table 4 uses dense VCHIP-ME as the baseline, so its dense MAPE is zero by construction; that section is circular, not evidence. Second, preprocessing adjusts point locations to enforce monotonicity before any method is applied, which undercuts the conclusion that monotonicity enforcement is critical—the input is already monotonicized. Finally, there are no confidence intervals or significance tests anywhere, so the ranking is statistically underdetermined even at the reported settings.

Bottom line: this paper is for practitioners and researchers working with bus AVL data who need practical reconstruction guidance. It deserves a serious referee, but a referee will ask for parameter disclosure, a non-circular baseline for Table 4, and significance testing. If those are fixed, it could be a useful reference; as is, the conclusions are conditional.

Referee Report

4 major / 5 minor

Summary. This manuscript compares thirteen spline-based trajectory reconstruction methods on bus AVL data from two high-frequency routes in Austin, using sparse (16.49-second mean gap) and dense (5.96-second) versions of the same 7,620 complete trips. The methods include position-only baselines (LSEG, PCHIP, LOCREG, LOCREG-PCHIP) and nine velocity-aware approaches, several introduced here (VCHIP-ME, PCHIP-VCHIP, LOCREG-V, LOCREG-PCHIP-V, V-SPLINE-MP, V-SPLINE-ME). Evaluation comprises held-out 5% point RMSE/MAE for position and velocity, physical realism checks (acceleration bounds, stop detection), intersection-level operational metrics, and computation time. The main findings are that velocity-aware methods outperform position-only methods, smoothing is generally not beneficial on this congested, preprocessed dataset, monotonicity enforcement is important, dense data substantially improves accuracy, and VCHIP-ME offers the best balance of accuracy and efficiency. The paper recommends VCHIP-ME or V-SPLINE-ME for practice and argues for investment in denser AVL data.

Significance. The study addresses a practical, under-tested question with a large real-world dataset, and the core hold-out comparison is internally consistent. Including computation time, monotonicity, and physical plausibility metrics alongside RMSE/MAE is a strength, as is the emphasis on a parameter-free method (VCHIP-ME) that could be readily deployed. If the findings hold, they provide actionable guidance for transit agencies. The authors also appropriately caveat that smoothing may be more useful in less congested settings. However, the central ranking is undermined by missing tunable-parameter disclosures for all smoothing methods, by an absence of statistical uncertainty quantification, and by a practical-value analysis that benchmarks methods against VCHIP-ME itself. These issues are fixable but currently weaken the generalizability of the headline claims.

major comments (4)

[§2.2, Table 1, §4.1, Table 2] The tunable parameters of every smoothing method are defined but never reported. LOCREG requires k; PCHIP-VCHIP requires α; LOCREG-V and LOCREG-PCHIP-V require (kx, kv); V-SPLINE and V-SPLINE-ME require (γ, η); V-SPLINE-MP requires (γ, η, μ). Table 1 counts these parameters, but Section 4.1 gives no values, no tuning/selection procedure, and no sensitivity analysis. This is load-bearing because the top-method gaps are small: sparse position RMSE is 61.55 ft for VCHIP-ME versus 58.33 ft for V-SPLINE, and dense RMSE is 14.55 versus 14.34 ft, while trajectory-level standard deviations are 34–50 ft. Different but equally defensible parameter choices could reorder the ranking and change the recommendation. Please report the parameter values and the tuning/validation protocol, or provide a sensitivity analysis demonstrating that the conclusions are stable over reasonable settings.
[§4.3, Table 4] The practical-application comparison is circular with respect to VCHIP-ME. Table 4 states that results are 'using VCHIP-ME as baseline,' so VCHIP-ME's dense-data MAPE of 0.00% for travel time, speed, speed volatility, and deceleration is true by construction. The text's claim that 'VCHIP-ME achieves travel time MAPE of 5.63% (sparse) and 0.00% (dense)' and that this demonstrates its practical value is not an independent validation. The relative ranking of other methods against this baseline is still informative, but the absolute MAPEs should not be interpreted as accuracy evidence. Please either compute errors against an independent reference (e.g., dense raw measurements or a separate baseline model) or explicitly reframe Table 4 as relative deviations from VCHIP-ME rather than as support for VCHIP-ME's superiority.
[Table 2, §4.1] No uncertainty quantification accompanies the central ranking. Table 2 reports means and standard deviations of errors across trajectories, but no standard errors, confidence intervals, or paired significance tests. With 7,620 trajectories, pairwise comparisons are feasible and would make the ranking much more informative. The absence is consequential because several differences between top methods are only a few feet (e.g., dense position RMSE: VCHIP-ME 14.55 vs. V-SPLINE 14.34) while the reported standard deviations are an order of magnitude larger. Without such statistics, the statement that VCHIP-ME is 'optimal' or that smoothing methods are 'generally unhelpful' is statistically underdetermined. Please add appropriate uncertainty measures or temper the conclusions to the observed sample.
[§3.1, §4.1] The preprocessing pipeline is aggressive: forward jumps over 500 ft are removed, all backward jumps over 200 ft are removed, smaller backtracking points are adjusted to force monotonicity, and trips with gaps over 10 minutes or 1 mile are discarded. Consequently, the methods are evaluated on cleaned, idealized inputs rather than raw AVL data. The authors themselves note in Section 4.1 that the lack of benefit from smoothing 'may indicate that there were relatively few outliers in our dataset,' which makes the conclusion that smoothing methods 'can degrade overall performance' conditional on this cleaning. Please quantify how many points or trajectories were affected by each preprocessing step and, if possible, include a sensitivity analysis with less aggressive cleaning. This would clarify whether the recommendation of VCHIP-ME over smoothing methods is specific to the cleaned dataset.

minor comments (5)

[Algorithm 4, §2.2.6] Algorithm 4 appears to use future values: inside the loop, the branch that sets ui uses yi+1 before that value has been computed, and ui may remain undefined for i = n if a monotonicity correction occurs. Please revise the pseudocode to reflect the actual two-pass or iterative procedure, or clarify the update order.
[§4.3, paragraph 3] There is a typo: 'VCHIP, VHIP-ME, and V-SPLINE-ME' should presumably read 'VCHIP, VCHIP-ME, and V-SPLINE-ME.'
[Eq. (34), §2.2.7] The notation 'nθ T Ωθ' is ambiguous; please format as n θ^T Ω θ with clear parentheses. Similarly, Eq. (42) would benefit from spacing to distinguish matrix products.
[Table 3, §4.2] The 100.00% acceleration adherence reported for LVMI is an artifact of the method's piecewise-linear velocity, whose acceleration is zero almost everywhere and undefined at knots. This should be noted in the table discussion or excluded from the physical-realism comparison, since it does not indicate physically plausible acceleration behavior.
[Table 2, §4.1] LSEG and PCHIP achieve 100% monotonic success partly because the preprocessing already enforces monotonic positions. The text should clarify that monotonicity success here measures whether the interpolation preserves the preprocessed monotonicity, not whether the method can recover monotonicity from raw data.

Circularity Check

1 steps flagged · score 3.0 of 10

Table 4's VCHIP-ME dense MAPE is zero by construction, but the central hold-out accuracy comparison is independent.

self definitional [Section 3.2 (Evaluation Framework) and Section 4.3 / Table 4]
"Then, using one model selected as the baseline from the dense dataset, we calculated percent error for the other metrics on both datasets to compare the value of each model. Using VCHIP-ME as the baseline, this analysis calculates mean and mean absolute percentage error (MAPE) of performance metrics aggregated across all intersections."

The baseline is the dense VCHIP-ME trajectory itself, so VCHIP-ME's dense MAPE is identically 0.00% for every metric in Table 4 by construction. The text then cites these identities as evidence: 'VCHIP-ME achieves travel time MAPE of 5.63% (sparse) and 0.00% (dense)' and 'maintains a reasonable speed volatility MAPE of 44.16% (sparse) and 0.00% (dense)'. This is not a measurement against an external ground truth; it is a tautology. It does not affect the independent held-out-point RMSE/MAE comparisons in Table 2, but it does circularly support the practical-application claims for VCHIP-ME.

full rationale

The core evaluation is not circular: the first test in Section 3.2 holds out 5% of each trajectory's points and compares reconstructed versus recorded location and velocity, and Table 2's RMSE/MAE numbers are external to the fitting of every method. The central recommendation of VCHIP-ME rests mainly on those independent holdout errors and on computation time, not on a self-referential benchmark. The one clear by-construction element is Table 4, where dense VCHIP-ME is chosen as the baseline, making its dense MAPE 0.00% for all metrics; citing that zero as accuracy is tautological. This affects secondary practical-application claims only. The monotonicity-success columns are also entangled with preprocessing that moves backtracking points forward until the input is monotonic (Section 3.1), and the paper itself notes LSEG guarantees monotonicity only after such preprocessing; however, the VCHIP vs VCHIP-ME comparison still shows an independent effect of monotonicity enforcement for velocity-aware methods. The unreported tuning parameters for smoothing methods are a serious reproducibility and validity threat, but they are not circularity: no fitted parameter is renamed as a prediction. There is no load-bearing self-citation; prior algorithmic work cited is external. Overall score: one secondary by-construction benchmark and a partially definitional monotonicity metric, while the central accuracy ranking is independent.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central ranking rests on the standard math of Hermite interpolation and on several domain assumptions about preprocessing, physical bounds, and evaluation honesty. The largest unstated burdens are the unreported tuning parameters and the treatment of dense VCHIP-ME as a reference standard. No new physical entities are introduced.

free parameters (9)

k (LOCREG neighborhood size)
Chosen neighborhood for local regression; not reported.
alpha (PCHIP-VCHIP velocity weight)
Blend between observed velocity and PCHIP derivative; not reported.
kx, kv (LOCREG-V neighborhoods)
Position and velocity neighborhood sizes; not reported.
kx, kv (LOCREG-PCHIP-V neighborhoods)
Position and velocity neighborhood sizes; not reported.
gamma, eta (V-SPLINE)
Velocity weight and adaptive smoothness penalty; not reported.
gamma, eta, mu (V-SPLINE-MP)
Velocity weight, smoothness penalty, monotonicity penalty; not reported.
gamma, eta (V-SPLINE-ME)
Velocity weight and smoothness penalty for the initial V-SPLINE pass; not reported.
Data-cleaning thresholds = 500 ft jump, 45 mph, 200 ft backward, 200 ft adjustment, 10 min, 1 mile, 20 deg bearing, 200 ft route distance
Hand-chosen filtering thresholds that shape the dataset and therefore all results.
Physics evaluation thresholds = tight accel (-5.79, 4.26), loose accel (-7.77, 5.43), stop speeds 2/5/10 ft/s
Hand-chosen criteria used to score reconstruction realism.

assumptions (6)

standard math Cubic Hermite basis and Fritsch-Carlson tangent constraints produce monotone, differentiable interpolants (Eqs. 3-16).
Classical numerical analysis results the paper uses without proof.
domain assumption Route distance along the matched path is a valid scalar coordinate and buses never legitimately move backward along it.
Justifies removing all backward jumps and enforcing monotonicity as a success criterion; stated in Section 3.1.
domain assumption Preprocessing that removes gaps and outliers does not remove behavior that would change the ranking of methods.
The 500 ft/45 mph and 200 ft rules re-shape the data before evaluation (Section 3.1).
domain assumption The stated acceleration bounds are realistic for buses.
Tight and loose bounds are imposed without source in Section 3.2.
domain assumption Held-out 5% point errors and trajectory-level aggregation are representative, and any parameter tuning did not leak into the holdout set.
The evaluation protocol in Section 3.2 presupposes clean separation; no nested tuning is described.
domain assumption AVL stop labels and dense VCHIP-ME can serve as reference for intersection-level metrics.
Table 4 baseline choice assumes dense VCHIP-ME is close enough to truth to compute MAPE for other methods (Section 4.3).

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Pith. "Pith review of A Comparative Study of Spline-Based Trajectory Reconstruction Methods Across Varying Automatic Vehicle Location Data Densities." pith.science (2026). https://pith.science/paper/IY4EUAVM

@misc{pith2026250900119,
  author       = {Pith},
  title        = {Pith review of: A Comparative Study of Spline-Based Trajectory Reconstruction Methods Across Varying Automatic Vehicle Location Data Densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IY4EUAVM}},
  note         = {Machine review of arXiv:2509.00119}
}

read the original abstract

Automatic vehicle location (AVL) data offers insights into transit dynamics, but its effectiveness is often hampered by inconsistent update frequencies, necessitating trajectory reconstruction. This research evaluates 13 trajectory reconstruction methods, including several novel approaches, using high-resolution AVL data from Austin, Texas. We examine the interplay of four critical factors -- velocity, position, smoothing, and data density -- on reconstruction performance. A key contribution of this study is evaluation of these methods across sparse and dense datasets, providing insights into the trade-off between accuracy and resource allocation. Our evaluation framework combines traditional mathematical error metrics for positional and velocity with practical considerations, such as physical realism (e.g., aligning velocity and acceleration with stopped states, deceleration rates, and speed variability). In addition, we provide insight into the relative value of each method in calculating realistic metrics for infrastructure evaluations. Our findings indicate that velocity-aware methods consistently outperform position-only approaches. Interestingly, we discovered that smoothing-based methods can degrade overall performance in complex, congested urban environments, although enforcing monotonicity remains critical. The velocity constrained Hermite interpolation with monotonicity enforcement (VCHIP-ME) yields optimal results, offering a balance between high accuracy and computational efficiency. Its minimal overhead makes it suitable for both historical analysis and real-time applications, providing significant predictive power when combined with dense datasets. These findings offer practical guidance for researchers and practitioners implementing trajectory reconstruction systems and emphasize the importance of investing in higher-frequency AVL data collection for improved analysis.

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Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Abdelhalim, A

Huang, Y ., A. Abdelhalim, A. Stewart, J. Zhao, and H. Koutsopoulos, Reconstructing Transit Vehicle Trajectory Using High-Resolution GPS Data. In 2023 IEEE 26th Interna- tional Conference on Intelligent Transportation Systems (ITSC), 2023, pp. 5247–5253

work page 2023
[2]

Cui, Y . and S. S. Ge, Autonomous Vehicle Positioning with GPS in Urban Canyon En- vironments. IEEE Transactions on Robotics and Automation , V ol. 19, No. 1, 2003, pp. 15–25

work page 2003
[3]

Khuanpet, W

Sirisombat, T., J. Khuanpet, W. Chancharoen, P. Thanartthanaboon, P. Ngamkajornwiwat, P. Achararit, W. Mahikul, and P. Singkham, Electronic Tracking Device of Mass Public Transport Vehicle for Evaluating Driving Performance in Thailand. Engineering Journal, V ol. 28, No. 1, 2024, pp. 13–22

work page 2024
[4]

Feng, S., Q. Yang, A. C. Hughes, J. Chen, and H. Qiao, A Novel Method for Multi- Trajectory Reconstruction Based on LoMcT for Avian Migration in Population Level. Ecological Informatics, V ol. 63, 2021, p. 101319

work page 2021
[5]

Guo, S., J. Mou, L. Chen, and P. Chen, Improved Kinematic Interpolation for AIS Trajec- tory Reconstruction. Ocean Engineering, V ol. 234, 2021, p. 109256

work page 2021
[6]

Wang, Y ., C. An, J. Ou, Z. Lu, and J. Xia, A General Dynamic Sequential Learning Frame- work for Vehicle Trajectory Reconstruction Using Automatic Vehicle Location or Identi- fication Data. Physica A: Statistical Mechanics and its Applications , V ol. 608, 2022, p. 128243

work page 2022
[7]

Ni, D. and H. Wang, Trajectory Reconstruction for Travel Time Estimation. Journal of Intelligent Transportation Systems, V ol. 12, No. 3, 2008, pp. 113–125

work page 2008
[8]

Sun, Z. and X. J. Ban, Vehicle Trajectory Reconstruction for Signalized Intersections Using Mobile Traffic Sensors. Transportation Research Part C: Emerging Technologies, V ol. 36, 2013, pp. 268–283. 24

work page 2013

Show all 28 references

[9]

Robbennolt, J. and J. Hourdos, Data-Driven Evaluation Methodology for Active Traffic Management Systems Utilizing Sparse Speed Data. Transportation Research Record, V ol. 2678, No. 4, 2024, pp. 90–105

2024
[10]

Rao, W., Y .-J. Wu, J. Xia, J. Ou, and R. Kluger, Origin-Destination Pattern Estimation Based on Trajectory Reconstruction Using Automatic License Plate Recognition Data. Transportation Research Part C: Emerging Technologies, V ol. 95, 2018, pp. 29–46

2018
[11]

Li, and J

Mo, B., R. Li, and J. Dai, Estimating Dynamic Origin–Destination Demand: A Hybrid Framework Using License Plate Recognition Data. Computer-Aided Civil and Infrastruc- ture Engineering, V ol. 35, No. 7, 2020, pp. 734–752

2020
[12]

Chien, and A

Patnaik, J., S. Chien, and A. Bladikas, Estimation of Bus Arrival Times Using APC Data. Journal of Public Transportation, V ol. 7, No. 1, 2004, pp. 1–20

2004
[13]

Kuipers, R. A. and C.-W. Palmqvist, Passenger V olumes and Dwell Times for Commuter Trains: A Case Study Using Automatic Passenger Count Data in Stockholm. Applied Sci- ences, V ol. 12, No. 12, 2022, p. 5983

2022
[14]

Gupta, J

Bin Zulqarnain, A., S. Gupta, J. P. Talusan, D. Freudberg, P. Pugliese, A. Mukhopad- hyay, and A. Dubey, Addressing APC Data Sparsity in Predicting Occupancy and Delay of Transit Buses: A Multitask Learning Approach. In2023 IEEE International Conference on Smart Computing (SMA...

2023
[15]

Yu, B., S. H. Kim, T. Bailey, and R. Gamboa, Curve-Based Representation of Moving Object Trajectories. In International Database Engineering and Applications Symposium, 2004, 2004, pp. 419–425

2004
[16]

Fritsch, F. N. and R. E. Carlson, Monotone Piecewise Cubic Interpolation. SIAM Journal on Numerical Analysis, V ol. 17, No. 2, 1980, pp. 238–246

1980
[17]

Fritsch, F. N. and J. Butland, A Method for Constructing Local Monotone Piecewise Cubic Interpolants. SIAM Journal on Scientific and Statistical Computing , V ol. 5, No. 2, 1984, pp. 300–304

1984
[18]

Sun, C., C. Wu, D. Chu, L. Xie, L. Liu, and H. Li, Vehicle Trajectory Restoration Based on V ondrak Filtering and Cubic Spline Interpolation, 2016, pp. 235–248

2016
[19]

Li, J. and C. Liu, Cubic Trigonometric Hermite Interpolation Curve: Construction, Prop- erties, and Shape Optimization. Journal of Function Spaces , V ol. 2022, No. 1, 2022, p. 7525056

2022
[20]

Bryant, T

Cao, Z., D. Bryant, T. C. A. Molteno, C. Fox, and M. Parry, V-Spline: An Adaptive Smoothing Spline for Trajectory Reconstruction. Sensors, V ol. 21, No. 9, 2021, p. 3215. 25

2021
[21]

Wang, and P

Wei, L., Y . Wang, and P. Chen, A Particle Filter-Based Approach for Vehicle Trajectory Reconstruction Using Sparse Probe Data.IEEE Transactions on Intelligent Transportation Systems, V ol. 22, No. 5, 2021, pp. 2878–2890

2021
[22]

Kwan, and T

Dong, Y ., K. Kwan, and T. Arslan, Enhanced Pedestrian Trajectory Reconstruction Using Bidirectional Extended Kalman Filter and Automatic Refinement. In 2024 14th Interna- tional Conference on Indoor Positioning and Indoor Navigation, 2024, pp. 1–6

2024
[23]

Huang, Y

Chen, Q., H. Huang, Y . Li, J. Lee, K. Long, R. Gu, and X. Zhai, Modeling Accident Risks in Different Lane-Changing Behavioral Patterns. Analytic Methods in Accident Research, V ol. 30, 2021

2021
[24]

Zhao, J., J. Lu, X. Chen, Z. Yan, Y . Yan, and Y . Sun, High-Fidelity Data Supported Ship Trajectory Prediction Via an Ensemble Machine Learning Framework. Physica A: Statis- tical Mechanics and its Applications, V ol. 586, 2022, p. 126470

2022
[25]

S., Robust Locally Weighted Regression and Smoothing Scatterplots

Cleveland, W. S., Robust Locally Weighted Regression and Smoothing Scatterplots. Jour- nal of the American Statistical Association, V ol. 74, No. 368, 1979, pp. 829–836

1979
[26]

Wand, M. P. and M. C. Jones, Kernel Smoothing. Chapman and Hall/CRC, New York, 1994

1994
[27]

Green, P. J. and B. W. Silverman, Nonparametric Regression and Generalized Linear Mod- els: A Roughness Penalty Approach. Chapman and Hall/CRC, New York, 1993

1993
[28]

Kano, H. and H. Fujioka, Velocity and Acceleration Constrained Trajectory Planning by Smoothing Splines. In 2017 IEEE 26th International Symposium on Industrial Electronics (ISIE), 2017, pp. 1167–1172. 26

2017

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Abdelhalim, A

Huang, Y ., A. Abdelhalim, A. Stewart, J. Zhao, and H. Koutsopoulos, Reconstructing Transit Vehicle Trajectory Using High-Resolution GPS Data. In 2023 IEEE 26th Interna- tional Conference on Intelligent Transportation Systems (ITSC), 2023, pp. 5247–5253

work page 2023

[2] [2]

Cui, Y . and S. S. Ge, Autonomous Vehicle Positioning with GPS in Urban Canyon En- vironments. IEEE Transactions on Robotics and Automation , V ol. 19, No. 1, 2003, pp. 15–25

work page 2003

[3] [3]

Khuanpet, W

Sirisombat, T., J. Khuanpet, W. Chancharoen, P. Thanartthanaboon, P. Ngamkajornwiwat, P. Achararit, W. Mahikul, and P. Singkham, Electronic Tracking Device of Mass Public Transport Vehicle for Evaluating Driving Performance in Thailand. Engineering Journal, V ol. 28, No. 1, 2024, pp. 13–22

work page 2024

[4] [4]

Feng, S., Q. Yang, A. C. Hughes, J. Chen, and H. Qiao, A Novel Method for Multi- Trajectory Reconstruction Based on LoMcT for Avian Migration in Population Level. Ecological Informatics, V ol. 63, 2021, p. 101319

work page 2021

[5] [5]

Guo, S., J. Mou, L. Chen, and P. Chen, Improved Kinematic Interpolation for AIS Trajec- tory Reconstruction. Ocean Engineering, V ol. 234, 2021, p. 109256

work page 2021

[6] [6]

Wang, Y ., C. An, J. Ou, Z. Lu, and J. Xia, A General Dynamic Sequential Learning Frame- work for Vehicle Trajectory Reconstruction Using Automatic Vehicle Location or Identi- fication Data. Physica A: Statistical Mechanics and its Applications , V ol. 608, 2022, p. 128243

work page 2022

[7] [7]

Ni, D. and H. Wang, Trajectory Reconstruction for Travel Time Estimation. Journal of Intelligent Transportation Systems, V ol. 12, No. 3, 2008, pp. 113–125

work page 2008

[8] [8]

Sun, Z. and X. J. Ban, Vehicle Trajectory Reconstruction for Signalized Intersections Using Mobile Traffic Sensors. Transportation Research Part C: Emerging Technologies, V ol. 36, 2013, pp. 268–283. 24

work page 2013