ScaleTraj: Scale-Decomposed Human Trajectory Generation

Yulin Zeng; Yiheng Zhao; Jun Yan; Yuhong Liu

cs.LGNov 3, 2026

ScaleTraj: Scale-Decomposed Human Trajectory Generation

Authors: Yulin Zeng, Yiheng Zhao, Jun Yan, Yuhong Liu

Organizations: Santa Clara University, Santa Clara, CA, USA · Concordia University, Montreal, Canada

Abstract

Individual-level trajectories are essential for applications that rely on user-level mobility behavior. Existing mobility-generation approaches face a fundamental tension between individual-level and population-level statistical fidelity: top-down models reproduce aggregate mobility statistics but struggle to guarantee realistic individual trajectories, while bottom-up generators synthesize individual trajectories yet often fail to preserve key population-level mobility regularities after aggregation. In particular, many bottom-up generated populations fail to reproduce the displacement distribution, whose heavy tail is well summarized by a truncated power law. We formalize this phenomenon as the Population-Level Displacement Gap. To investigate its prevalence, we conduct a systematic evaluation of representative model-driven and data-driven trajectory generators. Our results provide evidence that the Population-Level Displacement Gap is prevalent across diverse generation paradigms. To diagnose this gap, we introduce the Scale Fidelity Profile (SFP), a scale-decomposed metric that quantifies discrepancies in trip allocations across spatial scales. SFP indicates that population-level displacement mismatches co-occur with systematic errors in how trips are allocated across spatial scales. Motivated by this observation, we develop ScaleTraj, an autoregressive Transformer that decomposes location prediction into a Scale Selection Head that decides the movement scale and a Scale-Aware Cell Selection Head that predicts the destination conditioned on the selected scale. Experiments on two datasets show that ScaleTraj reduces the Population-Level Displacement Gap versus model-driven and data-driven baselines, while preserving fidelity. Our code is available at https://github.com/SCU-TrustworthyAI/ScaleTraj.

Figures & tables

Figure 1. Power-law diagnostic of trajectory generators. The vertical axis $P(\Delta r\geq x)$ gives the fraction of trips with displacement at least $x$ . (a, b) Complementary cumulative distribution functions (CCDFs) of displacement distances on the GeoLife and Tencent Beijing test splits. The collected real-world data are shown in black, with the literature-consensus truncated-power-law family ( González et al., 2008 ) shown as external reference: the GeoLife curve lies largely within the band, whereas the Tencent Beijing curve approaches and falls below its lower edge in the tail. (c) Same diagnostic applied to CoPB’s officially released dataset ( Shao et al., 2024 ) ; its maximum observed displacement is $38.3$ km, indicating a substantially shorter tail than the Tencent Beijing test distribution ( $\kappa\!=\!129$ km). Three log-log CCDF panels of displacement distance. Panels (a) and (b) show GeoLife and Tencent Beijing test-split real data as a black curve, with several colored baseline curves deviating from the band, mostly in the tail toward longer distances. Panel (c) shows the same diagnostic on the CoPB released dataset: the curve drops sharply to zero at 38.3 kilometers, far short of the gold band which extends past one thousand kilometers.

Dataset	Split	Users	Days	$\beta$	$\kappa$ (km)
GeoLife	Train	121	4,457	1.707	524
GeoLife	Test	121	1,171	1.730	356
Tencent Beijing	Train	500	28,957	1.674	213
Tencent Beijing	Test	500	7,507	1.686	129

Table 1. Truncated power-law fit (

\beta

\kappa

) for train and test splits. The exponent

\beta

is consistent across splits (

|\Delta\beta|\!<\!.03

on both datasets);

\kappa

shows expected sampling variation on the shorter test window, especially on Tencent Beijing (

213\!\to\!129

km), but remains within the same order of magnitude.

	GeoLife		Tencent Beijing
Method	$\|\Delta\beta\|\!\downarrow$	$\|\Delta\kappa\|\!\downarrow$	$\|\Delta\beta\|\!\downarrow$	$\|\Delta\kappa\|\!\downarrow$
SemiMarkov	.433	7,937	.304	6,629
TimeGeo	.077	4,703	.052	3,960
LSTM	.373	7,898	.260	5,922
TrajSynVAE	.382	6,566	.296	5,967
MoveSim	.343	10,046	.337	5,752

Table 2. Power-law diagnostics on the test set.

|\Delta\beta|

|\Delta\kappa|

(km): deviation of fitted parameters from real data; lower is better.

Figure 2. Gaussian Mixture Model ( $K=4$ ) fitted to log-displacement distances on the training set. Both datasets exhibit multi-modal structure in log-space, with four displacement bands from neighborhood to region. Vertical dotted lines indicate the dataset-specific scale boundaries used for the SFP diagnostic; the fitted GMM components provide the scale-conditioned distance structure used by ScaleTraj. Two side-by-side histograms of training log-displacement distances, one for GeoLife and one for Tencent Beijing. Each histogram is overlaid with four fitted Gaussian components in different colors, and vertical dotted lines indicate the dataset-specific boundaries used to partition displacement into neighborhood, city, urban, and region bands for evaluation.

Figure 3. Per-scale proportion ratio (sim/real) on log scale for seven baselines on the test set. The black line marks ratio $=1$ , and the gray band spans $\pm 20\%$ . Many methods show substantial scale-level misallocation, with short-range and long-range movements often deviating in different directions. These patterns expose compensating errors that are not visible from aggregate power-law parameters alone. Two log-scale line plots showing simulated-to-real per-scale proportion ratios for the evaluated baselines across the neighborhood, city, urban, and region bands. A horizontal line marks perfect agreement and a gray band indicates a tolerance region. The profiles show substantial over- and under-allocation across scales.

Figure 4. ScaleTraj architecture. Past visits are embedded and encoded by a causal Transformer (

L{=}4

H{=}4

d{=}128

) into

h_{i}

. The Scale Selection Head combines a frozen training-set prior

\hat{\pi}_{k}

with a learned context MLP to sample

k_{i}

(here “urban”). The Scale-Aware Cell Selection Head routes the current model state through a scale-specific MLP and combines it with a scale-conditioned Gaussian distance prior; an optional composite inference-time reranking stage

\mathrm{Reranker}(c)

is applied before sampling

c_{i}

. Duration and arrival heads complete the visit, and

(k_{i},c_{i},\tau_{i},a_{i})

is fed back autoregressively. Colors: orange = frozen; blue = learned. A flow diagram of the ScaleTraj architecture. Past visit tokens are embedded and passed through a causal Transformer encoder into a hidden state. The hidden state feeds a Scale Selection Head, shown combining a frozen training-set scale-frequency prior with a learned MLP to sample a scale band, and a Scale-Aware Cell Selection Head, shown routing the hidden state through one of four per-scale MLPs and adding a scale-conditioned Gaussian distance prior plus an optional inference-time reranker before sampling a destination cell. The duration head predicts duration outputs and intermediate duration features, which are also provided to the arrival head. The sampled scale, cell, duration, and arrival are fed back into the sequence for the next autoregressive step. Frozen components are colored orange and learned components are colored blue.

	Tencent Beijing											GeoLife
Method	SD	RG	SI	STAY	DTD	DUL	Visit	IRank	SFP	$\|\Delta\beta\|$	$\|\Delta\kappa\|$	SD	RG	SI	STAY	DTD	DUL	Visit	IRank	SFP	$\|\Delta\beta\|$	$\|\Delta\kappa\|$
SemiMarkov	.089	.199	.282	.101	.405	.487	.319	.440	.067	.304	6,629	.234	.351	.294	.142	.575	.572	.387	.586	.182	.433	7,937
TimeGeo	.056	.143	.032	.110	.127	.037	.269	.039	.040	.052	3,960	.158	.219	.111	.174	.207	.284	.435	.136	.057	.077	4,703
LSTM	.081	.197	.182	.106	.467	.548	.318	.502	.057	.260	5,922	.174	.202	.014	.030	.327	.227	.118	.264	.121	.373	7,898
TrajSynVAE	.106	.169	.088	.021	.284	.256	.112	.234	.080	.296	5,967	.204	.251	.020	.038	.345	.160	.070	.207	.155	.382	6,566
MoveSim	.231	.262	.178	.091	.318	.210	.158	.258	.186	.337	5,752	.197	.129	.122	.123	.140	.057	.143	.061	.117	.343	10,046

Table 3. Full evaluation on both datasets. All metrics

\downarrow

(lower is better); per-dataset best in bold.

Figure 5. Displacement CCDF comparison of ScaleTraj (red), the real test-set distribution (black), and TimeGeo (blue dashed), the strongest baseline in terms of $|\Delta\beta|$ . The literature-consensus band is shown as external reference. Two log-log CCDF plots, one for GeoLife and one for Tencent Beijing. Each panel compares the real test-set distribution, ScaleTraj, and TimeGeo, with the literature-consensus band shown as external reference. ScaleTraj more closely follows the real-data CCDF, while TimeGeo exhibits a larger deviation in the tail.

Variant	SFP $\downarrow$	$\|\Delta\kappa\|\!\downarrow$	$\|\Delta\beta\|\!\downarrow$	SFP $\downarrow$	$\|\Delta\kappa\|\!\downarrow$	$\|\Delta\beta\|\!\downarrow$
	Tencent Beijing			GeoLife
$K\!=\!1$ , shared MLP	.038	74	.169	.043	343	.869
$K\!=\!4$ , shared MLP	.009	58	.050	.002	19,644	.121
ScaleTraj	.003	24	.021	.003	131	.026

Table 4. Ablation on the cell head’s two knobs:

K

and residual-MLP sharing. Bottom row is the full ScaleTraj system.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Band	$e^{\mu_{\log}}$ (km)	$\sigma_{\log}$	weight	tokens
GeoLife	neighborhood	0.37	0.41	0.247	45.6%
	city	1.26	0.55	0.368	24.3%
	urban	8.23	0.70	0.360	26.3%
	region	165.27	0.70	0.025	3.8%
Tencent Beijing	neighborhood	0.51	0.19	0.362	53.3%
	city	1.46	0.59	0.261	20.5%

Appendix

Table 5. Fitted

K{=}4

GMM components on log-displacement: per-component center (km), log-space standard deviation

\sigma_{\log}

, mixture weight, and per-band token share after boundary assignment.

	Tencent Beijing			GeoLife
$K$	SFP $\downarrow$	$\|\Delta\beta\|\!\downarrow$	$\|\Delta\kappa\|\!\downarrow$	SFP $\downarrow$	$\|\Delta\beta\|\!\downarrow$	$\|\Delta\kappa\|\!\downarrow$
4 (default)	.003	.021	24	.003	.026	131
5	.007	.078	146	.002	.033	237
6	.004	.059	407	.004	.058	102

Appendix

Table 6.

K

-sweep on the ScaleTraj configuration; lower is better.

K{=}4

is the default.

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