Inferring missing locations in human mobility data using memory effects

Pablo Rosillo-Rodes; Antònia Tugores; David Sánchez; José J. Ramasco

cs.LGOct 31, 2026

Inferring missing locations in human mobility data using memory effects

Authors: Pablo Rosillo-Rodes, Antònia Tugores, David Sánchez, José J. Ramasco

Organizations: Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca, Spain

Abstract

Empirical datasets used to characterize human mobility are sparse in space and time, owing to technological factors such as coverage and battery, or to the erratic nature of online interactions. Missing data can bias the conclusions drawn from them, so filling the gaps in incomplete timelines is a key challenge in computational social science. Here, we propose a simple imputation method in which the informativeness of an observation decays with its bidirectional temporal distance from the missing entry, at a rate set by a single parameter per individual. On synthetic and real Twitter data, a power-law decay, selected by Bayesian model comparison, yields accurate imputations (macro-AUROC above 0.9 overall, and above 0.8 for the most mobile users) and matches a gap-aware continuous-time Markov chain, outperforming it among the most mobile users. The method is robust to moderate noise and generalizes across countries and European administrative regions.

Figures & tables

Figure 1: Sketch of the method . A human travels along different locations ( a ), which are mapped onto a timeline diagram ( b ). While certain locations are included in the mobility dataset (depicted as blue dots), others are missing (depicted as a gray rectangle or gray dots). Our goal is to infer those missing locations from a simple model. c We introduce a temporal weighting scheme that models the influence of past and future observations on a missing location by assigning weights based on their temporal distance $\Delta t$ , according to a specified function $f_{\lambda}\left(\Delta t\right)$ , where $\lambda$ is a decay parameter. d The negative log composite likelihood of the validation days, $-\log\mathcal{C}$ , as a function of $\lambda$ ; its minimum defines $\lambda_{*}$ . Three characteristic shapes appear. In Case 1 the minimum lies at $\lambda_{0}$ , where the model reduces to predicting the most frequently visited location. In Case 2 the minimum is interior, and the timeline is best described by a genuine decay of informativeness with temporal separation. In Case 3 the curve decreases monotonically towards $\lambda_{\infty}$ , where the model reduces to copying the location of the temporally nearest observed day (persistence).

Figure 2: The empirical dataset. Distributions over the

10^{6}

country-level timelines of a the number of observed days

n_{i}

, b the number of distinct countries

N_{c}

, c the fraction of observed days spent at the modal country, and d the change rate

p

. e Per-individual median temporal distance from a test day to its nearest training day, under random hold-out (median over individuals

3.0

days) and under

14

-day block hold-out (

14.0

days).

Figure 3: Synthetic generators. a–d One analysed trajectory from each generator at noise

p\approx 0.13

: a SG1, alternation between two locations with fixed residence intervals (

P=5

\Delta t_{1}=18

\Delta t_{2}=54

); b SG2, heavy-tailed inter-event times, with a switch of location at each observation with probability

p

; c SG3, four locations with a change of home location and jittered excursions; d SG4, four locations with log-normal, non-periodic residence durations. Colors and vertical positions indicate the location. e–h Median test-fold macro-AUROC as a function of the noise parameter

p

for the three kernels, the nearest observed day and the gap-aware Markov chain; bands span the interquartile range over the 500 trajectories at each value of

p

, and the dotted line marks chance level. The meaning of

p

differs between generators (Table V).

Generator	Stratum	$n$	PL	EXP	GAUSS	MostFreq.	LastObs.	NearestObs.	Markov	MarkovCT
SG1	$0.02\leq p<0.10$	1103	0.981	0.980	0.978	0.500	0.913	0.944	0.913	0.993
	$0.10\leq p<0.30$	2447	0.839	0.840	0.821	0.500	0.748	0.780	0.748	0.828
	$p\geq 0.30$	6450	0.550	0.541	0.500	0.500	0.545	0.556	0.535	0.557
SG2	$p<0.02$	166	1.000	1.000	1.000	0.500	1.000	1.000	1.000	1.000
	$0.02\leq p<0.10$	1479	0.962	0.963	0.948	0.500	0.889	0.929	0.889	0.989
	$0.10\leq p<0.30$	3743	0.800	0.754	0.636	0.500	0.717	0.758	0.715	0.833

Table 1: Performance on the synthetic generators. Median test-fold macro-AUROC of the three kernels and the five baselines, stratified by the realized change rate

p

;

n

is the number of trajectories in each stratum (500 trajectories at each of 20 values of the noise parameter per generator, random hold-out). Only SG2 populates

p<0.02

. MostFrequent assigns the same distribution to every day, so its macro-AUROC is 0.5 by construction. The best value in each row is in bold. The fraction of correct inferences, the log-loss and the remaining kernels are reported in Supplementary Note 5.

Generator	Stratum	$n$	EXP	GAUSS	STREXP
SG1	$0.02\leq p<0.10$	1103	1.62 (0.76)	4.06 (0.77)	$-$ 0.01 (0.47)
	$0.10\leq p<0.30$	2447	2.95 (0.98)	6.93 (1.00)	0.61 (0.73)
	$p\geq 0.30$	6450	3.36 (1.00)	6.83 (1.00)	0.95 (0.93)
SG2	$p<0.02$	166	$-$ 0.01 (0.26)	$-$ 0.01 (0.28)	0.00 (0.17)
	$0.02\leq p<0.10$	1479	1.41 (0.63)	3.81 (0.70)	0.00 (0.46)
	$0.10\leq p<0.30$	3743	3.41 (0.93)	7.34 (0.97)	0.87 (0.77)

Table 2: Kernel comparison on the synthetic generators. Median over the trajectories in each stratum of

\log\mathrm{PsBF}^{\mathrm{PL}}_{j}

, the log pseudo-Bayes factor of the power law against kernel

j

(Eq. ( 14 )), stratified by the realized change rate

p

. In parentheses, the fraction of trajectories favoring the power law. Positive values favor the power law. The truncated power law and the log-normal decay are reported in Supplementary Note 2.

	Random hold-out				Block hold-out
Method	$T$	micro-AUROC	macro-AUROC	log-loss	$T$	micro-AUROC	macro-AUROC	log-loss
PL	1.000	0.996	1.000	0.050	1.000	0.960	0.864	0.072
EXP	1.000	0.994	0.994	0.057	1.000	0.957	0.818	0.072
GAUSS	0.996	0.988	0.974	0.061	1.000	0.938	0.500	0.077
STREXP	1.000	0.996	1.000	0.054	1.000	0.960	0.850	0.072
MostFrequent	0.970	0.909	0.500	0.194	1.000	0.882	0.500	0.160

Table 3: Comparison with baselines on real trajectories. Median over individuals of test-fold metrics at country level, for random hold-out and for

14

-day block hold-out. The memory kernels use the point estimate

\lambda_{*}

; for STREXP the shape parameter is fixed at

\beta=0.5

. MostFrequent and NearestObserved are the

\lambda\to 0

and

\lambda\to\infty

limits of the memory model (Methods); MostFrequent assigns the same distribution to every day, so its macro-AUROC is

0.5

by construction.

Figure 4: Comparison with baselines. Test-fold macro-AUROC of the three memory kernels and the four baselines on the country-level timelines, a under random hold-out and b under

14

-day block hold-out. Boxes span the interquartile range across individuals and whiskers extend to

1.5

times that range. A dagger marks the baseline that corresponds to a limiting case of the memory model; the other, MostFrequent, is omitted because, by construction, its macro-AUROC is 0.5. c Median macro-AUROC and log-loss when the kernel sum of Eq. ( 3 ) is restricted to observations within

W

days of the imputed day, for the three main kernels, and “full” denotes the unrestricted timeline.

Figure 5: Performance stratified by mobility. Median test-fold fraction of correct inferences

T

(top) and median macro-AUROC (bottom) for selected kernels and baselines, stratified by the change rate

p

( a ), the number of distinct countries

N_{c}

( b ) and the number of observed days

n_{i}

( c ).

	Random hold-out				Block hold-out
$j$	$p<0.02$	$0.02$ – $0.10$	$0.10$ – $0.30$	$p\geq 0.30$	$0.10$ – $0.30$	$p\geq 0.30$
EXP	$0.00$ (0.30)	$0.00$ (0.46)	3.30 (0.74)	4.73 (0.93)	3.82 (0.74)	4.66 (0.93)
GAUSS	$0.00$ (0.32)	$0.00$ (0.47)	6.95 (0.76)	8.61 (0.94)	7.12 (0.75)	7.71 (0.94)
STREXP	$0.00$ (0.33)	$0.00$ (0.43)	0.85 (0.67)	1.54 (0.86)	1.16 (0.70)	1.54 (0.89)

Table 4: Kernel comparison on real trajectories. Median over individuals of

\log\mathrm{PsBF}^{\mathrm{PL}}_{j}

, the log pseudo-Bayes factor of the power law against kernel

j

(Eq. ( 14 )), stratified by change rate

p

, at country level. In parentheses, the fraction of individuals favoring the power law. Positive values favor the power law.

Figure 6: Kernel selection across mobility strata. Median log pseudo-Bayes factor of the power law against each competing kernel, stratified by a the change rate

p

, b the number of distinct countries

N_{c}

and c the number of observed days

n_{i}

. Positive values favor the power law. The kernels can be discriminated only among mobile individuals, where the light-tailed kernels are clearly disfavored and the stretched exponential, which requires an additional shape parameter, remains close to the power law. The truncated power law (TPL) and the log-normal decay (LOGNORM), discussed in Supplementary Note 2, are shown for reference.

$p$	Meaning
SG1	Chance that any individual day nominally assigned to a location is replaced by the other.
SG2	Probability that the user switches location relative to the previous observation.
SG3–4	Chance that any observed day is replaced by a location drawn uniformly among the $K$ locations.
Real data	Fraction of consecutive observations that imply a change of location.

Table 5: Meanings of the noise parameter

p

across generators and in the real data.

Incomplete Data Imputation

Incomplete data imputation estimates unobserved values while accounting for data structure, missingness patterns, and uncertainty.

See all 79 papers

Apr 26, 2026

Transferable Human Mobility Network Reconstruction with neuroGravity

Accurate modeling of human mobility is critical for tackling urban planning and public health challenges. In undeveloped regions, the absence of comprehensive travel surveys necessitates reconstructing mobility networks from publicly available data. Here we develop neuroGravity, a physics-informed deep learning model that reliably reconstructs mobility flows from limited observations and transfers to unobserved cities. Using only urban facility and population distributions, we find that neuroGravity's regional representations strongly correlate with socioeconomic and livability status, offering scalable proxies for costly surveys. Furthermore, we uncover that spatial income segregation plays a key role in model transferability: mobility networks are most reliably reconstructed when target cities share similar segregation levels with the source. We design an index to quantify this segregation and accurately predict transferability. Finally, we generate mobility flow proxies for over 1,200 cities worldwide, highlighting neuroGravity's potential to mitigate critical data shortages in resource-limited, underdeveloped areas.

Jinming Yang, Shaoyu Huang, Zongyuan Huang +4

MoE Key Laboratory of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University, Shanghai 200240, China · Data-Driven Management Decision Making Lab, Shanghai Jiao Tong University, Shanghai 200240, China · Department of City and Regional Planning, University of California, Berkeley, CA 94720, USA +2

Geospatial Representation Learning Cross-Domain Transfer Learning

Ask PDF

Jun 14, 2026

IBAD: Interpretable Behavioral Anomaly Detection on Human Mobility Data

Human mobility appears highly diverse, yet much of a person's daily mobility can be explained by a small set of recurring behavioral templates, such as commuting, school-centered activities, caregiving, nightlife, or errand patterns. We present IBAD (Interpretable Behavioral Anomaly Detection), a framework that learns interpretable daily mobility templates and represents each individual as a distribution over mixtures of these templates. Rather than focusing on specific locations, IBAD characterizes activities that individuals perform across locations. This approach first discovers global behavioral templates using Latent Dirichlet Allocation (LDA), then employs a hierarchical self-supervised model to learn normal behavior of individuals from their soft behavioral templates. We also introduce a splicing benchmark that creates controlled behavioral mismatches between an individual's historical profile and injected mobility patterns. Experiments on real-world and synthetic datasets show that daily behavior can be effectively decomposed into a small number of interpretable templates. Crucially, we show that the learned behavioral archetypes transfer across distinct geographic and demographic contexts. Furthermore, IBAD maintains a robust competitive performance across all settings. For reproducibility purposes, the code is accessible at ~https://github.com/USC-InfoLab/IBAD.

Bita Azarijoo, John Krumm, Cyrus Shahabi

University of Southern California Los Angeles, California, USA

Interpretable Anomaly Detection Anomaly Detection

Ask PDF

Aug 3, 2026

Secrets Everywhere: Auditing Memorization in Mobility Prediction Models

Human mobility prediction models, which forecast the next location in a user's trajectory, are increasingly deployed in urban analytics, navigation, and personalized services. Yet, little is known about their potential to memorize and expose sensitive user trajectories from training data. While memorization has been extensively studied in language models, mobility prediction poses unique challenges: training sequences encode human behavior at various spatial and temporal scales, creating privacy risks at different granularities. In this paper, we conduct the first systematic audit of memorization in mobility prediction models. While prior work has shown that privacy leaks can arise from such models, we systematically assess and quantify memorization risks at scale. We identify key challenges, including the lack of a randomness space, the multi-scale structure of trajectories, and user-specific behavioral diversity. To address these challenges, we introduce a framework to quantify mobility memorization at different levels of granularity: individual locations, anchor pairs, and subtrajectory segments. We also develop user-grounded reference sets to assess how likely a model is to prefer training data over realistic alternatives. Our evaluation across multiple models and datasets reveals pervasive memorization patterns that correlate with user regularity and increase the risk of data extraction at inference time. Our findings call for mandatory privacy auditing in mobility prediction models.

Anne Josiane Kouam, Hristo Boyadzhiev, Konrad Rieck

Inria & TU Berlin Palaiseau, France · BIFOLD & TU Berlin Berlin, Germany

Privacy Auditing Human Behavior Prediction Trajectory Prediction

Ask PDF

Abstract

Incomplete Data Imputation

Explore similar work

Transferable Human Mobility Network Reconstruction with neuroGravity

IBAD: Interpretable Behavioral Anomaly Detection on Human Mobility Data

Secrets Everywhere: Auditing Memorization in Mobility Prediction Models